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Non-Linear Vibrations

Non-LinearVibrationsG. Schmidt

A. Tondl

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Preface

It has not been an easy decision to choose the general title of Non-linear vibrations.Recently there has been an overwhelming development of the theory and many appli-cations of non-linear vibrations, development reflected in numerous books, manyspecialist journals and the Equa-Diff conferences and ten International Conferenceson Non-linear Oscillations.

In view of these development, we would like this book to be considered not as anattempt to survey the state of the art and extent of current knowledge of non-linearvibrations, but rather as an account of some methods and results in the field of non-linear vibrations we have obtained or encountered in our personal experience. We donot believe in a single, comprehensive theory of non-linear vibrations, not even in thesense of linear vibration theory. On the contrary, we see the specific difficulty - andthe attraction - of non-linear vibrations in the non-existence, more than in the exi-stence, of certain rules of order. The nature of our endeavour may perhaps best beconveyed in Rilke's words: Uns iiberfiillts. Wir ordnens. Es zerfallt. Wir ordnenswieder. ... `.

We believe that the material presented here can be used for many kinds of vibrationproblems, although translating it into their terms is no simple task. Various methodsand examples are traditionally associated with different notations, and we have nottried to make the notation uniform throughout this book.

The first author is responsible in particular for chapters 5, 6, 10, 11 and 12, thesecond for chapters 3, 4, 7, 8 and 9.

We wish to thank those who by their interest in our results encouraged, or perhapsseduced, us into writing this book. We especially thank Prof. G. WALLIs, Dr. R.SCHULZ, Prof. W. EBELING, and Dipl.-Ing. D. HAJMANOVA for many valuable com-ments. We also thank Cambridge University Press and Akademie-Verlag Berlin forediting this book.

Die Autoren

Contents

Introduction ............................................................... 9

1. Basic properties and definitions ...................................... 12

1.1. Non-linear vibration, non-linear characteristics and basic definitions ..... 121.2. Some examples of excited and self-excited systems ..................... 171.3. Basic features of excited systems .................................... 261.4. Basic features of self-excited systems ................................. 291.5. Stability .......................................................... 31

2. Methods of solution ................................................ 35

2.1. The harmonic balance method ....................................... 352.2. The Van der Pol method ........................................... 382.3. The integral equation method ....................................... 41

2.4. Stability conditions ................................................. 442.5. The averaging method ............................................. 46

3. Auxiliary curves for analysis of non-linear systems ..................... 48

3.1. Characteristic features of auxiliary curves, particularly the backbone curvesand the limit envelopes ............................................. 48

3.2. Use of auxiliary curves for preliminary analysis ....................... 573.3. Use of auxiliary curves for preliminary analysis of parametrically excitedsystems .......................................................... 593.4. Auxiliary curves of higher-order systems .............................. 643.5. Use of auxiliary curves in analysis of systems with several degrees of free-dom ............................................................. 713.6. Identification of damping ........................................... 74

4. Analysis in the phase plane .......................................... 77

4.1. Fundamental considerations ........................................ 774.2. Practical solution of the phase portraits .............................. 904.3. Examples of systems of group (b) .................................... 914.4. Examples of systems of group (c) .................................... 964.5. An example of a system of group (a) .................................. 99

5. Forced, parametric and self-excited vibrations .......................... 1125.1. Amplitude equations ............................................... 1125.2. Resonance curves, extremal amplitudes, and stability .................. 1175.3. Non-linear damping ................................................ 1245.4. Forced and self-excited vibrations ................................... 1295.5. Parametric and self-excited vibrations ................................ 1385.6. Forced, parametric and self-excited vibrations ......................... 1415.7. Non-linear parametric excitation. Harmonic resonance ................. 1465.8. Non-linear parametric excitation. Subharmonic resonance ............... 152

8 Contents

6. Vibrations of systems with many degrees of freedom .................... 1546.1. Single and combination resonances ................................... 1546.2. Stability of vibrations with many degrees of freedom ................... 1626.3. Vibrations in one-stage gear drives ................................... 1666.4. Torsional gear resonance ............................................ 1.706.5. Combination gear resonances ........................................ 1736.6. Internal resonances in gear drives .................................... 1786.7. Torsional vibrations in N-stage gear drives ............................ 1846.8. Strong coupling between gear stages ................................. 1936.9. Application of computer algebra ..................................... 196

7. Investigation of stability in the large .................................. 1997.1. Fundamental considerations ........................................ 1997.2. Methods of investigating stability in the large for disturbances in the initial

conditions ........................................................ 2017.3. Investigation of stability in the large for not-fully determined disturb-

ances ............................................................. 2067.4. Examples of investigations concerning stability in the large for disturbances

in the initial conditions ............................................. 2127.5. Investigation of stability in the large for other types of disturbances ..... 2327.6. Other applications of the results ..................................... 2357.7. Examples ......................................................... 236

8. Analysis of some excited systems ..................................... 2478.1. Duffing system with a softening characteristic ......................... 2478.2. Some special cases of kinematic (inertial) excitation .................... 2568.3. Parametric vibration of a mine cage ................................. 270

9. Quenching of self-excited vibration ................................... 2789.1. Basic considerations and methods of solution .......................... 2789.2. Two-mass systems with two degrees of freedom ........................ 2829.3. Chain systems with several masses ................................... 3029.4. Example of a rotor system .......................................... 313

10. Vibration systems with narrow-band random excitation .................. 32310.1. Application of the quasi-static method ............................... 32310.2. Application of the integral equation method. Probability densities ....... 326

11. Vibration systems with broad-band random excitation ................... 33611.1. The amplitude probability density .................................... 33611.2. Statistical properties of the vibrations ................................ 34311.3. Non-stationary probability density, transition probability density and two-

dimensional probability density ..................................... 353

12. Systems with autoparametric coupling ................................ 36212.1. Basic properties ................................................... 36212.2. Internal resonance ................................................. 37012.3. Narrow-band random excitation .................................... 37512.4. Broad-band random excitation ...................................... 38012.5. Fokker Planck Kolmogorov equation ................................. 38412.6. Behaviour of the solution ........................................... 39112.7. Application of computer algebra ..................................... 400

Appendix ................................................................. 401

Bibliography .............................................................. 405

Index .................................................................... 415

Introduction

Finding examples for vibrations of systems which are linear is not a trivial matter;there is something in the remark by R. M. ROSENBERG that dividing vibrations intolinear and non-linear is like dividing the world into bananas and non-bananas. Never-theless many important vibration problems must be and can be treated in the sameway as linear ones, and the book Lineare Schwingungen by MuLLER and SCMEHLEN,for instance, is very helpful for all those grappling with vibration problems.

However, numerous vibration phenomena which are theoretically surprising aswell as practically important can only be understood on the basis of non-linear vibra-tion. For instance, the wide field of self-excited, parametric and autoparametricvibration, to which we give special consideration in this book, demands non-lineartreatment from the very beginning.

The mathematical theory of non-linear vibrations, the numerical and experimentalmethods for their evaluation and their many applications in mechanics, in mechanicaland civil engineering, in physics, electrotechnology, biology and other sciences, haveall developed so rapidly of late that it would require many volumes and many specia-lists to give a comprehensive picture of non-linear vibration.

The difficulties faced by those engaged in the study of the subject are concerned,on the one hand, with the broad scope of problems and the diversity of systems in-volved, and on the other hand with the fact that some general laws, such as the prin-ciple of superposition and the principle of proportionality, which can be used toadvantage in solutions of linear systems, do not apply to non-linear ones. Again ana-lytical or qualitative methods are nearly always approximate and hence even a phys-ical analysis of vibration systems is more difficult if they are non-linear; such methodsof solution must be supplemented by digital or analogue computing techniques. Onthe other hand, numerical methods have led to an understanding of new phenomenain the field of strongly non-linear systems. Similarly, investigations concerning thestability of the solutions are also more difficult because the existence of several steadysolutions is a common occurrence in non-linear systems. Consequently, in detailedanalysis, it is necessary to investigate stability even for disturbances which can nolonger be considered small. Such an investigation of stability in the large is needless incase of linear systems.

Non-linear vibration theory can be divided into three main parts. The first comprisesanalytical methods dealing with approximate procedures chiefly of steady solutions.Several alternative methods have been developed, based on Poincare's idea of ex-pressing the solution in the form of an expansion with respect to a small parameter.Their most recent modification is the multiple scale method (see NAYFEH and MooK(1979)). These procedures, as with the averaging method of Krylov and Bogoljubov,are widely used and thoroughly grounded mathematically. The same can hardly be

10 Introduction

said of the harmonic balance method, which can nevertheless be a very effective toolwhen employed by an experienced dynamicist. The integro-differential equationmethod, less well known than the averaging method although it has some advantages,will be dealt with in detail in this book.

The seond main part of the theory encompasses qualitative methods, stabilityinvestigations, bifurcation problems, attractor analysis and determination of thedomains of attraction. The analysis of trajectories and singular points in the phaseplane (space) is among the first representatives of this group of methods. The deter-mination of the domains of attraction forms an important part of analyses of non-linear systems with several steady stable solutions. So far, this problem has beenstudied in only a few books. The pioneering work was done by HAYASHI (1964) onsystems with one degree of freedom. Using different approaches, the problem of do-mains of attraction will be discussed in this book without restriction as to the num-ber of degrees of freedom. Investigations of strange attractors and the use of cata-strophe theory in solutions of bifurcation and stability problems represent recentcontributions to this part of the theory.

The third main part of non-linear vibration theory includes investigations which,using the methods developed in the first two parts, chiefly aim at physical analysisof specific non-linear phenomena and effects or analyses of particular physical sy-stems. The synchronization phenomenon, the effect of a limited exciting energy sourceand the effect of tuning a system into internal resonance, can be cited as examples ofthis kind of problem. There are numerous books and monographs which have beendevoted to a special class of non-linear problems or to non-linear problems in parti-cular systems, for example in electronic circuits, different machine elements, gyros-copic systems, systems with impacts, self-excited vibrations in machine tools caused bythe action of cutting forces, problems of aero- and hydroelasticity, of non-linear con-trol systems, hydraulic systems, astronomy and aerospace engineering, etc.

A survey of the most important books and monographs published during the post-war period, dealing with vibration in non-linear systems which develop a substantialbody of non-linear vibration theory, would include the following works. To the oldeststudies of this time which attempt to give a general view or to describe some analyticalmethods, belong those which deal largely with systems with one degree of freedom :books by MINORSKY (1947), McLACHLArr (1950), STOKER (1950), BULGAKov (1954),PUST and TONDL (1956), KAUDERER (1958), HAAG and CHALEAT (1960), etc. Thetreatise by MALKIN (1956) discusses the Poincare method, that by BoGOLJUBov andMITROroL'sKIJ (1963) the asymptotic averaging method of Krylov and Bogoljubov.Of the many publications dealing with this method, the study of MITROPOL'SKIJ(1971) deserves special mention. Books published at a later date for purposes ofphysical and technical sciences contain numerous examples of analyses used in differ-ent fields: BLAQUIERE (1966), EvAN-IwANowsKx (1976), HAGEDORN (1978), VoJTA.-sEK and JANAc (1969)), etc. The book by NAYFEH and MooK (1979) contains a greatmany examples and also tackles non-linear continuum systems; it is a very compre-hensive work which well represents this class. The newly published third edition ofthe book by KLOTTER (1980) analyses basic results as well as the applicability of dif-ferent approximative methods.

The book by ANDRONOY, VrrT and CHAiKIN (1959) was the first publication toconcentrate on qualitative analysis and topological methods. Of the remaining studiesof this kind, mention should be made of the book by BUTENfl (1962) and especially ofthat by NEJMARK (1972), dealing with mapping. Publications devoted to the theory

Introduction 11

of stability, which plays an important role in the analyses of non-linear systems, forma special group. Besides the famous work of LJAPUNOV (1950) and the outstandingstudies of MALKIN (1952) and ICETAEV (1955), one can name, for example the publi-cations of KARACAROV and PIL.TuTNiK (1962) and BoGUsz (1966) as important contri-butions in this field. Dynamic stability in connection with parametrically excitedvibrations is comprehensively investigated by BOLOTIN (1956). The investigationof the domains of attraction is a relatively new problem, originally tackled by HAY-ASHI (1964) and then elaborated by NISHIKAwA (1964), UEDA (1968), TONDL (1970a,1973 a, 1973b), ZAKRZEVSKIJ (1980), GUCKENHEIMER and HOLMES (1983) and others.

This brief survey may be concluded by referring to some books on particular classesof non-linear vibrations and on applications to engineering practice. The first of theseis treated in monographs on self-excited systems (CHARKEVIC (1953), TONDL (1970b,1979c, 1980b), RUDOWSKI and SZEMPLINSKA-STUPNICKA (1977), RuDOwsKI (1979),LANDA (1980), on systems with parametric excitations (MCLACHLAN (1947), SCHMIDT(1975), EICHER (1981)), and on non-linear random vibrations (CRANDALL and MARK(1963), BOLOTIN (1979), M. F. DIMENTBERG (1980)). The second - applications ofnon-linear vibration theory to engineering - has received considerable attentionfrom many authors. A few of the pertinent studies and fields to which they referinclude: electronics (CHARKEVI6 (1956), KOTEK and KUBIK (1962), PHILmPow(1963)), machine elements (GRIGORIEV (1961)), rotor dynamics (GROBOV (1961), TONDL(1965, 1974a), MERKER (1981)), systems with impacts (PETERKA (1973, 1981),A. E. KOBRINSKIJ and A. A. KOBRINSKIJ (1973), RAGUL'SKENE (1974), BABICKIJ(1978)), synchronization effects (BLECHMANN (1971, 1981)), mutual effect of self-excitation and of external or parametric excitation (TONDL 1976, 1978)), non-lineardamping problems (PISARENKO (1955, 1970), SERGEEV (1969), PANOVKO (1960)),non-linear problems of vibration isolation (RAGUL'sKIS (1963), KoLOVSKIJ (1966),FROLOV and FURMAN (1980), phenomena and problems due to the limited source ofenergy (KoNONENKO (1964)), etc.

Both classes are given equal treatment in STRATONOVICH (1961), a book on self-excited random vibrations in electrotechnology which widely stimulated researchwork on random vibrations in mechanics too.

A comprehensive handbook on linear and non-linear vibrations in various branchesof technology is Vibracii v technike (1978ff.) in six volumes.

1. Basic properties and definitions

1.1. Non-linear vibration, non-linear characteristics andbasic definitions

As mentioned in the Introduction, vibration of linear systems is only a special caseof vibration of non-linear systems. Two important principles, which are valid forlinear systems, do not apply to non-linear systems:

(I) The principle of superposition,(II) The principle of proportionality.

In linear systems described by linear differential equations (all terms are functionsof dependent variables and their derivatives) the response to the various componentsof excitation can be added up; if the amplitude of harmonic excitation is increased ntimes, the amplitudes of steady vibration increase n times. It has been found advan-tageous to divide linear vibration into two large groups:

(a) Vibration of systems with constant parameters described by linear differentialequations with constant coefficients. The term linear vibration is usually understoodto mean vibration of systems of just this type. The theory of this vibration is essentiallycomplete and will not be discussed in the present book.

(b) Vibration of systems with non-constant parameters described by linear differ-ential equations whose coefficients are generally functions of a dependent variable,chiefly time. An important separate group is formed by systems whose parametersare periodic functions of a dependent variable. Such systems are called rheo-linearand their theory is usually studied together with that of non-linear vibration. Oneof the reasons for this is the fact that an investigation of stability for small disturb-ances of steady periodic solutions of non-linear differential equations generally leadsto an investigation of stability of the trivial solution of systems of differential equationswith periodically variable coefficients.

Using as criteria differential equations which describe non-linear systems and usingthe character of stationary solutions, non-linear systems can be assigned to the follow-ing groups:

(1) Systems governed by homogeneous non-linear differential equations with con-stant coefficients

y+f(y,y)=0.These systems can be divided into subgroups according to the character of the solutionexpressing free damped vibration, free undamped vibration, divergent vibration orself-excited vibration of periodic, quasiperiodic or even of chastic nature.

1.1. Basic definitions 13

(2) Systems governed by homogenous differential equations whose coefficients arefunctions of an independent variable (time)

y+f(y,y,wt)=0.The term consisting of the product of a periodic coefficient and a dependent variablerepresents the linear parametric excitation. Non-linear parametric excitation is re-presented by the terms where the dependent variable is replaced by a non-linear func-tion of this variable. When the coefficients are not periodic but random oscillatoryfunctions of time, we speak about stochastic parametric excitation. Moreover, thecharacter of the solutions can be used for the classification into different subgroups.

(3) Systems governed by non-homogeneous differential equations

y + f(y, y) = F(wt)where F(w, t) is a periodic or stochastic function of time. Of course, a further criterionis whether or not the homogeneous system is capable of self-excitation.

More general are systems governed by differential equations of the type

y + /(y, y, (ot) = F(cot)representing combinations of the above groups.

Another classification of non-linear systems can be achieved by estimating how faran analysed system differs from a linear one. It need not be stressed what difficultiescan be come across because different points of view, dividing limits and criteria canbe used. As an example of one of the different approaches the use of extreme valuesof non-linear and linear terms in differential equations can be :mentioned; as anothercriterion the characteristic features of a solution can serve. It is very difficult to definethe dividing limits because it depends very much on what characteristic features aretaken as criteria when comparing a non-linear system with a corresponding linear one.For example, when comparing the vibrations of the systems the vibrations can beestimated as similar although the resonance curves differ substantially from eachother (the hysteresis and jump phenomena occur in non-linear systems). As a quali-tative difference the occurrence of such a non-linear resonance, typical of non-linearsystems only, could be considered - for example, the occurrence of subharmonicresonance. When comparing the vibration of a nonlinear system with that of thecorresponding linear undamped system from a broader point of view the vibrations ofsuch systems can be considered to be similar. The vibration at subharmonic resonanceof order 1 /2 characterized by the dominant half-frequency component and by theexcitation frequency component can serve as an example. The first component canbe considered to be close to the natural vibration of a linearized undamped system.From the above it can be seen that a certain system can be assigned to different clas-ses when using different criteria.

We principally exclude in this book investigations of the global behaviour of non-linear systems, which cannot be described by approximations starting from the linearsystem. For the remaining non-linear systems we can distinguish between stronglynon-linear ones, especially important for applications and mainly dealt with in thisbook, for which the solutions, although found by approximations starting from thelinear solution, in some respect or other substantially - quantitatively or even quali-tatively - differ from the linear case, and between weakly non-linear ones, for whichsuch a substantial difference does not occur. As an example of such a substantial andqualitative difference a case can be mentioned where, by the action of periodic exci-tation (external or parametric), the vibration character is non-periodic and stochastic.

14 1. Basic properties and definitions

Note : The mere non-periodicity of the vibration in self-excited systems with severaldegrees of freedom cannot be taken as the decisive criterion because a multi-frequencyvibration can occur which, though non-periodic, is quasi-periodic in quasi-normalcoordinates; the frequencies of the components are close to the natural frequenciesof the components of the linearized system vibration. Then the vibration of the non-linear system can be considered to be similar to natural vibration of the linearizedundamped system. For that very reason this similarity is utilized in solution methods.These systems cannot be considered strongly non-linear; they should be regarded asweakly non-linear systems.

The various terms (or groups of terms) of non-linear equations which describe themodel of an actual system have a physical meaning in mechanical systems (for exam-ple, the restoring force of a spring or an elastic element, damping, etc.). The relation-ship between such quantities and the dependent variables and their derivatives istermed the characteristic (for example, the characteristic of a damper spring). Char-acteristics cannot in every case be expressed in terms of functions of a single variable.Some of the simpler characteristics which can be expressed as functions of a singlevariable are discussed below.

Characteristics can be divided into weakly non-linear and strongly non-linear(expressed by functions which differ substantially from linear relationship). Considerfirst the characteristics of springs which can be expressed in terms of a function ofa single variable, for example, the deflection y. If the function expressing the charac-teristic can be described by the equation

(1.1, 1)f(-y) = -AY)the characteristic is called symmetrical. If d/(y)/dy is an increasing function, thecharacteristic is a hardening one; if df (y)/dy is a decreasing function, the characteri-stic is a softening one (Fig. 1.1, 1).

Fig. 1.1, 1

If a spring with a symmetrical non-linear characteristic is acted on by a constantforce, the equilibrium position undergoes a change and, relative to the new workingpoint, the characteristic becomes asymmetrical for deflections from the new equili-brium position. Such a case is a common occurrence in mechanical systems - theconstant load is produced by the weight of the masses, or by a constant torque intorsional systems. The restoring force of a pendulum (Fig. 1.1, 2) can serve as an

1.1. Basic definitions

a) b)

Fig. 1.1, 2

15

example of a softening symmetrical characteristic. A symmetrical characteristic maynot have a monotonic character within the whole range of deflection y; the springmay be softening in one interval and hardening in another. Fig. 1.1, 3a shows thecharacteristic for the case when the restoring force is negative in a certain intervalof deflections and when several equilibrium positions exist. Fig. 1. 1, 3 b shows a sys-tem containing a bar loaded by an axial force S which is larger than the Euler buck-ling force; for transverse deflections the characteristic of the restoring force is asshown in Fig. 1.1, 3a. A system with a similar characteristic is shown in Fig. 1.1, 3c.

Fig. 1.1, 3

Various broken-line characteristics consisting in part of linear segments form a.large group, some examples of which are shown in Figs. 1.1, 4-1.1, 6: Fig. 1.1, 4 - asystem with clearance, Fig. 1.1, 5 - a system with stops, Fig. 1.1, 6 - a systemwith a prestressed (prestress Po) spring.

Damping which is a function of velocity has similar characteristics. The characteris-tic of dry friction (Fig. 1.1, 7 b) differs substantially from the usual ones and is oftenreplaced by that of idealized Coulomb friction (Fig. 1.1, 7 a).


Fig. 1.1, 4

Y

Fig. 1.1, 5

9(j )

Fig. 1.1, 6

Y I 1'

1.2. Examples of excited and self-excited systems 17

Characteristics cannot always be expressed in terms of a function of a single variable.Both elastic and damping forces are apt to be functions of several variables or to beexpressed in terms of vector functions. In cases of this sort it is more to the pointto speak of force or damping fields. Fields which can be described by a function whosevariable is a two-dimensional vector are very frequently encountered in mechanicalsystems. If such a field is described by the function

F(z, z) _ [/1(lzl, lzI) + a12(Ijj, lzI)] sgn z (i = -1)or by the function

F(z, z) = [fi(Izl, Izl) + it2(Izl, IzI)] sgn z

where z is a two-dimensional vector and fl(IzI, IiI) and f2(Izl, Ill) are scalar functions,the field is central symmetrical. Such a field is characterized by the fact that the abso-lute value I F(z, z) I and the radial and tangential components remain unaltered if theabsolute values Izi and jzI do not vary; the magnitudes of the radial and tangentialcomponents do not vary, either. Detailed information concerning two-dimensionalfields may be found in a monograph by TONDL (1967 c).

Mention should also be made of some of the most important differences betweenvibrations of linear and non-linear systems. In linear systems, steady vibration produc-ed by external periodic excitation has the same character as the excitation; in non-linear systems this is usually not so. Under harmonic excitation, for example, thesteady vibration of a linear system is also harmonic, whereas the response of non-linear systems is either periodic or quasi-periodic. In addition to the basic component,vibration produced by such excitation is apt to contain not only higher but also sub-harmonic components, i.e. the period of the response can be an N-multiple (where N isan integer) of the excitation period. The difference between linear and non-linearsystems becomes more marked still when, under periodic or even harmonic excitation,the response is not at all periodic and sometimes not even quasi-periodic. Differ-ences are found to exist in the character of the vibration course as well as in the occur-rence of resonant vibration. In linear systems with constant coefficients, resonancesoccur only for excitation frequencies which are identical with the naturual frequenciesof the system. Since this stipulation need not be satisfied in non-linear systems, theseare likely to be richer in resonances than linear systems.

In linear damped systems without external excitation, steady vibration with finiteamplitudes never occurs. Consequently, steady self-excited finite-amplitude vibrationis always a property of non-linear systems. Similarly, in the case of linear parametricexcitation, finite-amplitude vibration can be obtained only by the action of non-linear terms.

The possible existence of more than one steady solution for equal values of para-meters of the system and excitation may be set down as another significant propertyof non-linear systems.

1.2. Some examples of excited and self-excited systems

Mechanical systems are mostly linear so far as inertia forces are concerned. Thisapplies especially to discrete systems in which concentrated masses can execute onlyrectilinear motion. In rotating bodies in which the deflections of the gyroscope axiscannot be qualified as small, the gyroscopic effect is expressed by non-linear terms.2 Schmidt/Tondl


Inertial forces assume a non-linear character in the case of reduction of the masses inthe mechanism, or are functions of time if a particular member rotates uniformly(for example, in a crank mechanism). Damping is almost always non-linear; linearviscous damping defined by the product of a constant coefficient of damping and thevelocity is more or less an idealization, although a very popular one.

The quantity which has the greatest effect of all on the "non-linear" behaviour ofdiscrete systems with concentrated masses is spring non-linearity. A non-linear springis, for example, a coil steel spring with unequal lead (pitch) whose coils graduallycome to rest on one another as the spring deforms. Restoring force non-linearity canalso be produced by using several parallel springs of unequal length, some of whichcome into action only at a definite deflection of the mass. The characteristic of therestoring force thus produced is of the broken-line, in part linear type - Fig. 1.2, 1.Elements whose material has linear elastic properties but which are curved either bythe effect of prestress or geometry, have a non-linear character of the restoring force.These elements are examples of geometrical non-linearity in the same way as systemscontaining a mechanism with one or several components. The simplest system ofthis kind is a pendulum (Fig. 1.1, 2).

Fig. 1.2, 1

In systems with continuously distributed parameters the system non-linearity canbe caused by non-linearity in the boundary conditions. A whole class of these isformed by systems described by linear partial differential equations whose only non-linearity is that introduced by the boundary conditions. A beam supported by non-linear springs, or a shaft rotating in journal bearings whose analysis cannot be restrict-ed to small deflections of the journals in the bearings, serve as examples of suchsystems.

Non-linear systems having masses which are acted on by periodic forces, describedby non-homogeneous non-linear differential equations with constant coefficients, areso common as not to require detailed discussion of particular examples. Attention willtherefore be concentrated on systems with parametric excitation or combined exter-nal-parametric excitation, which are of less frequent occurrence.

Most vibration textbooks contain the classical example of a parametrically excitedsystem, i.e. vibration of a swing, which can also be modelled by a pendulum havinga periodically varying length (Fig. 1.2, 2). Another very popular example is a pendu-lum whose suspension moves harmonically in the vertical direction (Fig. 1.2, 3),which was first analysed by KLOTTER and KoTOwsKI (1939). Denoting by y the angulardeflection of the pendulum, the equation of motion takes the form

Oy + mgs sin y + msaw2 cos wt sin y = 0 (1.2, 1)


Fig. 1.2, 2 Fig. 1.2, 3

where O = mk2 is the moment of inertia about the suspension, m is the mass of thependulum, 1 = k2Js is the reduced length of the pendulum, and s is the distance be-tween the centroid and the suspension. For small deflections for which sin y = y,(1.2, 1) turns into the well-known Mathieu equation

2

y+ C g+ a l cos wt y= 0. (1.2, 2)

If the suspension moves in a direction deflected from the vertical by angle a, (1.2, 2)changes to

2 2

y + Z + al cos a cos wt y = al sin a cos wt (1.2, 3)

which describes a case of combined external and parametric excitation. For a = 0°,the excitation is pure parametric, fora = 90°, pure external. This system is an exam-ple of kinematic excitation when a periodic motion is specified for a definite point ofthe system rather than a force acting on the mass. The system is thus made to vibrateby the action of inertial forces.

Fig. 1.2, 4 shows a model of a proposed device for generating energy from tidalmotion: a conical buoy carries a crank mechanism whose connecting rod turns the

crank mechanism

Fig. 1.2, 4

prestressed

model

2`


generator rotor. Any equilibrium position, for example that with the connecting rodin horizontal position (Fig. 1.2, 4, right), can be obtained by application of prestress.Denoting by V the angular deflection, by m, M the concentrated masses according toFig. 1.2, 4, by c/2 the spring stiffness, by I = More the moment of inertia of the ge-nerator, and by r, l the crank radius and the connecting rod length, the vibrationabout the equilibrium position is described by the equation (for its derivation, a de-tailed description and analysis of the system see the paper by PARKS and TONDL(1979))

r2[Mo + M + m(cos V- 2 A sin 2V)2]

- V - 2 A sin 2p) (sin y' + A cos 2V)

+ Ki - raco2 sin cot [(M + m) cos ip - 2 Am sin 2V]

+ rg[(M + m) cos V - 2 Am sin 2y,]

+ cr { r[sin 1P - A(1 - cos 2yp)] - x0} (cos V -- s A sin 2y) = 0 (1.2, 4)

where A = r/l, xa is the prestress and K is the coefficient of the generator load. Itmay be seen that the excitation of this system is a combination of external and para-metric, linear and non-linear, excitations. In consequence the vibration about theequilibrium position grows steadily larger until, when certain conditions are fulfilled,the swinging motion of the crank changes to the desired rotation.

As pointed out by TONDL (1984), combined excitation in the case of kinematicexcitation can also be obtained for simpler systems provided the excitation occursvia a non-linear spring. Only for the simplest systems, such as that shown in Fig. 1.2, 5,

can the equation of motion be converted to a non-linear one with constant coefficientsby introducing the relative deflection z = x - a cos wt (x is the absolute deflectionand a cos wt is the motion of the spring suspension). Systems for which this cannot bedone and whose equations of motion are generally non-homogeneous non-linear differ-ential equations with periodically varying parameters are shown schematically inFig. 1.2, 6 - (a) system I, (b) - system IT. In either system, a body having mass m

non-linear springs a coscit o cos (cd t+yo)

non-Linear springs

a cos(ct-y)a) b)Fig. 1.2, 6


is placed between two non-linear springs and dampers. To simplify, the spring char-acteristics are assumed to be symmetrical, the non-linearity being expressed by acubic term; the dampers are linear viscous. In system I the excitation is provided bythe motion of the end of one of the springs; in system II both ends move but the mo-tions are generally shifted in phase. The equations describing the two systems willbe written for the harmonic motion a cos cot. Denoting by x the deflection of the mass,by x the damping coefficient, by k the coefficient of the linear portion of the restoringforce of the spring (assuming the linear terms of the two springs to be identical), andby yl and y2 the coefficient of the cubic term of the restoring force of the left-hand andthe right-hand spring, respectively, then system I is described by the equation

mz + kx + ylxs + xi + k(x - a cos cot) + Y2(x - a cos (ot)3

+ x(i + aco sin wt) = 0 (1.2, 5)

and system II by the equation

mi+x[i+awsin ((ot+g2)+x+awsin ((ot-qq)]+2kx- ak[cos (cot + 99) + cos (cot - qq)] + y,[x - a cos ((ot + q')]3

+ y2[x - a cos (cot - qq)]3 = 0 . (1.2, 6)

A detailed analysis of both systems is presented in Chapter S.A frequently discussed example of a parametrically excited system which has been

analysed by several authors is a beam subjected to a periodically varying axial loadP = Po + S cos cot (Fig. 1.2, 7). The lateral vibration of the beam is described bythe equation

EIa4y+(Po+Scoswt)a2y+m_Y=08x4 ax2 8t2

(1.2, 7)

Fig. 1.2, 7


where F is Young's modulus, I the cross-sectional moment of inertia, m the massper unit length, and y(x, t) the lateral deflection of an element at distance x from thesupport at time t. A very detailed analysis of this case is contained in the books ofBO OTUc (1956) and SCHMIDT (1975). The torsional vibration of beams under periodi-cally varying boundary conditions is also included in this group of systems.

Another well-known class of systems encompasses rotor assemblies whose shafts -due to the effect of slots - have unequal cross-sectional moments of inertia in twomutually perpendicular directions, or unequal mass moments of inertia. Rotor sys-tems of this kind have been analyzed by DIMENTBERG} (1959) and TONDL (1965).

Parametric excitation is also known to occur in torsional vibrating systems contain-ing gear trains, in which both the restoring and the frictional forces during uniformrotation are made to vary periodically by the effect of changes at the point of engage-ment of two gears. The periodic variation of the forces is one of the main reasons fora deterministic excitation of these systems. They are investigated in Chapter 6.

Several less conventional cases of parametrically excited systems of engineeringimportance will be dealt with next. Parametric excitation arises, for example, inconveyor belts on which the material being transported is deposited not uniformly butin sizeable amounts at definite time intervals, i.e. the conveyor carries equally spacedheaps of material (Fig. 1.2, 8). As the belt moves at a uniform speed, the distribution

Fig. 1.2, 8

A d77a DTn

of masses, and thus also the natural frequency of lateral vibration of the belt, under-goes a change which gives rise to parametric excitation. For certain intervals of thebelt speed, the equilibrium position becomes unstable, and the belt vibration eithergrows increasingly larger or, due to the effect of the non-linear terms, becomes stabi-lized at a finite amplitude. Such vibration is spoken of as parametric resonant and canbecome very intensive in lightly damped systems.

A similar situation "is found to exist in chain conveyors having belts stretched overdrums shaped like regular polyhedrons (Fig. 1.2, 9). As shown by ToNDL (1967 b),these conveyors are parametrically excited even when the transported material isdistributed uniformly. Assuming uniform rotation of the driving drum and takinginto consideration the stiffness of the belt as well as that of the chains, the authorexamined the torsional vibration of the tension drum and found the restoring momentper unit torsional deflection to have a periodic - pulsating - waveform (Fig. 1.2, 9b).

If the elasticity of the guide bar (a wooden beam) and of the whole structure sup-porting the guide bars in their mountings is considered in the analysis, the restoringforce for the lateral deflection of the mine cage guide bar support at the point ofcontact with the guide bar (see Fig. 1.2, 10 - a schematic view of the pit cross-


V

O)

M

b)

Fig. 1.2, 9

Fig. 1.2, 10

c)

section and of the mine cage in guides) is found to be periodically varying at a con-stant speed of travel. Details of the analysis of this system may be found in Chapter 8.In the discussion that follows, at least a few cases of self-excited systems will be

taken up and the main origin of self-excitation will be pointed out. One of the causesof self-excitation in mechanical systems is the flow of liquid or gaseous media pastan elastic, or a rigid but elastically mounted body, when the hydrodynamic or aero-dynamic forces have a destabilizing effect in, at least, a certain region of the systemparameters. If a coupling exists between the hydrodynamic or aerodynamic forcesand the elastic forces, the self-excitation is termed hydroelastic or aeroelastic, andits topical problems are treated in the separate disciplines of hydroelasticity or aero-elasticity. It is usually very difficult to express forces produced by a flowing medium.Frequently, therefore, artifical models are devised and their parameters adapted toexperimental results. In this connection, one often speaks, for example, of bluff bodymodels (DowELr., 1981).


In fairly simple cases the basic self-excitation effect can be expressed in terms ofnegative damping - usually linear viscous; finite amplitudes are ensured by progres-sive positive damping. The best known of all such systems is the Van der Pol oscilla-tor described by the equation

y" -(/9-by2)y'+y=O. (1.2,8)

Another group contains systems (discrete as well as those with continuously distri-buted parameters) in which the self-excitation effect is caused by relative dry frictionarising on the contact surface between an elastic (or an elastically mounted) body anda uniformly moving body. In a very simple example an elastically supported body(c - the spring constant) having mass m rests on an endless band moving with veloc-ity vo (see Fig. 1.2, 11). Fig. 1.2, 12 shows the curve of the friction force function

Fig. 1.2, 11

VO

Fig. 1.2, 12

V,=X- VO

H(v,) versus the relative velocity yr = x - v0, where x is the deflection of the massfrom the equilibrium position (at zero band velocity vol. The motion of the system isdescribed by the equation

mx+cx+H(x-vo)=0. (1.2,9)

Expanding function H(z - vo) into a Taylor's series, ignoring terms of orders higherthan the third,

H(x - vo) = -H(vo - x) _ -H(vo) + aHa2

2x2 +-a s2a0

Vv0 0


and introducing the new coordinate

x=xo+y (1.2, 10)

where xo = H(vo)/c, (1.2, 9) takes the form

y+Qty+hly-h$y2-}-h3y3=0 (1.2, 11)

where

Q2- c _ 1 aH 1 a2H 1 a3Hm

hlm av

h2m av2

h3m av30 0 0

So long as aH/avo < 0, then also hl < 0, and the equilibrium position x = xo is un-stable; equation (1.2, 11) becomes a system anlogous to the Van der Pol system. Asthe foregoing discussion reveals, a decrease of function H(vr), at least in a certain re-gion of the values of the relative velocity, is of substantial importance for the self-excitation effect.

A large group of self-excited systems is formed by rotors in which the self-excita-tion effect is provided either by the action of forces in journal bearings and forcesarising in gap flows (as in the case of labyrinth glands) or by the action of internaldamping, i.e. damping which is proportional to the relative velocity of the coordinatesystem rotating correspondingly to the rotor rotation. Internal damping may alsoinclude damping forces coming into existence due to material damping during shaftdeformations, or forces of friction arising on the contact surfaces of composite rotorsduring deformations (such as the friction between the shaft and the disk hub). Denot-ing by z the vector of the absolute deflection (of the disk, shaft centre, etc.), andby 4 the vector of the deflection in the coordinate system rotating with the angularvelocity of rotor rotation w, the relation between the two vectors may be written as

z = g exp (i wt)

where z = x + iy, _ + iq. The relative velocity is then given by

iwz) exp (-iwt) .

Expressing the damping force of internal damping in the form

Pr = 191)

ICI

or in the simplest form (linear viscous damping)

Pr = -ht

(1.2, 12)

(1.2, 13)

(1.2, 14 a)

(1.2, 14b)

then the forces in the absolute (non-rotating) coordinate system are given by therelations

-P. = -h(IzI, izl)

z- iwzIz - iwzI

or

(1.2, 15 a)

Pa = -h(z - iwz) . (1.2, 15 b)

Relations similar to (1.2, 15b) also hold for damping acting on a bladed disk (mount-ed on a flexible shaft) of an agitator's rotor mixing a viscous liquid in a vessel. Due tothe effect of rotor rotation, the liquid rotates in the vessel with a mean velocity not


identical with the angular velocity of rotor rotation. Accordingly, in the absolutecoordinate system, the damping force has two components : the radial component(hill) and the tangential component (-h(w (zj) which is proportional to the angularvelocity of rotor rotation co.

All types of self-excitation forces of rotor systems discussed in the foregoing haveone property in common, i.e. that, as the rotor whirls with natural frequency, thevector of the self-excitation force has a tangential component whose direction ina certain interval of rotor rotation agrees with that of the whirling motion and actsin opposition to the external positive damping. In all types of self-excited vibrationof rotors, the direction of the whirling motion agrees with that of rotor rotation.

Problems of self-excited vibration of rotors have been in the foreground of interestof many scientists. They have been discussed in numerous papers and comprehensivelytreated in books (DIMENTBERG (1959), TONDL (1965), Ku§uL (1963)), monographs(ToNDL (1961, 1973, 1974)) and lectures (RIEGER (1980)).

1.3. Basic features of excited systems

An essential distinction should be made between the variously excited systems withregard to their type of excitation (external, parametric, combined). Unlike linearsystems, externally excited non-linear systems do not follow the principles of super-position of the solution and of proportionality; sometimes, their response to periodicexcitation may differ even qualitatively from the excitation. In a non-linear systemexcited, for example, harmonically with frequency cs, the response may be periodichaving period 2Nrt/co (N is an integer), the component with frequency co/N being thedominant one in a particular interval of the excitation frequency. In excited systems,the feature of primary interest, especially from the practical point of view, is the reso-nant vibration. When dealing with it in connection with non-linear systems, one shouldalter the conventional "linear" approach to the concept of resonance. In linear systems,resonance is usually defined by a definite value of the excitation frequency (ordinarilyby that identical with the natural frequency of the system). In non-linear systems,resonance should be regarded as a phenomenon which brings about a substantialincrease of the amplitudes in a certain interval of the excitation frequency. Theresonance curve (the amplitude-excitation frequency dependence) of non-linear sys-tems may sometimes have more than one peak typical of the resonance curve oflinear systems (Fig. 1.3, la). Resonance curves featuring several peaks are oftenobserved in non-linear systems tuned to so-called internal resonance (the ratio oftwo or several natural frequencies is close to a ratio of, small integers) (Fig. 1.3, lb).

A

W

c) b)

Fig. 1.3, 1

CO

1.3. Basic features of excited systems 27

It may be useful to establish a basic classification of resonances. Consider a systemgoverned by the differential equations

n

Xs + eskxk + E[hs(l xl, l xl) + gs(x)] = Ps((ot) (s = 1, 2, ... , n) (1.3, 1)k=1

where csk are constants, s is a small parameter. h8(Ixl, l xl) x, gs(x) are analytic functionsof xk and Xk, and Ps((ot) are periodic or harmonic functions having period 27t/co.Functions h,(lxl , l xl) z representing damping are assumed to be such that no self-excitation can exist. Using the linear transformation

xs = askyk

wherek=1

(s= 1, 2, ... , n) (1.3,2)

a =1 ' a3k=k49(Qk)

ldl(S2k)

and dj(S2k) is the minor of the determinant of the characteristic equation

det I lcsk - ask`22Il = 0

(k, s= 1) 2, ...,n), 888=1, ask=0 for (s4k)corresponding to the j-th term of the first row of the determinant for 9 = 9k, system(1.3, 1) can be converted to the quasi-normal form

ys + Slay + EFs(i, y) = gs(wt) , (s = 1, 2, ... , n) . (1.3, 3)

S2k are the natural frequencies of the abbreviated system (1.3, 1) (for E = 0) whichare all assumed to be real.

Note: Details which are of use in practical analyses, relating to the coefficients oftransformation (1.3, 2) for s = 2, may be found in ToNDL's book (1965) and mono-graphs (1974a), (1976a).

According to the general theory of quasilinear differential equations of the type of(1.3, 3) (see, for example, MALKIN (1956)) the resonant solutions can be expected toexist in the vicinity of those values of the excitation frequency w for which

nrw = E NkQk (1.3, 4)

k=1

where r = 1, 2, ...; Nk = 0, ±1, ±2, ...Assuming the excitation to be harmonic or periodic with a dominant first harmonic

component, the classification of resonances can be set up as shown in Table 1.3, 1.If a non-linear system is excited by several periodic components having frequencies

w1, w2, w3, ... , (OM whose ratios differ from the ratios of integers, (1.3, 4) can bereplaced by the relation

M n

Z nkwk = E N9Q,k=1 j=1

(1.3, 5)

where nk, Nf = 0, ±1, ±2, ...In systems with only a linear parametric excitation, parametric resonances can be

conveniently classified by means of the instability intervals of the linearized system.Let the linearized system converted to the quasinormal form be described by theequations

n

ys + 0'y. + E E [gsk(wt) yk + psk(wt) yk] = 0 (s = 1, 2, ... , n)k=1

(1.3, 6)


Table 1.3, 1

Type of resonance Excitation Ratio of naturalfrequency close to frequencies Q1/S2k

Note

differs

fromis close

to

(a) Sz1 r/k - j = 1, 2, ... , nPure main resonance r,k=1,2,...,(b)Subharmonic resonance

NQ1 r + kn,N=2,3,...

(c)Superharmonicresonance

1

n

(d)Subsuperharmonic orsubultraharmonic reso-nance

NS2n

for (d):

,N

1, 2, ...

(e)Main internal resonance

Q1 r/k

(f) 1 n r/kQN

j = 1,2,...,nNon-periodic combina- 1

1

j 1

N1= :1, ±2, ...tion resonance r, k = 1, 2, ...

(g)Periodic combinationresonance

r/kr = k

Note: Superharmonic resonance has been called "ultraharmonic" by many authors.

where qsk, psk are periodic functions of period 2rc/w. As the general theory of equationsof the type of (1.3, 6) suggests (see, for example, MALKIN (1956)), the intervals ofinstability of a trivial solution may be expected to occur in the vicinity of such valuesof co which are close to

wo=I (j,k=1,2,...,n;N=1,2,...). (1.3,7)

For j = k, the instability intervals are said to be of the first kind and order N; forj + k, of the second kind and order N. Parametric resonances (the vibration becomeslimited due to the action of the non-linear terms) resulting from the instability inter-vals of the first kind and order N are called parametric resonances of the first kindor simple parametric resonances (for N = 1, main parametric resonances) ; thoseresulting from the instability intervals of the second kind are simply called combina-tion parametric resonances. The characteristic feature of parametric resonancesinitiated by the linear parametric excitation is their occurrence at a definite limitedinterval of exciting frequency. Outside this interval such a system does not vibratewhen acted on only by linear parametric excitation.

If an additional non-linear parametric excitation is also present in the system, otherresonances than those mentioned above may occur, similarly as they do in externally

1.4. Basic features of self-excited systems 29

excited non-linear systems. TONDL (1976 b) showed that of these, subharmonic resonan-ces are of significance in most cases.

The resonances of systems under combined (external and parametric) excitation donot lend themselves readily to classification. The reason for this is the possible inter-action of the two types of excitation (they can either combine to increase the vibrationamplitude or work in opposition to decrease it) and the difficulty of deciding which ofthem has the dominant effect. The situation becomes more straightforward if one typeprevails over the other, and the dominant excitation is harmonic. The resonances canthen be classified according to either Table 1.3, 1 or the instability intervals of para-metric resonances. An interesting but less known fact pointed out by several authorsis the "non-linear" character of resonances of damped linear systems under combinedexternal and parametric excitation. It should be stressed, however, that systems featur-ing such resonances must be subjected to damping whose level is such as to ensurecomplete quenching of all instability intervals associated with the linear parametricexcitation (PUST, TONDL (1956) and, particularly, TONDL, BACKOVA (1968), TONDL(1967 a) - rotor systems). A system with one degree of freedom, for example, whoseharmonic parametric excitation has double the frequency of the harmonic externalexcitation is likely to have resonance curves of double-peak character (Fig. 1.3, 1(a))depending on the phase shift between the two excitations.

The findings established above reveal an important feature characteristic of non-linear systems, namely that resonances can be produced at a vibration frequency whichis close to one of the natural frequencies of the system or, if the vibration containsseveral components, at frequencies which are close to the natural frequencies of thesystem (the frequency of the response being, however, always related, i.e. varyingin proportion, to the excitation frequency) by excitation whose frequency may beeven far remote from any natural frequency of the system. This cannot happen inlinear externally excited systems.

1.4. Basic features of self-excited systems

Self-excited systems described by non-linear homogeneous differential equations withconstant coefficients have the characteristic property that there exists, for any initialconditions or only a particular region thereof, a steady vibration which is periodic,quasiperiodic or possibly, non-periodic. This characteristic enables the systems to bedivided into two basic groups:(a) Systems with so-called soft self-excitation in which self-excited vibration arises if

and only if the equilibrium position is unstable.(b) Systems with the so-called hard self-excitation whose equilibrium is stable only

for definite limited disturbances or a definite limited region of the initial conditionsin the neighbourhood of the equilibrium position. If the limits are exceeded, thevibration - after a sufficient length of time - converges to a steady self-excitedmotion.

Consider first a system with one degree of freedom described by a second-orderdifferential equation. An analysis of systems of this kind may conveniently be carriedout by mean of phase-plane trajectories which provide a comprehensive picture of thebehaviour of the system and stability of steady solutions. The term "phase plane"refers to a plane whose coordinates are deflection and velocity, or, alternatively, thedependent variables obtained in transformation of the original equation to a system


of two first-order differential equations. In analyses of self-excited systems the mostimportant of the phase-plane trajectories are the limit cycles. These are closed tra-jectories which other trajectories in the neighbourhood either approach (a stable limitcycle) or move away from (an unstable limit cycle). The phase-plane analysis becomesparticularly simple when dealing with systems having a single equilibrium position,represented by a singular point in the phase plane. In such cases, the limit cyclessurround the singular point, the stable cycles alternating with the unstable ones ifcertain conditions of solution uniqueness are satisfied. As an example of systems witha single equilibrium position consider the system described by a differential equationof the type

y"+y+1(y)+F(Iyl,Iy)y=0 (1.4,1)

where 1(y) + 0 if y = 0. If function F( Iy I , Iyi) is a polynomial, the equilibrium positionis stable for a positive constant term and unstable for a negative one. If however thefunction also contains terms of the order 1 /n (n > 1), the stability of the equilibriumposition is decided about by the lowest term (for the largest n). The analysis of phase-plane trajectories is dealt with exhaustively in Chapter 4.

In systems with several degrees of freedom the situation is more complicated. Con-sider a system described by a set of homogeneous non-linear differential equationswhose trivial solution represents a single equilibrium position; let all the non-linearterms be of higher degree than the linear terms, and let the characteristic equation ofthe linearized system contain n pairs (n is the number of degrees of freedom) of com-plex roots. Further, let k pairs have real positive parts ak ± iQk where ak > 0.Assuming that the non-linear terms of the equations expressing damping are all posi-tive, i.e. the damping is positive progressive, the equilibrium position is unstable withrespect to k natural modes of the system; this is a necessary (but not sufficient) con-dition for the system to oscillate in r (k > r > 1) modes, the frequency of one vibra-tion mode being close to the frequency Stk. When the system vibrates in only onemode of frequency S2k, the vibration is termed single-frequency self-excited vibration.Assuming that the characteristic equation of the linearized system has k roots, thesteady vibration can have as many as k components of different frequencies. Suchvibration is called multi-frequency (two-, three-, ..., k-frequency) self-excited vibration.As the value of k increases, the number of possible multi-frequency vibrations growslarger. Since the ratios of the frequencies are generally different from the ratios ofsmall integers, multi-frequency vibrations have the distinctive quality of being ge-nerally non-periodic. On the other hand, vibrations expressed in terms of quasinormalcoordinates are found to be largely quasi-periodic. For further details on the subjectrefer to a monograph by TONDL (1970b).

Analysis of systems with several degrees of freedom, especially those capable ofoscillating with multi-frequency vibrations, is difficult for a number of reasons, one ofthem being the complications which are apt to arise in connection with the graphicalrepresentation and interpretation of the trajectories in the phase space.

Even less amenable to analysis are self-excited systems which are subjected toadditional external or parametric excitation. At resonance, i.e. when the external orparametric excitation is at resonance with respect to the abbreviated (undampedlinearized) system, a phenomenon called vibration synchronization occurs. The termrefers to a situation in which the component of the excited vibration fully synchronizeswith the corresponding component of the self-excited vibration, and the frequency iscontrolled by the self-excitation frequency. Details are given in Chapter 5.

1.5. Stability 31

1.5. Stability

In nearly all stability investigations of deterministic systems, stability is understoodin the sense of Ljapunov as uniform continuity: A solution y(v) of the vibrationaldifferential equation

y" + Ay = O(y, y',r) (1.5, 1)

with 0 being continuous in y, y', z is stable if for given positive e any (not necessarilyperiodic) solution')

Y(T) = Y(T) + z(z)

of (1.5, 1) satisfies the condition

Iz(Z)I, Iz (z)I < v

for allz>zoif

Iz(zo)I, Iz'(zo)I < 17

holds with a positive n = r7(e), in other words, if the solutions keep a given maxi-mum distance for all time when they have a certain positive maximum distance foran initial time z = zo. A solution is unstable if this condition is not fulfilled. A solutionis asymptotically stable if additionally

lim {Iz(z)I, Iz'(v)I} = 0T-00

holds, that is, if every solution Y(T) tends to y(z) for r -> oo.If we subtract the differential equations (1.5, 1) written down for Y and y we get

the variational equation

z" + Az = 0(y + z, y' + z', z) - 0(y, y', z)

which can be expressed (under the assumption of continuous partial derivatives) inthe form

z" + Az = u(z) z + v(z) z' + P(z, z', z) (1.5, 2)

where['(z, z', z) Iz'I},0->0 as Max{Izj , .

{I I I I}Max z , z

Neglecting the non-linear expression W we get the linear variational equation

z"+Az=u(t)z+v(T)z'. 1.5, 3)

The stability of a solution y(z) can be determined by inserting this solution in thevariational equations (1.5, 2) or (1.5, 3). If we use the linear variational equations(1.5, 3), we get infinitesimal stability. In general infinitesimal stability also determinesthe stability of the non-linear equations (1.5, 2). This is denoted stability in the large orpractical stability (compare for instance CESARI (1963), MALKIN (1952)).

The differential equation

z" = V(z) z + v(z) z' (1.5, 4)

1) We assume that the solution Y(T) as well as y(a) exist for every r > r0, compare Cesari(1963), p. 4.


whose coefficients are real continuous periodic functions with period 2n,

V(T + 2ic) = V(z) , v(z + 2ir) = v(r) (1.5, 5)

has, following for instance HORN (1948), a fundamental set of two linearly independentsolutions zl, z2, and any solution of (1.5, 4) can be written as a linear combination ofzl and z2. In particular, the functions

zl(r) = z1(r + 27c) , z2(r) = z2(r + 2n) , (1.5, 6)

which arise from the solutions z1, z2 if r is replaced by r + 27r, are solutions of (1.5, 4)because of (1.5, 5) and are therefore linear combinations of zl and z2,

zl = C11z1 + C12z2 , Z2 = C21z1 + c22z2 . (1.5, 7)

They are also linearly independent and therefore a fundamental set of solutionsbecause a linear dependence for every r and (1.5, 6) would yield the linear dependenceof z1, z2, hence

C11 C12 0 . (1.5, 8)C21 C22

The solutions of (1.5, 4) are in general not periodic, but there exists at least onesolution which only multiplies with an (in general) complex constant k if 'r is replacedbyr+2Tr:

= 1c . (1.5, 9)

To show this, we writerite C as a linear combination of z1, z2,

S =1121+1222 (1.5,10)

Substituting in (1.5, 9) gives

1121 + 12z2 = k(11z1 + 12z2)

which is, because of (1.5, 7), a linear relation between zl and z2, the coefficients ofwhich are zero because zl, z2 are linearly independent :

(cn-k)11+c2112=0, (1.5,11)c1211 + (c22 - k) 12 = 0 J

Because both quantities 11, 12 do not vanish simultaneously, the equation

C11-k c21

c12 c22-k=0 (1.5, 12)

holds; this is called the characteristic equation. The roots of this equation

011 + C22 1k1, 2 = 2 2 k(Cll - C22)2 + 4c12C21 (1.5, 13)

are non-zero because of (1.5, 8).If the roots k1, k2 are unequal, (1.5, 11) yields two pairs of values 11, 12 which lead

by means of (1.5, 10) to two solutions Sl, y2 for which

: = S2 = k2S2 (1.5, 14)

1.5. Stability 33

hold. These two solutions are linearly independent and therefore constitute a funda-mental set of solutions because a linear relation

x24_2 = 0 (1.5, 15)

with non-vanishing coefficients would, if r were replaced by r + 27c, lead to k1 e14l+ k2x24_2 = 0 because of (1.5, 14), thence to k1 = k2 because of (1.5, 15).

Setting

i = ecir Z, (T) , b2 = eeaT Z2(r) , (1.5, 16)

substituting in (1.5, 14), choosing Q1, e2 such thate2ne1 = kl , e2nei = k2 , (1.5, 17)

and substituting in (1.5, 14), we find

Z1(r + 27t) = Z1(r) , Z2(t + 27c) = Z2(r)

i.e., the functions Z1, Z2 are periodic with period 27r. The quantities Q11,02 defined by(1.5, 17) are called characteristic exponents, they, and the functions Z1, Z2, are in generalnon-real. The imaginary parts of the characteristic exponents are chosen in theinterval

-s<Im(el),Im{(J2}<2 (1.5,18)

which is possible because e2ni = 1.If the roots k1, k2 are equal, they are (because of (1.5, 13)) equal to (c11 + 622)/2 and

therefore real. In the case 612 = 621 = 0, the equations (1.5, 7) are already of the form(1.5, 14). If on the other hand at least one of these quantities, 621 say, is not zero,a solution of (1.5, 11) is 11 = 621,12 = kl - ell, and the corresponding solution (1.5,10),

b1 = 62121 + (k1 - 611) z2 , (1.5, 19)

yields, when r is replaced by r + 2z,

b1 = k14l . (1.5, 20)

As now a second solution of this kind does not exist, we choose as second solution2 = z2. Because of (1.5, 7), (1.5, 19),

S2 = z2 = 62121 + 62222 = S1 + (611 + 622 - k1) 4_2

that is

S2= klb2+S1 (1.5,21)

holds. We choose

ee,r Z1(r) , 42 = eOyz 1Z2(r) +2 v

Zl(t)1L 1

Substituting in (1.5, 20), (1.5, 21) and choosinge2nn, = k

1

shows that the (now real) functions Sl, 42 are periodic with period 276. Thus we haveproved the following theorem.

Floquet theorem. The differential equation (1.5, 4) with continuous and periodiccoefficients with period 21r has a fundamental set of solutions of the form (1.5, 16)3 Schmidt/Tondl


or the formi = eQ2T Z1(T) , 2 = e°'i [Z2('r) + KrZ1(r)l (1.5, 22)

with in general complex constants 0, 02 satisfying (1.5, 18), continuous and in generalcomplex periodic functions Z1, Z2 with period 27r and constant K which can be zero.

A result of the form (1.5, 16) or (1.5, 22) of the solutions is the following corollary.

Corollary. If the real parts of the characteristic exponents are negative, all solutionstend to zero for r --)- oo, in other words, we have asymptotic stability. If on the otherhand the real part of at least one characteristic exponent is positive, solutions existwhich are unbounded for v -> oo, that is, we have instability.

For a stability investigation in the critical cases when characteristic exponents withreal parts equal to zero, but no characteristic exponent with a positive real partexist, compare MALKIN (1952).

The method based on these results and on the knowledge of solutions of the vibra-tional differential equations is called Ljapunov's first method. With this method thestability will be determined in Section 2.4 and Chapter 5. Extensions to systems ofdifferential equations are given in Chapter 6.

A method based not on knowledge of solutions of the differential equation but onthe construction of Liapunov functions with certain properties is known as the secondmethod of Ljapunov, compare for instance MALKIN (1952).

If several steady stable solutions exist, infinitesimal stability is often called localstability. When the solution is asymptotically stable, only one steady solution existsand divergent vibrations are absent, we speak of absolute stability.

In connection with the periodic motions of planets, another definition of stabilityhas been introduced, orbital stability. A periodic motion is orbitally stable if thismotion and disturbed periodic motions correspond with (closed) phase curves in they, y plane which differ slightly if the disturbances are small. For an orbitally stablemotion, the period can be slightly different from the periods of the disturbed motionsso that after a long time the distance of the solutions increases infinitely. This showsthat an orbitally stable solution can be unstable in the sense of Ljapunov, but anorbitally unstable solution is always unstable in the sense of Ljapunov. Very often thesetwo forms of stability coincide (compare for instance STOKER (1950), KLOTTER (1980)).

As an example of a system, which is orbitally stable and unstable in the sense ofLjapunov, consider the free motion of an undamped pendulum. The motion in thephase plane is represented by closed trajectories. If two sets of slightly different initialconditions are applied, the trajectories lie close together. At a certain time, however,the distance between the corresponding points is no longer small because the periodof free vibration of a pendulum depends on its initial deflection.

It is only possible to mention recent investigations of non-periodic, irregularlyoscillating behaviour of deterministic systems the trajectories of which are verysensitive to small variations of the initial conditions and which is termed chaoticbehaviour, also interpreted as a transient motion of infinite duration. Connected withsuch a behaviour are attractors (limit sets in the phase space against which everysolution tends after long time) which are neither a point nor a limit cycle nor a surfaceand which are called strange attractors. A general view of these investigations can begot from YuNG-CHEN Lu (1976), TROGER (1982, 1984), HOLMES (1980), HAKEN (1982),PoPP (1982), SPARROW (1982), GUCKENHEIMER and HOLMES (1983), SCHMIDT (1986).In this book, strange attractor problems are not treated ; attractors are always under-stood in the sense of simple attractors.

2. Methods of solution

2.1. The harmonic balance method

This method is based on a very simple idea and that is why it may seem less rigorousmathematically and not exact enough in approach. However, when combined withexperience and sound engineering judgment on the part of the analyst it can be turnedinto a very efficient tool for solving non-linear problems, sometimes even excellingprocedures which are grounded more thoroughly in mathematics. It may, for examplebe usefully employed in solution of resonant vibration of systems excited by a harmo-nic force, in which one or two harmonic components can be assumed to predominatein the response. In problems of excited vibration, the solution of a particular type isalways restricted to a definite interval of the excitation frequency. Proceeding froman analysis involving considerations of the symmetry of the characteristics, the typeof resonance and the form of the response, the engineer assumes a certain form of thesolution, substitutes it in the equations of motion and compares the coefficients of thesame harmonic components; from this last step the method derives its name.

To illustrate the application of the method, consider a buffing system described bythe equation

(2.1,1)mx+hx+ex +ex3=Pcoscot.where m is the mass of the system, h is the damping coefficient, c is the stiffness ofthe linearized spring, and s is the coefficient of the non-linear term of the restoringforce. It is usually advantageous to convert the equation to the dimensionless form.Denoting by Plc = xo the static deflection of the linearized system produced by theaction of force P, writing

x c w h/m Ex0=y, =coo, -=f7, x , =Yxo fm coo coo c

and using the time transformation

coot=x (2.1,2)

(2.1, 1) becomes

y"+xy' + y+Yy3=cosgb. (2.1,3)

The task now is to find the solution in the main resonance, i.e. 1. Since the charac-teristics of the damping and restoring forces are both symmetrical and the excitationis harmonic without a constant term, this solution can be approximated (for y < 1) by

y = a cos rya + b sin ?IT = A cos (fix - q,) (2.1, 4)

3.

36 2. Methods of solution

in which the following relations hold:

Acos(p=a, Asingq=b, A2 = a2 + V. (2.1,5)

Substituting (2.1, 4) in (2.1, 3) and comparing the coefficients of cos 7jr and sin q-rgives the following algebraic equations for determining a and b, or possibly, A and c :

(1+4yA2-112)a+x'qb=1,-xiqa+(1+4b=0. (2.1,6)

Using (2.1, 5), equations (2.1, 6) after rearrangement (the first multiplied by cos 99is added to the second multiplied by sin 99; the first multiplied by sin 92 is added to thesecond multiplied by -cos q)) take the form

(1+4yA2)A=cosg9,x7A = sin T (2.1,7)

from which we get the equations[(1 +. 4 yA2 - 12)2 + -I x2n2} A2 = 1 , (2.1, 8)

tan 99 = xv]/(1 + 4 yA2 - q2) . (2.1, 9)

Equation (2.1, 8) readily yields the function i' (A)

12 4

yA2) x2 + ' x41 (2.1, 10)

and hence also the inverse function A(71). Equation (2.1, 9) is used to determine P asa function of rj.

Equations (2.1, 7) can also be obtained directly. Shifting the time origin bycpln, (2.1, 3) becomes

y" + xy' + y + yy3 = cos (,qv + T) (2.1, 3a)

whose solution can be approximated by

y = A cos riz . (2.1, 4a)

The stability of the solution can be examined by means of the procedure used in thecase of small disturbances, i.e. by substituting the approximate solution for the exactone. In the example being considered this involves substituting y = A cos nv +for y in (2.1, 3a). The variational equation thus obtained, i.e.

" + s yA2(1 + cos 21T) = 0 (2.1, 11)

is the well-known Mathieu equation. The boundary of the instability interval of thefirst order is approximately determined as follows.

The solution on the boundary of the instability region which is represented by theMathieu function of the first order is replaced by a harmonic function, i.e. approximat-ed by

= u cos nz H- v sin q7 .

Substituting (2.1, 12) in f2.1, 11) gives the equations

(1 + 4yA 2 - q2) v = 0 .

(2.1, 12)

2.1. The harmonic balance method 37

The condition of non-trivial solution/ (u + 0, v + 0) leads to the equation

(1 + 2 yA2 - 712)2 + (xr1)2 - (4 yA2)2 = 0

and in turn, to the equation for obtaining the function q(A) :

(12)1,2 = 1 + 2 7A2 2x2 [(4 yA2)2 - (1 + + yA2) x2 + q X41/2 .

(2.1, 13)

In the (A, r1)-plane, this curve forms the boundary between the stable and the un-stable solutions.

Fig. 2.1, 1 shows the A(i) curve drawn for the case of x = 0.05 and y = 10-2 (heavyfull line - stable solution, dashed line - unstable solution) as well as the bound-ary of the solution stability calculated from (2.1, 13) (light solid line). Since sub-sequent chapters contain a number of diverse examples solved with the aid ofthis method, it is unnecessary to continue here with illustrations of its applicability.However, mention should be made at this point of the suitability of the method fordealing with the steady vibration of self-excited systems.

Fig. 2.1, 1

If the system to be examined has one degree of freedom and the solution can beapproximated by a single harmonic component, the procedure is even simpler than inthe previous case. Since the phase shift between excitation and response need not betaken into account, the solution can be approximated by the simple equation

y=AcosQt. (2.1,14)

Substituting this solution in the equation of motion and comparing the coefficients ofcos Dr and sin Q-r results in two algebraic equations which readily yield the unknownamplitude A and the unknown frequency Si.


Chapter 9 will show an expedient procedure for solving more complicated self-excited systems with several degrees of freedom. In the case that the solution involvedis of single-frequency type, no transformation to the quasi-normal form (such as thatrequired for multi-frequency solutions) is necessary and the equations can be pro.cessed directly. If, moreover, the single-frequency vibration is close to a harmonicmotion, the harmonic balance method yields very accurate results.

2.2. The Van der Pol method

As in the harmonic balance method, the solution is approximated by a definite form ;in the vicinity of the stationary solution, the coefficients of the various terms of thisform are assumed to be functions slowly varying with time. To illustrate this, considera system governed by the differential equation (already transformed to the dimension-less form)

Y" + my, + f(y', y) = cos rpt. (2.2, 1)

In the case of parametric excitation, f might also be a periodic function of time(f = /(y', y, nz)), and the right-hand side of (2.2, 1) absent. Assume that the coefficientsx and o//ay' are small, and that the function f satisfies the conditions of the resonantvibration being close to the harmonic one. The solution in the main resonance canthen be sought in the form

y = a cos77T+bsinigr=Acos(,q -(p) (2.2,2)

where a, b or A, 99 are slowly varying functions of time for transient vibration andconstants for the steady solution. If the function/ has an asymmetric non-linearcharacteristic, the solution is approximated by the form

y = a cos it -}- b sin qz -}- Y. (2.2, 3)

Note: It is advantageous in some cases to use a solution of the form

y = U exp (iir) + V exp (-ijt) (2.2, 4)

where

U+V=Acosqp=a, i(U-V)=AsinT=b. (2.2,5)

On the assumptions put forward' above, the problem of solving (2.2, 1) may bechanged to an analysis of the system

a' = F1(a, b) , Y = F2(a, b) (2.2, 6)

or (in case of an asymmetric characteristic) the system

a' = F1(a, b, Y) , b" = F2(a, b, Y) , F3(a, b, Y) = 0 . (2.2, 7)

The stationary solution is represented by the singular points

Fj(a, b) = 0, F2(a, b) = 0 (2.2, 8)or

F1(a, b, Y) = 0 , F2(a, b, Y) = 0 , F3(a, b, Y) = 0 . (2.2, 9)

Since the procedure for obtaining solutions in the phase plane is discussed separatelyin Chapter 4, no analysis of system (2.2, 6) or (2.2, 7) will be presented at this point.

2.2. The Van der Pol method 39

Suffice it to say that for the same form of approximation, the steady solution isidentical with that obtained by the harmonic balance method. For the Duffing system,for example, equations (2.2, 8) are identical with (2.1, 6).

Before turning to an examination of the stability of steady vibration for smalldisturbances, a general note on the process of solution will be in order. Frequently,especially when trying to find solutions in secondary resonances (or when the excita-tion is periodic but non-harmonic), approximation (2.2, 2) or (2.2, 3) is not satisfactoryand additional harmonic components, for example, the terms c cos 2?p, d sin 2rla, etc.,should also be considered. The literature contains numerous references to the case ofa system with n degrees of freedom defined by n second-order differential equationswhose solution is approximated by a form having K terms with slowly varying coeffi-cients (K > 2n) ; the problem is reduced to an analysis of a system of K differentialequations of the first order. Any solution of this new system is thus defined by a grea-ter number of initial conditions than that of the original system, and this is physicallyinadmissible (for further details refer to ToNDL (1970 a)). Some authors claim that theinitial conditions of the original system form a subset of the set of initial conditionsof the new system of K first-order differential equations. A procedure which appearsmore correct physically consists of choosing only 2n time-variable coefficients andassuming the remaining K - 2n coefficients to be constant. This leads to 2n differen-tial equations of the first order and to K - 2n algebraic equations.

The stability of the steady solution for small disturbances is determined using thecharacteristic equation of the linearized perturbation system

aFl aFlas ab

aF2 aF2

aa ab

or

=0,

OF, - 8F1 aFlas ab ayaF2 aF2 _ Aas 8b

OF,

as

aF2

ay

aF3 aF3 - Iab aY

=0,

(2.2, 10)

(2.2, 11)

i.e. on the basis of an examination of the stability of singular points (see Chapter 4 forthe types and stability of the singular points).

As shown in Appendix I of ToNDL's monograph (1970), the vertical tangent ruleapplies to system (2.2, 6) and others, i.e. the points of the A(q) curve at which thetangent is vertical form the boundary between the stable and the unstable solutions.

Proof: Recalling the relations

a=Acosg9, b=Asinip, A=(a2+b2)112

the slope of the tangent to the A(ii) curve is defined by the equation

dA_aAda+aAdb (2.2,12)drl as drl ab drl


where

as= a(a2 + b2)-,l2 , ab = b(a2 + b2)-112 . (2.2, 13)

Eliminating the derivatives daldrt and db/drt from the system

dF,_aF,da aFldb aFl0

dF2_aF2da aF2db aF2

drt as drt + ab drt + art ' drt as drt + ab drt + apt0

(2.2, 14)and using the notation

da =

leads to

aF, OF1

art ab

aF2 aF2- art ab

db =

aF, aF,as - aqaF2 aF2

as - apt

da da db dbart -d' art=4'

Substituting (2.2, 16) and (2.2, 13) in (2.2, 12) gives

dAa

= [A (a' + b2)111-1 (ada + bdb) .

7-7

The characteristic equation (2.2, 10) can have the form

)2+al;[-+ -a2=0,where

4=

aF, OF,

as ab

aF2 aF2as ab

__

-- (apl aF2

) Ia2 = A -

In systems with positive damping, the first condition of stability

(2.2, 15)

(2.2, 16)

(2.2, 17)

(2.2, 18)

_ aF, aF2al - (aa + ab > 0 (2.2, 19)

is always satisfied and it is only the sign of the term d that is decisive; consequently,the limit of stability is at

a2 = d = 0. (2.2, 20)

At this point, dA/drt increases beyond all bounds (see (2.2, 17) provided that (unlikein an extraordinary case) the term ada + bdb is not simultaneously equal to zero). Itfollows from the above that:

The necessary and sufficient condition for applying the criterion of vertical tangentsto the resonant amplitude curve of the system (2.2, 1) is

sgn (aal +0F2\ - -1 .

As shown in Chapter 8, the vertical tangent rule is not a sufficient criterion in somecases as it provides only partial information about the system's stability. In the

2.3. The integral equation method 41

example presented there, owing to the softening characteristic of the restoring forceof the spring, divergent vibration is likely to exist next to the steady one in somedomain of the parameters, and the slope of the characteristic can even become negativefor larger deflections.

2.3. The integral equation method

The method widely used in this book is the method of non-linear integro-differentialequations (in short: ,integral equation method).

Vibrations of various kinds can be modelled by a differential equation

y" -F- Ay - 0 (2.3, 1)

where dashes denote derivatives with respect to a dimensionless time % and

0 = O(y, Y" --,;,r) = O[t]

is given by a power series in y, y' and parameters e, (p = 1, 2, ... , P) which may alsodepend on z.

Restricting ourselves for simplicity to continuous solutions with continuous firstand second derivatives, we can prove the following theorem (compare SCHMIDT (1961,1975)) : Every periodic solution of the differential equation (2.3, 1) is a solution of theintegro-differential equation

2,1

y(x) = f G(-r, a) c[a] da + &?1(r cos nz + s sin nvt) (2.3, 2)0

where

G(T'a)

n f vECos (z V2a)

is the corresponding generalized Green's function,

8A'= (1it

0

for A = n2, n being an integer,otherwise

is the Kronecker symbol, and

so that the denominator does not vanish. The bifurcation parameters r, s appearing inthe resonance case

2=n2are to be determined by the bifurcation or periodicity equations

2n 2x

r = r y(z) cos nt dz , s = I I y(Z) sin nv dr

0 0

which are equivalent to2n 2n

f O[r] cos me dr = f O[r] sin nz dr = 0 .0 0

(2.3, 3)

(2.3, 4)


The solutions of (2.3, 2) and (2.3, 3) respectively (2.3, 4) can be found by the methodof successive approximations based upon the following equations for the approximatesolutions yj(r), j = 1, 2, 3, ...,

2.

yj(r) = f G(t, 6) Oj_1[6] da 6' (r cos nz + s sin ni)0

where

00['c] _ 0(0, 0, en;,r) ,

0s['c] _ O(yf, y;, en; 'r) , j = 1, 2, 3,...

The convergence of the method can be proved under the assumption that the para-meters s , are less than certain upper bounds. In most cases, the actual evaluation ofthe radius of convergence is a very complicated problem, and one has to contentoneself with a verification of the solution for special values by direct numerical inte-gration of the differential equation. A second derivative y" in 0 can be handled in thesame way, for the proof of convergence it can be substituted successively by means ofthe differential equation.

As a simple example for the integral equation method, we once more consider theDulling equation (2.1, 1). We introduce a dimensionless time by

Z=cot

- derivatives with respect to z are denoted by dashes - instead of (2.1, 2) and write COwith a fixed frequency w0 and the small frequency variation a in the form

cu=cu0(1+a).

As in Section 2.1 we relate the deflection x to the static deflection of the linear systemby setting

cY= px

assume the main resonance cuo = cam and use the abbreviations

_ h P28X

' Y - c3 .

mco0

Instead of (2.1, 3) we get

y" + y = -(2 + a) ay" - (1 + a) xy' - Yy3 + cos r =) . (2.3, 5)

Substituting the first approximation

y1=rcosz+ssinrinto (2.3, 4) and taking into consideration only the first power of the small coefficientsa, x yields

(jA2_2cc)r+s=1-xr+(4yA2 -2a)s==0instead of (2.1, 6). Elimination of r

respectively``

s in the first equation by help of thesecond equation gives for the resonance amplitude

A = Ir2+82

2.3. The integral equation method 43

the formula

`1t 3y A2 - 2a)2

) + x2 A2 = 1 (2.3, 6)

instead of (2.1, 8) which can be written

2a=34 A2 ± fI2-x2(2.3, 7)

instead of (2.1, 10)Formulae (2.1, 8), (2.1, 10) seem to be a better approximation than (2.3, 6), (2.3, 7).

If we had also taken into consideration the second power of a, x in first approximation,we would have got as easily the formula

31 + (1 + a)2 x21 A2 = 1 (2.3, 8)y A2 - 2a - 2a2

2

which exactly corresponds with (2.1, 8) because il = 1 + a. But the second approxi-mation terms

y2 = -2a (r cos T + s sin T) + x(s cos T - r sin T)

yield the additional terms

0$ _ (-4a2 + x2) (r cos T + s sin T) + 4ax(s cos T - r sin T)

of second order in a, x, that is the complete second order amplitude formula

2r3y A2 - 2a + a2 - x21 + (1 - 3a)2 x2 A2 = 1 . (2.3, 9)

It shows that the partial consideration of second order terms by a one-step method in(2.1, 8) respectively in (2.3, 8) can not determine the real second order correctionswhich are given by (2.3, 9).

The main advantages of the integral equation method are:

1. The use of small parameters leads to solutions to every degree of accuracy needed.

2. The use of several independent small parameters saves unnecessary evaluations(in our example the evaluation of the complete second approximation).

3. The simple and clear mechanism of finding the successive approximations enablesus to estimate their influence without explicitely evaluating them.

4. The method also permits us to investigate different multiple resonances (Chapters 6and 12), stability (Section 2.4, Chapter 5), and systems with many degrees of free-dom where it is often necessary to adapt the method to the special problem in orderto confine oneself to a reasonable number of evaluations (Chapters 6 and 12), aswell as narrow-band random excitations (Chapters 10 and 12).

5. The method is especially suitable for combination with computer algebra methodsin order to transfer extensive analytical evaluations susceptible to errors to a com-puter. Hints on computer algebra methods are given in Sections 6.9 and 12.7.


2.4. Stability conditions

The integral equation method together with the Floquet theorem of Section 1.5 leadsto conditions for stability. To find them, we consider the linear variational equations(1.5, 3)

z" -- Az = u(r) z + v(t) z' . (2.4, 1)

The coefficient functions u(v), v(T) can be assumed to be real, continuous, periodicwith period 2n and sufficiently small in modulus, with the absolutely and uniformlyconvergent Fourier series

U(T) = uo + 20Y, (u1 cos jt + U, sin jt) ,j=100

v(r) = vo + 2 1 (v1 cos jr + V, sin jv) .j=1

Corresponding to the Floquet theorem, we write the solution of (2.4, 1) in the form

z = e QT Z ,

where a is the characteristic exponent, and get the differential equation

Z" + AZ = 0,I

(2.4, 2)0 = [u(v) + eV(t) - e2] Z + [v(t) -2,o ] Z'

for the function Z(r) which is periodic with period 2n and in general complex.The integral equation method of Section 2.3 is applicable to (2.4, 2) which can be

shown by applying it to the real and the imaginary part of (2.4, 2) and then againcombining the real and imaginary parts (compare SCHMIDT (1969b, 1975)). The firstapproximation is

Z,(t) = 8x'(R cos nt + S sin nr) ,

hence for the resonance case A = n2,

01 = (uo + v0e - e2) (R cos nv + S sin ni)00

+ Y_ [njR cos (n - j) t + ujS sin (n - j) v + ufR cos (n + j) v + ujS sin (n + j) rjetUIS cos (n - j) ,r - U1R sin (n - j) ,r - UjS cos (n + PT + UIR sin (n + j) v]

00

+ e Z [vjR cos (n - j) v + v1S sin (n - j) r + vjR cos (n + j) r + v1S sin (u + j) vj=1

+ VIS cos (n - j) v - V1 R sin (n - j) v - VIS cos (n + j) T + V1R sin (n + j) v]

+ n(vo - 2e) (S cos nv - R sin nv)00

+ n [v1Scos(n-j)t-v1Rsin (n-j)r+v1Scos(n+j)v-vjRsin (n+j)rj=1

- VIRcos(n-j)v-VISsin (n-j)t+ViRcos(n+j)v+VISsin (n+j)v].The periodicity conditions (2.3, 4) now read

(u+ v- e2) 1 + (u2l + v2n- nV2)/R

(_)+ (U2n + V2ne + nv2n) (R) + n(v0 - 2e) (-R) = 0,

2.4. Stability conditions 45

the determinant condition for non-vanishing (in general complex) solutions R, S yields

(u0 + v0e - p2)2 - (u2n + v2,, - nV2n)2

-(U2n + V2,2 + nv2n)2 + n2(vo - 2Q)2 = 0 . (2.4, 3)

The characteristic exponent e is of no higher order of magnitude than the Fouriercoefficients appearing because otherwise approximately the equation

e4 + 4n2Q2 = 0

that is e2 = -4n2 would hold, in contradiction to (1.5, 18). Therefore the terms v0e,v24, V2,, and e2 in (2.4, 3) can be omitted in the same way as products of the Fouriercoefficients have not been taken into consideration in the above approximation, and(2.4, 3) yields

2n2 = nv0 ' Y (u2n - nV2n)2 + (U2n + nv2n)2 - u0 .

Because of the Corollary in Section 1.5, we have asymptotic stability i/ and only i f theinequalities

v0 < 0 (2.4, 4)

and

n2v0 > (u2n - nV2n)2 + (U2n + nv2n)2 - u0 (2.4, 5)

hold, but certainly instability if at least one of these inequalities holds with the oppo-site sign.

As an example we investigate the stability of the Duf Ping equation (2.3, 5). Thecorresponding linear variational equation is in the approximation used for the so-lution (2.3, 7)

Using

we get

z"+z=2az-xz' -3yyiz.

2y1 = A2 (r2 - s2) cos 2-c + 2rs sin 2r ,

uo=2a- 32A2,

u2= -4 (r2 -s2), U2 - 2yrs, v2= V2=0

The first stability condition (2.4, 4) holds with x > 0, the second stability condition(2.4, 5) requires that

x2 > 9Y 2 (r2 - s2)2 +4 2

2 r282 - (2a - 3 2A2)2 = 9 62 A4 - (2a - y A2)2

or, after inserting (2.3, 7),

3A

-- x2 + 2> 0 .

Differentiation of (2.3, 7) yields

2A1 - x2 da = 3y A2

A2 dA 2


a comparison with (2.4, 6) shows that the solution A of (2.3, 7) with the upper sign(the upper branch of the resonance curve) is asymptotically stable if da/dA < 0, thelower solution if da/dA > 0 whereas the opposite sign causes instability. The pointsof the resonance curve with vertical tangent correspond to the boundary points of theinstability region.

2.5. The averaging method

The well-known averaging method of Krylov and Bogoljubov (compare BOGOLJUBOVand MITRorOL'SKI (1963), KLOTTER (1980)) will be introduced here by means of theDuffing equation (2.3, 5) which we write in the form, up to small terms of higher orderof magnitude,

y" + y = 2xy -xy' -yy3+cost.In connection with the solution

y = ao cos (tt + 0a) (ao, Oo constants)

for vanishing right-hand side, a solution of (2.5, 1) is sought for in the form

y = a(Z) cos [t + t(a)] = a(2) cos qu(a)

(2.5, 1)

(2.5, 2)

with the slowly varying functions a = a(a) and 0 _ 9(r) for which a condition canbe fixed:

a' cos p - a#' sin 99 = 0 . (2.5, 3)

Differentiation yields

y'=a'cosT-a(l+0')sing9=-asinp (2.5,4)

from (2.5, 3), hence

y" = -a' sin q, - a(l + 0') cos T

so that the differential equation (2.5, 1) reads

-a' sin T - aO' cos 99 = 2ay - xy' - yy3 + cos a .

Solving this equation and (2.5, 3) and substituting (2.5, 2), (2.5, 4) in the right-handsides gives the equations in standard form

a' = (-2aa cos T - ra sin 97 + ya3 cos3 99 - cos a) sin(2.5, 5)

at9' _ (-2aa cos 99 - xa sin ip + ya3 cos3 99 - cos a) cos ,

two differential equations of first order for a, 0 instead of the differential equation ofsecond order (2.5, 1).

Both the right-hand side of (2.5, 1) and the right-hand sides of (2.5, 5) are smallin comparison with the left-hand sides. The basic idea of the averaging method con-sists in averaging the terms on the right-hand side containing 99 over one period (theslowly varying term cost being assumed as constant during this period) :

a'=-2a--sin0, a#'=-aa + 3g a3- -cos 0.

2.5. The averaging method

For a stationary solution, a' = t9' = 0, there follows the amplitude formula

2a = 4Y a2 ±V

as - x2

which is the same as (2.3, 7), and

Xtan,O =2a - 3Y

a24

47

The advantage of the averaging method lies in the possibility of evaluating not onlystationary, that is periodic solutions but also non-stationary ones such as transitionprocesses or random vibrations. The drawback lies in a comparatively complicatedway of evaluating higher approximations and of estimating the accuracy. We willuse this method, also in higher approximation, in Section 10.1 and in Chapters 11 and12 in connection with random vibrations.

3. Auxiliary curves for analysisof non-linear systems

3.1. Characteristic features of auxiliary curves, particularly thebackbone curves and the limit envelopes

In harmonically excited systems, especially in systems with one degree of freedom,the specific characteristics of the auxiliary curves have been found to be very usefulfor preliminary qualitative analyses of stationary vibration as well as for identifica-tion of the various elements (e.g. damping) of the system being examined on the basisof experimental results. In the former case they enable the analyst to make a promptestimate of the basic properties of the system and of the effect of various parameterson its behaviour, in the latter, to identify the specific properties of the system andin turn to formulate a suitable analytic expression of the forces acting in a particularelement for the purpose of a mathematical model.

Let us first consider the characteristics of the so-called backbone (or skeleton) curve,and of the curves connecting the points at which sin cp = const (q9 is the phase anglebetween response and excitation). The limiting case of the latter curves (sin 99 = ±1;the minus sign has no meaning except in special cases stated farther on) is the so-calledlimit envelope (this term was proposed by ToNDL (1973d)). The backbone curves havebeen used in routine analyses for a long time; the application of the second type ofcurves, the limit envelope and the sin 99 = const curves, is less common. The limitenvelope was used for the first time by ABRAMSON (1954) and later applied to quali-tative analyses by KoLOVSKIJ (1966). Assuming linear viscous damping, NovAB(1963) employed the sin (p = const curves for the construction of the backbone curvefrom the measured vibration amplitude-excitation frequency dependence. Otherpossible uses of the properties of these curves have been investigated and reported byTONDL (1971b), (1973d), (1975c), (1979a), (1980c). TONDL was the first to proposethe use of the limit envelopes for the identification of damping and to extend the fieldof application of these curves to parametrically excited systems (1975 c), higher-ordersystems (1974b), (1977a) and to systems with several degrees of freedom (1978a).

To facilitate the subsequent explanation, the basic properties of the auxiliary curveswill be described using the example of a single-degree-of-freedom system having massm excited by a harmonic force whose amplitude is either constant (P) or proportionalto the square of the excitation frequency (inertial excitation - emo(o2). To simplifymatters, it is assumed that the damping and the restoring force have both symmetriccharacteristics and can, therefore, be expressed in terms of the functions c[I + f (IyI)] y,x(Iy), IyI) sgn y where y is the deflection and iy the velocity of the mass. A system ofthis kind is described by the equation

my + x(1 y1, IyI) sgn y + e[1 + f (IyI )] yP cos cut

(3.1, 1)emow2 cos cut

3.1. Characteristic features. Backbone curves and the limit envelopes 49

Using the harmonic balance method (described in Chapter 2) and introducing thetime shift t0 = c/w, i.e. substituting, in (3.1, 1) cos ((ot + 99) for cos wt, and approximat-ing the stationary resonance solution by

y = A cos wt (3-1,2)

the following equations are obtained for determining A and ry:

M(02]1f P cos

91(3.1,3)

E m0w2 cos q)

P sin 99K(coA, A)

em0w2 sin cp

where2a/w

F(A) r /(JA cos wtl) cos2 wt dt

02n/(o

2

K(wA, A) =

n

x(I -wA sin wtj, (A cos wtj) Isinw(ot

tldt .

0

The backbone curve is defined by the equation obtained from (3.1, 3) in whichcos 99 = 0, i.e. T = n/2, viz.

c[1 + F(A)] - mw2 = 0 ;this equation can be given the form of the dependence of frequency on vibrationamplitude A (which is identical with the dependence of the natural frequency of theundamped vibration on the initial deflection at zero initial velocity), viz.

m.12

w = [1 + F(A)] _ = S2(A) . (3-1,5)

As shown schematically in Fig. 3.1, 1, the various points of the resonance curveA(w) can also be obtained as the points of intersection of the curve defined by (3.1, 4)in which sin 99 = const = S

K((oA, A) _ PS (3.1, 6)CM,(02S

A

ci

Fig. 3.1, 1

4 Schmidt/Tondl

60 3. Auxiliary curves for analysis of non-linear systems

and the pair of curves defined by (3.1, 3) in which cos 92 _ C

A[c + cF(A) - mw2] = J ±PCl tsmau)2C

If

(3.1, 7)

K(wA,A)>0 (3.1,8)

for any A, then

0<sin99<1 ; (3.1,9a)

if the function K(wA, A) assumes negative values in a particular interval of A,

-1 < sin q2 <I . (3.1, 9b)

For the extreme values, i.e. sin T = 1 or sin T = t1, equation (3.1, 6) becomes

K(wA, A) = j 1 P (3.1, 10a)Emow2

or (if (3.1, 8) does not apply to all A),

K((oA, A) _-I-P

(3.1, 10b)±Empw2

These equations define the limit envelope.The limit envelope divides the (A, co) plane into two regions: one for which

K((o, A, A) < P (3.1, l l a)EMO(4)2

or

-P(3.1,11b)

-Emow2Emow2

the other, in which inequalities (3.1, lla, b) are not satisfied.The response curve A(w) can lie only in the first region. The limit envelope thus

forms the envelope of all possible response curves which touch it at the points atwith op = ±n/2. These points are the points of intersection of the limit envelo-pe AL(w) and the backbone curve As((o) (Fig. 3.1, 2) (Subscripts L and S are used todistinguish these curves from the resonance curve A(w)). At these points the limitenvelope and the response curve A(w) have a common tangent.

A

AL

As

Fig. 3.1, 2


The following fundamental characteristics of the backbone curve and the limitenvelope can be deduced from the equations defining these curves :

The position of the backbone curve is affected by the linear term of the restoringforce of the spring and by the magnitude of the mass. The form (curvature) is influenc-ed by the non-linear term of the restoring force characteristic. If this characteristicis linear, the backbone curve is a straight line parallel to the axis of amplitude A,i.e. the natural frequency is independent of initial deflection or velocity. The backbonecurve is not altered by change of damping.

The limit envelope is affected only by damping and the excitation amplitude. It isinvariant to changes of the mass, i.e. all the resonance curves obtained for differentmasses m touch the same limit envelope (Fig. 3.1, 3), as well as to changes of thespring characteristic - both its linear and non-linear terms (Fig. 3.1, 4).

Fig. 3.1, 3 Fig. 3.1, 4

Another feature of interest is the relation between the points of intersection of thecurve defined by (3.1, 6) and the backbone and the resonance curves. For the curvedefined by (3.1, 6) we can write the co - A dependence for a certain value of sin 99= const in the form

w=w*(A;q,);the reciprocal to function (3.1, 12a) is

A=A*(w;q').

(3.1, 12a)

(3.1, 12b)

The point of intersection of the above curve and the backbone curve has the w-coor-dinate denoted by Q9) which is defined by the equation

D2 = m [1 + F(A)]. (3.1, 13)

Accordingly, for the alternative with a constant excitation amplitude, equation(3.1, 7) takes the form

A((4 - w2) m = ±PICI = ±PV1 - S2 . (3.1, 14)

Equation (3.1, 14) is satisfied by both T = const and 7C -4*


Consider first the case of viscous (generally non-linear) damping (xflyj) iy wherex(1i1) > 0). For a constant excitation amplitude, (3.1, 6) has the form

coAK(wA) = PS (3.1, 15)

from which it follows that

wA = const = ko(9) (3.1, 16)

i.e. the curves for sin 4p = const and thus also the limit envelope are rectangularhyperbolas. The points of the resonance curve A (co) are obtained by establishing thepoints of intersection of this curve for a given q2, with the curves defined by (3.1, 14).Substituting A = koftw in (3.1, 14) and some rearrangement gives

co2+P

- -V1 -82w-S1"=0;0

from this follows the relationjQ2

m = wlwIl (3.1, 17)

where wI, wll are the coordinates of the corresponding points of the resonance curve(Fig. 3.1, 5).

A

A

A

Fig. 3.1, 5

Consider now the case of damping defined by the function x(jyj) sgn y wherex(jyI) > 0. In this case, (3.1, 15) becomes

K(A) = PS . (3.1, 18)

The (sin q7 = const)-curves are straight lines parallel to the co axis, i.e. A = const.It follows then from (3.1, 4) that

Q2 -'w2 = ±H0 (3.1, 19)

where Ho =Am

1 - S2 = const for sin p = const. Equation (3.1, 19) leads to

D92, =z

(w1 + C021)(3.1, 20)


where (vi, w1l are again the w coordinates of the points of intersection of the A*((0)curve and the resonance curve A(cv). Relations applicable to other types of dampingcan be derived in a similar way.

For the specified types of damping, Table 3.1, 1 shows the equations applicable tothe case of a constant excitation amplitude, Table 3.1, 2 those used in connection withan excitation amplitude proportional to (o2.

Table 3.1, 1

Type of damping Applicable equation

x(Iyi) y (3.1, 17)x(lyl) sgn y (3.1, 20)

Table 3.1, 2

Type of damping Applicable equation

xoy (3.1, 17)X41 (3.1, 17)xolyl sgn (3.1, 20)

xo = const

It has been assumed in the foregoing that the damping and the restoring forcecharacteristics are both symmetric, and that the system is not loaded with a constantforce. An analysis will now be made of a system described by the differential equation

my + /(y) y + h(y) + [x(y, y) + y(y, y)]PO + P cos wt

y =P0 + masw2 cos wt

(3.1, 21)

where m is the mass, P0 the constant load; the term (/(y) y + h(y)) represents the re-storing force, i.e. the symmetric and the non-symmetric part of the characteristicbecause the functions 1(y) and h(y) satisfy the relations

f(-y) = 1(y) , h(-y) = h(y) (3.1, 22)

Similarly, the functions x(y, y) y and y(y, y) ii represent the symmetric and the non-symmetric terms of the damping and satisfy the relations

x(-y, y) = x(y, -y) = x(y, y) , (3.1, 23)

Y(-y, y) _ -Y(y, Y) , Y(y, -y) = Y(y, y) (3.1, 24)

Assuming (3.1, 21) to be quasi-linear, the stationary solution can be approximated by

y= Y -}- A cos (wt - (p) . (3.1, 25)

Just as in the preceding case, the introduction of the time shift to = cp/w where isthe phase shift makes it possible to approximate the stationary solution by

y = Y + A cos wt (3.1, 25 a)


(the term cos cot in (3.1, 21) having been replaced by cos (cot + 99)). Using the harmonicbalance method, the following equations are obtained for determining Y, A and 99:

[F(Y, A) - mco2] A =1 P cos 99

lmoacu2

c

osgp'

F0(Y,A)Y+H(Y,A)+G(Y,A,coA)wA=PO, (3.1,26)

cAK(Y, A, coA) =j P sin og

m0sw2 s

where27c/w

YF0(Y, A) =2-x

r f (AY + A cos cot) (Y + A cos wt) dt ,

0

27c/w

H(Y, A) _ r h(Y+Acoscot) dt,0

27c/w

G(Y, A, wA) 2fr y(-wA sin wt, Y + A cos wt) sin wt dt ,

0

27c/w

YF(Y, A) _ r I (Y + A cos wt) (Y + A cos wt) cos2 wt dt,

27C/w

K(Y, A, wA) _ r x(-wA sin wt, Y + A cos wt) sin2 wt dt.

0

(3.1, 27)

(3.1, 28)

In view of (3.1, 22) to (3.1, 24), the following equations will apply:

27c/w

f f (Y + A cos wt) cos cot sin cot dt = 0 ,0

27c/w 27c/w

f h(Y + A cos wt) cos cot dt = 0 , f h(Y + A cos cot) sin wt dt = 0 ,

27c/w

f x(-wA sin cot, Y + A0

2n!w

f x(-wAsin cot, Y+A0

cos wt) sin wt dt = 0 ,

cos wt) sin wt cos wt dt = 0 ,

27c/w

f y(-wA sin wt, Y + A cos cot) sin2 wt dt = 0,0

in ry

27c/w

f y(-coA sin (ot, Y + A cos cot) sin cot cos wt dt = 00


as well as

Fo(-Y, A) = FO(Y, -A) = FO(Y, A), (3.1, 29)

H(-Y, A) = H(Y, -A) = H(Y, A), (3.1, 30)

G(-Y, A, wA) = G(Y, -A, wA) = G(Y, A, wA), l(3.1, 31)

G(Y, A, -(oA) = -G(Y, A, wA),K(-Y, -A, wA) = K(Y, A, -wA) = K(Y, A, (oA) . (3.1, 32)

With cos p = 0 substituted in (3.1, 27), the backbone curve is defined by the equa-tions

FO(Ys, As) Ys + H(Ys, As) + G(Ys, As, wAs) wAs = PO

,(3.1, 33)

F(Ys, As) = mw2

and the limit envelope by the equations

Fo(YL, AL) YL + H(YL, AL) + G(YL, AL, COAL) COAL = PO

(oALK(YL, AL, COAL) =2

3.1, 4)l mOsw

(subscripts L, S refer to the backbone curve and the limit envelope, respectively).If K(YL, AL, COAL) > 0 does not apply in some ranges of YL, AL, the minus signmust also be considered in the right-hand side of the last of equations (3.1, 34).

As in the case of systems with symmetric characteristics of the restoring force anddamping, the limit envelope is not altered by changes of the mass. It is, however, ingeneral no longer independent of changes in the restoring force non-linearity.

Condition (3.1, 32) implies the validity of the equation

(AL)P = -(AL)-P or (AL)E = -(AL)_.. (3.1, 36)

On the strength of this one can deduce the following :Upon determining the two extreme values of the deflection y ([y]ma", [y]min) as

functions of the excitation frequency for various masses, and drawing the correspond-ing limit envelopes ([y]L max, [Y]L min) (Fig. 3.1, 6), one finds that, within the scope ofthe present approximate approach, the following equations apply:

AL = 2 GAL max - [y]L min) , (3.1, 36)

YL = a ([y]L max + [Y]L min) (3.1, 37)

If function x(y, y) is not a function of the deflection or if it depends solely on the de-flection amplitude (x(y, amp y)) function K is only a function of AL and the limitenvelope depends neither on the constant load nor on the restoring force non-lineari-ty. Consequently, the course of the limit envelope AL(w) can be used for identificationof the symmetric component of damping (see Section 3.6 for further details). Identifi.cation of the non-symmetric component of damping is more complicated by far.The considerations which follow should aid in obtaining information concerning thepresence or absence of the non-symmetric component of damping on the basis of ex-perimental results.

The simplest of all is the case of zero constant load and a symmetric restoring forcecharacteristic, i.e. Po = 0 and h(y) = 0, when the first of equations (3.1, 34) be-comes

Fo(YL, AL) AL + G(YL, AL, COAL) = 0 .


[Y] Lmax

rYJLmin

LY1max

fYlmin

Fig. 3.1, 6W

If, according to experimental results, YL = 0 in the whole range of w, it must ne-cessarily be

G(YL, AL, (AL) = 0

and hence also

Y(y, y) = 0 .

If for a constant load PO and - PO

(YL)P, = -(YL)-P, (3.1, 38)

in the whole range of co, then, as (3.1, 29) and (3.1, 31) imply,

H(YL, AL) + G(YL, AL, wAL) = 0 (3.1, 39)

for all w. This identity cannot be satisfied in the whole range of w except for H(YL, AL)= 0 and G(YL, AL, WAL) = 0. This means that h(y) := 0, y(y, y) = 0. Since it iseasy to determine whether or not h(y) = 0, equation (3.1, 38) is a reliable guide toestablishing the presence (or absence) of damping asymmetry.

It may be of interest, in concluding the discussion, to set down some of the equationsapplicable to a simple system harmonically excited by a force with a constant ampli-tude for which

}(y) = c = coast , h(y) = 0 , Po = 0, Y = Y(y)(3.1, 40)

The limit envelope of such a system is described by the equations

CYL + G(WAL) wAL = 0 , WALK(WAL) = P . (3.1, 41)

Considering the accuracy of the approximate solution (3.1, 25), one can write

VL = WAL (3.1, 42)

3.2. Use of auxiliary curves for preliminary analysis 67

where VI, is the limit envelope of the velocity resonance curves. On the strength of(3.1, 42), equations (3.1, 41) can be written in the form

CYL + G(VL) VL = 0, VLK(VL) = P . (3.1, 41 a)

As the second equation implies,

VL = '(P) (3.1, 43)

and the AL((O) curve is a rectangular hyperbola. For a definite value of P, VL isindependent of w; consequently, YL is also independent of w and constant. Differentvalues of YL, however, are obtained for different values of P.

In the case of the excitation amplitude proportional to w2, (3.1, 43) no longer appliesand YL as well as VL depend on co.

What remains is to ascertain how well the equations derived above on the basis ofthe approximate solution, describe the real behaviour of the system. So far as thelimit envelopes are concerned, this problem was investigated by SVA61NA and FIALA(1980) who, for a number of systems, compared the limit envelope determined ana-lytically by application of the theory presented above, with the limit envelope ofa set of resonance curves obtained by analogue modelling for a gradually varyingmass. They found the quantitative difference between the analytic and the real limitenvelope to be very small for systems with symmetric characteristics of the restoringforce and damping (a Duffing system with hard and soft characteristics of the restor-ing force was solved for different types of damping), and somewhat larger for systemswith non-symmetric characteristics, especially those with non-symmetric damping.The fact that no qualitative difference exists between the approximate analytic andthe analogue computer results, is very important from the point of view of otherapplications of the limit envelopes, particularly that of damping identification.

3.2. Use of auxiliary curves for preliminary analysis

It has been found expedient in investigations of various systems to obtain first thebackbone curve and the limit envelope, and to form, on their basis, a rough idea ofthe system's behaviour before undertaking a detailed analysis. Thus, for example,knowledge of the course of the limit envelope and the backbone curve enables theanalyst to estimate the height of the resonant peak and the form of the resonancecurve and to determine whether the phase angle for the maximum amplitude is greateror smaller than 90°. If the limit envelope is a decreasing function of increasing fre-quency w, the amplitude reaches its maximum value for 99 < rr/2; in the case of asoftening spring characteristic, this maximum is larger than for a hardening spring.The reverse happens when the limit envelope is an increasing function of increasingfrequency w. It is, therefore, incorrect and naive to assume that the resonance ampli-tude will always be reduced by the use of a non-linear spring.

The backbone curve and the limit envelope of non-linear systems are sometimesapt to have more than one point of intersection, and a resonance curve which is nolonger a simple continuous line but consists of several branches. A case of this sort isshown in Fig. 3.2, 1 (the limit envelope is an increasing function of excitation fre-quency co and the backbone curve changes twice its sense of curvature). Systems inwhich the backbone curve is not unique in a certain interval of the excitation fre-quency, include those in which the spring characteristic changes from softening to


Fig. 3.2, 1ci

hardening as the deflection increases, and those with a hardening spring characteristicadditionally loaded with a constant force. A limit envelope which is not unique in thewhole range of the excitation frequency and hence has more than one points of intersec-tion with the backbone curve, belongs, for example, to systems for which the condi-tion K((oA, A) > 0 is not satisfied for all A. These systems, however, are counted in theclass of self-excited systems in which the interaction of self-excitation and parametricexcitation must be taken in consideration. They will be treated in greater detail inChapter 5.

The courses of the limit envelope and the backbone curve or the equation describ-ing them make it possible to decide whether or not a specified damping is capableof ensuring a limited resonance amplitude. As an example, consider the Duffing sys-tem with a progressive characteristic, excited by a harmonic force having an ampli-tude proportional to the square of the excitation frequency and with linear viscousdamping. This system is governed by the differential equation

y+xy+y+ey3=w2cosCot (3.2,1)

where e > 0, x > 0. In this case the limit envelope is a straight line passing throughthe origin of the coordinates, described by the equation

AI (3.2,2)M

the backbone curve is defined by the equation

w2=l+;EA2. (3.2,3)

The coordinate A of the point of intersection of the two curves is given by the equa-tion

A2 = (x2 - 3 E)-14

as this equation implies, the condition of existence of a point of intersection and inturn, a finite value of amplitude A is the fulfilment of the inequality

x2>is. (3.2,4)

3.3. Preliminary analysis of parametrically excited systems 59

As another example, consider a system with dry friction and damping defined by thefunction xI yI sgn y. This system is defined by the differential equation

y + y -}- ($ + x IYO sgn y = P cos wt (3.2, 5)

and its limit envelope, by the equation

AL =

-P-2t52

(3.2, 6)X

So long as 2 P- 20 > 0, the limit envelope is a straight line parallel to the w axis,

i.e. independent of frequency. It follows from this fact that for a damping which isindependent of frequency and a constant-amplitude excitation, the maximum resonan-ce amplitude is independent of the spring characteristic as well as the magnitude ofthe mass; the phase angle at resonance is always T = t/2. For x --> 0, lim AL = oo,

x-.0i.e. dry friction alone cannot ensure a limited amplitude at resonance so long as thecondition

4P> (3.2, 7)

is satisfied. Failure to satisfy this inequality means that the excitation force amplitudeis so small as to be unable to make the system vibrate even in resonance.

3.3. Use of auxiliary curves for preliminary analysisof parametrically excited systems

This section will show that the backbone curve and the limit envelope are also veryconvenient tools of analysis of parametrically excited systems. The way in whichthese curves are applied to such systems will be illustrated by way of the followingexample :

Consider a system with a periodically variable stiffness described by a non-lineardifferential equation of the Mathieu type, viz.

my + x(1 j1, Iyi) y + [c(1 + It cos 2(ot) + 1(y)] y = 0 (3.3, 1)

where m is the mass of the system, x(IjI, I yI) y - the damping force, c - the meanvalue of the restoring force stiffness, and the function /(y) y - the non-linearity ofthe restoring force. To simplify, assume that l(-y) = 1(y). For convenience, (3.3, 1)is usually rearranged so as to make the coefficients dimensionless. As in Section 3.1,no such rearrangement will be made here in order to show clearly the effect of thevarious parameters of the system.

The solution is sought in the main resonance (in the interval of instability of thefirst-order trivial solution), i.e. in the vicinity of w = c/m. Introducing the shift ofthe origin of time, q9/w, the term cos 2w t is replaced by cos 2(wt - 99) and the solutionapproximated by

y = A cos wt. (3.2, 2)


Using the approach outlined in Section 3.1, the following equations are obtained forthe determination of A and (p:

A[-mw2 + c(1 + -!p cos 299) + F(A)] = 0 ,

A[-wK(wA, A) + 2 uc sin 2q7)] = 0 ; (3.3, 3)

functions F(A) and K(w, A, A) are derived from ft) and x(IyI, jyj). In a certain inter-val of w, the equations can, in addition to a trivial solution, have a non-trivial solu-tion given by the equations

mw2 - c - F(A) = a pc cos 2qi , (3-3,4)

wK(w, A, A) = a ptc sin 29 . (3.3, 5)

On eliminating p (by squaring and adding (3.3, 4) and (3.3, 5)) one obtains the follow-ing equation for determining the dependence A = A(co):

(w2 -X2)2 + [_K(coAto, A)12 = (3.3, 6)

in the above

w2 =e

, Q2 [c + F(A)]m m

and c is obtained from the equation

tan 9) _

"-' K(wA, A)M

(3.3, 7)1

2,uwp - [22 + w2

As (3.3, 4) and (3.3, 5) also imply, in the interval (0, 27c) two values of the angle <p(i.e. 99 and n + T) belong to every value of A. For cos 292 = 0 (or ,u = 0), (3.3, 4)yields the following equation for the determination of the backbone curve As = As(w) :

w = m [c +F(As)]J1/2.

(3.3, 8)

Equation (3.3, 8) is wholly analogous to that for the backbone curve of systems excitedby an external force.

The limit envelope is obtained from (3.3, 5) by putting sin 292 = 1 (on the assumptionthat K(wA, A) > 0 for any co, A) ; this is the extreme value of the right-hand side of(3.3, 5), viz.

wK(wA, A) = 2 ,uc . (3.3, 9)

Equation (3.3, 9) makes it possible to determine the dependence AL = AL(w). Theresonance curve can only lie on one side of the limit envelope. As in systems excitedby an external force, the limit envelope is independent of the mass of the system and thespring non-linearity but depends (unlike in those systems) on the stiffness c and thecoefficient fc, which is the amplitude of the parametric excitation.

The various points of the resonance curve can also be obtained as the points ofintersection of a set of curves described by (3.3, 4) and (3.3, 5) for a particular 99.The right-hand side of (3.3, 5) is the same for both q) and it/2 - 99. In (3.3, 4), on the


other hand, the sign of cos 292 changes if 99 is replaced by Is/2 - T. For specified valuesof q' and n/2 - 99, the corresponding points of the resonance curve can thus be ob-tained as the points of intersection of the curves

wz = Q2 a y coo cos 2T

and the curve

m K(w, A, A) = 2 ,uw sin 299 .

(3.3, 10)

(3-3,11)

Denoting by Q, the coordinate co of the point of the backbone curve which also lieson the curve defined by (3.3, 11), one can write the equation

w2 = Q +2

,uco cos 292

from which it follows thatDT

= z (w1 + wit) (3.3, 12)

As in section 3.1, wI, w11 denote the coordinates w of the points of intersection of thelimit envelope and the resonance curve. Fig. 3.3, 1 shows schematically the dependenceof the amplitude A in parametric resonance on co (heavy solid lines - the stable so-lution, dashed lines - the unstable solution) ; the backbone curve is drawn in dot-and-dash lines, the curve defined by (3.3, 11) for a particular value sin 2q3 = const,in light solid lines.

Fig. 3.3, 1C)I "' 'jr C)

Equation (3.3, 12) is identical with (3.1, 20) and, so far as parametric excitation isconcerned, applies to a broad class of damping.

As an example, consider a system which (after rearrangement to the dimensionlessform of the linearized part of the equation) is described by the following differentialequation:

y+8y2y+0sgny-}-[1 -}-ycos2(wt-q)}y+YJ3=0. (3.3,13)

Substituting the approximate solution (3.3, 2) and comparing the coefficients ofcos cot and sin wt results in the following equations for determining A and T :

w2 - (1 + 4-/A2) = 2 ,u cos 299 ,

,u sin 2q .1

dwA2 + n 6 A =1

(3.3, 14)

(3.3, 15)


The amplitude in parametric resonance, A, as a function of co is obtained from theequation

((02-522)2- \4&oA2+ 4OA =4IU 2

7c )where

(3.3, 16)

Q2 = 1 + 1 rA2 .

In the case of i = 0, equation (3.3, 16) is biquadratic in co; hence the dependence ofco on A and in turn A = A((o) are readily determined. The dependence 99 = 99(0,)) isobtained from the equation

ll , (3.3, 17)tan 99 = 4 8wA2 -f- 4

Al (2 /u - 522 -}- w2

i

the backbone curve from the equation

w=S1=(1+ yAs)1'2and the limit envelope from the equation

Since

4 1 4

A2L 2/Z n A

(3.3, 18)

(3.3, 19)

lim (co) = 00, lim (w) = 0 ,AL-+0 AL-00

(o=0 for AL= 8 -,w=camas for AL=12 0

7C 1A 7C 1A

the limit envelope looks like the curve shown in Fig. 3.3, 2 a (0 = 0) or that in Fig.3.3, 2b (i = 0). The region in which the curve A = A(co) cannot exist, is hatched in.The backbone curves for y > 0 are drawn using dot-and-dash lines.

7 7t1l, I /////a) b)Fig. 3.3, 2

Q

The course of the auxiliary curves enables the analyst to draw some interestingconclusions. For systems without dry friction (0 = 0), provided that there exists aninterval of instability of the trivial solution, the first-order parametric resonance alwaysarises spontaneously because the resonance curve A(w) must start from the co axis.


For systems with dry friction (? r 0), the trivial solution is stable in the wholerange of w and parametric resonance can arise if there exists a point of intersectionof the backbone curve and the limit envelope; the process, however, is not spontaneousbut takes place only on application of definite initial conditions. If the backbone curvedoes not intersect the limit envelope but lies whole outside the region of the possiblepoints of the resonance curve (in the hatched region in Fig. 3.3, 2 a), parametric vibra-tion cannot arise at all. The course of the limit envelope also provides information asto which spring characteristic (softening or hardening) is the more advantageous forthe system in question. The maximum amplitude of the system being discussed hereis smaller for y > 0 than for y < 0. Fig. 3.3, 3a shows the curves A(w), AL(w),As(w) for 9 = 0, 6 = 0.01, ,u = 0.2 and y = 0.01, Fig. 3.3, 3b, the curve ry((o).Figs. 3.3, 4a, b show these curves for the same value of S, ,u and y but for 0 = 0.1.The sections of the curves A((o) and 99(a)) to which corresponds an unstable solution(for small disturbances) are drawn in dashed lines. As seen in the figures, the dependen-ce A = A(w) forms a closed curve. If non-trivial solutions of A exist for a definite w,

6

A-

0

9110

45

0

AL

j.

A .'AS /A

1.1

it

0.9

Fig. 3.3, 31.1

a)

b)


6

A

AL

As

4

2

0

0

900

45

AL

As1

AL

1 1.1

0.9

Fig. 3.3, 4

01 1.1 c,)

a)

b)

1.2

then one value (to every A belongs a value of 4p and it + 92) corresponds to a stable,the other to an unstable solution. Every stable solution is only locally stable becausethere exists the locally stable trivial solution A = 0. A non-trivial solution cannotbe obtained except on application of definite. initial conditions, i.e. there exist, forcertain w the domains of attraction of the trivial and non-trivial solutions.

3.4. Auxiliary curves of higher-order systems

Consider a system governed by a higher-order differential equation (i.e. one containingderivatives of an order higher than the second; the derivative of the n-th order isdenoted by y(n)), viz.

120(2n) + f2n-1y(2n-1) + ... + fly + foZJP cos (wt + T)

(3.4, 1)8MOw2 cos (cot + 99)

The coefficients of (3.4, 1) are either constants or functions of ly(2n)l, ... , lyl, and asbefore, cp is the phase angle between response and excitation. Equation (3.4, 1) doesnot have to be of an even order, i.e. fen may be equal to zero. The even order is used

3.4. Higher-order systems 65

here to facilitate formal expression. The stationary solution at resonance is approxi-mated by

y = A cos cot. (3.4, 2)

Using the harmonic balance method, the following equations are obtained for deter-mining A, 99:

A[(- 1)n w2nF2n + (-1)n-1CO

2(n-1) w2F2 + Fob = {P Cos (P

Emow2 Cos 99

(3.4, 3)

+ (-1) + ...+ (OF,A[(-1)n w 2n-1F2n-1 n-1 w2n-3F2n-3 ={ P sin 97

Emow2 sin 97

(3.4, 4)

The coefficients F2n, ... , FO are generally functions of A, wA, ... , w2nA2n which areestablished from functions f2n, , fo by the averaging process.

For cos T = 0, the equation(_1)nw2'F2n+...+FO =0 (3.4,5)

defines the backbone curve which, as will be shown later, may consist of severalbranches. The equation

A[(_1)nw2n-1F2n-1 + ... + wF11PP

(3.4, 6)= ± 2

defines the limit envelope which is also apt to consist of several branches. If the left-hand side of (3.4, 6) is positive for all A, only the plus sign on the right-hand sidemakes sense. Hence the resonance curve A(w) must lie only in the region between thew axis and the limit envelope and touch the latter at its points of intersection with thebackbone curve. The method of solution is outlined below.

Discrete mechanical systems are usually described by a set of second-order differen-tial equations. In some cases it is expedient - if it can be done at all - to convertthe set to a single differential equation of higher order. The following example willshow how to proceed in such cases.

A mechanical system containing two masses, m1, m2 (Fig. 3.4, 1) with linear viscousdamping of the motion of mass m2 is excited by a harmonic force w2 cos wt. The spring

5 Schmidt/Tondl


connecting the two masses is linear, with a stiffness coefficient c1. The other spring isprogressive non-linear, and its restoring force is expressed by a linear term with coeffi-cient c2 and a cubic term. After rearrangement to the dimensionless form, the equa-tions describing the system are:

x;+x1-x2=0,X; - M(x1 - x2) + xx2 + 82x2(1 + ex2) = w2 cos wt (3.4, 7)

where M = m1/m2 is the ratio of the masses, q =C2/m2/1/2

the tuning coefficient ofCl/ml

the restoring force linear component, x - the damping coefficient, and e - the coeffi-cient of the non-linear term of the spring characteristic.

The first of (3.4, 7) gives

x2=xi+x1. (3.4,8)

Substituting (3.4, 8) in the second of (3.4, 7) and using, as before, the shift of the timeorigin results in the following single fourth-order differential equation

xiV + xxl/ + {1 + M + q2[1 + e(x1 + xi)2]} x1 + V1x1

+q2[1 +E(xi +x1)2] x1 =w2COS(Cot +9?) (3.4, 9)

If the stationary solution is sought in the form of (3.4, 2) (i.e. x1 = X1 cos wt in thecase being analyzed), equations (3.4, 3) and (3.4, 4) take the form

X1[w4 - (1 + M + q2) w2 + q2 + a Eg2(1 - w2)3 X) = w2 COS 99 , (3.4, 10)

xwX1(1 - w2) = w2 sin 99 . (3.4, 11)

X1 and 99 are obtained from the equations

Xi{ I0)4 - (1 + M + q2) w2 + q2 + a eg2(1 - w2)3 X ]2 + [xw(1 - (02)]21 = w4,(3.4, 12)

tan 99 = xw(1 - w2) [ o4 - (1 + M + q2) w2 + q2 + a eg2(1 - w2)3 X2j-1l

(3.4, 13)The backbone curve is defined by the equation

X13 = 8 [8g2(1 - (02)3]-11 [w4 - (1 + M + q2) w2 + q2] . (3.4, 14)

In the case being considered, the curve has two branches, both starting on the co axis.The coordinate co of the starting points is given by the roots of the equation

w4-(1+M+g2)w2+q2=0, (3.4,15)

i.e. by the natural frequencies of the abbreviated (linearized, damping-less) system.Denoting these frequencies by 521, 522 and considering that Q1 < 1 < Q2 for any M, q,it is clear that for s > 0, the real values of X15 are obtained in the interval Q1 < co < 1,522 < to. As co is increased from Q1 to 1, X1s (the lower branch) becomes a continuallyincreasing function of co and grows beyond all bounds for co - 1. As w is increasedfrom 522 (the upper branch), X15 first increases in a certain interval of w, reaches itsmaximum and then becomes a decreasing function because lim X15 = 0.

W-00For e < 0, Xis is defined in the intervals w < 521, 1 < w < 522. For decreasing co,

lim X15 = oo for the lower branch, and lim X15 = oo for the upper branch. X1S is anW+1 W-.oincreasing function of decreasing co.


Recalling that only the positive values of X1 have a meaning, the equation of thelimit envelope can be written as follows:

X1L = w(x 1(02 - 11) -1. (3.4, 16)

Clearly, the course of the limit envelope depends only on the damping coefficient xand the type of excitation, and is independent of other parameters of the system. Itshould be stressed that it is independent of the non-linearity coefficient e. It is foundthat

lim X1L = 0 , lim XIL = oo , lim XIL = 0 . (3.4, 17)W-.0 W-.1 W-00

Using (3.4, 8) one can write

x2 = (1 - (02) x1 (3.4, 18)

dan

XX = (1 - Co') X (3 4 19)1 .2 . ,

Let us now examine the backbone curve and the limit envelope corresponding tovibration in the x2 coordinate. The backbone curve is defined by the equation

X28 = 4 I--g2(1 - (02)]-1 [0)4 - (1 + M + q2) w2 + q2] (3.4, 20)

The definition intervals are the same as for X15; for e > 0, however, X2s increases withincreasing to for both branches. For e < 0,decreases.

lim X2s = 0 and lim X2s = oo as coW-.0 W-.1

The equation of the limit envelope is

X2L = 04M, (3.4, 21)

i.e. as in the case of one-mass systems with linear viscous damping, (Section 3.1), thelimit envelope is a straight line.

The stability for small disturbances can be established by analyzing the variationalequation. The equation used for the purpose is the linearized equation obtained aftersubstituting xl + u for x1 in (3.4, 9) and neglecting the non-linear terms. xl is thesolution of (3.4, 9), i.e. xl = Xl cos wt in the case being considered. The differentialequation thus obtained is

u1v+xu"'+(1+M+q2)u'+xu'+g2u+3eg2(x1+x1)2(u"+u)=0;

(3.4, 22)

after substituting xl = Xl cos cot, it becomes

u1' + xu" + (1 + M + q2) u' + xu' + q2u + g sg2X1(1 - w2)2

(1 + cos 2wt) (u" + u) = 0 . (3.4, 23)

This is a linear differential equation with periodically variable coefficients. Its approxi-mate analysis is effected in two steps: The first involves a check of the stability forthe average constant values of the coefficients. Applying the Routh-Hurwitz criterionto the characteristic equation

24+x13+(1 +M+g2)22+xA+ eg2X2(1 -w2)2(,12+1)=05*


and assuming that x > 0, it is found that the following inequalities must be satisfied:

q2[1 + 2 sXi(1 - w2)2] 0 (3.4, 24)

x2{ 1+ M + q2[1 + 2 e(l - c)2)2 X2] - 1 } - x282[1 + 2 e(1 - (o2)2 X 1]

= x2M > 0 . (3.4, 25)

For e > 0, both inequalities are satisfied in all cases. For e < 0, (3.4, 24) implies thenecessity of satisfying the condition

X1 < 2[3(-e) (1 - (o2)2]-1 . (3.4, 24a)

The second step of the stability investigation concerns the effect of the variabilityof the coefficients. An approximate solution on the boundary of the main instabilityintervals can take the form

u=acoscot+bsincit. (3.4,26)

Substituting this solution in (3.4, 23) and comparing the coefficients of cos cot andsin not results in two homogeneous equations for a and b. The condition that the solu-tion should be non-trivial leads to the equation

H(i) + e eg2(1 - co2)3 Xi , xcu(1 - Cot) = 0 (3.4, 27)-xco(1 - c)2) , H(co) + a:sg2(1 - w2)3 Xi

where H(w) = a04 - (1 + M + q2) w2 + q2. X1 is obtained from (3.4, 27), viz.

8 H(w) _ 1 x22(1 - X2)2 1/2

11

1/2

X19 eg2(1 - 0)2) 1 + 2 (1 - 3

[H(w)]2) (3.4, 28)

By calculating X1 as a function of co, one obtains the boundaries of the regions in the(X1, co) plane in which the amplitudes X1 determined from (3.4, 12) belong to unstablesolutions. For e < 0, inequality (3.4, 24a) must naturally be also taken into account.Applying (3.4, 19) to (3.4, 28) one obtains the regions in the (X2, (o) plane in which theamplitudes X2 correspond to unstable solutions.

The system described by the differential equation (3.4, 7) was also solved for severalsets of parameters by help of an analogue computer. The extreme values of vibrationin the x1 and x2 coordinate were plotted as functions of w by means of a graph-plotter.The extreme values corresponding to X1 and X. are denoted by [x1] and [x2].

The alternative presented below as an example has the following parameters:M = 0.2, q = 0.8, e = 0.05, x = 0.25. For information about other cases, the readeris referred to a paper by TONDL (1977).

Fig. 3.4, 2 shows the diagrams of X1 (heavy solid lines - stable solutions, heavydashed lines - unstable solutions), X1L (light solid lines) and X18 (dot-and dashlines) as functions of frequency co. The boundaries of the regions in which the solutions(obtained by using (3.4, 28)) are unstable are drawn in light solid lines, and the regionsthemselves are hatched. The upper branch of the backbone curve is seen to differfrom that obtained by means of the approximate method based on the normal modesof the abbreviated system (the latter has the character of a parabola, i.e. is an increas-ing function of increasing c)). The qualitative difference between the results of thetwo approaches is particularly noticeable at higher resonances for which the proposedmethod yields a function which is in part increasing, in part decreasing. As analoguecomputing proves, the proposed method leads to more correct results than does the


10

X2

X2L

X2S

0

Fig. 3.4, 3

3600,

cP

270

180

90

0.5 1 1.5 ci 2

0 0.5 1 1.5 Cd

Fig. 3.4, 4


quasi-normal mode approximation. Fig. 3.4, 3 shows analogous results correspondingto coordinate x2, Fig. 3.4, 4 the phase angle 99 as a function of frequency o). A com-parison of analytical and analogue computer results is presented in Figs. 3.4, 5 and3.4, 6. [x1] and [x2] are drawn as functions of co (heavy solid lines) and the diagrams

10

Cx21

X2

5

0

Fig. 3,4, 60.5 1 1,5 Q 2

are complemented by the curves X1(oj) and X2(co) obtained analytically (stable solu-tions only - light solid lines). The results of the analogue and analytic solutions areseen to be in close agreement, the existing differences being largely due to the presenceof higher-order components in actual vibration. Considering, however, that at theresonant peak the contribution of the non-linear component of the restoring force atmaximum deflection is several times larger than that of the linear component, it canbe said that the proposed method is equally effective in cases of systems with com-paratively strong non-linearities.

Fig. 3.4, 7 shows the results of an analogue solution of a system which differs fromthe former only by a slightly smaller value of x (x = 0.24). As shown by the correspond-ing vibration records, the motion is quasi-periodic in a certain interval in the vicinityof co = 1.2. Reducing x still further causes violent non-periodic vibration to arise inan interval of co beyond the second resonant peak.

The following conclusions can be drawn on the basis of the foregoing analysis :

3.5. Systems with several degrees of freedom 71

For some systems with two or more degrees of freedom, particularly for those whichare no longer quasi-linear, the proposed method is apt to yield more realistic resultsthan do the methods using quasi-normal mode approximation. As shown by way ofan example, there exists a limit to the system's parameters for which the methodensures good agreement between the analytic solution and actual vibration. Outsidethis limit, the response is no longer periodic. A shortcoming of the proposed method isthat it can be applied to a particular class of systems only.

3.5. Use of auxiliary curves in analysis of systems with severaldegrees of freedom

The backbone curve and the limit envelope find useful application in analysis ofsystems with several degrees of freedom whose equations of motion cannot be convert-ed to a single higher-order differential equation. The class of systems for which ananalysis using these curves is feasible, is, however, very small. The backbone curvescan be obtained without difficulty for systems with masses interconnected chain-wiseby dampers and springs, only one of which is non-linear. In the simplest case the non-linear spring is connected to the first or the last mass of the chain. If it is only thisterminal mass which is acted on by harmonic excitation, one can (with some restrictionconcerning the damping non-linearity) also obtain the limit envelope and on the basisof it estimate the position of the main resonances and the amplitudes at the resonancepeak. If the system has symmetric spring and damping characteristics, is not loadedwith constant forces or tuned into internal resonance, the possibility of occurrence ofother than the main resonances is small. A system of this kind is described by differen-tial equations (after rearrangement to the dimensionless form; to simplify, linearviscous damping is assumed throughout) of the type

+ + + 99)xl T 1x1 + x2(xl - x.,) + x1[1 + f(x1)] r

g22(x1

- x2)cos (L)E

_CU2 cos (wt +

xk xk(xk - xk-1) + xk}1(xk - xk-1) + gk(xk - xk-1) + qk 11(xk - xk 11) = 0

99),

xn + xn(x - xn-1) + g(xn - xn-1) = 0 (3.5, 1)

where f (-x1) = f (x1).


The stationary solution can be approximated by

x1 = X1 cos cut,(3.5, 2)

xk=AkcosWt +Bksin wt=Xkcos(cot-TO (k=2,...,n).Substituting (3.5, 2) in system (3.5, 1) and using the harmonic balance method leads

to the following non-linear algebraic equations:

-(,)2X,, + X1 [l + F(X1)] + g2(Xl - A2) - x2cuB2 = j cosw 2 COS 9>

2B2 sin 99co[x1X1 + x2(X1 - A2)] + g2 B2 = 21 w sin 99 "

-w2Ak + g2(Ak - Ak-1) + q22+l(Ak - Ak+1) + w[xk(Bk - Bk-1)

+ xk+1(Bk - Bk+1)J = 0 ,

-w2Bk + gk(Bk - Bk-1) + qk+1(Bk - Bk+1)+ xk+1(Ak - Ak+l)] = 0

wLxk(Ak - Ak-1)

(3.5, 3)

where F(X1) is a function resulting from function /(x1), k = 2, ... , n, qn+1 = 0,xn+1 = 0.

Writing, as before, cos 99 = 0 and sin 9' _ -}-1, the last 2k equations of (3.5, 3)together with the equation

-cu2X1 + X1[1 + F(X1)] + g2(X2 - A2) - x2coB = 0 (3.5, 4)

define the backbone curve (consisting of n branches), and the last 2(n-1) equationsof (3.5, 3) together with the equation

x1coX1 + x2co(Xl - A2) + g2B2 = t 1(02 (3.5, 5)

define the limit envelope.In digital computations the values of X1 are increased in steps starting from zero.

The system of 2(n -1) algebraic equations is linear with respect to coefficients Ak, Bk.Fora given X1 these coefficients are obtained as functions of co; their substitution in(3.5, 4) or (3.5, 5) results in a single equation in co. This equation is used for determin-ing the values of co which correspond to the specified X1 of the backbone curve andthe limit envelope. The data thus obtained are then plotted in the form of diagramsrepresenting X1$(co) and X1L((o).

In systems of the kind discussed, the backbone curve is generally no longer indepen-dent of damping; neither is the limit envelope dependent solely on damping and theexcitation amplitude of the system.

As an example, consider a two-mass system which differs from that shown inFig. 3.4, 1 only by a linear damper introduced between the two masses. The governingdifferential equations of this system are

xl+x1(31 -32)+x1 x2=0

4 - M[x1 - x2 + 31(x1 - x2)] + 32x2 + q2(l + Exn) x2 = (02 COS (wt + 9))

where, as before, M = m1/m2, q2 =ccl1/mM

(3.5, 6)

3.5. Systems with several degrees of freedom 73

If the stationary solution is sought in the form

x1 = A cos wt + B sin wt = X1 cos (wt - cc) ,

I(3.5, 7)

x2 = X2 cos wt

the proposed method leads to

(1 -w2)A+xcoB=X2,-x1wA + (1 - w2) B = -x1wX2 ,

23.5, )

) - (02] X2 - Mco2A = w2COS9,[q2(1 + 4 _*X

x2wX2 + Mw2B = w2 sin 4p .

The first two equations give

X2 22 2A = +x1w ),4(1 -w

(3.5, 9)

B = 2 xiw

where

A = (1 - (02)2 + xlw2 .

Substituting for A, B in the third of (3.5, 8) and setting cos T = 0 yields the follow-ing equations for X28 and X1s:

1/2

q2 [1 + M1 - w2 +

xlw2]_i}A

Y2$Xls =d

[(1 - w2 + x1Oo2)2 + xlw8]1/2

The equations for the limit envelope are

Mx1w4 -1X2L=Co Y.1+

A

Y1L = AL [(1 - w2 + xl o2)2 + xlwg]1/2

(3.5, 10)

(3.5, 11)

(3.5, 12)

(3.5, 13)

Fig. 3.5, 1 shows the curves X1L(w) (light solid lines) and X1S((0) (dot-and-dashlines) complemented by the curve of the extreme deflections [x1] as function of co,obtained by analogue modelling. Fig. 3.5, 2 shows analogous curves for the x2 coordi-nate. The parameters of the corresponding system are: q = 0.5, M = 0.5, E = 0.1,x1 = x2 = 0.1. The backbone curve of the higher resonance in the x1 coordinate re-sembles that of the example discussed in the preceding section. In this case also, theshare of the non-linear term of the non-linear spring characteristic is several timeslarger than that of the linear term for the extreme deflection at resonance. This factnotwithstanding, the error of the resonant peak estimate is comparatively small.

Additional examples may be found in a paper by ToxDL (1978a).


15

5

X,5

X1 L r--

1

/ I CX

0 1

W2

Fig. 3.5, 1

15

5

X25\'

X2L

10,

l I

i CX21

0 1

Fig. 3.5, 2

C)2

3.6. Identification of damping

It was established on the basis of several equations derived in Section 3.1 that theform of the limit envelope is affected by damping and the type of excitation. Thisfinding can be put to good use whenever damping identification is required in order

3.6. Identification of damping 75

to provide a more realistic expression of the damping forces in the mathematicalmodels of real systems. The theoretical considerations presented in the foregoing canfind most useful application in the field of simple elements, for example, rubberelastic elements which represent a spring and a damper as well. With many suchelements one is at a loss for an expression of damping capable of describing realityat least approximately. If an elastic element of this kind is to be replaced simplyby a damper and a spring, the frequency interval of excitation must be limited toa range (lower frequencies) in which the element need not be regarded as a continuum.

The test facilities should be designed so as to satisfy the assumptions made in thetheoretical considerations (one degree of freedom, harmonic excitation, etc.). Since thestiffness of most elastic elements cannot be changed without altering the damping,it is best to obtain the limit envelope as an envelope of the set of resonance curvescorresponding to different masses (which can be changed more readily). The testdevice is best designed as a torsion system in which, by varying the distance betweentwo masses on its arm, one can change the moment of inertia quadratically (andhence the natural frequency linearly). Since the effect of prestress is not examined,the element is not subjected to a constant load in order not to complicate the identi-fication. The test device should preferably incorporate means of providing more typesof excitation (with a constant amplitude, with an amplitude proportional to thesquare of the excitation frequency). The possibility of varying the amplitude of theharmonic excitation over a sufficient range should be ensured in all cases.

Because of the limited scope of the book, only the salient points of the identificationprocedure are presented. The first point to be established is whether or not the damp-ing characteristic is symmetric. The equations put forward at the end of Section 3.1should be of help in this respect. If a symmetric characteristic or the symmetric partof a characteristic is involved, it is recommended to ascertain whether the dampingof the element being tested belongs to a fundamental class of damping, is linear ornon-linear.

If, for any excitation frequency, the amplitudes of the limit envelopes correspondingto two arbitrary excitation amplitudes Pl and P2 satisfy the relation

AM Pl(3.6, 1)

AL2 P2the damping is linear (but not necessarily linear viscous).

The characteristic features of the limit envelopes for the fundamental types ofdamping are as follows:

(a) If damping is described by the function x(l yl) egn y, the limit envelope is a straightline parallel to the w axis whenever the excitation force has a constant amplitude P.If (3.6, 1) is not satisfied, the values of P represent directly the values of functionK(AL) (cf. (3.1, 11 a) on the consideration that, in this case, function K depends onlyon amplitude A). Function K(AL) is obtained by means of regression and from therefunction x(lyl) sgn y taking into account the relations (resulting from the averagingprocess) between the corresponding coefficients of, for example, the polynomial ofthe two functions.

(b) If damping is described by function x(IiJ) y, the limit envelopes correspondingto different values of the constant amplitude of the excitation force are regular hyper-bolae. If (3.6, 1) does not apply to all P, function K(WAL) cWAL is non-linear. As in theformer case, it is obtained by means of regression using different values of P whichagain represent the values of K(WAL) cJAL for a particular w.


(c) If damping is described by function x(jyj) y, the limit envelope is defined by theequation

K(AL) COAL = P (3.6, 2)EmOw2

Taking the inverse to function AL(w) for both types of excitation one obtains theequations

PALK(AL)

fromviz.

= wp(AL)w =

A(3.6, 3)

L K(AL) = we(AL)Em0

which it follows that the product of the two functions is constant for any AL,

COP(AL) ws(AL) = const . (3.6, 4)

The curve representing the function K(AL) can be obtained as follows. The limitenvelopes corresponding to the first type of excitation are determined for variousvalues of P. For a chosen co, the values of AL are read from the diagram and thevalues of P/ALW corresponding to them are calculated. The latter determine thevalues of function K(AL) whose course is obtained by means of regression.

The more complicated types of damping and their combinations will not be treatedhere.

A brief mention, at least, should be made of the following important fact. In systemswith visco-elastic elements, the actual resonance is shifted against that obtained onthe basis of the static characteristic of the restoring force. Hence springs of visco-elastic materials exhibit a difference between the static and the dynamic characteristicof the restoring force. The dynamic characteristic of the restoring force is determinedby the experimental backbone curve.

4. Analysis in the phase plane

4.1. Fundamental considerations

In the present context, the term "phase plane" denotes a plane whose coordinatesare the dependent variables of a system of two coupled first-order differential equa-tions of the type

X(x, y)(4.1, 1)

y= Y(x,y)where the functions X(x, y) and Y(x, y) are to satisfy the conditions of existence anduniqueness of the solution at any point of the (x, y, t) space.

In interpretation of the results of an analysis made in the phase plane, the analystshould know the way in which (4.1, 1) were obtained. Knowing this, he can dividethe various systems into the following three groups :

(a) Equations (4.1, 1) describe directly the mathematical model of the system beingstudied.

(b) Equations (4.1, 1) were obtained by rearrangement, e.g. introduction of the va-riables y = v, y, of the homogeneous second-order differential equation of the type

y+f(y,y)=0 (4.1, 2)

which describes the model of a self-excited system with one degree of freedom.

(c) Equations (4.1, 1) were obtained by transformation of a second-order differentialequation, either non-homogeneous with the right-hand side representing a periodicfunction of time (the independent variable), or homogeneous with coefficientswhich are periodic functions of time (the independent variable). The correspondingsystems are those with external or parametric excitation, and equations (4.1, 1)are obtained when these systems are solved by means, for example, of the methodof slowly varying amplitudes, or of amplitude and phase (the van der Pol orKrylov and Bogoljubov methods).

On the above assumptions it holds for the integral curves - the phase plane tra-jectories - of system (4.4. 1) which are obtained by solving the equation

dxX(x,y) (4.1,3)dy Y(x, y)

that, except for the singular points defined by the equations

X(x, y) = 0 , Y(x, y) = 0 (4.1, 4)

only a single trajectory passes through every point of the phase plane.

78 4. Analysis in the phase plane

Points which lie on the curve defined by the equation

X(x,y)=0

but fail to satisfy the second of equations (4.1, 1) are characterized by the fact thatthe tangent to the trajectory is parallel to the y-axis. Similarly, the tangent to thetrajectory at points lying on the curve

Y(x, y) = 0

is parallel to the x-axis.The singular points define the equilibrium state and represent the non-oscillatory

solution of (4.1, 1), i.e. the real equilibrium state in the case of systems of the groups(a) and (b), and steady periodic solutions of a certain type (e.g. in the main or sub-harmonic resonance) for systems of the group (c).

In analyses in the phase plane it is important to know the position of the singularpoints as well as their type and stability. The last two properties can be determinedfrom the roots of the characteristic equation

ax - d axax ay

aY aY - Aax ay

= 0 (4.1, 5)

where the coordinates of the singular points are substituted for x and y in the expres-sions ax/ax, ax/ay, aY/ax, aY/ay. The characteristic equation (4.1, 5) can also begiven the form

22+fA+y=O (4.1,5a)

where_ ax ay _ aX aY aX ay

- ( ax + ay y ax ay ay ax

Whenever the conditions

#>0, y>O (4.1,6)

are satisfied, the singular point in question is also asymptotically stable.The singular point is a node if the roots of the characteristic equation are real and of

equal signs, viz.

#2 > 4y, V>0. (4.1, 7)

The singular point is a saddle if the roots of the characteristic equation are real butof opposite signs, viz.

fj2>4y, Y<0. (4.1,8)

The singular point is a focus if the roots of the characteristic equation are compleximaginaries with non-zero real part, viz.

N2<4y, P+0. (4.1,9)

4.1. Fundamental considerations 79

A complete discussion of the singular points should also include the so-called centre,i.e. a singular point for which the roots of the characteristic equation are pure imagi-naries, viz.

y>0, fl=0. (4.1,10)

The corresponding solution is stable but not asymptotically.Knowledge of the type of the singular point enables the analyst to estimate the

character of the trajectories in the point's immediate neighbourhood. In the neigh-bourhood of a node, the trajectories form a bundle (Fig. 4.1, 1) ; the node is stable ifthey move towards it (Fig. 4.1, 1 a), and unstable if they move away from it (Fig. 4.1,1 b). A saddle point (Fig. 4.1, 2) features two pairs of trajectories, one entering, theother leaving it. The trajectories of each pair have common tangents at the saddlepoint. In the neighbourhood of a focus, the trajectories are spiral shaped and moveeither towards (stable focus - Fig. 4.1, 3a) or away from (unstable focus - Fig. 4.1,3b) the point. In the neighbourhood of a centre, the trajectories form closed curves(Fig. 4.1, 4); this case has a more or less theoretical meaning (systems without damp-ing).

a)

Fig. 4.1, 1

Fig. 4.1, 2

b)

(a) (b)

Fig. 4.1, 3 Fig. 4.1, 4


If there exist - apart from the singular points - additional steady but oscillating(periodical) solutions, their trajectories in the phase plane are closed curves - thelimit cycles. A limit cycle is stable (Fig. 4.1, 5 a) if the trajectories in its neighbourhoodconverge to the cycle, and unstable (Fig. 4.1, 5b) if they diverge from it. The case ofsystems without damping whose set of closed trajectories surrounds the singularpoint - the centre - is not considered. If a single singular point is surrounded byseveral limit cycles, their stable and unstable forms alternate regularly - see Fig. 4.1, 6where the stable limit cycles marked Ls are drawn in heavy lines, the unstable limitcycles, LN, in heavy dashed lines. If a particular parameter of a system is varied soas to cause the stable limit cycle to approach the unstable, than at a definite value ofthe parameter, the two cycles become one. The resulting limit cycle is called semi-stable and in practice its presence is revealed by close spacing of the trajectories in itsneighbourhood (Fig. 4.1, 7).

a)Fig. 4.1, 5

Fig. 4.1, 6

b)

Fig. 4.1, 7

Other important properties of the phase plane trajectories are presented below.Introducing the time transformation (negative time)

t = -a (4.1, 11)

causes the system (4.1, 1) to take the form

-X(x,y),(4.1, 12)

y = -Y(x, y)It can readily be shown that a phase portrait in the phase plane (x, y) of system

(4.1, 1) is identical with that of system (4.1, 12) except for the opposite sense of thereference point motion. This finding has the following important consequences:


If the point (Xk, yk) is a stable (unstable) focus in the (x, y) plane for system (4.1, 1), itis an unstable (stable) focus for system (4.1, 12).

If the point (Xk, yk) is a stable (unstable) node in the (x, y) plane for system (4.1, 1),it is an unstable (stable) node for system (4.1, 12).

If the point (Xk, yk) is a saddle for system (4.1, 1), it is also a saddle for system (4.1, 12) ;the motion of the reference point in the two systems is, however, of opposite sense.

A stable (unstable) limit cycle for system (4.1, 1) is an unstable (stable) cycle for system(4.1, 12).

These properties can be used to advantage when drawing the phase portraits bymeans of a digital or an analogue computer by alternating the solution of system(4.1, 1) with that of system (4.1, 12) to speed up the process of plotting the mostimportant phase plane trajectories.

An analysis using the phase plane portraits is advantageous in that it providescomprehensive knowledge of both the stationary and the non-stationary solutions andenables the stability of steady solutions to be investigated not only for small but forany disturbances in the initial conditions ; this is the way in which the stability in thelarge is investigated and the domains of attraction for various steady solutions aredetermined.

A few notes concerning systems more complicated then (4.1, 1) seem to be necessaryat this point. In some cases the two first-order differential equations are supplementedby additional algebraic relations (as, for example, in systems of the (c) group for whichthe steady solution must be approximated by yet other terms, such as that of a con-stant deflection, etc.). As an example consider the system of the type

x=X(x,y,z)

y= Y(x,y,z),Z(x, y, z) = 0 II

(4.1, 13)

in which even other algebraic relations are apt to exist. The singular points in the(x, y) plane which exist in this case, also, are similar to those of system (4.1, 1) andobtained from the roots of the characteristic equation

aA2 -}- A -{- y = 0 (4.1, 14)

where

az

az

ax axax ay

az azax ay

aY ayay az

az az

ay az

, v=

ax ax axax ay az

ay ay ayax ay az

az az az

ax ay az

The type and stability of the various singular points (assuming a > 0) are shownin table 4.1, 1.6 Schmidt/Tondl


Table 4.1, 1

Type of Satisfying the Stable for Unstable forsingular point conditions

The node R2>4ay,y/c > 0 fi>0 #<0The saddle 142 > 4ay, yfa < 0 always

The focus R2 < 4ay, fi + 0 j > 0 P<0The centre y > 0, P = 0 always

Let us turn to systems governed by a set of three differential equations of the firstorder

x = x(x, y, z) ,

iy= Y(x,y,z),z=Z(x,y,z)

(4.1, 15)

Denoting by x8, y, z8 the coordinates of a singular point and introducing x = x8 + $,y = ys + 71, z = z8 + C the perturbation equations are obtained. As critical caseswill be excluded from our considerations the stability of singular points can be deter-mined according to the roots of the characteristic equation

23+a22+a22+a3=O (4.1,16)

where

ax a Y aZ_a

- ( + +(4 1 17)l

ax ay az. ,

'

a2 =

ax axax ay

ax axax Or

+

aY aYay Or

(4.1, 18)aY ayax ay

az az

ax Or

az az

ay Or

ax ax ax

a3 = -

ax ay az

ay ay ay(4.1, 19)

ax ay Or

az

ax

x=x8,

az az

ay Or

Y=Y8, z=z8.Let us denote by Al, 22, 23 the sought-for roots of the characteristic equation. Then itholds that

a1 = - (A1 + A2 + A3)

a2 = 2122 + A1A3 + A2A3 '

a3 = -212223 .

(4.1, 20)


In our discussion of the quality and signs of the roots we shall make use of thediscriminant of the equation

D3 = -4(a3 - s a2)3 - 27(a3 - s ala2 + 2 ai)2 (4.1, 21)

as well as of Hurwitz's condition

a1>0, a3>01 a1a2 - as > 0 (4.1, 22)

and finally, of Descartes' theorem :The number of positive roots of an algebraic equation is equal to, or smaller by an

even number than the number of sign changes in the sequence

1, al, a2, a3 .

Let us note that this theorem may be reversed to apply to the number of negativeroots and sign changes in the sequence

-l,a1, -a2,a3i.e. to those of the equation

-R3 + a122 - a22 + a3 = 0 (4.1, 16 a)

Let us now classify the singular points. The ones most likely to occur are :

Stable node. Hurwitz's conditions are satisfied for (4.1, 16); D3 > 0. All the rootsare real negative. All the trajectories enter the singular point (see Fig. 4.1, 8).

71

k

Fig. 4.1, 8

Unstable node. Hurwitz's conditions are satisfied for (4.1, 16a); D3 > 0. All theroots are real, positive ; a1 < 0, a2 > 0, a3 < 0. All the trajectories start from thesingular point (see Fig. 4.1, 9).

4

8


Stable focus. Hurwitz's conditions are satisfied for (4.1, 16); D3 < 0; a pair ofcomplex roots with negative real parts, one negative real root. All the trajectoriesenter the singular point, but in contrast to the node are spiral-shaped (see Fig. 4.1, 10).

Fig. 4.1, 10

Unstable focus. Hurwitz's conditions are satisfied for (4.1, 16a); D3 < 0; a pair ofcomplex roots with positive real parts. All the trajectories - spiral shaped - startfrom the singular point (a1 < 0, a2 < 0, a3 < 0) (see Fig. 4.1, 11).

Saddle of the first kind. Two real negative roots, one positive root; D3 > 0. Thetrajectories entering the singular point form a surface - the separatrix. Only twotrajectories leave the singular point, and they have a common tangent there (seeFig. 4.1, 12).

Fig. 4.1, 12


Saddle of the second kind. Two real positive roots, one negative root; D. > 0. Onlytwo trajectories enter the singular point. The trajectories leaving that point form asurface (see Fig. 4.1, 13).

t

Fig. 4.1, 13

Saddle-focus of the first kind. A pair of complex roots with negative real parts, onepositive real root. All the trajectories entering the singular point form a surface -the separatrix, and are spiral-shaped. Two trajectories leave the singular point (Fig.4.1,14);D3<0,a3<0.

Fig. 4.1, 14

Saddle-focus of the second kind. A pair of complex roots with positive real parts, onenegative real root. The sense of the course of the trajectories is opposite to that of thepreceding case (see Fig. 4.1, 15), D3 < 0, a3 > 0.

n


Unstable centre. A pair of pure imaginary roots, one positive real root (in this as wellas in the next case, a stable centre) ; this is of course only a necessary condition for theexistence of a root of this type. Only two trajectories leave the singular point; theremaining ones lie on parallel cylindrical surfaces (see Fig. 4.1, 16).

Fig. 4.1, 16

Stable centre. The sense of the course of the trajectories is opposite to that of theproceding case (see Fig. 4.1, 17). Here as well as in the critical case it is the terms ofhigher orders with respect to t; in the system of perturbation equations that decideon the stability. The characteristic equation has a pair of pure imaginary roots andone negative real root.

Fig. 4.1, 17

All cases with a single or multiple zero root are critical. Similarly as in case of twodifferential equations of the first order, our task will consist in the solution of theseparatrix and in turn in the examination of the points of the saddle and saddlefocus type, lying on those separating surfaces.

Let us turn now to systems governed by a set of four differential equations of thefirst order:


X(x,y,z,w),Y(x,y,z,w),

(4.1. 23)z=G(x,y,z,w),

iv=W(x,y,z,w).In a similar way as in the previous case the following characteristic equation can beobtained :

A4 + alA3 + a2A2 + a3A + a4 = 0 (4.1, 24)

whose coefficients are defined by the relations

ax ay aZ aWal_

ax + ay + az + aw) I(4.1, 25)

waax a-x -1Yx aw ay aw I aZ alza2 =

aW aWax aw

aW aW aW aWOr aw

a3 = -

a4 =

ay aw

ax ax ax axax ay ax Or

a Y a Y + aZ aZ +ax ay ax Or

ay ayay Or

(4.1, 26)aZay

aZOr

ax aX ax aX ax axax ay az ax ay aw

ay ay ay ay ay ayax ay Or ax ay aw

aZ aZ aZ aW aW aWax ay Or ax ay aw

ax ax ax ay ay aYax Or aw ay Or aw

aZ aZ aZ aZ aZ azax az aw ay Or aw

aW aW aW aW aW aWax az aw I I ay az aw

ax ax ax axax ay Or aw

ay ay ay ayax ay Or aw

aZ aZ aZ aZax ay Or aw

aW aW aW aWax ay Or aw

, (4.1, 27)

(4.1, 28)


Denoting by 21, 22, A3, 24 the roots of (4.1, 24), it holds that

a1 = - (2__11I__+ 22 I+-- 23 + 24) ,

a2 = 2122 + A123 + 2124 + A2A3 + A2A4 + 2324 ,

a3 = -2122(23 + 24) - 2324(21 + 22) ,

a4 = 21222324 .

a1 1 01

In the discussion of the roots we shall again make use of the discriminant of (4.1, 24)

D4 = 27 (12a4 + a2 - 3a1a3)

- a; (27a2 - 72a2a4 + 2a2 - 9a1a2a3 + 27aia4)2 , (4.1, 30)

of Hurwitz's conditions for (4.1, 24)

a1>0, a2>0, a3>0, a4>0,

a1 1

a3 a2

>0, a3 a2 >0,a1

(4.1, 29)

(4.1, 31)

0 a4 a3!

of Hurwitz's conditions for equation

-14 + a123 - a222 + a32 - a4 = 0 (4.1, 24a)

and finally, of Descartes' theorem concerning the changes of sign of the sequences

1, al, a2, a3, a4

appertaining to (4.1, 24), and

-1, a1, -a2, a3, -a4 (4.1, 32 a)

appertaining to (4.1, 24a).Compared with the previous case a higher number of unstable singular points of the

saddle or saddle-focus type exist, mainly because a distinction must be made between

sgn Re (21) = sgn Re (22) = sgn Re (A3) = -sgn Re (24) (4.1, 33)

and

sgn Re (A1) = sgn Re (22) = -sgn Re (23) = -sgn Re (24) . (4.1, 33 a)

The critical cases are again excluded. The singular points can be classified as follows :

Stable node. All the roots of the characteristic equation are negative, real; D4 > 0;the stability conditions are satisfied for (4.1, 24). All the trajectories enter the singularpoint.

Unstable node. All the roots are real, positive; D4> 0; Hurwitz's conditions aresatisfied for (4.1, 24a). All the trajectories leave the singular point.

Stable focus. Two pairs of complex roots with negative real parts; D4 > 0; Hurwitz'sconditions are satisfied for (4.1, 24). The trajectories are spiral-shaped and all enterthe singular point.

Unstable focus. Two pairs of complex roots with positive real parts ; D4 > 0; Hur-witz's conditions are satisfied for (4.1, 24 a). All the spiral-shaped trajectories leavethe singular point.


Stable focus-node. A pair of complex roots with negative real parts, two real negativeroots; Hurwitz's conditions are satisfied for (4.1, 24); D4 < 0. All the trajectoriesenter the singular point and are again spiral-shaped.

Unstable focus-node. A pair of complex roots with positive real parts; two realpositive roots; D4 < 0; Hurwitz's conditions are satisfied for (4.1, 24a). All spiral-shaped trajectories leave the singular point.

Saddle of the first kind. Three real, negative roots; one positive root; D4 > 0;sequence (4.1, 32 a) has three changes of sign. Two trajectories leave the singular point;the trajectories that enter it form a three-dimensional separatrix surface.

Saddle of the second kind. Two real negative roots; two positive roots; D. > 0.;sequence (4.1, 32) has an even number of sign changes. The trajectories entering andleaving the singular point form two two-dimensonal surfaces.

Saddle of the third kind. A real negative root, three real positive roots; D. > 0;sequence (4.1, 32) has three sign changes; a4 < 0. Only two trajectories enter thesingular point ; those leaving it form a three-dimensional surface.

Saddle-focus of the first kind. Two real roots: one positive, one negative root; a pair ofcomplex roots with negative real parts; sequence (4.1, 32) has an odd number of signchanges; a4 < 0. Saddle foci of the first or fifth kind are by their nature analogous tosaddles of the first or third kind, similarly as saddle-foci of the second, third andfourth kind are analogous to saddles of the second kind; they differ from saddles onlyby the spiral character of the trajectories at the singular point.

Saddle-focus of the second kind. Two real positive roots, a pair of complex roots withnegative real parts; D4 < 0; sequence (4.1, 32) has an even number of sign changes.

Saddle-focus of the third kind. A pair of complex roots with negative real parts, apair with positive real parts; D4 > 0; sequence (4.1, 32) has an even number of signchanges.

Saddle-focus of the fourth kind. Two real negative roots, a pair of complex roots withpositive real parts; D. < 0; sequence (4.1, 32) has an even number of sign changes;a4 > 0-

Saddle-locus of the fifth kind. Two real roots: one positive, one negative ; a pair ofcomplex roots with positive real parts; D4 < 0; sequences (4.1, 32) has an odd numberof sign changes. From our point of view the cases of a multiple complex root with non-zero real part that occur in addition to the points just enumerated, represent neitheran exception nor new types.

Unstable centre-node. A pair of pure imaginary roots, two positive real roots; D4 > 0;sequence (4.1, 32) has an even number of sign changes.

Unstable centre-focus. A pair of pure imaginary roots, a pair of complex roots withpositive real parts; D4 < 0; sequence (4.1, 32) has an even number of sign changes.

Centre-saddle. A pair of pure imaginary roots, one real positive, one negative root;D4 > 0; sequence (4.1, 32) has an odd number of sign changes.

Additional information can be found in the monograph by TONDL (1970a).


4.2. Practical solution of the phase portraits

In order to be sufficiently clear and comprehensive and to provide enough informationabout the behaviour of the system being investigated, a phase portrait must containthe various singular points and the most important trajectories in the phase plane.Whenever several singular points are present, it is particularly necessary to establish(approximately at least) the position of all saddle points which are important for apractical solution. The most important trajectories are the limit cycles (if they existat all) - both stable and unstable - and the trajectories entering and leaving thesaddle points. The trivial case of a single steady solution represented by a singularpoint in the phase plane is not considered.

The cases which are apt to come up can be divided into three groups:

(1) A single singular point surrounded by one or more limit cycles.(2) Several singular points but no limit cycles.(3) Several singular points, one or more limit cycles.

Stable singular points and stable limit cycles represent the simple attractors of thesystem. In practical solutions it is well to keep in mind the following important conse-quences of the fact that, on the assumptions made in connection with system (4.1, 1),no trajectory can intersect another at a regular point (all points in the phase planeexcept the singular ones). At least one singular point must lie inside each limit cycle.A single singular point can be surrounded by several limit cycles, the stable cyclesalternating with the unstable ones. If a single singular point, but no other limit cyclelies inside an unstable (stable) limit cycle, the singular point is stable (unstable). Theunstable limit cycle then represents the separatrix which bounds the domain ofattraction of the stable singular point lying inside the unstable limit cycle. In othercases the unstable limit cycle always forms part of the separatrix which bounds thedomain of attraction of a particular stable steady solution. An unstable limit cyclecan be obtained in the same way as a stable one by solving (4.1, 1) in "the negativetime", i.e. by solving (4.1, 12) instead of (4.1, 1).

A solution should proceed from the determination (approximate at least) of theposition of the singular points and identification of the saddle points. As will beshown in Section 4.5, this can be done quite generally for some systems. In the major-ity of cases, a practical analysis is not essentially concerned with finding out whethera singular point is a focus or a node.

The simplest procedure of all is that applicable to systems of group (1) (a singlesingular point). It starts with the initial conditions corresponding, approximately, tothe coordinates of the singular point; depending on whether the singular point isstable or unstable, the equations are solved in the positive or negative time. The limitcycle is drawn as soon as a steady solution (a closed trajectory) has been obtained. Inthe next step the solution starts from a point lying outside the limit cycle drawn forthe opposite time (i.e. the negative time if the preceding solution was effected in thepositive time). Notice is taken of the trajectory (the motion of the plotter pen) tosee whether or not it converges to a new limit cycle. In this way the limit cycles aredetermined one after another.

In the case of systems of group (2) (several singular points but no limit cycles) itshould be realized that - as the trajectory geometry in the phase plane implies -n saddle points must exist among 2n + 1 singular points. Accordingly, the solutionshould always start from a saddle point and continue by drawing the trajectories

4.3. Examples of systems of group (b) 91

issuing from it in both the positive andnegative times. The character of the two pairsof trajectories thus obtained frequently provides enough information about thebehaviour of the system being investigated. If a divergent solution cannot exist, thetrajectories issuing from the saddle point in the positive time must terminate in astable singular point. Only the stable singular points are the attractors of the system.

In the case of systems of group (3), one or both trajectories issuing from the saddlepoint in the positive time are apt to terminate on a stable limit cycle. An unstablelimit cycle or an unstable singular point must lie inside this cycle. If the two trajecto-ries issuing from the saddle point in the negative time are continuously moving awayfrom this point, they form a separatrix, i.e. a boundary between the different domainsof attraction of a particular steady solution. The trajectories issuing from the saddlepoint in the negative time can, however, also terminate in an unstable singular pointor on an unstable limit cycle. Section 4.5 will discuss these possibilities using actualexamples. The procedure just outlined makes it possible to obtain comprehensiveand informative phase portraits within a comparatively short time.

Early phase portraits were frequently obtained by application of the graphico-numerical method of Lienard (see, e.g. KLOTTEB (1980)). It has been found, howeverthat an analogue or a digital computer fitted with a graph plotter for direct drawingof the trajectories is much more effective for this task. When a digital computer isused, the systems of first-order differential equations can be solved by one of thespecific methods, such as the Runge-Kutta method or a simple procedure based on theidea of the Lienard method; the latter, proposed by TONDL (1978 c) has been used forsolving the examples presented in this chapter and its principle is explained inthe Appendix.

The next three sections will deal with examples of different systems belonging tothe groups defined at the beginning of Section 4.1. The simpler cases of groups (b)and (c) will be discussed first. The following notation will be used throughout:FS(FN) stable (unstable) focusNs(NN) stable (unstable) nodeSP saddle pointLs(LN) stable (unstable) limit cycle (drawn in solid (dashed) heavy lines)s trajectories converging to the saddle point, provided that they form a separatrix

(drawn in dot-and-dash lines).

4.3. Examples of systems of group (b)

The first example refers to a self-excited system governed by the equation

y+(x-fly2+ey4)y+y+µy3=0 (4.3,1)

where x, fl, E and It are positive constants. Introducing the new variables y = v, y,equation (4.3.1) takes the form

y-v,1.

3(4.3,2)v= -(x-fly2+Ey4)v+y+µy.

Although an analytical solution of steady vibration is not the principal subject ofthis chapter, it will be recapitulated here in order to provide as complete a pictureas possible. Approximating the steady solution of (4.3, 1) by

y=AcosQt


and applying the harmonic balance method for the determination of A and S2, leadsto the following equations

Q2 = 1 + ; uA2 , (4.3, 3)

K(A) = x -4

#A2 -F l eA4 = 0 (4-3,4)

For the amplitude A, (4.3, 4) implies

= tE + \E/ 2- 8 E

1/2 1/2

A1,2So long as

#2 > Sxe

(4.3, 4a)

(4.3, 5)

the equation yields two values of A, i.e. two limit cycles exist in the phase plane. Ifcondition (4.3, 5) is not satisfied, no real values of A are obtained. Since function K(A)represents the damping (positive for K(A) > 0, negative for K(A) < 0), it can beassumed that the equilibrium position y = 0 represents a stable solution, that theunstable limit cycle in the phase plane corresponds to the solution with the amplitudeAl (A1 < A2) and the stable limit cycle to the solution with the amplitude A2. Thecurve of function K(A) is shown in Fig. 4.3, 1 for x = 0.3, e = 10-4 and two valuesof 13, i.e. 0.0175 and 8xs. In the first case (13 = 0.0175) there exist two limit cycles inthe phase plane; in the second (a border-line case) the two limit cycles become a singlesemi-stable limit cycle. From the practical point of view, the latter case may be con-sidered to belong to those for which condition (4.3, 5) is not satisfied and the trivialsolution (y = 0) is absolutely stable. The phase portrait of the first case (13 = 0.0175)is shown in Fig. 4.3, 2 while Fig. 4.3, 3 shows the records of the vibration y(t) correspond-ing to the limit cycles. The system belongs to the class with the so-called hard self-

K(A) = a3-1 (3A2+

e10 4A4

0.5

0

J3=0.0175

-0.5t-

Fig. 4.3, 1

4.3. Examples of systems of group (b) 93

-30 -20

Fig. 4.3, 2

yj20

-20

Fig. 4.3, 3

-10 0 10 20 y 30

excitation because - so long as the initial conditions (v(0), y(0)) correspond to thepoints lying inside the unstable limit cycle - the solutions converge to the equilibriumposition y = 0. The unstable limit cycle forms the boundary of the domain of attrac-tion of the trivial solution, and outside it lies the domain of attraction of the steadyself-excited vibration.

The second example refers to a self-excited system with three equilibrium positions,governed by the equation

(4.3,6)

When self-excitation is not in effect (fl = 0), two equilibrium positions, y = -f-(a/y)112,are stable and one, y = 0, is unstable. The solution is carried out for constant fl = 0.2


and a = y = 1 and alternatively varying 6. For fi > 0, all equilibrium positions areunstable and the values of the coefficients fl and 6 decide (given a and y) whether thesteady vibration vibrates with a smaller amplitude about the equilibrium positiony = ± {1a-/y (y = ±1 in the case being considered) or with a larger amplitude aboutthe equilibrium position y = 0. If the first case takes place, the analytic solution canbe approximated by

y = Y+AcosQt (4.3,7)

where Y is the constant deflection (for A -> 0, Y ---> 11a/y). If the second, the solutioncan be approximated by

y=AcosQt. (4.3,8)

Applying the procedure used in the first example, equation (4.3, 6) (for the givenvalues of the coefficients fl, a and y) can be written in the form

yv,(0.2 -bv2)v+y(1 - y2)

(4.3,9).

It can easily be shown that the singular points y = ±1 and v = 0 are unstable foci,and the point v = 0, y = 0 is a saddle. The phase portraits for three different valuesof S are shown in Fig. 4.3, 4 (b = 0.2), Fig. 4.3, 5 (b = 0.4) and Fig. 4.3, 6 (b = 0.7).In the first two cases, all three singular points are surrounded by a single stable limitcycle which represents the only absolutely stable steady solution. The trajectoriesissuing from the saddle point in the negative time terminate in the singular points withcoordinates y = ±1. The trajectories issuing from the saddle point in the positivetime terminate on the limit cycle. In the third case (b = 0.7) each of the foci is sur-

-2 1 0 1 YFig. 4.3, 4

2

4.3. Examples of systems of group (b)

-2

Fig. 4.3, 5

0 1

s

.\.L-2 -1 0 1

Fig. 4.3, 6

Y 2

Y 2

95

rounded by a stable limit cycle. The trajectories issuing from the saddle point in thenegative time form the separatrix which separates the domains of attraction of thesteady solutions represented in the phase plane by the above-mentioned limit cycle.Fig. 4.3, 7 shows the records of vibration corresponding to the limit cycles of thecases discussed.


Fig. 4.3, 7

4.4. Examples of systems of group (c)

As the first example consider the Duffing system (whose steady resonance solution wasanalyzed in Chapter 2) described (in the dimensionless form) by the equation

y+xy+y+ay3=cosgr (4.4, 1)

where x and y are dimensionless coefficients, and q is the relative excitation frequency.Applying the van der Pol method to the investigation of the steady solution and itsstability one obtains the following system of equations

a = 2 -x1a + [1 - n2 -{- 4 y(a2 + b2)] b}

b =2,q

{ 1 - xnb - I1 - X72 + 4 y(a2 + b2)] a}

(4.4, 2 )

The steady solutions of (4.4, 1) are represented by the singular points. The problemof stability of the steady solutions to any disturbances in the initial conditions is solvedby determining the domains of attraction of these singular points. The phase portraitsare drawn for the parameters x = 0.05, y = 10-2 and for three values of the relativeexcitation frequency ri lying in the interval for which three steady solutions (twostable, one unstable to which a saddle point corresponds in the (a, b) phase plane)exist. The stable solutions differ from one another by the magnitude of the vibrationamplitude (A = (a2 + b2)112) : small A - non-resonant solution, large A - resonantsolution. The phase portraits are shown in Fig. 4.4, 1 (rl = 1.2), Fig. 4.4, 2 ()7 = 1.3)and Fig. 4.4, 3 (iq = 1.5). As increases, the amplitude of the resonant vibrationgrows larger, that of the non-resonant vibration, smaller. The reverse is true of the

4.4. Examples of systems of group (c) 97

1

-15 -10 -5Fig. 4.4, 1

-15 -10

Fig. 4.4, 2-5

0

! 7) =1.3

0

5

5

10 a 15

10 a 15

domains of attraction: as q increases, the domain of attraction of the resonant solutiongrows smaller, that of the non-resonant solution, larger (in Figs. 4.4, 1 to 4.4, 3, thelatter domain is shown in hatching).

This finding has some important consequences: Consider a device whose motion isgoverned by the Duffing equation and which is desired to vibrate with a large amplitude(e.g. a vibratory compressor). Although this can be achieved in a broad interval of theexcitation frequency in consequence of the nonlinearity of the restoring force with ahardening characteristic, it should be borne in mind that there exists an interval ofthis frequency in which two stable steady solutions (one resonant, the other non-resonant) are possible. The more the resonance peak is approached, the more readilycan the effect of disturbances cause the device to "fall out" from the resonance regime;consequently, the device is incapable of being run in a regime corresponding to thevicinity of the resonance peak.7 Schmidt/Tondi


-15 -10

Fig. 4.4, 3

-5 0 5 10 a 15

As an example of a parametrically excited system consider the system whose analy-tic solution of steady vibration was offered in Section 3.3 and which is governed byequation (3.3, 13). Using the van der Pol method one obtains the following set ofdifferential equations:

a2w

J-I -

O,(a2 + b2)-1/2 + 1 Sco(a2 + b2)1 a

b+[I - 1

2u + 3

4 y(a2 + b2) - w2111

1 3 1- 1 +2

u+4

y(a2 + b2) - w2] al.J

(4.4, 3),

The phase portraits for three different values of (o and the parameters 6 = y = 0.01,0 = 0.1 and ,u = 0.2 (cf. the diagram in Fig. 3.3, 4) are shown in Fig. 4.4, 4 (w = 1)

5

b

0

-50 a 5 U a

Fig. 4.4, 4 Fig. 4.4, 5

4.5. An example of a system of group (a) 99

Fig. 4.4, 5 ((o = 1.05) and Fig. 4.4, 6 (w = 1.1). Three domains of attraction arefound to exist: that of the trivial solution, which grows larger with increasing w(shown in hatching), and two domains of parametric resonance vibration. These vibra-tions have the same amplitude and differ from one another only by the phase shift.Their domains grow smaller with increasing w.

A number of additional examples of systems of this group may be found in a bookby HAYASHI (1964) and in a monograph by TONDL (1970a).

15

b

10

S

FS

-10

-15 -10

Fig. 4.4, 6

S

5 0 5 10 a 15

4.5. An example of a system of group (a)

The example does not represent a mechanical system but a model of a system with aflowing medium pumped into a pipe line by a centrifugal compressor, a pump ora blower. This simplified model is governed by a set of two first-order differentialequations describing the time variation of the pressure difference P and the volumetricflow rate of the conveyed medium, Q. The example has been chosen not only becauseit represents an interesting case of (4.1, 1) when both the singular points and the limitcycles can exist in the phase plane but also because with its aid one can show quitegenerally some interesting properties of the system and the application of the phaseportraits to its analysis. Only the salient points of the analysis will be discussed inthis book. For a thorough treatment of the topical problems the reader is referred toa monograph by ToNDL (1981 a) discussing, among others, the effect of the system para-meters, machine characteristics, etc.

The system used in the example is governed by the equations (whose derivationmay be found in a book by KAZAKEVIC (1974); MOQuEEN (1976), SKALICKY (1979)and other authors have used similar equations)

(4.5, 1)

7.


where P is the inlet-outlet pressure difference, Q - the volumetric flow rate of themedium, t - the time, V(P) or the inverse of it, P = cp(Q) - the load characteristic(pipeline resistance), P = I (Q) - the machine characteristic, and the parameters C(acoustic compliance) and L (acoustic mass) are defined by

eCc2 LSwhere V is the total volume of the medium in the whole pipe system, P - the densityof the medium, c - the sound velocity in the medium, 1 - the length and S - thecross-sectional area of the piping.

The machine characteristic P = I (Q) is specified for definite rpm (revolutions perminute) and obtained in the process of loading the machine by throttling at theoutlet. Typical machine characteristics feature two local extremes. As shown schema-tically in Fig. 4.5, 1 a the transient portion between the local extremes is less steepfor centrifugal pumps and blowers than for centrifugal compressors (Fig. 4.5, 1 b) ;in the latter case, and especially in the case of axial compressors, the machine cha-racteristic is apt to consist of two branches (it is not described by a unique continuouscurve in the whole range of Q - Fig. 4.5, 1 c) and hysteresis phenomena (marked witharrows in the diagram) take place. The load characteristic is usually expressed by aquadratic relation of the type

P=kQ2sgnQ+d (4.5,3)

where d = 0 so long as the machine is not pumping to overpressure (e.g. a pumpconveying water to an elevated reservoir); accordingly, the load characteristic isalways an increasing function of Q.

P

f(Q )

.Q

a) b)

Q

Qc)

Fig. 4.5, 1

4.5. An example of a system of group (a)

Introducing the relative quantities

_ Q _ Pq Q0, p-Po.

where P0, Q0 are chosen constant values, and the time transformation

dt=LQOdzPO

equations (4.5, 1) can be converted to the dimensionless form

- = A[q - c(p)] , dq = F(q) - p

where

101

(4.5, 4)

(4.5, 5)

(4.5, 6)

2A=C(QO)oisa dimensionless parameter, and the functions 0(p), F(p) are derived from the func-

tions p(P) and f (q).The points of intersection of the machine and the load characteristics are the singu-

lar points of (4.5, 6) in the (p, q) plane and represent the steady state - the non-oscillatory solutions. It is desirable that the steady state should occur for sufficient-ly large values of p and q, i.e. that it should be represented by a point on the right-hand portion of the characteristic. This means that the machine is supplying a largequantity of the medium at a high enough pressure. For a machine to operate reliablythe steady state should be stable and that not only for very small disturbances (inthe ideal case, the steady state would be absolutely stable). Such a case is apt to occurwhenever the load characteristic intersects the machine characteristic at severalpoints, i.e. whenever several steady states exist (if the limit case of the load characteris-tic only touching the machine characteristic is not considered, the usual number ofsuch states is three).

Accordingly, the problem consists in establishing the initial conditions for whichthe different steady states occur, or the disturbances which are apt to lead to transitionfrom the desirable steady state - represented by the singular point on the right-handpart of the machine characteristic - to the undesirable ones, i.e. the non-oscillatorysteady states represented by the other singular points, or even the oscillatory ones(self-excited vibration) represented by the limit cycles in the (p, q) plane. This self-excited vibration - called the surge - manifests itself by violent fluctuations of thepressure and the flow rate and always constitute a serious hazard for safe operation.

In the analysis which follows the stability and the type of the singular points willbe dealt with first. They can both be established by using the characteristic equation

-Aa -A Aap

aF-2aq

= 0 (4.5, 7)

where the coordinates of the singular point are substituted for p and q in expressionsaO(p)/ap = dO/dp, aF(q)/aq = dF/dq. Denoting by q(q) the reciprocal to function


O(p), one obtains

dO 1

dp d99

dq

and the conditions for the singular point stability take the form

where

(4.5, 8)

(4.5, 9)

(4.5, 10)

'=dqcP , F'=dq

Since always A > 0, qi > 0 (as already mentioned, 99(q) is an increasing function),it follows from conditions (4.5, 9) and (4.5, 10) that a sufficient condition for the sta-bility of a singular point is a negative slope of the tangent to the machine characteris-tic at that point (F' < 0).

In the next step it will be generally shown which of the singular points are saddles(a proof concerning this matter was offerend by ToNDL (1979 b)). Arrange the singularpoints according to the magnitude of the coordinate q:q1 < q2 < q3 < ... and markthem with the numerals 1, 2.... (Fig. 4.5, 2). Since in the case being considered the

P

Fig. 4.5, 2q

inequality F(q) > 92(q) is satisfied for q < q1, it always holds for the odd singularpoints that

9),> p

and for the even singular points that

9''<F'.

(4.5, 11)

(4.5, 12)

As condition (4.5, 10) implies, all even singular points are always stable.

Theorem: The even singular points are always saddles, and an odd singular pointcan never be a saddle.


Proof: In this case, conditions (4.1, 8) for the existence of a saddle take the form

\A-F'12 >4A 1-F//

(4.5, 13)

1 - F <0. (4.5, 14)T

The inequality (1 - Y IT' < 0) always applies to the even singular points; conse-quently, the above conditions are always satisfied. For the odd singular points, onthe other hand, it always holds that (1 - F'199' > 0) and the condition (4.5, 14) is,therefore, never satisfied.

The theorem is thus proved.

Mention (without proof) will also be made of another property of the system beingstudied. As TONDL (1980 c) has shown the boundary - when the stability of thesingular point is defined by the condition (4.5, 9) while the condition (4.5, 10) isalways satisfied - is represented in the phase plane by a straight line passing throughthe point q = 0, p = 92(0) and having a slope of 1 A. If the singular point lies abovethis line, the condition (4.5, 9) prevails; this means that the stability of the singularpoint is affected by the magnitude of the parameter A (so long as a change of Adoes not cause the singular point to move below the limit straight line). As the analysisof a number of examples reveals, condition (4.5, 9) is the decisive one in most cases,i.e. the stability of the singular points is affected by the parameter A.

The more complicated case of a machine characteristic consisting of two brancheswill not be treated here. TONDL'S (1979b) (see also (1981 a)) approach to the solutionof this problem is as follows : Divide the phase plane by the dividing curve into twodefinition domains; in each domain define the machine characteristic in terms ofa function obtained by means of regression of the corresponding experimentallydetermined branch. As proved by the author, the point of intersection of the dividingcurve and the machine characteristic can - for practical purposes - be regardedas a saddle point. The solution then proceeds in the way outlined in connection withthe continuous machine characteristics.

As mentioned in Section 4.2, in the most effective procedure for obtaining the phaseportraits, the solution starts from the saddle point and continues by drawing thetrajectories in the positive as well as the negative time. If both characteristics are spe-cified in the phase plane (p, q), the saddle points can readily be determined on thebasis of the considerations put forward in the foregoing. The subsequent strategy iscontrolled by the course of the trajectories issuing from the saddle point in the positiveand the negative time.

Below is a discussion of two sets of phase portraits obtained in the case of a conti-nuous compressor characteristic expressed by the function

F(q) = 0.8 tan-1 (100q - 20) + 1.8 - q + 25q2 - 50q3 (4.5, 15)

and a load characteristic defined by the function

p = 99 (q) = Kq2 sgn q (4.5, 16 a)

i.e. for the function i(p)/p 112

O(p) = I K) sgn p (4.5, 16 b)


Each set of phase portraits is introduced by the diagrams of the characteristics.The individual portraits correspond to different values of the parameter A (shown inthe top right-hand corner). The first set corresponds to the load characteristic para-meter K = 40 (Fig. 4.5, 3), the second to K = 50 (Fig. 4.5, 4). The notation is thesame as that used in the preceding examples, but for better clarity the trajectoriesleaving the saddle point in the negative time (i.e. those approaching the saddle point inthe positive time) and not forming the separatrix are drawn in light dashed lines. ForK = 40 and the lowest value of A = 50, both trajectories issuing from the saddlepoint (in the positive time) terminate on the stable limit cycle. One of the trajectoriesissuing from the saddle in the negative time is moving towards the unstable focus,the other terminates on the unstable limit cycle which surrounds the stable focus.It will be seen that a sufficiently clear phase portrait is obtained by simply drawingthe trajectories issuing from the saddle point in both the positive and the negativetime. Two locally stable steady solutions exist in this case: one, non-oscillatory,represented by the stable focus lying on the right-hand part of the compressor charac-teristic, the other, oscillatory, represented by the stable limit cycle. A comparativelysmall unstable limit cycle bounds the domain of attraction of the desirable steadystate. The whole remaining area of the phase plane constitutes the domain of attrac-tion of the dangerous oscillatory solution. The oscillations are so violent that theminimum value of the instantaneous relative pressure p tends to zero and the valueof the relative flow rate q is, in fact, negative in a certain interval of time, i.e. themedium flows in the opposite direction. An increase of the parameter A causes thearea bounded by the stable limit cycle to grow smaller and the area bounded by theunstable limit cycle (which, for A = 100, reaches as far as the saddle point) to growlarger. For greater values of A (150 to 200) the stable limit cycle no longer exists andboth foci are stable. The trajectories issuing from the saddle point in the negativetime form the separatrix which divides the domain of attraction of the foci corres-ponding to the operating points on the right-hand and the left-hand part of the com-pressor characteristic. The latter point is unfavourable not only because the compres-sor delivers a small quantity of the medium at a low pressure but also because of thedanger of its blades becoming overheated.

The phase portraits shown in Fig. 4.5, 4 for K = 50 are similar to those of the preced-ing set except that, for lower values of A (A < 150), the singular point on the right-hand part of the compressor characteristic is unstable (F' > 0). As A increases, thepoint on the left-hand part of the compressor characteristic becomes stabilized first(at an A as low as 75). The domain of attraction of the singular point on the right-hand part of the characteristic is comparatively small even for large values of A.Although the limit cycle no longer exists for A = 100, the singular point on the left-hand part of the characteristic is the only stable steady (i.e. absolutely stable) solu-tion.

As a study of the different shapes of the compressor characteristics reveals, a round-ed flat peak (the neighbourhood of the local maximum) when the slope of the tangentis positive (F' > 0) over a wide range of q, is not advantageous. In such a case thelimit cycles are present for a wide interval of the values of the parameters K and A.This finding is confirmed by the results obtained for a compressor characteristicdefined by the equation

F(q) = 0.7 tan-1 (50q - 10) + 2 + q + 20q2 -- 70q3 + 40g4 . (4.5, 17)

Fig.

4.5

, 3

100

150

SP

75

\20

0

0 OZ

Fig.

4.5

, 4

100

150

sP

75

200

Fig.

4.5

, 5

100

150

200


5

P

4

3

2

1

0

-0.17 -

-10

P

0.5

0.1

1.0

0.2 0.3

A=50

1.5

0.4

2.0 £ 2.5

Fig. 4.5, 6


5

P

4

3

2

1

0

7T0 0.1 0.2

0

0.3 0.4 0.5 q 0.6

225

Fig. 4.5, 7


5

P

4

0

f -I-K= 40 A=50

3

2

1

0

7--1-0.1 0.1 0.2 0.3 0.4 0.5 q 0.6

Fig. 4.5, 8

As the phase portraits shown in Fig. 4.5, 5 for K = 60 and different values of Asuggest, limit cycles representing an absolutely stable steady solution exist for awide range of A. Fig. 4.5, 6 shows the time relations p(z), q(r) for different values of Acorresponding to the limit cycles. The phase portraits drawn in Fig. 4.5, 7 (K ='80)and Fig. 4.5, 8 (K = 40) (cases of a single singular point) disclose the presence of


a limit cycle at low values of A in these cases, also. The effect of the parameter A isclearly seen in both figures. As this parameter is increased from a low value, one findsthat at a definite A the singular point becomes stabilized and an unstable limit cyclelying inside the stable limit cycle forms around it. At a further increase of A the twolimit cycles begin to approach one another until, at a definite value of A, they becomea single semi-stable limit cycle (Fig. 4.5, 8 for A = 87). For A larger than this value,the limit cycles no longer exist.

The most important results of this section can be summed up as follows:

(a) Stability of a non-oscillatory steady state is no guarantee that self-excited oscil-lations (the surge) will not arise.

(b) Increasing the parameter A has a favourable effect on stability of the odd singu-lar points lying on the right-hand and left-hand parts of the machine characteri-stic (provided that the rare case of condition (4.5, 4) being decisive is not consider-ed).

(c) The domain of attraction of the singular point lying on the right-hand part ofthe machine characteristic does not necessarily increase monotonously withincreasing A (Fig. 4.5, 3).

(d) The intensity of the self-excited oscillation decreases with increasing A. Althoughthe self-excited oscillation vanishes at a certain value of A, its amplitudes do nottend to zero -- the change occurs by a jump.

(e) A rounded peak (at the local maximum of the machine characteristic) along whichthe slope of the tangent to the characteristic is positive in a wide interval of q,is not advantageous.

5. Forced, parametricand self-excited vibrations

5.1. Amplitude equations

The methods introduced in Chapter 2 make possible quite systematic investigationsof vibrating systems under forced, parametric and self-excitation. In this chapter weinvestigate systems with one degree of freedom. The vibrations of a wide class of suchsystems can be modelled by a differential equation of the form

0"+Qx= (P1 cosj(ot+Q1sin jcot) -Bx-Cx2x-Dx4x=o- Ex3 - Fx5 - (G1 cos jwt ± H, sin jolt) x

CO- (K, cos jwt + L, sin jwt) x2 (5.1, 1)3=0

where dots denote derivatives with respect to time t, the coefficients P, and Q1 ageneral periodic forced excitation (containing a factor w2 in case of an inertial exci-tation; PO describes a constant prestress), E and F a symmetric restoring force com-ponent B, a linear and C, D an amplitude-dependent non-linear damping if positive(for the equivalence of a nonlinear damping -C'x3 -- D'x5 see KLOTTER (1980)).A soft self-excitation occurs for the van der Pol equation where

B<0, C>0, D=0 (5.1,2)

and for the generalized van der Pol equation whereB<0, C>0, D<0, (5.1,3)

a hard self-excitation forB>0, C<0, D>O. (5.1,4)

The coefficients G, and H, denote a general periodic (linear) parametric excitationK, and L, (j > 1) a periodic non-linear parametric excitation, Ko especially a non-symmetric restoring force component.

We will introduce a dimensionless time r = wt, derivatives with respect to whichare denoted by dashes, and a frequency variation

w - woa = , i.e. co = wo(1 + a)coo

which characterizes the distance from a fixed (circular) frequency wo, and write2

2=wo, x(t)=x )=y(v')

5.1. Amplitude equations 113

Then the differential equation (5.1, 1) reads

1 00

y" + Ay = 2 E (P1 cos jr + Q1 sin jr) - a(2 + a) y"W0 j=0

(1 +a) B (1 +a) C2

, (1 +a) D E- y yy - y4y -2y3Coo BOO wo (00

F 1 °°

2Y - 2 E (G7 cos jr + H1 sin jr) ya)0 w0.7=1

12 jr + L1 sin ft) y2 . (5.1, 5)(OA

We assume the coefficients P1, Q,, a, B, G1, H1 on the right-hand side as sufficientlysmall. If A is not a square of an integer (non-resonance case), the integral equationmethod introduced in Section 2.3 leads to the first approximative solution

1 P1 cos jr + Q1 sin jrY10 2

w0i=0 y2 - j2

where the factor 0 = V f, now equal to 1 for every j, is added because of the later useof the formula. Inserting this approximation in the right-hand side of (5.1, 5), we getthe second approximation, and so on. The second and all higher approximationscontain only terms of the order of magnitude P1, Q1 multiplied by small terms. There-fore, the influence of non-linear restoring forces, damping, self-excitation and para-metric excitation on the solution is of smaller order of magnitude than that of forcedexcitation, which mainly determines the vibrations.

For the rest of this chapter, we investigate the resonance case where A is the squareof an integer,

A=n2.Then the first approximative solution is

y1=rCOSnr+ssin nr+y10 (5.1,7)

(the non-resonance case being included for r = s = 0). Omitting additional higherP1, Q1-terms mentioned above we get the second approximation

y2 = yi - 2a (r cos nr + s sin nr) + C- A2(s cos nr - r sin nr)4naw0

IC [s(3r2 - s2) cos 3nr - r(r2 - 3s2) sin 3nr] + D A4(s coS nr - r sin nr)

8naw0

3D+ 128oo0

A2[s(3r2 - s2) cos UT - r(r2 - 3s2) sin 3nr]

+ D[s(5r4 - 10r2s2 + s4) COS 5nr - r(r4 - 10r2s2 + 5s4) sin 5n-C]

240nw0

+ 4n 2w2 A2(r cos nr + s sin nr)0

+E

32n20J2[r(r2 - 3s2) cos 3nr + s(3r2 - s2) sin 3nr]

8 Schmidt/Tondl

114 5. Forced, parametric and self-excited vibrations

F+ 8n2cco2 A' (r cos nz + s sin nz)

0

5FA2 r r2 - 382 cos 3nz 2 - 82)+

128n2coo [ ( )+ s(3r s) sin 3n-c]

7 r4 - 10082 5s4 cos 5nr + 8(5r4 - 10r2s2 4

+ 240n2coo [ ( )+ s) sin 5n-r]

1 G,rcos(n-j)y+ssin (n-j)z+rcos(n+j)v+ssin (n+j)vc00 = On2 - (n - j)2 n2 - (n + j)2

1 HJIscos(n-j)z-rsin(n-j)z+-scos(n+j)r+rsin(n-{-j)t20009=1 L On2 - (n - j)2 n2 - (n + j)2

1 A2 °° K1 cos jr + L1 sin jr2 u>02 j=o On 2 - j2

1 [(r2 - s2) K1 + 2rsL1] cos (2n - j) r + [2rsK1- (r2 - s2) L1] sin (2n - j) z4coo On2 - (2n - j)2

1 [(r2 - s2) K1 - 2r8L1] cos (2n + j)z + [2r$K1 + (r2 - s2) L1] sin (2n + j)z4co02 j n2 - (2n f. j)2

(5.1, 8)

where we denote the amplitude of the resonance part of the first approximation by A,

A=Vr2+82,

which we simply call "amplitude", and also omit terms containing a2 because theyare small in comparison with the term containing a and the third approximation wouldlead to additional a2 terms.

The bifurcation parameters r and s have to be determined from the periodicityequations (2.3, 3). If they are known, the different approximative solutions describethe behaviour of the resulting vibrations. But as they are rather complicated, wehave to confine ourselves mainly to the most important quantity characterizing thevibrations, the amplitude A which we derive from the periodicity equations.

Using the second approximation (5.1, 8), the periodicity equations contain thelinear parametric excitation in form of the coefficients P2,,,, Q2n (main parametricresonance) only. In order to describe other parametric resonances, we have to use thelengthy parametric excitation terms of the third approximations. Thus we come to thetwo periodicity equations written in vector form

()+a(T) r1 -h (rl

/3r2 + $2 ` 2rs l r\2 /- s2L

2rs _ ( )- k !

2rs) - l (0 +

382) - K \ -2rs (r2 - s20 5.1 0

where for abbreviation

a=a-eA2-/A4+F, fi=b+cA2+dA4 (5.1,10)

5.1. Amplitude equations

and

P. 3E 5F=2n2coo

e= ,8n2wo

f16n2wo'

1 CIO [ 1 1 2 2 BI'= E + (G +H , b=8n2w09=1 O'n2 - (n - j)2 n2(n + j)2 a ) 2nwo,

Cc= -,8nwo

_ Dd

16nwo'

G2n9

4n2w0+ h=4Hw2+H,

0

1 0 rG1G2n-j + G1Gj-2n - H1H2n-j + H1H,-2nG [8n2w0 j=1 One - (n - j)2

+ GJGj+2n + H1Hj+2nI

n2-(n+j)2 '1

rG1H2n-j - G1H;-2n + H,02.-3+ H1G;-2n2

HL8n2wo 7=1 On2 - (n - j)2

+G1H;+2n - H1Gj+2n1

n2 - (n + j)2 J '

K. _ Ln K3n L3nk8n2wo ,

l- 8n2o)2

K=8n2wo

L =8n2wo

115

(5.1, 11)

The coefficient Q. may be assumed to be zero, Pn to be positive without loss of gene-rality by a suitable shifting of the time origin in the basic differential equation (5.1, 5) :

Pncosnz+Qnsin nr=Pncos(nz-T) (5.1,12)

with

Pn = I/p n + Qn , tan T = Q.

P.

Up to Section 5.7, we neglect the influence of the non-linear parametric excitationin the periodicity equations (5.1, 9), k = l = K = L = 0, which holds in any casewhen no non-linear parametric excitation exists at all. Then the periodicity equationsare, besides the expression A2 = r2 + $2, linear in r and s, their solution is

(a2+N2 -g2 -h2)r= -(a+g)p,(a2+P2-g2-h2)s=(j9-h)p

(5.1, 13)

Squaring them and adding leads to the frequency amplitude equation or brieflyamplitude equation

(a2 + NR2 -9 2 - h2)2 A2 = [(a + g)2 + (N - h)2] p2 (5.1, 14)

which contains the bifurcation parameters r, 8 in form of A2 only. It is of power fourin a, that is in the frequency variation a, and of power nine (five, if d = 0, resp. linear,if additionally c = 0) in A2.

In what follows, different vibration phenomena will be explained by help of thisamplitude equation. Some general conclusions are :8*


1. The relation

A if, if a2 > 32, g2, h2Ia N (5.1, 15)

holds, that is, A tends to zero (absolute minimum) with increasing jai.

2. In case of vanishing forced excitation, p = 0, and linear damping, c = d = 0, theamplitude A vanishes for a2 + g2 + h2 - R2 (which always holds for 132 > g2 + h2)whereas from the opposite relation a2 = g2 + h2 - (32 follows

eA2 + fA4 = a + I' ± g2 + h2 N- 2 ,

F'

an unlimited increasing of the amplitude.

3. If p = 0, no self-excitation occurs and damping is non-linear, d > 0 or c > 0,d = 0, the maximum amplitude is determined by

fl= b + cA2 + dA4 = j/g2 + h2 . (5.1, 16)

4. If p + 0 and damping is linear, the amplitude is limited for R2 > g2 + h2.

5. If p + 0, no self-excitation occurs and damping is non-linear, the amplitude islimited.

The number of coefficients can be diminished by introducing transformed quanti-ties. We introduce, if forced excitation p and linear damping b do not vanish,

A=L A, glbl hIbl

a=a-eA2- fA4+1', P=b+cA2+dA4with

A - T - b _ ep2a = IbI , 1' = Ibl , b = bl , e = Ib2l

4 e 2 d 4ibl cIbI d=bbl

(5.1, 17)

(5.1, 18)

The transformed quantities are also called amplitude, frequency variation, and so on.Thus (5.1, 14) reads

(a2+R2 -g2 -h2)2A2 = (a+g)2,+ IN -h)2 (5.1, 19)

where b = sign b = -I-1 and especially for linear damping (vanishing self-excitation)

C.= d = 0 , that is 13 = b = 1 (5.1, 20)

because then b > 0 is valid.The amplitude A is, as (5.1, 14) shows, generally of the order of magnitude of p

divided by the small parameters a, i4, g, h. A comparison with the non-resonance so-lution (5.1, 6) reveals that A is of higher order of magnitude than the amplitude ofYio so that we are justified in denoting only the amplitude of the resonance part of thesolution as "amplitude".

5.2. Resonance curves, extremal amplitudes, and stability 117

5.2. Resonance curves, extremal amplitudes, and stability

The resonance curves show the dependence of the amplitude on the exciting frequen-cy (variation).

If the system is linear, a and 8(=1) are independent of the amplitude A, and theresonance curves can be evaluated immediately from (5.1, 19), the only parametersbeing the parametric excitation coefficients g and h. As an example, we choose inFig. 5.2, 1 with g = 0 a special phase relation between forced and parametric excita-tion. The resonance curves give the (transformed) amplitude as a function of a = a+ F, that is of the frequency, for different values of the parametric excitation coeffi-cient h.

If damping is linear, but non-linear restoring forces with the coefficients e, f appear,the dependence of A on a is the same, only the ordinate axis a = a + P = 0 isreplaced by the backbone curve

a= a - eA2 - fA4 -{- P = 0 (5.2, 1)

from which the points of the resonance curve have equal horizontal distance. Fig. 5.2, 2gives an example corresponding to Fig. 5.2, 1 but now e = -0.5, / = 0 is valid. Ife and f have different sign, the backbone curve and in general the resonance curvestilt over, as Fig. 5.2, 3 shows for e = -0.5, f = 0.2.

A

0 1-2 -1

Fig. 5.2, 12 a


h=.Q97

-2 -1 1 a+rFig. 5.2, 3


It is apparent that already for linear, the more for non-linear systems the resonancecurves give information on the influence of the system parameters that is too complexand too complicated, at least as far as a combined forced and parametric excitation isinvolved. Very often, it is less important to know how the amplitudes depend on thefrequency (variation) than how the maximum amplitude depends on the other systemparameters.

In order to find this dependence, we differentiate (5.1, 19) by a,

[(a2+R2-g2-h2)2+4(a2+N2-g2-h2) i9@+2dA2)A2

+2(h-T)(c+2dA2)]A da = -2a(a2+N2-g2-h2)A2+a+9.(5.2, 2 )

Therefore the (necessary) condition dA/da = 0 for extreme values A = Aq of theresonance curve yields (if we exclude the amplitude values for which da/da = 0)the equation

2A2a3+[2(P2-g2-h2)A2-1] i -9=0. (5.2,3)

The extreme values can be found by solving the system (4.1, 19), (5.2, 3) for A and,what is of minor interest, a.

For the rest of this section, we assume a linear damping, d = 0, that is, f = bindependent of A. Then A can be eliminated in (5.2, 3) by help of (5.1, 19),

a3+39x2+ (P2-4flh+3g2+3h2)a+ (92+h2 -f2)g=0 (5.2,4)

or

a3 + 3 I YI cga2 + (1 -4j/1 - (p2 + 3y2) a + (Y2 - 1) IYI 4' = 0

with (5.1, 20) and the abbreviations

(5.2, 5)

y = sign h lg2 + h2 , that is IyH = 1g2 + h2 , and 4p = 9IYI

The real solutions (three at most) ii, al and a2 of (5.2, 5) depend only on the two para-meters y, T. The corresponding amplitude extreme values A,, follow from equation(5.1, 19), which now reads

(av+1 -Y2)2`4 (a,+IYI4')2+ (1 - yl'l 2, v=0,1,2.(5.2, 6)

The phase relation between the parametric excitation, with amplitude IyI, and theforced excitation is characterized by 99(-1 < 4, < 1). Changing the sign of g andtherefore of 99 is, as follows from (5.2, 5), (5.2, 6), equivalent to changing the sign of a,the amplitudes A being independent of this change. Therefore we can assume0 < 4, < 1 without loss of generality. Equation (5.2, 6) shows that A,, is finite forIyI < 1 whereas IyI = 1 leads to ao = 0 and A, tending to infinity at least for 9' + 0.The maximum amplitude in dependence on y and T can be found by numericallyeliminating a from (5.2, 5) and (5.2, 6).

Fig. 5.2, 4 gives the resulting maximum amplitude A as dependent on y for differentvalues of 9'. It reveals the phenomenon that the maximum amplitude is smaller for


a region of positive values of y (then the sine component of parametric excitation andthe forced excitation operating in (5.1, 9) have the same sign) than for y = 0, that isan additional parametric excitation diminishes the amplitudes caused by forced excitation.This effect occurs for all values of 99 and shows more distinctly the smaller qq is. When yis negative, no such effect arises and the maximum amplitude increases monotonicallywith M.

Formulae (5.2, 5), (5.2, 6) respectively Fig. 5.2, 4 give the maximum amplitude,the most important vibration characteristic, for forced and linear parametric excita-tion in full generality, by means of (5.1, 17), (5.1, 18) and (5.1, 11) depending onall system parameters, with the one exception that we omitted non-linear damping.Fig. 5.2, 4 shows that the maximum amplitude depends for I y I < 0.15 slightly, but fory < -0.5 and y > 0.25 essentially on the phase relation between forced and para-metric excitation expressed by T. _

The special case 99 = 0, that is g = 0, y = h, which we assume in what follows, canbe evaluated much more easily. In this case the amplitude frequency relation (5.1, 19)is biquadratic in a. The extreme value equation (5.2, 5) simplifies to

a3+ (1 -4y+3y2)a=0with the solutions

ao = 0 , ai,2 = ± V4y - 1 - 3y2 (5.2, 7)

the latter values being real for

<y< (5.2,8)

Equation (5.2, 6) gives the corresponding amplitude extremes

(1 -}- y)2 AO = 1

and

(5.2, 9)

8y(1 - y) Ai,2 = 1 (5.2, 10)

The amplitude extreme A1,2, drawn as a dashed line in Fig. 5.2, 5, represents, in theinterval (5.2, 8) of its existence, the maximum amplitude. The amplitude extreme Ao,


drawn as a full line, represents the maximum amplitude in the interval -1 < y S sand a (relative, compare (5.1, 15)) minimum in the intervals < y < 1. The maximumcurve in Fig. 5.2, 5 is identical with the curve 92 = 0 in Fig. 5.2, 4. Using (5.2, 9)and (5.2, 10), we can confirm the phenomenon explained above, for 99 = 0: The maxi-

A

1 1 i

-1 -'/z 0 1/3 '/z 1

Fig. 5.2, 5

mum amplitude decreases when y changes from 0 to + s and increases with increa-sing jyl only outside of this interval, especially for negative y. The minimum valueof the maximum amplitude is A1,2 = 11V2 = 0.7071 whereas AO = 1 for y = 0,that is an amplitude decrease of about 30% by additional parametric excitation.Figures 5.2, 1 to 5.2, 3 show examples for this phenomenon. The extension to non-linear damping is given in the next Section. _

In the intervals < y < 1, the minimum amplitude A0 lies on the backbone curve0, the horizontal frequency distance of the maximum amplitude from this curve

being given by (5.2, 7), sketched in Fig. 5.2, 6. The greatest possible distance al =occurs for y = 3.

Each of the resonance curves in Figures 5.2, 1 to 5.2, 3 corresponds to curve points(marked in the same way) in Figures 5.2, 5 and 5.2, 6 which give the maximum andminimum amplitudes and corresponding frequency values. Figures 5.2, 4 to 5.2, 6 donot show the special shape of the resonance curves, but reveal the dependence of themaximum and minimum amplitudes and the corresponding frequency values on thesystem parameters.

The forced excitation coefficient p influences, as (5.1, 17), (5.1, 18) show, besidesthe non-linear restoring force and damping terms, only the amplitude scale. If thesystem is linear, the (original) amplitude A is proportional to p because of (5.1, 14).

a1

Fig. 5.2, 6


The extremes A0, A1, 2 of the original amplitude, being independent of the non-linearrestoring forces, are proportional to p when only damping is linear, as follows from(5.2, 9), (5.2, 10) and (5.1, 17). If the forced excitation vanishes, A also vanishes fory < 1 (lower curve in Fig. 5.3, 5), but remains indefinite for the value y = 1, that isVg2 + h2 = b, which we denote as the threshold value because the parametric excitationreaches the damping threshold. If the forced excitation does not vanish, the amplitudeis unlimited for the threshold value. A parametric instability occurs, in other words,when the parametric excitation is equal to, or greater than, the damping threshold,then the linear damping model is not sufficient for the description of the real vibrationbehaviour.

Introducing instead of (5.1, 17), (5.1, 18)

A=hA(=IhI A), b=h (>0when (5.2,11)P

gives, because of y = h = h/b, instead of (5.2, 9), (5.2, 10)

(b+1)2Ao=1and

8(b-1)A1,2=1

(5.2, 12)

(5.2, 13)

The amplitude extremum Ao is the maximum for b < -1 and b > 3 and a relativeminimum for 1 < b < 3 whereas A1,2 is the maximum (drawn by dashed line inFig. 5.2, 7) in the latter interval. In the interval -1 < b < 1 the linear dampingmodel fails to describe the vibrations. The intervals (--1, 0) and (0, 1) in Fig. 5.2, 5now correspond with the intervals (-oo, -1) resp. (1, oo).

Af

-3 -2 -1 1 2 3 b

Fig. 5.2, 7

In order to investigate the stability of the vibrations described by the differentialequation (5.1, 5), we use the linear variational equation (1.5, 3). Omitting terms ofsmaller order of magnitude, in particular substituting -Az for z" on the right-hand


side, results in the linear variational equation of the form

z" + Az = 22o z - B z' - 0y2z' -

2Cyy'z - D y4z' - 4D y3y'z(Do (00 wo too (00

00- 3E2 y2z -

5F2 y4z - 12 f, (G, cos j r + H1 sin jz) z

wo wo woj=i2 00

- 2 E (K1 cos jr + Lq sin j r) yz . (5.2, 14)(0 oj=o

After inserting the resonance part, as in evaluating (5.1, 8), of the first approximativesolution (5.1, 7), the linear variational equation (5.2, 14) takes the form (2.4, 1). Thecoefficients appearing in the stability conditions (2.4, 4), (2.4, 5) now follow as

uo = 2Aa - 2n2(2eA2 + 3/A4 + 4kr + 41s) ,

vo = -2n(b + 2cA2 + 3dA4) ,

u2n = -2n2[g + e(r2 - s2) + 2fA2(r2 - s2) + 4crs + 8dA2r8

+2(kr -ls + Kr+Ls)],U2n = -2n2[h + 2ers + 4fA2rs + 2c(s2 - r2) + 4dA2(s2 - r2)

+ 2(lr + ks + Lr - Ks] ,

v2n = -2n[c(r2 - s2) + 2dA2(r2 - s2)]

V2n = -4n(cr8 + 2dA2r8)

where the transformed quantities (5.1, 11) have been introduced and non-linearparametric excitation has been included.

The first stability condition (2.4, 4) now reads

b + 2cA2 + 3dA4 > 0. (5.2, 15)

It holds for every amplitude if no self-excitation exists (b, c, d non-negative and b > 0or c > 0, A > 0 or d > 0, A > 0). The second stability condition (2.4, 5) can be written

(b + 2cA2 + 3dA4)2 > [g + (e + 2/A) (rz - s + 2(c + 2dA2) rs

+ 2(kr - is + kr + L$)]2+ [h - (c + 2dA2) (r2 - s2) + 2(e + 2/A2) rs + 2(lr + ks + Lr - Ks)]2-(a - 2eA2 - 3/A4 - 4kr - 41s)2. (5.2, 16)

These conditions will be the starting point for different stability results.Differentiating the amplitude equation (5.1, 14) by A and using

dA dA - 2A(e + 2/A2),dA

2A(c + 2dA2) ,

we get

[2(g2 + h2 - a2 - j92) aA2 + (a + g) p2]dA

= -2[2(a2 + #2 - g2 - h2) aA2

- (a + g) p2] (e + 2/A2) A + 2[2(a2 + #2 - g2 - h2) RA2 + (h - j3) p2]X (c + 2dA2) A + (a2 + R2 - g2 - h2)2 A. (5.2, 17)


At present we restrict ourselves to the linear case c = d = e = f = k = 1 = K= L = 0. Then (5.2, 16) reduces to

a2> g2 + h2 - b2

from which follows an instability region that vanishes for b2 > g2 + h2. After (5.2, 16),a vertical tangent, da/dA = 0, implies

a2=g2+h2 - b2

because otherwise A = 0 holds if p + 0. Consequently, the points on the resonancecurve with vertical tangents correspond to the boundary points of the instabilityregion.

5.3. Non-linear damping

If damping is non-linear the amplitude equation (5.1, 19) is of ninth (for d = 0 offifth) degree in A2 and of fourth degree in a, a numerical evaluation of the resonancecurves is possible for numerically given parameters.

In this section, we exclude a self-excitation by b > 0 (that is b = 1), c > 0, d > 0and assume the special phase relation g = 0 (that is g = 0, y = h) between forcedand parametric excitation. Then the amplitude equation (5.1, 19) is biquadratic in a,with the solution

a2= 1 -#2 +,y2±2A2

(5.3, 1)

which gives, together with (5.1, 17), vice versa the frequency variation as a functionof the amplitude.

The condition (5.2, 3) for amplitude extremes yields ao = 0, as for linear damping,and

1 /y2 - R2 + 1ai, 2 = ± U fl 2

2A1, 2

Substitution in (5.1, 19) leads to the corresponding amplitude extremes

and

(1 + y)2 AO = 1

(5.3, 2)

(5.3, 3)

8y(fl - y) A1,2 = 1 (5.3, 4)

which generalize (5.2, 9) and (5.2, 10). Using (5.3, 4), formula (5.3, 2) can be written

a1, 2 = ± V 4Ny - R2 - 3y2

which is real forF'

<y< (that isy<<3y).3

(5.3, 5)

In this domain, Al = A2 is also real and represents the maximum amplitude whereasAO is a (relative) minimum amplitude. For y < P/3, the amplitude AO represents themaximum amplitude.

5.3. Non-linear damping 126

The form

y= -1 - cAo - dAo± 1 (5.3, 6)

A0

of equation (5.3, 3) reveals the composition of the extremal curve (full lines whichasymptotically approach the y axis and the dotted "damping curve"

Y. = -1 - cAo - dAo (5.3, 7)

in Fig. 5.3, 1 where only non-linear damping is specialized by c = 0.2, d = 0). Forsufficiently great negative values of y, a second extremal curve appears which corre-sponds to a second detached resonance curve for smaller amplitudes with a medium-sized minimum amplitude and a second small maximum amplitude.

Analogously, (5.3, 4) can be written

y = 1 { cA1,2 + dAi1,2 -

8yAi, 2

or, solving for y,

2y = 1 -{- CA2 + dA4 -}- 1 /(1 -{- CA2 + dA4)2 - 1 .

V 2A2

(5.3, 8)

The boundaries of the domain of existence (5.3, 5) for the maximum amplitude givenby (5.3, 8) are drawn in Fig. 5.3, 1 as dashed-dotted lines, the maximum curve (5.3, 8)in this domain, which asymptotically approaches the upper boundary, is drawn as adashed line.

The influence of different values c of cubic damping on the maximum amplitudes isgiven in Fig. 5.3, 2 in greatest possible generality. Only the fifth order dampingcoefficient d (which would further diminish the maximum amplitudes) is put equalto zero.

Fig. 5.3, 2 shows:

1. The parametric diminishing effect (Section 5.2) holds for non-linear as well as forlinear damping with the smallest maximum amplitude not appearing for vanishingparametric excitation but for y = 0.5.

2. Non-linear damping has little influence on the maximum amplitudes for 0 < y <0.5but a decisive influence for y > 0.8 and y < -0.5.


3. If parametric excitation is equal to, or greater than, the damping threshold,jyj > 1, the maximum amplitude can only be determined by aid of non-lineardamping.

4. Restoring non-linearities do not influence the maximum amplitude.

Fig. 5.3, 2

A different representation of the amplitude extremes occurs when we use

, 2A=1/e Y=3h , b=3b d=_ dp cp2 Vcp2 c

(5.3, 9)

instead of (5.1, 17), (5.1, 18). Then (5.3, 6) and (5.3, 8) read

y= -b-Ao-dA0 y=b--A1,2+dA12- 1 .

Au 8yA1, 2

In case of a cubic damping non-linearity, d = 0, the corresponding amplitude extremalcurves can be composed by parabolae and hyperbolae where only the starting point(I) b of the parabolae and the second hyperbola depend on damping, as Figures5.3, 3 and 5.3, 4 show. If d is positive, the maximum amplitudes are smaller.

In order to determine the influence of forced excitation, we introduce

A=1/bcA 52(=52),

-a-

Fig. 5.3, 3

5.3. Non-linear damping

Fig. 5.3, 4

Fig. 5.3, 5

127

(the factors 5 and 0.2 enable the correspondence with the example of Fig. 5.3, 1) andget

y= -1 -0.2Ao-dAot p,AO

2

y= 1 +0.2Ai,2+dAi,2- p8yAi,2

instead of (5.3, 7), (5.3, 8). Fig. 5.3, 5 gives the corresponding maximum amplitudesfor different values of p under the only restriction d = 0. In case d > 0, the maximumamplitudes are smaller. The figure shows that the influence of forced excitation iscomparatively great if the parametric excitation is below the threshold value M = 1.

Using, in correspondence with (5.2, 11),

A= 7' A, =b+cA2+dA4, b= b, e=cp2

,

d=dp4

,P It h3 h6

(5.3, 10)(5.3, 3) and (5.3, 4) read

(i4+1)2Ao=1, (5.3,11)

8(8-1)A1,2=1 (5.3,12)

in generalization of (5.2, 22), (5.2, 13). Now Ao is the maximum amplitude in thedomain < -1 and j > 3 and a relative minimum for 1 < j9 < 3 whereas Ai 2 isthe maximum in the latter interval. When we write (5.3, 11), (5.3, 12) in the form

b= -1 -c"A4-dAo± 1, (5.3,13)

Au

I 1 -, (5.3,14)28A1 2


we find the composition of the extremal curve as given fore = 0.2 (that is h > 0) inFig. 5.3, 6. The dashed line represents a maximum curve, the full line a maximum orminimum curve. The equation (5.3, 14) is simpler than (5.3, 8) because now the right-hand side is independent of the left-hand side term. Only the amplitudes for b > 0in Fig. 5.3, 6 really appear, because b < 0 implies h < 0 (compare Section 5.6). Theextremum curves for different values of e > 0 are drawn in Fig. 5.3, 7 where againonly the amplitudes for b > 0 really appear. The corresponding case h < 0, that isb < 0, c < 0, d = 0 is sketched in Fig. 5.3, 8.

-3 -2 -1 1 2. 3 b

Fig. 5.3, 7

5.4. Forced and self-excited vibrations

Fig. 5.3, 8

5.4. Forced and self-excited vibrations

129

The amplitude equation (5.1, 14) specializes for vanishing parametric excitation in-fluence, g = h = 0, when we exclude the trivial case a = # = 0, to

(a2 + j92) A2 = p2 . (5.4, 1)

This basic equation is, contrary to (5.1, 14), only quadratic in a (that is in the fre-quency variation a) and of power five (cubic, if d = 0, resp. linear, if additionallyc= 0) in A2.

The assumption Q,, = 0, that is, the time shifting (5.1, 12), now brings no simplifica-tion because otherwise (5.1, 9), (5.1, 13) read

()+a(r))= with P = P. Q - Q.2n2wo

2n2wo

(a2+82)r=-aP-PQ, (a2+fl2)8 =OP - aQ, (5.4,2)

from what follows (5.4, 1) with

p = VP2 + Q2.

The influence of self-excitation will be discussed in this section independently ofthe somewhat more complicated analysis in Sections 5.2 and 5.3; compare HORTELand SCHMIDT (1983) and TONDL (1970b, 1976a, 1982).

To begin with, (5.4, 1) shows that the amplitude A has the same order of magnitudeas p divided by the small terms a, fi, that is, a greater order of magnitude than yioprovided that all P1, Q1 are of the same order of magnitude. This justifies denotingonly the amplitude of the resonance part of the solution as amplitude.9 Schmidt/Tondl


The amplitude equation (5.4, 1) is with the original terms (5.1, 10), because p + 0implies A + 0,

2

a=eA2+fA4± A2- (b + cA2 + dA4)2. (5.4,3)

The a distance of the two branches of the resonance curve given by (5.4, 3) from thebackbone curve

a=eA2+ fA4

is the same, it tends to p/A for small amplitudes. The resonance curves coalesce in thezeros

(b + cA2 + dA4)2 A2 = p2 (5.4, 4)

of the radicand which determine the maximum and minimum amplitudes. Equation(5.4, 4) is the extreme value equation.

Non-linear damping (vanishing self-excitation) diminishes the (unique real) solutionof (5.4, 4), that is, the maximum amplitude. On the other hand, self-excitation cancause one or two additional regions of greater amplitudes.

Particularly for d = 0, equation (5.4, 4) yields - in the irreducible case-b3/c > 27p2/4 of not too small self-excitation - three real solutions

2 2b f_ / -mb(1+cos3),Al=-

3

for

s-c<27p2,

A2= -3b(1 -cos3),

for

s > 27-c2p2A2, g = - 3b I 1 + cos (60° + 3).

with the auxiliary variable 92 introduced by

COST =

For example is

1+2bsp2l.

L, p V 27 2Al = 3 Il , A2 = As = 1.5Ib1 for - = 4 p (5.4, 5)

andV 27

2Al = 4.0981 i-fbj

, A2 = 3 Ibl , A3 = 1.0981 -fl for -c

=2

p .

(5.4, 6)

The transitition from (5.4, 5) to (5.4, 6) leads to the separation of an upper part of theresonance curve. When realized by diminishing Icl to half its value, Al and A2 increaseand A3 diminishes corresponding to the numerical factors. When realized by duplica-tion of the value of bs, the amplitudes Al and A2 increase by 8.42% and 58.74%respectively and As diminishes by 41.89% (transition from the points 0 to the points

5.4. Forced and self-excited vibrations 131

Q on the same curves in Fig. 5.4, 1, corresponding resonance curves in Fig. 5.4, 2).When at last realized by diminishing p2 to half its value, Al and As diminish by3.41 % and 48.24% respectively and A2 increases by 41.42 % (transition from 0 toin Fig. 5.4, 1).

The extreme value equation (5.4, 4) can be written

b = -cA2-dA4± .A (5.4, 7)

For d = 0, Fig. 5.4, 1 shows the composition of the values b from a (dashed) parabolaand the hyperbolae ±p/A expressed by (5.4, 7) with c = 1 and p = 0.002, 0.001, and0.001/V2 corresponding to the transition from (5.4, 5) to (5.4, 6). Without self-excita-

-0.05 -0.04 -0.03 -0.02 -0.01

Fig. 5.4, 1

tion, b > 0, only one maximal amplitude exists, which increases withforced excitation p and with diminishing linear damping b. In particular,

Pf'0 0.01 0.02 0.03 b

s,/pA=I/1

increasing

holds for b = d = 0. On the other hand, sufficiently great (van der Pol) self-excitationb < 0 effects, in addition to the maximal amplitude, also a medium-sized minimalamplitude and a second small maximal amplitude.

The resonance curves belonging to the points Q and A in Fig. 5.4, 1 by (5.4, 3)are drawn in Fig. 5.4, 2 where the (dashed) backbone curve is chosen by e = 0.5,f=-2.

If we choose a generalized van der Pol self-excitation (for which b < 0) by d = -9and the above values for c and p, Fig. 5.4, 3 shows the composition of the extremevalue curves corresponding to Fig. 5.4, 1. A hard self-excitation leads to an analogousfigure where only the right and the left side are permutated because then the signof b, c and d is changed.9


-0.06 -0.04.

Fig. 5.4, 2-0.02

-0.05 -0.04 -0.03 -0.02 -0.01 0 b

0.02 0.04 0.06 a

Fig. 5.4, 3


The extreme value curves given by (5.4, 7) reveal how A depends on b for givenparameters c, d, p. By a suitable transformation of A and b, two of the three para-meters can be eliminated so that a one-parametric family of curves describes thegeneral behaviour. When we put

A pFA,Iel 3

for c = 0, d $ 0, (5.4, 7) reads

p=

b= -sign c- A2 - sign p .

A(5.4, 8)

herefore Fig. 5.4, 3 represents the general case (not only an example) of a generalizedvan der Pol self-excitation, the corresponding figure with permutated right and leftside represents the general case of a hard self-excitation (5.1, 4) for the transformedparameters.

If by contrast we put, corresponding to (5.3, 9), for c 4 0, p 4 0

2A-p

A, b -FP-12

pd

Y II

d,2 5

(5.4, 7) reads

b= -sign 1 .

A(5.4, 9)

Fig. 5.4, 4 gives the corresponding extreme value curves for the general case of a(generalized for d < 0) van der Pol self-excitation in the region b < 0 respectivelyof vanishing self-excitation for b > 0, d = 0. The general case of a hard self-excitation(5.1, 4) is given by the left half, permutated to the right-hand side, of the figure (be-cause then b > 0 holds) with d < 0.

The extreme value equations (5.4, 7), (5.4, 8), (5.4, 9) and Figures 5.4, 1, 5.4, 2 and5.4, 4 lead to the following general results:

1. With augmenting forced excitation, the maximum amplitude augments, and an upperpart of the resonance curve separates only for greater self-excitation.

2. For van der Pol self-excitation, an upper part of the resonance curve separates if IbIexceeds a positive threshold value, and the maximum amplitude increases with I bi.

3. For generalized van der Pol self-excitation as well as for hard self-excitation, thecorresponding threshold value is zero, comparatively great amplitudes can arise evenfor small values of I bI , and the maximum amplitude descreases as I bI increases.

The stability of the solutions found can be evaluated by means of the two stabilityconditions (5.2, 15) and (5.2, 16) where now g = h = Ic = 1 = K = L = 0.

Using (5.4, 2), (5.4, 1), (5.4, 3) and (5.1, 10), the condition (5.2, 16) can be giventhe form

±2(e + 2fA2) A2 - (b + cA2 + dA4)2 - A4

IdI3 pIc!5 27

- 2(b cA2 + dA4) (c + 2dA2) < 0 .


ij

I

\ \\ \\ \ is

1-1N4\1_ - IN I

1%,- I I

-8

to16

5

4

3

0̀ 1 L

Fig. 5.4, 4


A comparison with (5.2, 17) reveals: The second stability condition (5.2, 16) holds forthe upper sign, that is, for the parts of the response curve on the right-hand side ofthe backbone curve if and only if da/dA < 0, in other words, till respectively fromthe point of the resonance curve with vertical tangent. For the parts of the resonancecurve on the left-hand side of the backbone curve, it holds correspondingly for da/dA

0.The first stability condition (5.2, 15) holds in case of van der Pol self-excitation for

A>f rb , (5.4,10)V 2c

in case of generalized van der Pol self-excitation (5.1, 3) if c2 > 3bd and A lies in theinterval I built by the positive values

V 31dl (c ± 62 - 3bd) ,

and in case of a hard self-excitation (5.1, 4) if c2 < 3bd or A lies outside of the inter-val I. Consequently, a self-excitation of the forms considered (with the exception oftoo great a hard self-excitation) always causes unstable amplitude intervals.

In investigating the vibrations in the amplitude intervals unstable because of self-excitation, we should first bear in mind the character of the solutions found so far.

The (circular) frequency n (na) regarding t) of the resonance part of the first approxi-mation coincides with the (circular) frequency of a component of forced excitation andthus of the non-resonance solution ("one-frequency solution"). It does not differmuch from the eigenfrequency Q = nwo of the system. In the neighbourhood of theresonance, the resonance part of the solution and through this its harmonic characterprevails.

If no stable solution exists, for instance if for van der Pol self-excitation the stabilitycondition (5.4, 10) does not hold, other solutions have to be sought. We keep thenotation n for the circular frequency of the resonance part of the first approximativesolution, but we presume in what follows that the forced excitation includes noresonance component, Pn = Qn = 0.

If there exists no forced excitation at all, (5.4, 1) yields the two equations

j=0 and a=0. (5.4,11)

The first one is only valid in case of self-excitation (otherwise A = 0 follows), for

A2 bc

(5.4, 12)

in case of van der Pol self-excitation. The second one determines the frequency (varia-tion) at which the self-excited vibration appears with the amplitude given by (5.4, 12).

We now investigate a system (5.1, 1) where forced excitation exists, but withouta resonance component, P = Q. = 0. In order to determine the influence of forcedexcitation on the periodicity equations and through this on the amplitude equation,we have to take into consideration also the forced excitation terms of the secondapproximation. For simplicity we assume forced excitation as harmonic of the formPk cos kawt + Qk sin kcwt (k = n) and the non-linearities of fifth degree as zero,D = F = 0, that is, we investigate van der Pol self-excitation. In the non-resonance


case, the first approximations are

Pk cos ka + Qk sin krYio = 2(A - k2)

Y20 = yio +2k2a

(Pk cos ka + Qk sin ka)CUo(A - k2)2

kB(Qk cos kr - Pk sin kr)

oJp(A - k2)2

kC 1Pk + Q2 (Qk cos kz - Pk sin kz)4aoo(A-k2)3 L A-k2

+ Qk(3Pk - Qk) cos UT + Pk(3Qk - Pa) sin 3ka2-9k2

E k2 2

4wo(), k2)3 3+

k2

, (Pk cos ka + Qk sin k)

+ Pk(Pk - 3Q2) cos UT + Qk(3P2 - Qk) sin UTA-9k2

(5.4, 13)

while in the resonance case A = n2, if k + 3n, 3k + n, additional to (5.1, 8) the terms

containingcos

na arise:sin

Y2 add

2 +QjI

(r COs na -{- s sin na) .(n 2(5.4, 14)

The periodicity equations now also yield the two equations (5.4, 11), which can bewritten, with the additional terms (5.4, 14) and the notations

Pk+`Q2 2N=Pk2n2o)20

in the form

k21 -

n2

Af=NPk,

b+c(A2+2Af)=0, a-e(A2+2Af)==0.

(5.4, 15)

(5.4, 16)

The quantity Al is, as (5.4, 13) shows, the amplitude of the forced excitation part ofthe first approximation which we denote by forced amplitude, in contrast to the self-excitation amplitude A = A8. Writing the first equation (5.4, 16)

2AB=-b-2Afe

and comparing with (5.4, 12), we realize the influence of forced excitation. A self-excited vibration of the form considered here only appears if the forced amplitudeyields

A, G

C p2 2+ Qk (s cos nr - r sin na)2yLW5 /n2 - k2)2

0 \


This condition is identical with the condition that the resonance solution, which hasthe same period as the forced excitation and the amplitude of which is given by(5.4, 1), is unstable.

By help of the transformed quantities

Ar ,AJ , A8 = AsPk Pk

the equations (5.4, 15) and (5.4, 16) for the forced and the self-excitation amplitudesimplify to

2Af I 1 - 'q2I , A, = T/2 (b2 - A?) .

The behaviour of these amplitudes is given in general form in Fig. 5.4, 5. The self-excitation amplitudes A$ are greater the greater b is. They diminish for q -> 1 andbecome zero for values r7 = 17b f, which are closer to the resonance value 17 = 1 thegreater b is. The corresponding forced amplitudes Af increase for q -> 1 hyperbolically.

15

IT

Fig. 5.4, 5

0.5 71 3_ %_ n5_n7 n10_1

107.V4.n3. 02.1.5

n1. 0.

The value A, = 0 corresponds to the limit AJ = b given by (5.4, 17). The combinationof forced and self-excited vibrations (two-frequency solution) leads to beating whichis stronger the greater the self-excitation b is and the less ri differs from one. For77 - ?7b±, the beating disappears because A8 --->. 0; in the interval (flb_, 77b+) only theformer one-frequency solution exists and it is stable in just this interval. The amplitu-des Al of the two-frequency solutions and those of the one-frequency solution areequal at the limit points 77bf. The suppression of the self-excited vibration in the


interval (?7a-, i70+) is called synchronization or frequency pulling (pulling of the fre-quency of the self-excited vibration by the forced excitation frequency), the intervalis called the pull-in range.

If we take into consideration an unharmonic forced excitation, vibration compo-nents with other frequencies which are not in the neighbourhood of the resonance,q = 1 are added the amplitudes of which are therefore smaller and which cause nobeating. Higher approximations can lead to additional beating parts of the solution,but they are small in comparison with the parts evaluated.

5.5. Parametric and self-excited vibrations

When parametric excitation and eventually self-excitation, but no forced excitationterms appear in the periodicity conditions, the amplitude equation (5.1, 14) yields,because of p = 0, besides the trivial solution A = 0

a2 + MR2-G2=0

or explicitely, using (5.1, 10),

a = eA2 + fA4 - T ± G2 - (b + cA2 + dA4)2 (5.5, 1)


G = g2 + h2

is the amplitude of the parametric excitation occuring in the periodicity equations.')Corresponding to Section 5.4 and to (5.1, 16), the two branches of the resonance

curve given by (5.5, 1) have the same distance from the backbone curve

a=aA2+ fA4-rwhich is for small amplitudes approximately constant, G2 - b2. The amplitude ex-treme values are found when

G = Ib+cA2+dA4I. (5.5,2)

Two of the five parameters in (5.5, 2) can be eliminated. For instance, dividingby JbI (if b + 0) gives

G = Ib+cA2+dA4Iwhere

IbI (>_0) b=sign b, sign c, A=V I A.

(5.5, 3 )

When no self-excitation exists, b = c = 1, d > 0, Fig. 5.5, 1 shows the generalbehaviour of the amplitude extremes: Only for G> 1 do positive amplitudes occur,in particular a positive maximum amplitude which decreases with increasing non-linear damping d.

1) As the time shifting (5.1, 11) for getting a vanishing sin-component of the forced exci-tation now does not take place, a corresponding time shifting would lead to h = 0(at least in first approximation, that is Hen = 0).

5.5. Parametric and self-excited vibrations

Ai

0.5

Fig. 5,5 1,2 5

139

In case of van der Pol self-excitation, b = -1, c = 1, d = 0, the general dependenceof the amplitude extremes is given in Fig. 5.5, 2 where, as in the figures to follow,the dashed line shows only the emergence of the upper curve. A positive maximumamplitude exists for every 6 -and an additional minimum amplitude for G < 1.

For generalized van der Pol self-excitation,-b = -1, c = 1, d < 0 (as well as forhard self-excitation (5.1, 4), b = 1, c = -1, d > 0), we get, as drawn in Fig. 5.5, 3,two maximum and two minimum amplitudes for Idl < 1/4 and every G, but onlyone maximum and one minimum amplitude if Idl > 1/4 and 0 is sufficiently great,G > 1 - 1/(41d1), and no positive amplitudes at all in the remaining case (when non-linear damping is too great respectively parametric excitation is too small). Thecases in Fig. 5.5, 1 and Fig. 5.5, 2 correspond to Fig. 5.5, 3 as limiting cases.

The result is that without self-excitation, vibrations arise only when the parametricexcitation exceeds a threshold value depending on the linear damping (G = 1),whereas self-excitation renders vibrations possible for parametric excitations belowthis threshold value, but with a positive minimum amplitude. The group of curvesreveals the form of the vibrations especially in dependence on d and G.

In order to construct the resonance curves for at least one example, we chooseG = 0.4, e = / = 0, F = IbI and write a = a/Ibl analogous to (5.5, 3). The resonancecurves coming from (5.5, 1) for different values of d are given in Fig. 5.5, 4, the extreme


A3

Fig. 5.5, 3

1

Fig. 5.5, 4

values are marked by the same signs as in Figures 5.5, 2 and 5.5, 3. It shows againthat the resonance curves give only an isolated result whereas the extreme valuecurves of Figures 5.5, 2 and 5.5, 3 reveal the tendency of building one or severalresonance curves.

An even simpler description is possible if we multiply (5.5, 2) by d2/c4 (if cd 4 0). Weget

G=lb+eA2+A41

5.6. Forced, parametric and self-excited vibrations 141

withG _ Idl 2G (] 0) b _ db "c = sign cd,

C2 '

For generalized van der Pol self-excitation as well as for hard self-excitation (5.1, 4),b > 0 and c = -1 hold. Therefore the different amplitude extreme curves are identi-cal, only shifted to the starting point A = 0, G = b (Fig. 5.5, 5).

Many results on self-excited and parametric vibrations can be found in ToNDL(1978b).

TA

Fig. 5.5, 542 3

5.6. Forced, parametric and self-excited vibrations

As in Section 5.3, we assume non-linear damping and restoring forces and restrictourselves to the special phase relation g = 0 between forced and parametric excitation,but now take into consideration an additional self-excitation (5.1, 2), (5.1, 3) or(5.1, 4). The relations (5.3, 1) to (5.3, 4) now are valid without change whereas for,B < 0, (5.3, 5) reads j9/3 > y > 8 and the equations (5.3, 6), (5.3, 8) for the amplitudeextremes are modified to read

y = -b - cA2 - dA4 1AO

y = b -}- cAi, 2 -}- dA1, 2 -8yA1, 2

that is,

(5.6, 1)

(5.6, 2)

2y = b -I- cAi 2 }4Z,4

2 } 1/ (b -}- cAi 2 + dA4, 2)2 - 2(5.6, 3)

VV 2A1, 2

with b = sign b.

For van der Pol self-excitation, b = -1, c > 0, d = 0, Fig. 5.6, 1 shows how theamplitude extreme curves given by (5.6, 1) (full line) and (5.6, 2) (dashed line, in theregion (5.3, 5) limited by the dashed-dotted lines) compose by parabolae y = ±(1-cA2)and hyperbolae for c = 1/9. Fig. 5.6, 2 gives the general dependence of the amplitudemaximum curves on the coefficient c of nonlinear damping, the influence of which


-2 -1 1 2r

Fig. 5.6, 1

proves to be essential. The influence of parametric excitation (including the amplitudediminishing effect found in Section 5.2) is greater the smaller the non-linear damping.

Resonance curves corresponding to Fig. 5.6, 1 are given in Fig. 5.6, 3 for e = / = 0,writing a = a/IbI as in Fig. 5.5, 4. The extreme values are marked by the same signsas in Fig. 5.6, 1. The resonance curves yield examples of the result, given in Figures5.6, 1 and 5.6, 2, that parametric excitation in general (disregarding the diminishingeffect) augments the maximal amplitudes but gives rise, in connection with self-excita-tion, to the possibility of isolated smaller resonance curves, that is, of much smallermaximal amplitudes (relative maxima, under the condition that no suitable disturb-ances bring the system to the higher amplitude level).

In case of generalized van der Pol self-excitation as well as hard self-excitation, atilting over of the amplitude extreme curves occurs. Fig. 5.6, 4 shows an example forthis behaviour with c = 1, d = 0.1.

5.6. Forced, parametric and self-excited vibrations 143

Fig. 5.6, 2-1 1 a-

The dependence on b becomes apparent for c 4 0 when we divide (5.6, 1), (5.6, 2)by Ic11/3 and introduce, corresponding to (5.3, 9),

A = ICI1/3A b = c = si n C d =

We get

, ICi/3, ICllls ,g ICI513.

-b-cAo-dA4± 1,

AO

-f-dA11y=b+cAi -

(5.6,

(5 6

5)

6)2,28yAi,2

. ,


dt

1 a

and a formula corresponding to (5.6, 3). In case of van der Pol self-excitation (b < 0,c = 1, d = 0) for instance, we get the case in Fig. 5.6, 5 for b = -1 and b = -2.Now the curves corresponding to (5.6, 5) differ only by the starting point -b of thebackbone curve, whereas the curves answering to (5.6, 6) exist in the interval (5.6, 3)differing with b. It shows that an augmenting self-excitation coefficient b on the onehand augments the maximum amplitude, on the other hand gives rise to the possibility ofa smaller relative maximum amplitude even for smaller values of y (even for y = 0 incase b = -2).

The notation (5.3, 10) leads to the equations (5.3, 13), (5.3, 14) where, in contrastto Section 5.3, for van der Pol self-excitation b < 0, Z'> 0 now holds. The left-handsides of Figures 5.3, 6 and 5.3, 7 show the behaviour of the amplitude extreme values.Self-excitation enlarges the maximum amplitudes whereas non-linear damping di-minishes these amplitudes.

5.6. Forced, parametric and self-excited vibrations

-2 -1

Fig. 5.6, 4

AI

Fig. 5.6, 5

1

1

2

145

10 Schmidt/Tondl


5.7. Non-linear parametric excitation. Harmonic resonance

We now consider a non-linear parametric excitation with the coefficients k, 1, K, L

in the periodicity equations (5.1, 9). Multiplying by (), (5.1, 9) yields the equation

p8 - jA2 - 2grs + h(r2 - 82) - kA2s + lA2r + Ks(s2 - 3r2) + Lr(r2 - 382)= 0 (5.7,1)

which enables, if for instance s is given, numerical evaluation of r and by this of Aindependently of the frequency variation a. The value a belonging to every triple A,r, s is given by one of the periodicity equations (5.1, 9) or by the equation

pr + aA2 - g(r2 - 82) - 2hrs - 3kA2r - 31A28 - Kr(r2 - 382)+ Ls(s2 - 3r2) = 0 (5.7, 2)

and is independent of damping andwhich follows from (5.1, 9) by multiplying (r)self-excitation.

In what follows we assume K = L = 0, that is, the third harmonic of non-linearparametric excitation vanishes (harmonic resonance) and for simplicity l = 0, thatis, the first harmonic has no since term. By help of (5.1, 9), the non-linear terms in(5.7, 1), (5.7, 2) can be written

2kr8 =a8 +8r+gs-hr,k(r2 - 82) = k(3r2 + 82) - 2kA2

= p + ar - #8 - gr - hs - 2kA2and so be eliminated. This way the equations (5.7, 1), (5.7, 2) become linear in r, s:

9,r+(x-V)8=Q,1(5.7,3)(x+V)r-9)8=a j

where

99=ha-gfl, x=kp -ga-hj3-2k2A2,V=g2+h2-k2A2, e=kgA2+2hkA2-hp,a = -kaA2 - 2gkA2 + gp.

Their solution is

(992+x2-V2)r=9,Q+(x-V)a,(992+x2-V2)8= (x +tp)e-99a

if the left-hand side expression in parentheses does not vanish. Squaring and addingleads to the amplitude equation

(T2 + x2 - ,2)2 A2 = (9'2 + x2 + V2) (e2 + a2) + 2x''(c2 - a2) - 49,y ea(5.7, 4)

which contains r and 8 only in form of the sum of the squares, A2. It is of power four ina, that is, in the frequency variation, and can be solved numerically for differentgiven values of A.

5.7. Non-linear parametric excitation. Harmonic resonance 147

In case of the phase relation g = 0 between forced and parametric excitation, theequation (5.7, 4) is biquadratic in a with the amplitude formula

2h2A2(k2A2 - h2) a2 = -(x - V)2 k2A4 + 2(x2 - aV2) h2A2 - 4ip hkA2 -Q2h2

± { (x - V)4 k4A8 + 81V(z - V)2 Qhk3A6 + 4(x - 1,)2 (jp2 - x2) h2k2A6

+ 4(x2 - V 2)2 h2k2A6 + 2(x - ip)2 Q2h2k2A4 + 16y1(V2 - x2) Qh3kA4

- 4(x + 02 e2h2k2A4 + 161p2Q2h2k2A4 + 4(1p2 - z2) Q2h4A2

+ 4(z + ip)2 y2h4A2 + 8tVQ3h3kA2 + Q4h4}112 .

In particular, for h = 0 (linear parametric excitation influences only T) the amplitudeformula follows (with (5.1, 10), for A + 0, A2 + p/k)

a = eA2 + /A4 - T -I- (3kA2 - )A 21

1 - (b )A4)27 5)(5p

2(k

pV. ,

or, if in addition p = 0 (no forced excitation influence)

a = eA2 + /A4 - T ± 3 Vk2A2 - (b + cA2 + dA4)2. (5.7, 6)

Formula (5.4, 3) is a special case of (5.7, 5) for k = T = 0.

The condition dA/da = 0 for amplitude extreme values requires the radicand in(5.7, 5) respectively (5.7, 6) to be zero. Correspondingly, real values a appear if theradicand is non-negative, in case (5.7, 5) for

(kA2 - p)2 > (b + cA2 + dA4)2 A2 . (5.7, 7)

If damping is linear, c = d = 0, the condition

(2k2A2 - 2kp - b2)2 > b2(b2 + 4kp)

for real amplitudes follows, which holds (Fig. 5.7, 1) in case b2 + 4kp < 0 for allamplitudes, in the opposite case if 2k2A2 lies outside of the interval defined by thepositive values

b2+2kp±bt/b2+4kp.For additional non-linear damping, the interval of real amplitudes lies in the interior

of that for linear damping and is bounded above because (5.7, 7) does not now holdfor A sufficiently great. The maximum amplitude is the greatest value of A for whichthe equality sign holds in (5.7, 7).

If self-excitation occurs, for instance hard self-excitation b > 0, c < 0, d > 0, theinterval of real amplitudes need not lie in the interior of that for c = d = 0, butbecause of (5.7, 7) it is bounded above as for non-linear damping.

In case (5.7, 6) of vanishing forced excitation, the condition for real a correspondingto (5.7, 7) reads

k2A2 > (b + cA2 + dA4)2 (5.7, 8)

which yields

A> b

k(5.7, 9)

10


forc=d=0and(2c2A2 + 2bc - k2)2 < k2(k2 - 4bc) (5.7, 10)

for d = 0 and therefore c > 0 (non-linear damping or van der Pol self-excitation). Forlinear damping the resonance curves coalesce only in the lower amplitude thresholdvalue (5.7, 9), a maximum amplitude does not exist. For non-linear damping, (5.7, 10)shows that the resonance curves are limited, if linear damping vanishes (b = 0) toamplitudes from zero to I kl /c and if b > 0 to an interval in the interior of this intervalwhich completely vanishes for 4bc > k2 (Fig. 5.7, 2).

The condition (5.7, 8) can be written with d = 0

IkIA>b+cA2, if A2>-bc

(especially for non-linear damping), from which follows

(2cA - (k!)2 < k2 - 4bc (5.7, 11)

and analogously

(2cA k 2 > k2 - 4bc, if A2b

(only possible for self-excitation). For non-linear damping, a is real in the intervaldefined by the positive values

2cA = Ik! ± Vk2 - 4bc .

For van der Pol self-excitation (left-hand side of Fig. 5.7, 2), the relations (5.7, 11)and (5.7, 12) imply that a is real in the greatest amplitude interval built by the threepositive terms

1 k2-4bc±lkI and b2c 2c c

The equation for the amplitude extreme values given by the equality sign in (5.7, 7)can be written

b = -cA2 -. dA4 f (kA - A l . (5.7, 13)

In case of linear damping, b >``0, c = d //= 0, Fig. 5.7, 1 gives these amplitude extremevalues for p = 1, k = -F-1, but also for any p (> 0) and k if (5.7, 13) is transformed by

Ikl_ bA= A, b= _p

p Ikl

_

to

sign k

A

For k < 0, no amplitude extreme values exist if damping is small, b2 < 4 1 ki p,whereas for greater damping and always for k > 0 we get a smaller maximum anda greater minimum, that is, an interval where no vibration amplitudes exist. Thisreveals the basic influence on the vibrations of the sign of k.

5.7. Non-linear parametric excitation. Harmonic resonance 149

2 4a) b)

Fig. 5.7, 1

Fig. 5.7, 2 shows the amplitude extreme values (5.7, 13) for c = 1, d = 0 (that isnon-linear damping for b ? 0, van der Pol self-excitation for b < 0), p = 0 (vanishingforced excitation) and different values of non-linear parametric excitation k. Thecorresponding minimum amplitude for linear damping, c = 0, is given by dotted lineswhereas in the non-linear case the maximum and minimum amplitudes are given bythe greater respectively (for b + 0 also positive) smaller value of the full-lined curve.Without self-excitation, for increasing linear damping the maximum amplitude de-creases and the minimum amplitude increases, whereas for self-excitation, withincreasing self-excitation coefficient the maximum as well as the minimum amplitudeincrease.

-6 -4- -2 2 4

Fig. 5.7, 2

Fig. 5.7, 2 can be used for every c > 0 if for instance b = b/c and k = k/c are insertedinstead of b, k. By transforming A, the value k also could be brought to ±1 (compare(5.7, 15)), so that only one curve would do, but it is just the dependence on k that wasto be illustrated.

For d + 0, (5.7, 13) with p = 0 can be transformed to the equation

b = -sign c A2 - sign d- A4 ± kA (5.7,14)

by setting

A k-,2

Fig. 5.7, 3 illustrates (5.7, 14) for sign c = 1, sign d = -1, that is, for generalizedvan der Pol self-excitation (therefore only b < 0 is of interest) and different values of k.The interval of real vibration amplitudes enlarges with non-linear parametric excita-tion A and also for diminishing Ibl, for small Ik! and [bl, for instance for IkI = 0.2,b = -0.08 it splits into two such intervals.


A hard self-excitation (5.1, 4) yields a corresponding behaviour; in Fig. 5.7, 3 onlyb has to be exchanged for -b, therefore again the full lines are of interest, becausenowb>0.

The interaction of forced and non-linear parametric excitation with non-lineardamping respectively self-excitation described by (5.7, 13) will be discussed now forvan der Pol self-excitation respectively cubic damping, c > 0 and d = 0. Using

4cA

16cbA=IkI, b=

k2

64c2P = IkI$ p (5.7, 15)

where the integers are chosen for comparison with I k! = 4 in Fig. 5.7, 2, (5.7, 13) nowreads

b = -Aa (4A - sign k p 1.A//

The extreme value behaviour is somewhat different for forced and non-linear para-metric excitation having the same sign (k > 0, Fig. 5.7, 4) and for the opposite casek < 0 (Fig. 5.7, 5). For k < 0 and self-excitation as well as damping (b < 0 respecti.vely b > 0), the possibility of detached resonance curves, that is, of smaller ampli-tudes, decreases with increasing forced excitation p, whereas for k > 0 this possibilityis nearly constant.

k> 0

-6 -4 -2

Fig. 5.7, 4

5.7. Non-linear parametric excitation. Harmonic resonance

-8. -6 -4

Fig. 5.7, 5

151

Up till now, in this section we have discussed the dependence of amplitude extremeson self-excitation respectively linear damping for different values of non-linear para-metric excitation (Figures 5.7, 2 and 5.7, 3) and of forced excitation (Figures 5.7, 4and 5.7, 5). To consider the dependence of amplitude extremes on forced excitation,we may write (5.7, 13) in the form

p = kA2 T (bA + cA3 + dA5) .

Assuming d = 0 (following c > 0) and transforming

A= k bp=b2p, c=k- c,k

bA,

so that p has the sign of k and o has the sign of b, we get

p= A2±(A+eA3).For different non-negative values of c (no self-excitation), Fig. 5.7, 6 gives the ampli-tude extreme values in dependence of forced excitation. Fig. 5.7, 7 gives the correspond-ing curves for van der Pol self-excitation, c < 0. It shows how increasing non-lineardamping diminishes, while self-excitation as well as a negative sign of p (different signof forced and non-linear parametric excitation) augments the maximum amplitude.

As an example of stability investigations in connection with non-linear parametricexcitation, consider the amplitude formula (5.7, 6). Differentiation by a shows thatdA/da >< 0 yields if

2(e + 2/A2) k2A2 - (b + cA2 + dA4)2

±3[2(b + cA2 + dA4) (c + 2dA2) - k2] (5.7, 16)

and the radicand is not zero. The first stability condition (5.2, 15) is independent ofthe special excitation at hand. Because

3kr = a, ks = -fl, 3k2rs = -a# ,3k2(r2 - $2) = a2 + 32 - 6k2A2 ,

the second stability condition (5.2, 16) leads with (5.1, 10), (5.7, 6) after a lengthyanalysis, to the condition

±2(e -t- 2/A2) V k2A2 - (b + cA2 + dA4)2

< 3[2(b + cA2 + dA4) (c + 2dA2) - k2] .


c=0.1

-4 -2

Fig. 5.7, 6

A

4 10

Comparison with (5.7, 15) gives stability for the parts of the resonance curve on theright-hand side of the backbone curve (upper sign) if dA/da < 0 and for the parts ofthe resonance curve on the left-hand side of the backbone curve (lower sign) ifdA f da > 0. In other words, the upper parts of the resonance curves up to the pointswith vertical tangent are stable (if also the first stability condition holds), while thelower points are unstable.

5.8. Non-linear parametric excitation. Subharmonic resonance

Now assume that with L = k = 1 = 0 a third harmonic component K of non-linearparametric excitation appears in the periodicity equations (5.1, 9). As in the case ofharmonic resonance as a first step the second order terms in (5.7, 1), (5.7, 2) are trans-formed to linear ones, and these are eliminated as a second step, now the third orderterms appearing in (5.7, 1), (5.7, 2) can be eliminated in three steps. We more easily

5.8. Non-linear parametric excitation. Subharmonic resonance 163

obtain an amplitude equation from the equations

ps + 2ars + i4(r2 - 82) - hA2 + KA2s = 0 (5.8, 1)

and

pr + a(r2 - s2) - 2#rs - gA2 - KA2r = 0 (5.8, 2)

which follow from (5.1, 9) by multiplication by (8) respectively (r) and are only

second order in r, s. Now (5.1, 9) gives the equationsof

K(r2-s2)=p+ar-fls-gr-hs,l2Krs = -as - 13r - gs + hr

by which (5.8, 1), (5.8, 2) yield linear equations of the form (5.7, 3) where now

p=ha - gfl, x =Kp - ga - hfl, V =a2+#2-K2A2,e=kKA2-p9, a=gKA2-pa.

The amplitude equation (5.7, 4) is now of power eight in a and so too in the frequencyvariation a. In the special case g = 0 it is of power four in a2, even if additionallyh = 0 or p = 0. Only for g = h = p = 0 (no influence of forced and linear parametricexcitation) a simple amplitude formula yields,

that is

a = eA2 + /A4 - F ± VK2A2 - (b + cA2 + dA4)2,

which can be considered in the same way as (5.7, 6). The condition dA/da C 0corresponds to

2(e + 2/A2) K2A2 - (b + cA2 + dA4)2

c ± [2(b + cA2 + dA4) (c + 2dA2) - K2] (5.8, 4)

when the radicand is not zero. The first stability condition is again (5.2, 15). In thesecond stability condition (5.2, 16), r2 - s2 and rs can be eliminated by (5.8, 3) fromwhich it follows that

K2(b + 2cA2 + 3dA4)2

[(e + 2/A2) (ar - js) - (c + 2dA2) (#r + as) + 2K2r]2

+ [(e + 2/A2) (jr + as) + (e d- 2dA2) (ar - js) + 2K2s]2- K2[a - (e + 2/A2) A2]2, (5.8, 5)

but now (5.1, 9) does not yield formulae for r and s. In the case in hand this does notmatter because (5.8, 5) can be shown to depend on r, s only in form of the sum ofsquares A2. Using (5.1, 10), the stability condition (5.8, 5) can be reduced to (5.8, 4)so that it coincides on the right-hand side of the backbone curve with the conditiondA/da < 0, on the left-hand side with dA/da > 0, from which it follows that there isstability for the upper parts of the resonance curves up to the points with verticaltangent and instability for the corresponding lower parts of the resonance curves.

Other results on non-linear parametric excitation can be found in METTLER (1965),,.SCHMIDT (1969 a, 1975), and ToNDL (1978b).

6. Vibrations of systems with many degreesof freedom

6.1. Single and combination resonances

Vibrations of systems with many degrees of freedom show such complex behaviourthat it seems impossible to develop a general theory for them. The analysis in this sectionshould be understood not as part of such a theory but only as a kind of general frame-work for the investigation of certain classes of vibrations with many degrees of free-dom. We shall discuss later and in more detail the vibrations arising in two typicalproblems of many degrees of freedom, the vibrations in gear drives (Sections 6.3 to 6.)and the autoparametric vibrations (Chapter 12).

First we formulate the non-linear integro-differential equations for a system ofcoupled differential equations

y + Atyt = 0t (i = 1, 2, ... , N) (6.1, 1)

where dashes denote derivatives with respect to a dimensionless time t and

Ot = Ot(y3, Yt, Ey,'r) _'0t[tlare given by power series in yt, ys (i = 1, 2, ... , N) and parameters ep (p = 1, 2, ..., P),which may also depend on time.

Restricting attention for simplicity to continuous solutions with continuous firstand second derivatives, we can prove the following theorem (compare SCHMIDT(1961, 1975)) : Every periodic solution of the differential equations (6.1, 1) is a solutionof the integro-differential equations

279

yt(z) = f Gt(t, a) Ot[al da + b (rt cos ntz + st sin ntr) (6.1, 2)0

where

Gt(t, a) = 1 r 1J+

cos v(t - v)T[ L2At ti=1 OA, - N2

are the corresponding generalized Green's functions, da is the Kronecker symbol, and

60=0', =1-a',so that the denominator does not vanish. The bifurcation parameters rt and s{ appear-ing in the resonance case

At=n;are to be determined by the bifurcation or periodicity equations

2n 2n

rt = f yt(z) cos ntt ft , st = f yt(Z) sin ntz dt (6.1, 3)7C 7r

0 0

6.1. Single and combination resonances 155

which are equivalent to

2n 2n

f $i[C] cos nix dz = f $i[z] sin nir d'r = 0.0 0

(4.1 6,)

The solutions of (6.1, 2) and (6.1, 3) respectively (6.1, 4) can be found by the methodof successive approximations based upon the following equations for the approxima-tive solutions yik(a), k = 1, 2, 3, ...

2,1

yik(Z) = f G(c, a)'Pi,k-1[6] da + Bar (ri cos niv + st sin nir)0

where

(Pio[-C] _'0, 0, 80;'0 , Oik([t] = Oi(yjk, yjk, Ep; Z) , k=1,2,3,...

The convergence of the method can be proved if the parameters are smaller thancertain upper bounds. Second derivatives yj in Ot can be handled in the same way,for the proof of convergence they can be substituted successively by means of thedifferential equations.

Vibrations with many degrees of freedom can very often be described by the equa-tions

00

xi + D xi = (Pij cos j(ot + Qij sin jwt) - B+jx1

- Y. Cijklxlxkxl - Y, Dijklpgxlxkxlxpxqjkl jklpq

- E Eijklxlxkxl - E Fijklpgxlxkxlxpxqjkl jklpq

00- (Gijk cos kcot + Hijk sin k(ot) xl (6.1, 5)j k=1

where, corresponding to (5.1, 1), the coefficients Pij, Qij denote general periodicforced excitations, Bij linear and Cijkl, Dijklpq non-linear damping respectively self-excitation, Eijkl, Fijklpq non-linear restoring forces and Gijk, Hijk general periodic(linear) parametric excitations. Summations run, if not indicated otherwise, from 1 toN.

Consider first a single (at most) resonance of the system, when only one (say, thefirst) of the values Qtfco is approximately an integer n,

2

=2 =v-2 for i = 1 (6.1, 6)0

with

w = coo(1 + a)

where

CO - w oa= 1(6 7)coo

is the frequency variation. With a dimensionless time

,.

,= wtaz ,

xl(t) = xi 1w = mw

156 6. Vibrations of systems with many degrees of freedom

(6.1, 5) can be written, divided by ow o,

1 00

yi + Aiyi = 2 E (Pij cos ft + Qij sin ft) - a(2 + a) yiW0j=O

1+ a- -- E Bijyj - 1 aCijklyiykyl

(O0 j coo jkl

- 1 } aE DijklpgYiykyiYPYq - 12 E Eijklyiykyl

ai0 jklpq jkl

1- 2 Z Fijklpgyiykylypyq

(00 jklpq

- W12 E E (Gijk cos ft + Hijk sin kt) yi . (6.1, 8)0 j k=1

The convergence of the iteration method demands the coefficients Pij, Qij, a, Bij,Gijk, Hijk to be sufficiently small.

The non-resonance case, when no Ai is the square of an integer, is included; then thefirst approximative solution of the integral equation method is

1 00 Pij cos ft + Qij sin j TYito = E

W j=0 &Ai - j2

All higher approximations contain only small terms multiplied by Pij, Qij, so that,as for single-degree-of-freedom systems, non-linear restoring forces, damping, self-excitation and parametric excitation influence the solution only to a smaller order ofmagnitude than forced excitation.

A single resonance (6.1, 6) leads to the first approximation

yii = 6 (r cos nt + s sin nz) + Yiio

The second and parts of the third approximation, corresponding to (5.1, 8), yield theperiodicity equations

(p\+a(r)_,(-,3)-g(-r)-h(r)=0 (6.1,9)

where in the abbreviations (5.1, 10), (5.1, 11)

instead of P B C D E F G,, HP K,, L,,

andnow substitute P1v B11 C1- D111111 E1111 Fin111 G11v 1111v

Q1nq

2n2(02

The solution of (6.1, 9) is

(a2 +N2-g2-h2)r= -(a+g)p-(N+h)q,(a2+fl2-g2-h2)s=_h)p-(a--g)q.

Squaring and adding gives the amplitude equation

(a2 + R2 -9 2 - h2)2 A2 = (a2 + R2 + g2 + h2) (p2 + q2)

+ 2(ag - flh) (p2 - q2) + 4(ah+flg)

pq

0 0

(6.1, 10)


which is of power four in a, that is, in the frequency variation a. The resonance curvescan be found by numerically solving for A or a. Particularly for vanishing forcedexcitation (p = q = 0), we get A = 0 or

a=±V92+h2-N2,that is

a = eA2 + fA4 - I' ± g2 -I- h2 - #2 . (6.1, 11)

The general investigation of systems with one degree of freedom in Chapter 5 basedon the amplitude equation (5.1, 14) can be transmitted to equation (6.1, 10). We willdo this when necessary in what follows.

We now take into consideration a two-fold resonance, when two (say, the first andthe second) of the values S2ilw are approximately integers,

Al -_ n2 (= n2) , A2 N n2 (< n2) (6.1, 12)

with the notation2

Ati = 2 . (6.1, 13)W

Instead of (6.1, 12) we write

A,. _ 2j(1 - 2ai) , Az = n2 (i = 1, 2) (6.1, 14)

with small parameters a1, a2. The other parameters ad (i > 3) are not used and there-fore assumed equal to zero. Using (6.1, 14), the integers m± = n ± n2 can be written

Al f A2

1 - 2a1 1 - 2c2or, developing the square roots,

m± = VT, ± VA2 + nal ± n2a2 + O(a1, a2) . (6.1, 15)

On the other hand, the approximative equation m± ,. VA1 ± VA2 can be madeexact when the variable excitation frequency w in (6.1, 13) is replaced by a suitablychosen value wo,

m± = VA2)w

, i.e. QUO mt = VA, -I- VA2.wo w

Subtracting from (6.1, 15) yields, up to terms O(a2, a2), the relation

m±a = na1 ± n2a2 (6.1, 16)

between the frequency variation (6.1, 7) and the parameters az.The dimensionless differential equations, divided by w2, now have on the right-hand

side2A;aiyj instead of -a(2 + (x) y (6.1, 17)

and the factors

1 1 1 1 + a'

instead of (6.1, 18)W2 w (00 coo

which are 1/(02, 1/0)o up to O(a).


The first approximative solution is now

yil = 61(r cos nz + 8 sin nz) + ba (r2 cos n2z + s2 sin n2 r) + yi1O .

Evaluating the second approximation and inserting into the periodicity equations, weget, if we assume for simplicity

Dijklpq = Fijklpq = Hijk = 0 , Gijk = 0 for only one j

and exclude the internal resonance 3n = n2 (compare Section 6.6) the four equations

(::) -S;+n'G2'\-sz/

-6j_n'G12j(rz1=0J ,

2 s2

(QP2..2..)

+ a2 (82!- N2 r2) - S'nsG22j (-82) - S' +n'G21f (A

-8n-nG21j(r0with the abbreviations

a = 2n2w2ai - a Eu11A2 - E12A2

a2 = 2n2w2a2 - E21A2 - 4 E2222A2 }

q4 = naw(B11 + a Cm1A2 + 2 C1122A2) ,

N2 = n2w(B22 + 2 C2211A2 + 4 C2222A2)

where

(6.1, 19)

(6.1, 20)

(6.1, 21)

2 2A2 = 62 + 82

is the partial amplitude of the resonance part of y21 and

Epv =2

(E,,,, + Eo, ,y + Eµ»mµ)

In what follows, we assume P1n = Ql,, = P2,,, = Q2n, = 0, that is, neglect the influ-ence of forced excitation and consider first the summed type combination resonance

j=n+n2.We first seek a relation between A and A2 only. Multiplication of (6.1, 19), (6.1, 20)

with (-s/ and (-s2l respectively givesr r2/

lA2 + + G12j(rs2 + sr2) = 0 ,

N2A2 + 2 G21j(rs2 + sr2) = 0

from which it follows that

NG21jA2 = N2G12jA2 (6.1, 22)

a relation (because of the dependence of N1#2 on A, A2 biquadratic) between A and A2,independent of the frequency parameters a1, a2.


In order to find, for A and A2 separately, formulae depending on the frequencyvariation, we use the determinant condition

a -8 1a G12j

a 0

G21j 0 a2

0 21

G21j #2 a2

= (aa2 + NN2 - a G12jG21j)2 + (aj2 - a29)2 = 0

for non-vanishing solutions r, s, r2, s2, which yields (for NN2 + 0)

a = + f4fl2 G12jG21j - N2 , a2 = V N G123G21j - #2 . (6.1, 23)

In these equations, in the normal (damping) case where i 2 > 0, equal signs beforethe roots correspond with one another because multiplication of (6.1, 19) and (6.1, 20)

with\ r /

respectively\s2 /

leads to2

a =.fla2 N2

For j = n + n2, equation (6.1, 16) reads

ja = na1 + n2a2;

using (6.1, 21), (6.1, 2/0),4 we find the amplitude frequency formula

2jnn2w2a = 3 Enu + nE21) A2 + (nEi2 + 34 E2222) A2(

, Q {{ G12jG21j 6.1, 24(n2N +nN2)4NN2 - 1 ( )

which determines, together with (6.1, 22), the partial amplitudes A1, A2 in dependenceof the frequency variation a.

If we assume a difference type combination resonance

j=n-n2, (6.1, 26)

an analogous analysis leads to the amplitude formulae

2jnn2w2a = (4 2 Eu 1 --nEz1) A2 + (usEi2 - 34 E2222) A2

± (n2N + nN2) 1/(6.1, 26)

4 2

andNG21jA2 = -N2G12jA2 . (6.1, 27)

Compare the resonance amplitudes given by (6.1, 24), (6.1, 22) respectively (6.1, 26),(6.1, 27) with the amplitudes (6.1, 11). The order of magnitude of A2 is in everycase equal to the parametric excitation, consequently it is not smaller than linear


damping. In the case of linear damping only, all the resonance curves are equidistantparabolae, while in the case of non-linear damping they are curves - with the samestarting-points A = 0 - which coalesce for

j92=g2+h2

in case of single resonance and for

4NN2 = ± G12jG21 j

in case (6.1, 24), (6.1, 22) or (6.1, 26), (6.1, 27) of combination resonance.An example of a difference resonance (6.1, 25) with n = 8, n2 = 3, w = 1, E;;iti

=4#2=0.04+0.4A2+0.2A2 (6.1, 28)

and G12j = G213 = 0.1 is given in Fig. 6.1, 1 for E21 = 1 and in Fig. 6. 1, 2 for E21 = 3.Because j9 and N2 are proportional independently of A and A1, these amplitudes arealso proportional (dashed line in Fig. 6.1, 3). If in contrast

fl=0.04, (6.1, 29)

the proportionality does not hold (full line in Fig. 6.1, 3). The corresponding resonancecurves for E21 = 1 in Fig. 6.1, 4 and for E21 = 3 in Fig. 6.1, 5 show that the partial

-0.002

Fig. 6.1, 10.002 a -0.004 -0.(102 0.002

Fig. 6.1, 2

amplitude A grows much more quickly than A2. If E21 = 0, but E2222 = 1/3 and (6.1,28), we get the resonance curve of Fig. 6.1, 1, and for E2222 = 1 Fig. 6.1, 2 holds;whereas for the damping (6.1, 29) the resonance curves are not given by Figures6.1, 4 and 6.1, 5 but, with a much steeper backbone curve, by Fig. 6.1, 6 for E2222 = 1/3and by Fig. 6.1, 7 for E2222 = 1. It shows how the restoring force coefficients signifi-cantly influence the form of the resonance curves whereas the maximum amplitudesare, as (6.1, 24), (6.1, 22) respectively (6.1, 26), (6.1, 27) show quite generally, inde-pendent of them.

A detailed discussion of further combination resonances can be found in SCHMIDT(1969, 1975) and EvAN-IwANowsKI (1976).

-0.006 -0.004 -0.002Fig. 6.1, 4

-0.015 -0,005 -0.002-0.010Fig. 6.1, 5

-0.002 0.002

Fig. 6.1, 6-0.004 -0.002

Fig. 6.1, 7

0.002

0.002 a

0,002 a

11 Schmidt/Tondl


6.2. Stability of vibrations with many degrees of freedom

The linear variational equations

zi + 2izi = E [uij('r) zf + vij(t) zj] (6.2, 1)j

of the system (6.1, 1) have, because of the Floquet theorem, which holds also forsystems (6.2, 1) (compare for instance METTLER (1949), MALKIN (1952) or CESARI(1963)), solutions of the form

zi = eez Zi (6.2, 2)

with generally complex characteristic exponents e and periodic functions Zi with theperiod 2ir. Inserting (6.2, 2) into (6.2, 1) gives the equations

Zi + 2iz, = 0i,

0i, _ i[u,(v) + Qvj(v) - 02] Z1 + [v1j(,r) - 2(.I] Z} .

First assume a single resonance of (say) the first equation and use the Fourier series00

un(t) = uo + 2 E (u1 cos iv + U1 sin jt) ,j=1

v11(Tr) = vo + 2 (v1 cos jt + V1 sin jr) .j=1

As in Section 2.4, the conditions for asymptotic stability prove to be (2.4, 4) and(2.4, 5), whereas there is instability if at least one of the inequalities holds with theopposite sign.

Let us investigate in what follows a double resonance (6.1, 14)

Al=n2 22= n2(<n2).

The linear variational equations corresponding either to the linearized differentialequations (6.1, 8) with (6.1, 17), (6.1, 18), or to the non-linear equations and thesolution y, = 0 (which exists if the forced excitation coefficients Pij, Q,j are zero), are

1 00

zi + ,Zizi = 22iaizi - - E Bijzj - - (Gijk cos kr {Hijk sin kt) z1UJ j CL) j k=1

where ai=0(i>3).The Floquet theorem yields (6.2, 2), with periodic functions Zi. As in Section 2.4

we can omit 02 and Qvij in the approximation at hand, from which it now follows that

Za + .1iZi = 0i , ]

1 00

BOi = 22iaiZi -

2Z (Gijk cos kt + Hijk sin kt) Z1 - E°j + 201 Zi .

CU j k=1 (CO /(6.2, 3)

Insertion of the first approximation

Z, = R cos nr + S sin nt,

Z2 = R2 cos nt + S2 sin nT

6.2. Stability of vibrations with many degrees of freedom 163

of the solution in (6.2, 3), (6.1, 8) gives, by aid of

2(G cos kr + H sin kz) (B cos nz + S sin nz)

_ (GR - HS) cos (k + n) ,r + (HR + GS) sin (k + n) z

+ (GR + HS) cos (k - n) z + (HR - GS) sin (k - n) z

and similar formulae, the four periodicity equations

and

2n2a1 (S) - n (+ 2e)(_)CO R

2w2 IG] 1, 2n (-S) + H11, 2n (R) + G12, n+n, (-S,22)

+ H12,n+n, (R)S2+G12,n_n2

(::) - H12,n_n2 (-S2)] 0R2

822n2a2 (S2) - n2 (Bzz + 2P)(- R2)

2I EG22,2n2 (-S2) + H22,2% (R) + G21,n+ns (-S)2 2

(6.2, 4)

+ H21,n+n2 ( ) + G21, n-n, (S) + H21, n-., (-R)J = 0 . (6.2, 5)41

We now investigate a summed type combination resonance, assuming

G11, 2n = H11, 2n = 022,2n2 = H22, 2na = Gij, n -n, = Hij, n -,n, = 0

and for simplicity

Hij,n+na = 0

for i = 1, j = 2 and i = 2, j = 1, and using the abbreviations

rl =2w2

G12, n+n2 , r2 =2w2

G21 n+n,

The determinant condition for non-vanishing solutions R, S, R2, S2 of (6.2, 4), (6.2, 5)leads to the equation

2 + n2 (22+20)21

+ J'l[4n4o21 + n2 (B11 + 20] C4n2a4 2 22

CO w

zz + 0 (6.2, 6)- 8n2n2a1a2F1r2 - 2nn2 ((0Bii +2eWO

It is not possible to determine, corresponding to a single resonance, a as a functionof the frequency variation a because of the unknown parameters a1, a2. In what followswe have first to determine these parameters. By scalar multiplication of (6.2, 4)

with (S) we get

2n2a1(R2 + S2) = r1(RR2 - SS2) ,

11*


multiplying by(_RS) gives, on the other hand,

-n (Bll +gel

(R2 + S2) = rl(SR2 + RS2) .`w

Squaring these two equations and adding yields the relation

+ 2e12] (R2 + 82)2 = ri(R2 + 82) (R2 S2)[4n4c1 + n2(B11

cv(6.2, 7)

quaring andCorrespondingly, multiplying (6.2, 5) by \R2/ respectively by (-R2)52, sadding leads to \'S2)

4 2 2 B22 2 2 2 2 g 2+S2)[nx2 + n2 ( + 2el ] (R2 + S2)22= r2(R + 5) (R2 2. (6.2, 8)

By multiplication of (6.2, 7) and (6.2, 8) we get

[442na1 + n2 (B"+ 2e)2] [irncx2 + n2 2 (B22 + 2e)2] = rir21\ww (6.2, 9)

noting that neither of the sums of squares on the right-hand side vanishes because,following (6.2, 4), (6.2, 5), the vanishing of one sum would imply the vanishing of theother one.

Just as for the relation (6.2, 6), the relation (6.2, 9) does not contain the quantitiesR, S, B1, S1. But these relations contain both the parameters al, a2.

We need separate relations for al and a2. To begin with, (6.2, 6) simplifies to

4n2n2a1 2 + nn2 (B" + 2e) \B22 + 2el = rlr2 (6.2, 10)w /

by inserting (6.2, 9) and dividing through 21'1F2. Subtracting the squared equation(6.2, 10) from (6.2, 9) yields

n2a2 B22 + 2e = nalif

Bu + gel (6.2, 11)

a relation between al and a2 only. Introducing this relation into equation (6.2, 10),

multiplied by (B +2e

, gives\ao )

2

n3n2ai B22 +2e/

= (-B"' + gel r1r2 - nn2 I Bn + 2e/\- + 2e/

This equation determines the parameter al from which follows by means of (6.2, 11)the parameter a2.

For the summed type combination resonance chosen, the relation (6.1, 16)

(n-f-n2)a=no(,+n2a2between the frequency variation a and the parameters al, a2 holds. With (6.2, 11)there follows

(n + - n2) a (Bu + gel = nal \Bu+ B22 + 4e/to CO CO

6.2. Stability of vibrations with many degrees of freedom

and by squaring, multiplying with (B22 + 20) and introducing (6.2, 12)

(n + n2)2a2IBll+2Q')(B22+2e

co j``co l(Bl+B22+4P)2Ln

x2 - \B11+20/Solving this equation gives the formula

2

4e (B11 + B22) + TV

T2 + (n + n2)2a2

(Bu - B22)2

where

T= 2nn2 + 2 - (B11 - B22)2 -

2(n + n2)2a2 .

2

165

From this formula we conclude: The solutions are asymptotically stable if and onlyif the conditions

B11 +B22>0and

w2(Bll + B22)2 T +

VT2 + (n + n2)2

u2(B11 - B22)2 ,

that is,

2(B11 + B22)2 - T> T2 + (n + n2)2

W2(BI, - B22)2

hold. The opposite sign in at least one of these conditions leads to instability.The first condition is met if damping does not vanish. The second condition can,

for Bu > 0 and B22 > 0, be expressed in the form

/

(n + n2) H>I

(B11 + B22) 1 /_(02_,11'2

(0 B - 1 .2 ll 22

(6.2, 13)

An analogous condition, only with the opposite sign before the first part of theradicand, can be found for difference type combination resonance.

Because of the product of damping terms in the denominator, the stability condi-tion (6.2, 13) can cause an enlargement of the instability regions when one of thedamping terms increases, that is, a destabilizing effect of damping.

The stability condition (6.2, 13) was found by SCHMIDT and WEIDENHAMMER(1961), independently once more by MASSA (1967), and generalized by VALEEV(1963) and SCHMIDT (1967 a, 1967b) with methods different from the one used here(compare, for the application of the integral equation method, ScHMIDT (1973)).Combined analytical, numerical and experimental investigations of combinationvibrations and their stability are given by BENZ (1965), BECKER (1972), where theabove effect is fully verified by experiment, and ScHMIEG (1976).


6.3. Vibrations in one-stage gear drives

As a first example of coupled vibrations in systems with many degrees of freedom,we shall investigate vibrations in gear drives. Consider a pair of meshing spur gears(Fig. 6.3, 1). The vibrations of one-stage and in many cases also of multi-stage geardrives can be modelled in this way because other gear stages are often of minor in-fluence on the vibrations of one stage, compare BoscH (1965), HORTEL (1968, 1969,1970).

Fig. 6.3, 1

The i-th (i = 1, 2) gear, on which acts a moment Mi, has rotating mass mi and iselastically supported in two orthogonal directions, deflections in which are denoted byx,, z,. The spring stiffness and damping coefficients are Cix, C12 (combining the stiff-ness of the bearings, the oil film and the shafts) respectively Di,,, Die. The meshingof the teeth of the two wheels is modelled by the torsional stiffness C. the damping D,the tooth error s which describes deviations from the ideal form of teeth and mounting,and the friction F which takes into consideration the sliding forces between the mesh-ing teeth and also causes additional moments MiF. Eccentricities of the wheels aredenoted by ei, the moments of inertia relating to the rotational axes Oi by Ji, a phaseangle between the torsional angles 92, of the wheels by d and time derivatives bydots.

The relative deflection of the meshing teeth in direction of the path of contact canbe shown (HORTEL (1968)) to be

X3 = x1 - X2 + 101T1 + Q2992 + E + s1 sin 971 - E2 sin (4 - 972) (6.3, 1)

where

e1 = R1 + s1 Cos -T1 , Q2 = R2 - E2 COS (d - 992)

and Ri are the pitch radii of the two wheels.

6.3. Vibrations in one-stage gear drives

The kinetic energy of the system is

Z' = IJ1 + j29*92 + m1(xi + zi) + m2(x2 + z2)1

the potential energy function

V = 2 \Cx3 + Clxxl + C2xx2 + C lzz1 + C2zz2)

the dissipation function

W = 2 (Dx2 + D1zx1 + D2xx2 + D1:z1 + D2zz2) ,

the virtual work of the remaining generalized forces

(M1 - M1F) 8991 + (M2 + M2F) 42 + F8z1 + F8z2 .

Lagrange's equations are, using the Lagrangian L = T - V,

dra fl -1)2Map,Iat azi/ aggi azi

d (LL aL a W

dt axi axi axi

d aL aL 3W Fdt azi - azi + azi =Insertion of (6.3, 2), (6.3, 3), (6.3, 4) yields taking into consideration

Pi -_ Ri = const, Ji , const

and

Ei,aS <ei ,

aggi

the equations

1 ac 2JA + CC1x3 + 2a9g1

x3 + Deix3 = Ml - M1F ,

J2

22 + CA QQ2X3 +1

2aca2 x3 + DQ2x3 =. M2 + M2F ,

m1x1 + C1x1 + Cx3 +2

dxlx xi +2

axx3 + D1xx1 + Ax, = 0 ,I ac

1 3

m2x2 + C2zz2 - Cx3 +2 dx2x x2 2 ax

x3 + D2A - Dx3 = 0 ,2 3

m1z1 + Clzzl + 2d dCzlz zi2 + D1zz1 =F,

1

m2z2 +C2zz2 + 2 d2z z2 + D2zz2 = F.2

167

(6.3, 2)

(6.3, 3)

(6.3, 4)

(6.3, 5)

The last two equations are not coupled with the other ones, they describe forcedvibrations z1, z2 and can be solved easily. We therefore confine ourselves to the remain-


ing coupled equations. Differentiating the equation (6.3, 1) for the relative toothdeflection, taking into consideration

921 At I 9'2 -_ At

and inserting the first two equations, we get

C 1

(Jjei ac X02 aC \ 2X3 + _ x3 + - _ '.+" - ) x3

µx3 ';'l x2

2 aq71 J2 42 1u

_ 1 (Ml - M1F) + 02 (M2 + M2F) + w2s1 sin £21t + (02E2 sin (d - At)Jl J2

where(6.3,6)

J1J2

iJ2 + e2J1is the reduced mass.

The torsional stiffness C substantially depends on the number of meshing toothpairs, it is periodic in time with the meshing frequency co = Q2S 2 where i is thenumber of teeth on the i-th wheel. It depends further on the relative tooth deflec-tion; we assume therefore (compare HORTEL (1968), LnciE (1970)

C = (y, cos jwt + vJ sin jwt)) (5G) cos jwt xJ sin jwt) x3 + cx3 (,6.3, 7)j=0 j=0

where e = const. The other stiffness coefficients as well as the damping coefficientare also assumed to depend (quadratically) on the deflection but not to depend on ti-me:

cix=fix+Cixxy, D=a +Sx3 Dix==6i+sixa (i=1,2)and corresponding equations for Ciz, Di,

Inserting these expressions into (6.3, 5), (6.3., 6) and introducing a transformedtime

T = cot,d

xi(t) = yi(p) (i = 1, 2, 3)

yields three equations of the form

yi + 2iyi = -ai(2 + oci) yi - (1 + ai) (Bi + Diya) yi - (1 + ai) (B1 + Diy3) ys00- Ed - Fiy2 - I (Gil cos j T + Hi, sin jt) y3

j=o00

- F, (Ki, cos j2 + Lit sin jt) y3 (i = 1, 2),j-0

00

Y3 + 23Y3 = (PJ COS jT + QJ Sin x3(2 -+- a3) y3 + (1 + x3)2 (yI - y2 )j=0

00- (1 + a3) (B + Dy3) y3 - Ey33 - (GJ cos jr '+ Hi sin ft) y3j=1

00

+ (G1o - G2o) y3 - F, (KJ cos jT + LJ sin jt) y3 (6.3, 8)j=0

6.3. Vibrations in one-stage gear drives 169

for the lateral and the torsional vibrations where

Hio=Lio=Lo=0,the frequency parameters

at =W -Wi

(1)1

(i = 1, 2, 3)

have been introduced describing the distance of to from fixed frequencies coi. Theabbreviations

Ai = ci2, A3 = Yo2 + G10 - G80

Yo

\2 I 1 + P +

mia)i W 3 1uW 3 m1 m2

Bi = Si, Bi = (-1)i-1

S Si, Di = ,

miWi micoi miWi

Di = (-1)i-1S

mia)i

mia)i miWi

c 2cix 2cE E 2, F,2,

/1(03

Gil i-1 Y1

miao1

H? = y12 ,

luW3

D=JW 3 /LW 3

1HitW2, Gq=W3In i z

3xKi!

2miwi

K11uw3

+ 12a)3 3

J1 W1 + J2W2 19Y1'1 2

xl_ 1 +3 '0'L12w3 J1W/003

Lit 3x' ,2miaw1

1+ J2w2 9Y1

for i = 1, 2 have been used, and the right-hand side of (6.3, 6) containing the tootherror e and the eccentricities ei has been written (compare HOSTEL (1968)) in the form

00

W3 (P, cos j + Q, sin jr)J-0

On the right-hand sides of these equations, the static prestress Pa predominates.We can take this into consideration by a suitable choice of the approximations.In the non-resonance case (At not square of an integer for i = 1, 2, 3), we therefore useas a first approximation

Yii = Yi1o = 0

P.

(i=1,2),

Y31=Ys1o= -23

For the second approximation we get

FiPo P0 °° Gil cos jz + Hit sin ft Po °° Kit cos ja + Lit sin'Fl 2Yi20 =

AiA3 As j 0 Ni - 92 As i OAi - j2

(i = 1, 2)

Y


where now 0 = 1, and by inserting this approximation

°° P, cos ft + Qj sin ft//

PO EPoY320 jE 79A3 - 2 m (G10 - G20)

33

- A3

PO 1

; =1 z9A3 - j2AS

[G, cos jr + Hj sin jr

22 iGij cos jt + Hil sin jr

i=1 OA,-j2

PO (K y +t L y r-I-A3

9 7 j 7 )

j2Po 2 Kij cos jr -}- Lij sin jtj2

This shows that in addition to the predominant static prestress, the time-dependentinput, output and tooth error as well as the tooth-pair stiffness have a major, anddamping has a minor, influence on the torsional vibrations, whereas only the staticprestress and the tooth-pair stiffness have a significant influence on the lateral vi-brations.

6.4. Torsional gear resonance

In this section we investigate a single resonance of the third (torsional) equation(6.3, 8) :

A3 = n2, Ai 4 n, (i = 1, 2; n, ni integer) .

Designating a3 = a as the frequency variation and setting a1 = a2 = 0, the firstapproximation is now

y31 = r cos nt + s sin nr + y3io,

Yi1=0 (i=1,2).For the second approximation we find

yi2 = Yi2onB.E- - (s cos nr - r sin nr)

1911 - n22Di (nA2+n(scosnr-rsinnr)

z9Ai-n2 4 n3

DiPO[2rs cos 2nr + (s2 - r2) sin 2nr]

n(OAi - 4n2)

nDi [s(3r2 - 82) cos 3nr+ r(3s2 -- r2) sin 3nr]4(4921 - 9n2)

3FiP0A2 Vi A2PO- ( + 4) (r cos nr + s sin nr)

2n2Aj i2-n2 4 n

6.4. Torsional gear resonance 171

- 3F,P° [(r2 - s2) cos 2nr + 2rs sin 2nr]2n2(99A1 - 4n2)

- 1 F, [r(r2 - 3s2) cos 3nr + s(3r2 - 82) sin 3n-c]4 #A1 - 9n2

- 1 (Giir + H11s) cos (j - n) r + (H11r - G11s) sin (j n) r2 j 0 n)2

1 (G11r - H,,s) cos (j + n) ,r + (Hijr + G1us) sin (j + n) r2 9 91A1 - (j -}- n)2

PO «. Oil cos jr A2 00 Kil cos jr + L1g sin jr- E - En2 9=0 91A1 - j2 2 9=0 j2

- Po (K1jr + Li1s) cos (j - n) ,r + (Li r - K11s) sin (j - n) ,r912 9=0 (j - n)2

- Po (K11r - L11s) cos (j + n) ,c + (Lijr + K1Js) sin (j + n) ,r

n2 9=0 '19'A1 - (j + n)2

- 1 [K1g(r2 - s2) + 2L11rs] cos (j - 2n) ,r + [L1g(r2 - s2) - 2Kijrs] sin (j - 2n) r

4 9=o 02i - (j - 2n)2

1 [Ki1(r2 - s2) - 2L1Jrs] cos (j + 2n) ,r + [L1Q(r2 - s2) + 2K1Jrs] sin (j + 2n) r

4 9 9921 - (j -I- 2n)2

where

A = Yr2 + s2,

the amplitude of the resonance part of the first approximative solution, is simplytermed amplitude. The real torsional amplitude consists (approximately) of A, theconstant static prestress y31o and the amplitude 52C of Y120' the latter depending mainlyon the variable tooth stiffness and the tooth errors. When A + % is greater than theconstant prestress, the tooth faces separate and the underlying equations are nolonger valid.

The second approximation 1/32 has been determined by introducing Y12 and Y3i'it cannot be given here because of its great length. It leads to the two periodicityequations

1ql+a r(_8)r-g(-s)-h(r)

3r2 + 82) 2rs r2 - s2-k(2r8

-l(r2+3s2)-K -2rs -Lgeneralizing (5.1, 9) where now /

( i11921

1)1

n2

HGi.

4l - \Q)PO() - Po

_i)

(-1)1 /K2.\- Po (Kn) - p2 24n L J n n2 L;,,

l'

(

2rsr2 - s.-)

0 (6.4, 1)


2 2 2

a = 2n2a - 3 (n2 toF1

n2 + ?9A

F2

n2) to 4 + n2)1 - 2 -+ ( OA,Gio - 1 - 2P0 (K° - _ K10 + K20 1 (6.4, 2)

WAl - n2 79A2 - n2 n2 NA1 -11n2 n2 J ,

Afl=n[B+Bl-B2+(D+Dl-D2)I 42+n2IJ

(g) 1 G2n n2 `2 (-1 aGi, 2n PO K2nh 2 \H2.) + 2 `l AAq - n2 (Hi, 2n) + n2 (L2n)

(-1)` Ki,2n+ PO Eii6Ai - n2 Li,2n

(k _ 1 n2 2 (-1)' Kin) 4

(K,)Ln4 #Ai - n2 Lin

(K Kinl n2 2 (-1)' Ki, 3"L) 4 \L3,/ + 4 i=19At - n2

Li,3n).

Assume a torsional stiffness (6.3, 7) such that the coefficients of non-linear para-metric excitation appearing in (6.1, 4) are zero,

k=l=K=L=0.(The influence of non-linear parametric excitation is discussed by HoRTEL and SCHMIDT(1979, 1981).) Then the periodicity equations (6.4, 1) are of the form (6.1, 9). The result-ing amplitude equation (6.1, 10) represents a simple formula for A as dependent on a iffor the coefficients of nonlinear damping, the equation

D+D1-D2=0 (6.4,3)

holds, and especially if nonlinear damping vanishes.We shall now give an example of how the results of Chapter 5 can be transmitted

to the gear vibration problem at hand.In order to find the maximum curves, we differentiate (6.1, 10) by a and put

dA/da = 0:2A2a3 + 2(j92 -9 2 - h2) A2a - (p2 + q2) a = g(p2 - q2) + 2hpq (6.4, 4)

Eliminating a from (6.1, 10) and (6.4, 4), we get the extreme values of A which dependonly on the parameters in 9, g, h, p, q, not on the restoring force parameters containedin a.

Assume the special phase relationg(p2 - q2) + 2hpq = 0 ,

that is,

g = 2pq if q + ±p and h = 0 if q = -}-pq2 - p2

between the parametric excitation by variable stiffness (represented by g, h) and theforced excitation, mainly by tooth errors (represented by p, q). Then the amplitudeequation (6.1, 10) is biquadratic in a,

((a2 + R2 - Y2)2 A2 = !a2 + (R - y)2] 2 (6.4, 5)

6.5. Combination gear resonances

if we abbreviate

n=` /p2+q2and

2

h ify=p2

2

+q2

so that

resp. y = -{-g if

Y2=g2+ h2.Solving (6.4, 5) for a yields

a22 R2+Y2± c2 2(N-Y)Y2A2 4A4 A2

q = fp

173

The extreme value equation (6.3, 12) now leads to ao = 0 and

l (/al,2 =V

Y2 - Ej2 +2

2A2(6.4, 6)

(6.4, 5) gives for the corresponding amplitude extremes

(6.4, 7)+ Y)2 Ao = n2and

8Y(# - Y) Ai, 2 = n2 (6.4, 8)

Insertion of (6.4, 8) into (6.4, 6) shows that al 2 is real if

3

<ythatist,ifIn this domain, Al = A2 is the maximum, AO a (relative) minimum amplitude, whereasAO is the maximum amplitude for y < fl/3.


q+±P

We now investigate a double torsional-lateral resonance :

23 = n2 , 2l = nl , 22 + n2 (n, n1, n2 integer) .

For the first approximation we get

yll =r1cosn1,r +slsinnlc Y21=0and through that,

y31 = r cos nz + s sin na + y31o + N(r1 cos n1T + s1 sin nla)where

N= 2n1

n2 - 0n2

Assuming for simplicity a cosine-shaped torsional stiffness (6.3, 7),

Y1=x1=xt=c=0,from which it follows that

E=F{=H1=Hi1=K1=Ks1=Lij=0.


We can now find the additional terms

Yi2,add - -Sy 2a1(rl cos nZ + sl sin nl'6) + S,,Bl

(sl cos nl2 - rl sin n12)nl

+ SiD1

A2(sl cos n,,r - rl sin n1'C)1

+ SZ 3Dn [sl(3r2 - si) cos 3n1t + r1(3$ - r2) sin 3n1t]1

z- n1NBi2 (sl cos n1z - rl sin n1a) -nN Di

Ails cos nx - r sin n2)A, - nl 2(OAi - n2)

n1NDi(2A2 N2 A1

4Po)4(9Ai-n2)+NA+ n4 (slcosnl2-rlsin nlz)

- n1N2DP° [2r s cos 2n s2 r2 sin 2 z1 1 1 + (1 - 1) ln2(Ai - 4,01)

3nN Di2

[s1(3r2 - si) cos 3n1z + r1(3si - r2) sin 3n1z]4(Ai - 9nl)

(n - nl) NDiPo[(sr1 - rsl) cos (n - n1) Z - (ssl + rr1) sin (n - n1) Z]

n2[i Ai - (n - %)2]

(n + nl) ND1P0[(sr1 + rs1) cos (n + n1) 2 + (ss1 - rr1) sin (n + n),r]

2n [Ai - (n + %)l

(n - 2n1) N2Di2 2f [s(rl - sl) - 2rr1s1] cos (n - 2n1) 2

4[$Ai - (n -- 2n.1)2]+ [r(si - r2) - 2srlsl] sin (n - 2n1) z}

(n + 2n) N2Dj { [s(ri 2 2

4[Ai - (n + 2n1)2) + 2rrls1] cos (n + 2nl) x- si

]+ [r(si - r2) + 2sr1s1] sin (n + 2n1) 2}

(n1 - 2n) NDi{ [sl(r2 - s2) - 2r1rs] cos (nl - 2n) ,r

4[021 - (nl - 2n)2]+ [rl(s2 - r2) - 2s1rs] sin (n1 - 2n) 2}

(nl + 2n) NDi{ [sl(r2 - s2) + 2r1rs] cos (nl + 2n) x

4[Ai - (n1+ 2n)2]+ [r1(s2 - r2) + 2slr8] sin (n1 + 2n) x}

+ S{14n2

Ai(rl cos nl'C + sl sin rLlx)1

+61

E,32n2

[r1(r1 - 3e) cos 3n1 + sl(3r1 - si) sin 3n1t]1

N °D r1 cos (n1 + j) t + sl sin (n1 + j) 7jZo2 Gig Ai - (nl + j)2

rl cos (721 - j),r + sl sin (n1 - j) xl+ IAZ-(n1-j)2

6.5. Combination gear resonances 175

where

i/ 2 -I-2Al = rl sl

is the partial amplitude of (the first approximation of) yl. Insertion of y12 and y12, addinto the periodicity equations yields

\Pl/+a.(ri)-fl1(-r1)-I2(

)-r8)

-sl)-92\-s/ -h2(r)

n n -R(r2

(r2 - s2) sl - 2rsr1 s(ri si) - 2rrsl l4 - 82) r1 + 2rss1) + (r(r2 - s2) + 2srlsl),

- Stn,Po

Dlrs1 - srl) + San, n

Dl -s(2 i -2i) + 2rrlsl

6n (rr1 768 (r(r2 - s1) + 2srlsl

2n Pa - 2rs 3n n - s(3r2 - s2)- Sn1

n D1 (s2 - r2) Sn' 4D1(r(3s2

- r2) = 0

where

p1=-POGln, 2 3 2a1 = 2n1a1 - 4 ElA1 - NG10 ,

3

Nl = nl (B1 + NB1 -}4

D1Ai -{4

D1Ai +2

D1A1 -I- n j oD1

i - 1 - P22 = 8n,n (Bl +

2D1Ai -f

4 D1A2 + n4 Dl) ,

91 =2

G1, 2n, 92 =2

01,n+n,

1 1h2 2G1.n-n,+ 2G1,n,-n

The second approximation y32, which results from the insertion of y;l, Y W M, addand y31 (with a3 instead of a) into the right-hand side of (6.3, 8), leads, after a lengthyanalysis, to the additional terms

2

gadd = -2

L,,A1

and

# add- 2

#add =2

DA 1

(6.5, 1)

(6.5, 2)

(6.5, 3)

rsl - rsl 2r1s1

(Sj2

2r1,1 3n 9nD (r2 - s2) Sl -

+ 1' (rr1 {ss11b (r2 - s216 - rl) - Sn' 32 (($2 - r2) r1 -

nD (r2 - 82) sl - 2rsr1 n nD s(r2 - si) - 2rrlsl+ Sn

4 r2 - s2 r 2rss + an' 4 r r2 s2 + 2sr s

f - 3si)n D - 3D, s(3r2 s2)

Sn9E r1(r

3n,32 sl(3ri - si)+ San,

32 (192 ) \r1(3si - r1))

)

2rsr1)

2rss1

(6.5, 4)


on the left-hand side of (6.4, 1) where

a = 2(Gn-n, + Gn,-n) + bn,(l - 2x3 - 2a1) 212 + 6n,(G2o - Gio)

8n,4EiAi

=6,-,nCB+B1-}2

(l)in2Bi D'Ai

i=1 n2 4

2\1+ - (2A2

+Al

+4Po

(L,e,-n - Ln_n1)

N-9= 2Gn+n,, 11=Ln+n,

13 =N

L2n+n,

N215 =

4Ln+2n,

with

12=NLn,,

14 = 2 (Ln,-2n - L2n-n,) + an;` 43 D ,

N2 n o1° =4 (L2n,-n - Ln-2n,) + 62n, 18n

2 (-1)'n2Dii=1 n2

Gv = Gv +2 (-1)i n2Giv

i=1 n2

(6.5, 5)

The equations (6.5, 1) coven the case of a simple lateral resonance A = n2, 22 + n2,23 = n2 of the gear drive. We have to set r = s = A = 0 in (6.5, 1) and find the ampli-tude formula

(ai+f1-91)2Ai [(at+x1)2+p1]p1f

which yields for pi = 0 especially Al = 0 or

a1=±Vgi- fi,that is,

2n1a1 = a E1Ai { vG10 ± 9i - #i

Now we exclude the influence of an internal resonance in the formulae found byassuming

v% + n, n1+vn (v=1,2,3). (6.5, 6)

In order to find closed amplitude formulae, we further neglect by assuming

p=q=p1=g=a=g1=h2=0certain components of forced and (linear )parametric excitation and by assuming

h=j9=11=12=13=14=1s=1o=0

P D


the influence of non-linear parametric excitation. Then the periodicitysimplify to

a(8)(-71

-g(-si)=0

al \S1) - N1 (-ru )- g2(-)=the coefficients beinggiven by (6.4, 2), (6.5, 3), (6.5, 2), (6.5, 5).

177

equations

(6.5, 7)

As in Section 6.1, (6.5, 7) leads because of (n + n1)a = nan + n1a1 to the formula

0102nn1(n + nl) a = 34 EjA1 + n2 (v - - 02Glon2 +

n1020

02, - 2 -

(ni9 + nni) /vQ G G 1

4NN1n+n, 1 n+n, -

for the frequency variation as dependent on the two partial amplitudes and to therelation

flGj,n+n,A2 = v#Gn+n,A2

between the partial amplitudes.If 9- = g2 = 0 holds instead of a = h2 = 0, a similar analysis yields because of

(n - n1) a = na3 - n1a1 the formulae

2nn1(n - nl) a

and

3n 2 v010 n1010 n1G2o4 L'lAl -

n2

n { 09 - n2 12 - n9

v(n1 + nf1)

OffGln-niIG1, In-n,l - 1

1

YG1,In-n,1A2

In all these formulae, Nadd from (6.5, 3) has to be taken into consideration.If for example we set n=8,n1=3,fl =4f1=0.04+0.4A2+0.2Ai,Gis=0.1,

Gb = - 1 = 55and G10 = G20 = 0 we get the resonance curves of Fig. 6.1, 1

ION 90for E. = 1/3 and of Fig. 6.1, 2 for E1 = 1, where Al has to be written instead of A2.Non-vanishing values of G10 and G20 only shift the resonance curve in direction a. Ifin contrast fi = 0.04, the resonance curves are give by Fig. 6.1, 6 for E1 = 1/3 andby Fig. 6.1, 7 for E1 = 1.

The amplitudes found are of the same order of magnitude as in case of single tor-sional resonance where for instance formula (6.1, 11) yields for

g= G2n=0.2, h=0, fl=0.04+1.2A2

the maximum amplitude Amax = 0.365.12 Schmidt/Tondl


6.6. Internal resonances in gear drives

We now investigate the internal resonances excluded by (6.5, 6) in the last section,beginning by assuming

nl = n .

In contrast to single torsional resonance (r1 = sl = 0), the parameters rl, sl neednot be small in comparison with r, s, and a first approximation of the periodicityequations (6.4, 1), (6.5, 3), (6.5, 4) is

`4') - n2 (8r)0

that is

p 4 62-Ferl =yG2

, si = - , Al = n2

Assuming a linear tooth damping, D = 0 (that is, also D, = D = 0), the periodicityequations (6.5, 1) read

N2 (_')

T g2(-s)

h2() - (Q )

where

n2 n2 n2

(6.6, 2)

(6.6, 3)

The solution

(=R),(N2-g2+h2)s=N2P+(g2+h2)Q (=S)

leads to the amplitude formula

(6.6, 4)

(9 - g2 h2)2 A2 =[N2 + (g2 - h2)2] P2 + LN2 + (g2 + h2)2] Q2 + 4N2g2PQ

(m R2+S2), (6.6,5)

where the coefficients are independent of A, and the right-hand side is quadratic in al,that is, in the frequency parameter al (and in Al which is given directly by (6.6, 1)).Because the constant tooth stiffness component yo in (6.3, 7) is greater than the varyingComponents, the inequality g2 < h-2 holds, so that the factor of A2 in (6.6, 5) does notvanish, and A has a positive minimum. In the approximation at hand, the resonanceamplitudes A and AI are independent of a, that is, of the detuning which is definedas the small distance between the two resonances,

al - a i-+ 0,)3 - WlOJi

In second approximation, we have to use in (6.6, 2), instead of (6.6, 3),

P = a1r1 - QQ#isl - girl + pl

Q = alsl + Nirl + gist ,

6.6. Internal resonances in gear drives 179

so that (6.6, 4) is now

(N2 -92+h2)r= o1r1+0281+ (h2 -92)pi+

(N2 - g2 + h2) 8 = 03x1 + 0481 + N2p1


of = 1N2 - (a1 -1.g/11) (g2 - h2)

02 = //-N2(a1 + 91) T

fl,

N1(92 - h2)

03 = N2 (al - g1) + (92 + h2) ,

04 = -/l/2+ (a1+91) (92+ h2)

I(6.6, 6)

Inserting (6.6, 6) into the periodicity equations (6.4, 1), (6.5, 4) and neglecting byassuming In = 0 (v = 1, 2, ... , 6) the influence of non-linear parametric excitation,we get two equations

o5r1 + 0681 =09

,

o7r1 + 0881 = 010

with

o5 = (a - g) 01 - (8±

h) 03 - (a+ 9) (N2 - g2 + h2) ,2 2 206=(aQQ-9)02-+h)o4-/M2-92+h2)2 2 207= (/N9 -h)o1+ (a+9)o3+N(N2

08= (8-h)02+ (a+9)04- (a9 (N2 -92+h2)09 = -(fl2-9i+h2)p+(.-9)(g2-h2)PI +(P+h)j2p1,010 -(N2-92+h2)q+(N-h)(g2-- h2)pl-(a+g)N2p1

(6.6, 7)

for r1, , only, with coefficients depending on A1, A2, ix and al. The solution of (6.6, 7)is

(0508 - 0607) r1 = 0809 - 06010

(0508 - 0607) 81 = 05010 - 0709 .(6.6, 8)

Squaring and adding gives the amplitude formula

(0508 - 0607)2 Al = (07 + 08) 09 + (05 + 06) 010 - 2(0507 + 0608) 09010. (6.6, 9)

A corresponding formula for the torsional resonance amplitude A results from (6.6, 6)by introducing (6.6, 8), squaring and adding:

(N2 - 92 + h2)2 (0508 - 0607)2 A2

_ [9+ (g2 - h2)2] [(a1 - 91) (0809 - 06010) - N1(05010 - 0709) + (0508 - 0607) p1]2

+ (g2 + h2)2]61(0809 - 06010) + (a1 + 91) (05010 - 0709)12

+ `j3292[(a1 - 91) (0809 - 06010) - N1(05010 - 0709) + (0508 - 0607) P11

X [N1(0809 - 06010) + (a1 + 91) (05010 - 0709)] . (6.6, 10)

12


In contrast to (6.6, 5) where only the right-hand side depends on al and Al isdirectly given by (6.6, 1), the coefficients of to 04 now depend linearly on fl, or al, thatis, on D1A2, E1A2 or al, whereas the coefficients o9, 010 depend linearly on fl or a, thatis, on DA2, DdA2 or a, and the coefficients o5 to 08 contain the product of both thesegroups of terms. Thus (6.6, 9), (6.6, 10) are coupled non-linear equations in A2, Al, aand al. They can be solved numerically for A2 and A2 as dependent on frequencyvariation and detuning. If D = Di = El = 0 holds (linear damping and restoringforces), they simplify because then all the coefficients are independent of A, Al sothat the amplitudes can be found immediately for every given value of frequencyvariation and detuning.

Fig. 6.6, 1 gives an example of the torsional amplitudes (6.6, 5) (full line) and thelateral amplitudes (6.6, 1) (dotted line) for n = nl = 3, N1 = 0.1, fl, = 0.2, P3 = 0.001,Q3 = 0, 6;0 = 0 and 2a1G3 < G13 where we write

a = al _ POG13

18P3

(an amplitude maximum would appear if a non-linear damping were considered), andfor comparison by dashed line the corresponding amplitude (6.6, 16) for single tor-sional resonance and P9 = 0.2, ft = 0.1, G3 = 0.04.

As a second internal resonance excluded in Section 6.5 by (6.5, 6), we consider nowthe internal resonance

n=3n1.Assuming also in this case D = 0 (from where follows Di = D = 0), a linear tooth

damping, the periodicity equations (6.4, 1) because #2 = 0, are

(h2+92)r = (qqa1 -91)rl -N1s1+p1,(h2 - 92) s = P. + (al + g1) sl

4 103

0.14

0.06

0.04-

0.02

(6.6, 11)

A,........................................

-0.01 0.01 a =-0.01 a =0 Z Z=0.01

Fig. 6.6, 1

6.6. Internal resonances in gear drives 181

Inserting into (6.4, 1), (6.5, 4) and assuming l = 0 (v = 1, 2, ... , 6), gives two equa-tions for r1, sl, the coefficients of which depend on A, A1, a, al:

wheretr1+us,=x, vr1+ws1=y (6.6, 12)

2 --t=(a-9)(a1-g1)(h2-g2)-flj91(h2+92)-(a+9) (h2 92)2

u = -(a - g) j1(h2 - 92) - (a1 + 91) fl(h2 + 92)

v (a+9)f1(h2+92)+(a, -g1)j(h2-g2),w=(a+g)(al+g1)(h2+92)-l9fl1(h2-92)-(a-9) (h2-92)

x = -(h22 2 2 2 1- 92) p - (a - g) (h2 - g2) pl +32

D1(h2 - 92) sl(3r2 - si)

+32

E1(h2 - 92) rl(r2 - 382)

y = -(h2 - 92) q - fl(h2 - g2) p1 +3n32

D,(h2 - 922) rl(3si - r2)

+ 32 E1(h2 - 92) sl(3r2 - 8) .

The equations (6.6, 12) are of third order and have to be solved numerically forgiven coefficients (that is, also for given a, a1 and A2). The corresponding values of rand 8 have to be determined numerically from (6.6, 11). If Dt = 0 (linear damping),the equations (6.6, 12) do not depend on A2. If further E1 = 0 (linear stiffness of thebearings), the system (6.6, 12) is linear in r1, 81 with the solution

(tw-uv)r1=wx-uy,(tw-uv)81=ty-vx

which leads to the amplitude formula

(tw - uv)2 A2 = (v2 + w2) x2 + (t2 + u2) y2 - 2(tv + uw) xy . (6.6, 13)

The small-letter terms are now independent of r{, ss, A, A1.A second formula for A can be found for D1 = 0 by solving (6.5, 1) for r1, sl :

(ii+ai-91)r1=(a1+g1)(h2+g2)r+191(h2-g2)s-(a1+9i)p1,(lei+ai-9i)sl= -f1(h2+g2)r+(a,-g1)(h2-g2)8+#3pl

Insertion into (6.4, 1), (6.5, 4) yields, if again Di = El = 0,

Tr+Us=X, Vr+Ws=y (66,14)with

T = (a - 9) (Ri ai - 9i) - (a + 9) (al + 9i) (h2 + 92)U = -#(9 + ai - 9i) - (a + 9) f1(h2 - 92)V = fl(#i + ai - 9i) + (a - 9) #i(h2 + 92)

W=(a+9)(fli+a2-92)-(a-9)(a1-91)(h2-92),X=-(#1+ai-92)p-(a+9) (al+91)pl,y= -(n+ai-9i)q+(a-9)filpl


The solution

(TW - UV) r = WX - UY,(TW - UV) s = TY - VX

of (6.6, 14) gives the amplitude formula

(TW - UV)2 A2 = (V2 + W2) X2 + (T2 + U2) Y2 - 2(TV + UW) XY.

(6.6, 15)

The torsional resonance amplitude found shall now be compared with the amplitude(6.1, 10) for single torsional resonance wdich tends to infinity, in the frame of thelinearized equations, for the parametric excitation reaching the threshold value,g2 + h2 = R2 (if we choose a = 0 and for instance g = 0, q + 0), which is approxima-tely

F

A2 = p2 + q2

a2 -{ NR2

for small parametric excitation, g, h < P in modulus.Formula (6.6, 15) gives for small parametric excitation,

N1 > g, g1, g2, h2, P1 in modulus,

approximately

(a1 + N1)2 p2 + [(a1 + 9) q (g - a) P1P1]22=A

(a2 + fl2) (a,2 + 3 2which is for every a1 approximately equal to (6.6, 16) if in modulus

(g - a) p1 < i1 VP2 + q2,that is

(6.6, 16)

(6.6, 17)

(6.6, 18)

2n2 fli p2 + q2G1n,(G4n, - G2.) < N P

O

otherwise (for smaller forced excitation) the torsional amplitude A can become greaterthan for single resonance. Fig. 6.6, 2 shows an example of the amplitudes (6.6, 18) forn = 3, n1 = 1, a1 = a (that is vanishing detuning), Po = 0.2, P3 = 0.001, Q3 = 0,P = N1 = 0.1, G1 = G2 = G3 = -G4 = 0.04, G;o = 0 in comparison with the corre-sponding single resonance (6.6, 16) for n = 3, Po = 0.2, P3 = 0.001, Q3 = 0, j9 = 0.1,03 = 0.04, G;o = 0 (dashed curve).

For the corresponding lateral resonance amplitude A1, equation (6.6, 13) and(6.6, 17) lead to the approximate formula

[(a2 + N2) (ai + fl) (a2 - fl2) (ai - j9) ha]2 Al(a2+ fl) [(a2 + N2) (g2 + h2) (x2 + y2) + 2(a2 - i2) g2h2(x2 - y2)l

+ 8af g2h2xy]where

x = (g2 -h2)p+a(g2 -h2)Plr

y=(g2-h2)q+fl(g2-h2)PI

6.6. Internal resonances in gear drives

Fig. 6.6, 2

183

If we choose a such that a = 0, we get

$2[(al + Nl) 92 4(ai -ff /i 2) h2]2 '91

+ ( t(a N) (92 - h2)2 1 (92 - h2)2 p2 + (g2 + h,2)2 [(g2 + h2) q + Npl]21

(6.6, 19)

from which it follows, for instance, for al = 0 and flpl < (g2 + h2) q because of(6.6, 17),

A1CA,corresponding with Al = 0 for single torsional resonance; but for I1pl > (g2 + h2) q orif the expression in brackets on the left-hand side of (6.6, 19) vanishes,

al(92 + h2) = Nl(h2 - 92) (+0),

the lateral resonance amplitude can become greater.If we dispense with the condition (6.6, 17) for parametric excitation being small

compared with damping, we can find threshold conditions for finite torsional ampli-tudes A. For instance if a = al = fl1 = q = 0, formula (6.6, 15) reads

Lfl29i + 92h2 - (991 - 992)2]2 A2 = [9291 + (99i + 9h2 - 992)27 (91p - 9pi)2

If the bracket on the left-hand side vanishes (corresponding to the threshold equationR2 = g2 + h2 in case of single resonance), the right-hand side is

2[(991 - 992)2 + 9h2(991 - 9g2)] (91p - 9p1)2 .

Because

N2 -9h2(991 - 992) = 16 G4.91,2% A. P1, 2n, - G4. P1, 4n,)

can be positive, the right-hand side need not vanish simultaneously.


It has been shown that an internal resonance 3n1 = n can enlarge the torsional andlateral vibrations. The excitation and damping parameters being given, it is thereforenecessary to evaluate the resonance amplitudes A, Al as dependent on the frequencyvariation and detuning by means of the formulae (6.6, 15), (6.6, 13).

The internal resonance n1 = 3n, which is of minor importance for practical gearvibrations, can be discussed in the same way.

6.7. Torsional vibrations in N-stage gear drives

In this and the following sections, we investigate the torsional vibrations in an N-stage gear drive, sketched in Fig. 6.7, 1. An essential contribution to this problem isdue to MOLERUS (1963) who investigated N-stage gear drives with rigid shafts, inparticular, the stability question. For the following results compare SCHMIDT (1984),SCHMIDT and ScHULZ (1983), SCHULZ (1986).

(1,2) j(2,1)

1

(2,2) 1(3,1)

X2>e2

V2

1 T

1

(N-1,2) (N,1)

vN-1

XN , EN

(N,2)

Fig. 6.7, 1

We denote by ink (n = 1, 2, ... , N; k = 1, 2) the torsional angle of the k-th wheelof the n-th stage, which can be divided into a stationary part with the constantangular velocity of the ideal rotation of perfect wheels, where Sl.l =and small supplementary terms

D._1,2'

Jnk (6.7,1)

The latter terms express the deflection of the real rotation from ideal rotation andare connected with the elastic displacements x of the teeth of the n-th stage and thetooth error functions e - characterizing the joint deviation of the teeth of the n-th

6.7. Torsional vibrations in N-stage gear drives

stage from the ideal form - by the relation

Qnl9'nl - en297n2 = -ft - En

185

(6.7, 2)

where enk is the radius of the base circle of the wheel (n, k). The relative deflection ofthe wheels of the n-th stage from ideal rotation is denoted by

Vn =Tn2 - Tn + 1, 1

The equations (6.7, 2) and (6.7, 3) yieldn-1 1 E1 - xj911 v

and

where

-Y,1)

El,n-1 j=11j+1,n-1 1,+1,nQj2

9111 + -,I - xj 'j11,n j=1 Zj+1, nQj2 j=1 Zj+1, n

Al _ ej2 _ j2Dj2 ejl bj1

are the transmission ratios, bik7k

tj,k= 71120

v=jwith

(6.7, 3)

(6.7, 4)

(6.7, 5)

being the number of teeth of the wheel (j, k), and

to+l,n = 1 .

The inertia moment of the wheel (n, k) is denoted by Jnk, the tooth stiffnessshaft stiffness coefficients by Cn and C. respectively, thedamping coefficients by Dn and D. respectively.

The kinetic energy is

N 1T _ 1 2

(Jn1T'2.2

n1 + JnOn2)

the potential energy function

1 N 1 N-1 -V = Fr

C.X.

+ - Cn n2 n=1 2 n=1

the dissipation function

W nZ Dnxn + 2 1 Dn1Vn= 2

and

tooth damping and shaft

and the virtual work of the external input and output moment

MA911 = - M0892N2

We assume, as usual, the tooth error functions e. and the tooth stiffness coefficients.C. - which, strictly speaking, both depend periodically on the torsional angles.(6.7, 1) - as periodic functions of time (we write

C. = C. + yn(t)


with the mean tooth stiffness cn), the shaft stiffness and shaft damping coefficientsCn respectively Dn as constants, whereas the tooth damping coefficients Dn maydepend quadratically on the elastic displacements xn.

The Lagrange formalism for the generalized coordinates qqu, xn (n = 1, 2, ... , N)and 'pn (n = 1, 2, ... , N - 1) leads, together with the equations (6.7, 4), (6.7, 5), tothe equation

null Il 11)1 M0+ (6.7,6)

n=1 tin \21n j=1 2j+1, nQj2 j=1 2j+1, nj 21n

which corresponds to the law of moment of momentum for the whole system, and theequations

n=m+l Zm+ii,n-1 [11,n-1

+j=1

2j+1 n-1 \E'Qj2 x1 - f/JJn2 VII- + n j, - x1 - La1 + Cmxm + Dmxm = _-

1110n=14 2m+1,n \21n j=12j+1,nQj2 j=1 2j+1,. I 2m+1,NQm2

(m=1,2,...,N) (6.7,7)and

N

n=m+1 m+iinn-1 12 n-1 + j=1 2j+7, n-1

\E1

Qj2

x1

N Jn2 -. + E E1 - x1 _ 1V 1 )+7mipm+muirn_ 0n=m-{-1112 - 2m 1,Nm+i,n (in j= j+1,nQj2 j= j+1,n +

(m= 1,2,...,N-1) (6.7,8)

describing the elastic vibrations of the gear drive system. Eliminate 47n by meansof (6.7, 6) and use the abbreviations

Jn=Jn1+Jn-1,2 (n=2,3,...,N)Ji=J11, JN+1=JN2,

in = Jnl + 2 Jn2n

(n=1,2,...,N),N 1

Io = s Jn+ln=0 21n

N J1If= .-."+ =1,2,...,N),n=j 21n2j +1, n

N

I1= E J. (j=1,2,...,N),n=j+1 21, n-12j+1, n-1

1mj =Qj2Q,n2

(21, min (m, j)I max (m, j) - Io ImI9 (m,j= 1,2,...,N),

(6.7, 9)

1 1 _ m-iI mj = - 21, min (M' j)Imax (m, j) I rI1 + biJm2

Qm2 IO l=1

(m=1,2,...,n;j=1,2,...,N-1)

6.7. Torsional vibrations in N-stage gear drives 187

and

1Imj = il,min (m, j)Imax(m, j) -

1 1-110

Then the differential equations (6.7, 7), (6.7, 8) can be written in the formN N-1 _I Imjxj + E Imj1Vj + Cmxm + Dmxm = P. ,j=1 j=1

N-1P. = Imjxj 1 Im I Ma -0

J+ M0

10 \ 11N im+1, Nem2

(m=1,2,...,N)and

N N-1 -Z Irjxj+ Ei

j=1 j=1N _ _

Qr = I,. lif + 1 Ir Mt - MO M0j=1 IO 11N Er+1, N

(6.7, 10)

(r = 1, 2, ... , N - 1) (6.7,11)

If the shafts can be assumed to be rigid, yj - 0 holds and the equations of motionreduce to (6.7, 10) :

NImjxj + Cmxm + Dmxm = P.

j=1

In this case we can write

21jkmKj ifImj =

ilmkjKm if

where

m>j,m<j

(6.7, 12)

N J 'v-1 Jlv µ+1 a+1k"= E 2 Kr= E 2

IO µ=v 2lµ µ=O 21µ

As (6.7, 12) shows directly, the symmetry condition Ijm = Imj holds for the inertiamatrix. If all transmission ratios are equal, i,. = i, we get

I0=Jl+2z+...+OTN+l

In the general case of rigid shafts, (6.7, 9), (6.7, 12) yield for a one-stage drive(N=1)

Io=Ji+ Jzi2,

1

zk1 =

i1JoK1=J,,

for a two-stage drive

IO=J1+J2+.22it 1122

ki(J2

-}-J3z ,i1o i2

ill = i1JJk1,

J3k2 = ili2l0

Jn = i1JLk1 , J12 = JS1 = iiJlk2 ,


and for a three-stage drive

J2 J3 J4I0 = J1 + .2 + .2.2+ 2.2.2,2] 1112 312223

k1 J2+.g +g.2.21

(J3 J4

)3110 22 2223

k2 -{ k3 =(J324)

J4

313210 33 31223310

ill = i1Jlk1 , J12 = J21 = 31J1k2 , J13 = J31 = i1Jik3 ,

J22 = 1122 1J1 + J2 J k2 , J23 = J32 = 3132 (j, + 2 J k331 11

J33 = 313233 (j, + 22 + 22) k3 .21 3132

Corresponding to the method of Sections 6.3 to 6.6, we first assume a weak couplingof the different gear stages in the sense that we can solve iteratively the equations(6.7, 10), (6.7, 11) by taking into consideration in a first approximation, besides theconstant components c.m, Cm, only the diagonal elements of the inertia matrix of theleft-hand sides. Therefore we can write these equations in the form

00

yn + 2nyn = E (pnv cos vv + qnr sin vv) - an(2 + an) ynP=O

2N-1- (1 + an)2 E 9.,,y',' - (1 + an) (bn + dnyn) yn

v=1(v+n)

00

- E (9nv cos vx + hnr sin vt) yn (n = 1, 2, ... , 2N - 1) .P-1

Here we have introduced a dimensionless time(6.7, 13)

where co is the greatest common divisor 3) of all (circular) frequencies appearing in(6.7, 10), (6.7, 11). Differentiations by r are denoted by dashes. For

n,v=1,2,...,2N-1,m,y=1,2,...,N and r,e=1,2,...,N-1

use the vectors

(yn(o) =xm(t)

, (An) _P.(t)

cm 1

2I mmwm

Cr

Irrwr

Pm

(p,,, cos yr + qnv sin v'r) _v=000

MImmw2

Qr

1) Common divisors exist because the meshing frequencies appearing in C. are proportio-nal to the numbers of teeth and the frequencies appearing in the error terms Pm, Q.are equal to, multiples or fractions of, these meshing frequencies.


where pno is the predominant static prestress,

em2

I mmwm

(bn + dnyn)Cr

2JIrrwr

00

( F.i (gnv cos yr + hnv sin vr)) ='v=1

and the matrix

Imp Imp

Imm Imm

(fnv) _

rr II'

The frequency parameters

a, -an =

0

wn

are, as will be shown, related to the frequency variation and express the small distanceof co from fixed frequencies (o..

In the non-resonance case, the first approximation is

Ynlo=pno (n=1,2,...,2N-1),A.

from which follows the second approximation

00pnv cos yr + qnv sin Yr pnOyn20 = F, E (gnv cos yr + lLnv sin vr)

v=0 t%tn - v2 An v=1

This shows that, besides the predominant static prestress, the error and the variablestiffness terms (the latter multiplied by the static prestress) influence the vibrationalready in second approximation whereas damping as well as inertia coupling of thedifferent gear stages by means of jnv (v 4 n) influence the solution only in thirdapproximation. The formulae for the third approximation can be derived easily.

In what follows, we investigate a single resonance

2k = K2 (K integer)

of the k-th stage, calling Ak an eigenvalue (leading to an eigenfrequency (0k) and ak = athe frequency variation and choose an = 0 for n 4 k.

The first approximation is

ynl = an(r cos Kr + s sin Kr) + ynto

Inserting this and the second approximation into the periodicity equations yields

(R)+2K2a(s) _r}-g(-s) -h(s) =0 (6.7,14)


with the abbreviations2N-1

PkogkKP = PkK + Y, 1bPvx

v=1 K2(v4k)

2N-1 pkohkgq= gkKT Lr lk,,gvK - 2

v=1 K( v$k)

= K(bk + 4 dkA2) ,

g = 2 gk, 2K

where

17h,)kv

v - 1K2

(6.7, 15)

(6.7, 16)

gives the coefficients of the secondary components in p, q and

A = r2 -+s 2

is the amplitude of the resonance part of the first approximative solution, designatedsimply as amplitude. The real amplitude of the elastic displacement xk consistsapproximately of A, the constant prestress yklo and the amplitude 91 of Yk2o. WhenA + 9 is greater than the prestress Yk1o, the tooth flanks separate.

The parametric excitation with the coefficients gk, 2K, hk, 2K of the periodic toothstiffness is called stiffness excitation. The forced excitation p, q stemming mainlyfrom the tooth errors (but containing also components of periodic tooth stiffnessmultiplied by static prestress) is called error excitation.

As in Section 6.1, the solution of (6.7, 14) leads to the amplitude equation

(4K2(X 2 + R2 -9 2 - h2)2 A2 = (4K2a2 + R2 + g2 + h2) (p2 + q2)

+ 2(2K2o

g'

- fSh) (p2 - q2) + 4(2K2ah + fly) pq. (6-7,17)

This equation can be treated in the same way as equation (6.1, 10). It is linear in A2if the non-linear damping coefficient dk vanishes.

The backbone curve is now the coordinate axis a == 0. Differentiating by a andsetting dA/da = 0 yields

16K6A2a3 + 4K2(fi2 - g2 - h2) A2a - 2K2(p2 + q2) a

= g(p2 - q2) + 2hpq. (6.7, 18)

The amplitude extreme values result from (6.7, 17) and (6.7, 18) by eliminating a.They - as well as the resonance curves given by (6.7, 17) - depend only on thedamping coefficients bk, dk and stiffness coefficients gk,2K, hk, 2K, gk, K, hk, K of the k-thstage, whereas they depend on the error excitation coefficients P,K, q,,K of all stages.A diminution of the amplitudes is possible by appropriately influencing these coeffi-cients.

Transferring the analysis in Sections 5.1 to 5.3 shows how the vibration amplitudesdepend on linear and non-linear damping as well as on stiffness and error excitationand the phase relation between them.

h =2

hk, 2K


The above coefficients depend significantly on the number k of the stage whereresonance occurs, on the order K of resonance and on the inertia moment Ikk. Theother frequencies A, inertia moments Ikti and by these the transmission ratios i,,influence only the secondary components in p, q containing %K (v = K). Thisinfluence can be discussed using the formulae derived above.

In order to make such a discussion easier to follow, we assume in the rest of thissection that the shafts are rigid, all transmission ratios are equal, i = i, and theinertial moments are proportional to i4,

J2 = (i4+1)J1,...,Jy=(j4+1)J1, Jx+1=i4J1 (6.7,19)

A two-stage gear drive yieldsi3 i

)12 = i4 + i2 + 1 , )21 = i4 + i2 + 1

a three-stage drive leads to2(84 + 22 + 1) 24

312- 86+4+282+ 1, )13 =86-}-84+282+.

i3 iY23= i4+i2+1 21- j4 +i2+I

8(84 -I-- 82 1) 82X32= is+2i4+2+ 1 )31 i6+2i4 + 22 =I 1

If for gearing down i > 1 holds, we get for two-stage drives and three-stage drivesrespectively 1121 123, j32 = 0(1/i), j13 = 0(1/i2), j21 = 0(1/i3), and j31 = 0(1/i4). If forgearing up i C 1 holds, we find analogously h2, 121, 232 = 0(2), 131 = 0(82), ,23 = O(83),

and 113 = O(i4). Already that suggests that the influence of the secondary componentsin p, q is small in comparison with that of pkK, qka.

In order precisely to evaluate the influence of the secondary components, we havealso to evaluate A,/K2. If we bear in mind that the quantities w,,, are approximatelyequal to w, we get, approximately,

A. A. _ CnJkk

Ak K2 CkJnn

that is, in the two-stage case for resonance of the first (k = 1) as well as of the secondstage (k = 2)

Al i2c1

A2 c2(6.7, 20)

in the three-stage casefor k = 1 (resonance of the first stage)A2 86+84+282+ 1 C2A,

(i4 + i2 + 1)2Cl

for k = 2 (resonance of theAl (24 + 82 + 1)2 C1

22 =i6 +i4+2i2+ 1 C2,

for k = 3 (resonance of the

A3 86 + i4 + 282 + 1 C3

Al.82(86 + 284 + 82 + 1) C1

second stage)

A3 (84 + 82 '7 1)2 C3

A2= i2(i6 + 2i4 + i2 + 1) C2

third stage)Al 82(86 + 284 + 82 + 1) Cla3 =

i6+i4+2i2+1 C3, A2 82(86+284+22+ 1) C2

A3(j4 + 22 + 1)2 C3


By means of these formulae, the eigenvalues Av and coefficients lxv of the secondarycomponents can be evaluated immediately. Here are some examples.

For a two-stage drive and resonance in the second stage, Fig. 6.7, 2 gives the second-ary components for c2 = ci (full line) and c2 = 1.5ci (dashed line) as dependent on the

Fig. 6.7, 2

transmission ratio. It shows that they are very small for i > 2 (for instance 121.= 0.004121 for i = 3, 121 = 0.0009768 for i = 4 and c2 = Cl)' If resonance occurs inthe first stage, i and c2/ci have to be replaced by the inverse values.

For a three-stage drive, i = 57/20, c2 = 1.5c1 and resonance in the third stage,132 = -0.9034 and 131 = 0.002439, whereas 112 = -0.3883 and 113 = -0.1153 if theresonance occurs in the first stage.

6.8. Strong coupling between gear stages 193

6.8. Strong coupling between gear stages

In this section we consider a strong coupling between several or all, say M > 2, ofthe 2N - 1 equations of motion (6.7, 13) in the sense that already in first approxima-tion the coupling terms are taken into account, in other words, that the starting pointis a transformation to principal axes. Without loss of generality we assume that thefirst M equations are strongly coupled. The equations (6.7, 13) are written in theform

yn + 7npyp + Anyn = n (n = 1, 2, ..., M) ,u=1(kin)

CO

n = (pn, Cos v'C + qnv sin vr) - IXn(2 + an) ynv=0

M 2N-1- IXn(2 an) 9nvyli - (1 + an)2 Jnv.7V

u=1 v=M+1(k +n) (v4n)

Coo

- (1 + IXn) (bn Clyn) Y. cos yr + hnv sin vr) yn ,v=1

Y. + Anyn = (pnv cos yr + qnv sin vr) - IXn(1 + an) yn00v=0

2N-1- (1 + an)2 9nvyv - (1 + an) (bn + dnyn) yn

v=1(v 4n)

00

(6.8, 1)

E (gnvcosyr+hnvsin vr)yn (n= M+ 1,...,2N - 1).V-1

The equations (6.8, 1) have to be decoupled by principal axes transformation, whichamounts to multiplication by such factors tvn (v, n = 1, 2, ... , M), so that the equa-tions given by summation are of the decoupled form

M

Vv(z + Avzv) =n=1

that is,M t,.n

zv+Avzv=99, 99v = vvn0n , vvn=n=1

Vv (v = 1,2,...,M)

(6.8, 2)

with new coordinate functions

M

zv = + T vnyn , (6.8, 3)

n=1

the original coordinates can be expressed as :

M

yn= E UnpZI, (n = 1,2,...,M). (6.8, 4)µ=1

13 Schmidt/Tondl


The expressions q,,, can - if (6.8, 2), (6.8, 4) are used and the frequency parametersa,, (v = 1, 2, ... , M) are defined by means of A,, instead of Av - be shown to be

M Co

Tv = E vvn E (pnm cos mr + qnm sin m'C)n=1 m=0

M 2N-1- o,(2 + a,,) zv - (1 + (Xv)2 E vvn E I.Y.

n=1 m=M+1(m4n)

M Ml

CMS

- (1 + NO E vvn [bn + dn( E Unxzx)2] Unzµn=1 x=1 µ=1

M 0o ME v. E (gn,n cos mr + hnm sin mT) Unµzµ

n=1 m=1

The first approximative solution of (6.8, 2) is, in the non-resonance case,

Myvnpno

zv1o = En=1 A,

while the second approximation isM 00

Y.z 20 = E vv vnn=1 m=0 iAv - m2

M Co gnm Cos mr + hnm sin mr M Mv,4xp,,o-

vvn Unµn=1 m=1 #Av - m2 µ=1 X=I A

For a single resonance

Ak = K2 (1 < k < M; K integer)

we get, putting ak = a as the frequency variation and choosing a = 0 for v + k, thefirst approximation

zvl = Sv (r cos Kr + s sin Kr) + zv10

The periodicity equations are with zv1 and the second approximation zv2 again of theform (6.7, 14) where now

M N Mpµo

P = E vkn (Pnli + E lnmpmx - E Unmvmµ gnlfn=1 m=M+1 2n,µ=1 A.M N M

pµoq = E vkn qn% + E lnmgmF - E Unmvmµ h.Kn=1 m=411+1 m,,1=1 Am

M

= E K(wnbn 4 4 WndnA2) ,n=1

9= 2 E wngn, 2h , h= 2 E wnhn, 2Ks=1 n=1

with

pnm cos ma + qnm sm mr

3wn =vknUnk , Wn =vknUnk

Using these terms, the amplitude equation (6.7, 17) and the ensuing discussion remainvalid, only now the damping and stiffness coefficients of all M stages strongly coupled

6.8. Strong coupling between gear stages 196

influence the resonance curves given by (6.7, 17) as well as the amplitude extremevalues. The real influence of the different system parameters occuring in (6.8, 5) isrevealed after determining the quantities Ak, vv,,, by principal axes transforma-tion.

We now take into special consideration a strong coupling of M = 2 gear stages.The analysis is simplified by choosing two of the values t,,,, (say, t12 and t22) equal to 1instead of normalizing z,,. Thus we derive the formulae

/ \2

21

i12tv1 = 1 -1

--- I 1 - Al I + 4L A 21A2 A2 \ A2 / A2

T12 = T22 = 1 , T11 = tvl , Vv = 1 + 712tv1

a2 tv1

V,, Vv

Un = -U12=1

T11-T21' U21 =T21

T21-TllU22 =

TitTn - T21

forv=1,2.For a two-stage gear drive (N = 2), equal transmission ratios i, inertia moments

(6.7, 19), c2 = 1.5c1, and resonance in the second equation (mainly in the second stage),using (6.7, 16), we get the following values:

i 121 V21 V22 w2 W2 W1 W1

1.5 0.3609 0.2443 0.9008 0.7216 0.4630 0.1195 0.028562 0.05714 0.05414 0.9794 0.9484 0.8895 0.01169 0.0005323 0.00659 0.00656 0.9981 0.9957 0.9910 0.00038 0.000 0014

These values have to be inserted into formulae (6.7, 15) for weak coupling respectively(6.8, 5) for strong coupling, which if for simplicity the last terms in p, q containingp.o are neglected, can now be written:

weak coupling strong coupling

P = 121P1K + P2K , = V21p1% + V22P2K ,

q = 121 q1K + q2K, = v21g1H + v22g2X ,

K= b2 + 4 d2A2 , = w2b2 +4 W2d2A2 + wlbl + 4 W1d1A2 ,K

2g = g2,2K , = w2g2,2s + w1g1,2% ,2h = h2,27f , = w2h2,2K + wlhl,2K

This shows that for transmission ratio i = 3, the influence of any (weak or strong)coupling on the left-hand side parameters determining maximum amplitude andresponse curve is smaller than 1 per cent. For i = 2 the influence of coupling at all(of the first stage) on error excitation p, q expressed by v21 is about 5.4 per cent (thelatter being fairly well taken into consideration by 121, that is by weak coupling), theinfluence of strong coupling on the error excitation in the second stage (compare v22)is about 2 per cent. The influence of (strong) coupling on the stiffness excitation g, has well as on the linear damping is (compare w2) for i = 2 more than 5 per cent butthe influence on the maximum amplitude is much smaller because only the quotient13*


of stiffness excitation and linear damping determines the maximum amplitude, asfollows from (5.1, 17), (5.3, 3), (5.3, 4). For i = 1.5 the corrections of weak and strongcoupling are greater.

The corresponding values for the quotient of strong and weak coupling eigenvaluesand of T21 and T11 - characterizing, because of (6.8, 3) and T,9 = 1, the share of ylin the resonance coordinate z2 and in z1 respectively - are

i A11A1 A2/22 T21 T11

1.5 1.1979 0.9008 0.4069 - 1.63842 1.0595 0.9794 0.1474 - 4.52273 1.0118 0.9981 0.0395 -16.8912

We see that the resonance in the second equation leads to pronounced vibrationsnot only in the second but also in the first stage. The eigenvalues, consequently alsothe eigenfrequencies, differ for i = 3 by about 1 per cent respectively 0.2 per centfrom those found by the equations of weak coupling, for i = 2 by about 6 per centrespectively 2 per cent.

We have seen that the weak coupling model, which avoids the main axes transfor-mation and therefore can lead to general and relatively simple formulae for more thantwo degrees of freedom, represents a far better approximation than the model withone degree of freedom.

6.9. Application of computer algebra

Under certain simplifying assumptions, Sections 6.7 and 6.8 give results on the reso-nance vibrations of N-stage gear drives. If we dispense with these simplifications orinvestigate more complicated gear drives, the amount of analytical evaluation furtherincreases. On the one hand, the basic differential equations of motion become muchmore complicated; on the other hand, often a complete transformation to principalaxes cannot be avoided. Thus solving the problem analytically becomes very toilsome,and the possibility of mistakes is great.

The question arises if such complicated analytical evaluations can be performed byaid of computers. In fact, during the last two decades activities in computer-aidedanalysis have led to several different computer languages and program systems.Starting in general from certain problems of theoretical physics, computer languageshave been developed which permit analytical evaluations by reducing them to alge-braic ones and which are therefore termed computer algebra languages. Examplesare the languages MACSYMA, REDUCE and FORMAC.

For certain types of multy-body systems, several programs have been developedwhich allow to find the differential equations of motion by help of computers. Howeverup to now much less is known on the implementation of analytical approximationmethods with computers although this is of highest importance for problems as thosediscussed in this chapter.

R. SCHuLZ (1983,1985) has developed a program system ASB(,,Automatisierte Schwin-gungsberechnung") for the automatic evaluation of vibrations by means of the com-puter algebra language FORMAC. This system combines the generation of differentialequations of motion (taking into consideration especially gear drives) with the con-

6.9. Application of computer algebra 197

struction of iterative analytical solutions and the numerical evaluation of these itera-tive solutions.

The program system ASB consists of four parts which can be used also separately.

1. The form of the gear drive leads to formulae for the kinetic and potential energyfunction and the dissipation function of a linear model describing the torsional vibra-tions of the gear drive. It is possible to change the number of degrees of freedom aswell as to introduce non-linearities and other modifications. Only this part of thesystem ASB is especially adapted to gear problems, the other parts are universallyapplicable.

2. From the terms found in the first part (or from corresponding terms given by theuser), Lagrange's equations of motion are derived by means of the computer.

3. In order to solve vibration problems modelled by ordinary differential equations,the iterative determination of periodic solutions by help of the integral equationmethod widely used in this book has been realized in form of a FORMAC program.Thus the advantages of this analytical method is used for the automatic evaluationof vibrations. The results are firstly the eigenfrequencies and eigenfunctions, secondly(taking into consideration also the time-varying terms) the resonances of differentkind, in the above case of gear drives the critical torsional frequencies, thirdly theiterative solutions in form of finite Fourier series and at last the periodicity equations,a system of algebraic equations which determine the bifurcation parameters occuringin the Fourier series. As has been shown in Chapter 5 and this chapter, the periodicityequations give manyfold general analytical information on the system behaviour; forfurther information by numerically evaluating the iterative solutions and the perio-dicity equations, PL/1 programs are generated from these Fourier series and perio-dicity equations.

4. In the last part the analytical results are evaluated numerically. The PL/l pro-grams generated in the third part are used for numerical solution of the periodicityequations and numerical evaluations of the Fourier series.

The influence of the stiffness of the shafts on the resonance vibrations of a two-stage gear drive has been investigated by means of the system ASB by R. SCHULZ.The parameters taken into consideration were

il =i2=i,J12 = "J11J 1 1 , J 2 1 = i 2 J11 J22 = i 2 J11 ,

MO = -i2M1,

C1=c2pp=c,

Y1 = Y(b11Q11t')

whereY2 = Y(b21Q21t)

Y(T) = 41 cos t + y2 cos 2T) ,

further

D1=D2=d, D1=0,?11 =Q21 =Le1' e12=e22=L2,£1 =E2=0.


In the numerical evaluations the values

i = 3 , J11 = 0.1 [kg m2] , c = 109 [N m-1]y1=0.1, y2=0.02, d = 2. 104 [N seem-1]

X01=0.1[m], P2= 0.3 [m], MO = 1000 [N ml

were chosen. The results for the resonances in the second stage are

Cl [Nm] Du xl ''2

1011 410.5 1.38 2.90(approximately rigid shaft)

108 392.1 1.51 3.00

3 107 356.0 1.54 3.23

where Qll is the angular velocity of the wheel (1, 1), which leads to the main resonancein the second gear stage, and

xi, max

Xi, stat

xi, max being the maximum value of the solution xi and xi, stat being the mean valueof the quasistatic solution,

MOxi, stat =

e11c

It shows that the results of the simplified model with rigid shaft hold approximatelyif the rigidity of the shaft is taken into consideration.

The ASB system has also been used successfully for higher-stage gear drives(SCHULZ (1986)), for planetary gear drives (SCHULZ and FRIEDRICH (1985)) and forother problems (compare Section 12.7).

7. Investigation of stability in the large

7.1. Fundamental considerations

Many non-linear systems of engineering interest possess more than one stationarysolution stable against small disturbances, that is, more than one attractor. Suchsolutions are termed locally stable. In practical applications usually only one ofthese is desirable; the others, for the most part, are unwelcome because they signaldanger to safe and reliable operation of the device whose model is being analyzed.The problem is to establish the conditions which lead to a particular steady state, oralternatively, to examine the disturbances which are apt to cause one stationarystate to change to another (for example, small-amplitude stationary vibration tolarge-amplitude motion, a non-oscillatory state to an oscillatory condition, etc.).

The solution to the first aspect of the problem, i.e. establishing the domains of theinitial conditions which lead to different stationary solutions, is well known. Thesedomains are called the domains of attraction of a particular solution. For systemswhich are directly described by a set of two first-order differential equations of thetype (4.1, 1) or whose set of such equations is identical with the original second-orderdifferential equation the problem is solved by means of phase portrait analysis outlinedin Chapter 4. The differential equations of motion of one-degree-of-freedom systemsexcited harmonically by an external force or parametrically can be converted, byapplication of known procedures (van der Pol or Krylov and 3ogoljubov methods) toa system of two first-order differential equations of the type (4.1, 1) which, however,are not identical with the original equation of motion. Consequently, an analysis madein the phase plane (see Section 4.4) is only approximate.

This chapter reviews some exact methods of stability-in-the-large investigationseffected by means of digital or analogue computers. The best known of these is thestroboscopic method which has been found very useful for analyses of one-degree-of-freedom systems. The analyst obtains a sequence of points of the separatrix, which -on being connected - yield the boundaries of the domains of attraction of the varioussteady solutions. The TV scanning and the circular scanning method proposed andused by TONDL (1970 a, b) (see also the Russian translation (1973 b)) are applicable tosystems with several degrees of freedom as well; their author used them in combinationwith the fast repetition procedure of analogue computers.

For systems determined by two initial conditions both the solution and its graphicalrepresentation are comparatively simple. Difficulties begin to arise with graphicalrepresentation of the boundaries of the domains of attraction when the system beingexamined is governed by more than three initial conditions ; in four- and multi-dimen-sional spaces intuitive orientation is no longer unambiguous and charting can onlybe effected by means of sections. This is the reason the problem has been solved,

200 7. Investigation of stability in the large

almost exclusively, for systems with two initial conditions (the work of HAYASHI,especially his book (1964), constitutes without doubt the greatest contribution to itsanalysis). In the case of more complicated systems the initial conditions are subjectedto restrictive assumptions (for example, are assumed to be such that only a singlenormal mode vibration can be initiated - see BENZ (1962), BEcKER (1972)). Somesystems governed by three or four initial conditions are analyzed in TONDL'S mono-graphs mentioned above.

A solution to the second problem, the investigation of the resistance of a particularsteady solution to disturbances of a certain type or establishment of the disturbanceswhich do not lead to a qualitative change of a steady solution is less well known. Inthe discussion which follows special attention is accorded to the class of problems inwhich the disturbances acting on the system satisfy definite assumptions.

In real systems, the actual disturbances cannot always be expressed in terms of theinitial conditions. This is particularly true of disturbances which act on the systemduring a definite time interval. Consider a disturbance (an external pulse, a change ofa system's parameter) expressed in terms of a function which is wholly determined bya small number of parameters (for example, two: the amplitude and the time ofapplication of a sinusoidal pulse). If both the beginning of the disturbance action andthe initial steady state of the system are known, we can determine the domain ofdifferent values of the parameters defining a certain type of disturbance, for which thetransient solutions lead to the given steady state. Accordingly, the meaning of theterm "domains of attraction" can also be extended to include, as the coordinates ofthe boundaries of the domains of attraction, the parameters which define a certaintype of disturbance. When a disturbance is described by two parameters, the domainof attraction can be represented by a plane diagram irrespective of the number ofinitial conditions which govern the system being examined.

In the discussion which follows, the solution to this problem is restricted to casesfor which the time interval between two successive disturbances is long enough topermit the analyst to assume that the transient motion of the system can be regardedas stationary before the next disturbance begins to take place. This assumption enablesa solution to be obtained even in the case when the disturbance or the time of thebeginning of its application is stochastic. The probability that a disturbance will leadto a particular stationary state can be determined using a finite number of determin-istic solutions. This approach is similar to that of the Monte Carlo method.

Disturbances can be defined in various ways and modified to represent thoseoccur-ring in real systems. A disturbance may be represented by a force pulse, for example,a sinusoidal (Fig. 7.1, 1 - light lines) or an oscillatory decaying pulse (Fig. 7.1, 1 -heavy lines), a step load, a constant load acting during a definite time interval, etc. A stepchange of an internal force or of a system parameter, such as a change of damping or thefriction force, of the mass of the system or the stiffness of an elastic element, etc. canbe counted among other types of disturbances. A change of a parameter (denoted Q)can be expressed in terms of a step change from QI to QII (Fig. 7.1, 2 a) or as a stepchange which exists for a definite time interval (Fig. 7.1, 2b). The basic types ofdisturbances are sufficiently defined by two parameters. This fact is advantageousfor graphical representation of the results, which no longer depends on the numberof initial conditions of the system. The results can also be used in connection withother problems, for example, in considerations relating to the stochastic characterof the disturbance or optimization of a particular parameter.

7.2. Methods for disturbances in the initial conditions

P(t)

Fig. 7.1, 1

on

t tjt2 ta) b)Fig. 7.1, 2

201

Modern analogue computers, particularly those provided with fast repetition andlogic circuits, have been found very useful for analyses of the kind described. Althoughan analogue computer was employed in the solution of the examples presented in thischapter, the principles of the methods proposed here do not exclude the applicationof a fast digital computer.

7.2. Methods of investigating stability in the large fordisturbances in the initial conditions

(a) Stroboscopic method

Consider a system governed by the differential equation of motion

y + /(y, y) = P cos wt (7.2, 1 a)

(external excitation), or by the equation

y+ /(y, y, (0t) = 0 (7.2, 1 b)

(parametric excitation) where the function /(y, y, (ot) is a periodic function of timehaving a period of 2n/w. Writing y = v one can convert (7.2, 1) to an equation of the


formV(v,y,t),

y = Y(v, y, t)(7.2, 2)

where the functions V, Y are periodic functions of time having a period of 21c/w.Trajectories obtained in the (v, y) phase plane for definite initial conditions do not,generally, obey the rule that a single trajectory passes through each point - although,as it is assumed, this rule holds for trajectories at any point of the space (v, y, t).Follow now the projection of the reference point of a certain trajectory onto the(v, y) plane during the time intervals spaced 2n/w apart, that is at times t = 0,27r/co, 4n/w, ... , n 2rc/co, as though the reference point were illuminated strobosco-pically during those time intervals and projected onto the (v, y) plane. Alternatively,imagine the trajectories in the (v, y, t) space to be intersected by a system of planesparallel to the (v, y) plane and spaced 2rc/w apart, and the points of intersection to beprojected onto the (v, y) plane (Fig. 7.2, 1). The trajectories obtained by connectingthe corresponding projections are termed trajectories in the stroboscopic plane (inthe sense of the well-known Poincare mapping) ; for the purpose of differentiation,their coordinates are denoted by (v)*, (y)* and the phase plane is spoken of as stro-boscopic.

Considering that the functions V(v, y, t) and Y(v, y, t) are periodic functions of timehaving a period of 2is/w, and consequently, that the functions V[(v)*, (y)*, 2nrlw]

V

Fig. 7.2, 1

7.2. Methods for disturbances in the initial conditions 203

and Y[(v)*, (y)*, 2nn/w] are invariant with respect to n, it can readily be shown thatthe stroboscopic trajectories are controlled by conditions of uniqueness which arewholly analogous to those applying to system (4.1, 1).

Rather than using approximate equations, the stroboscopic method solves directly(7.2, 1 a) (or (7.2, 1 b) or (7.2, 2)). Using the track-hold memory circuits records aremade, from the solutions of y(t) and y(t), of the values of y(2nn/w), y(2nn f w) (n = 0, 1,2, ...) or of the values of y(to + 2nn/w), ,(t0 + 2nn/w) when examining the separatrixor another important trajectory for the initial conditions at time t = to (instead of attime t = 0). In practical applications the signals of y(2nn/(O) and y(2nnjw) are broughtto the automatic plotter whose stylus jumps at intervals from point to point thusdrawing a broken line; alternatively, the stylus is dipped at intervals by means of arelay and records only individual points. The procedure of drawing the phaseportraits is analogous to that described in Chapter 4.

The procedure of obtaining the separatrix is again based on the idea of realizing thenegative time. In the (y, --iy) plane the system

y + f(-y, y) = P cos wt (7.2, 3)

(or possibly ft y, y, -(ot) for parametric systems) has trajectories which are identicalwith those of the system (7.2, 1) in the (y, y) plane; the reference point, however,moves in opposite direction.

When the points of the set thus drawn are not close enough for the purpose ofdrawing a trajectory, the procedure is repeated; the starting point is a point whoseposition can readily be determined from the first set. In selecting the starting pointuse is usually made of the fact that in the neighbourhood of a singular point (forexample, a saddle), the points are spaced close together. Thus, for example, if in thefirst solution the analyst obtains a sequence of points 1, 2, ... where the first pointsare spaced close together (see Fig. 7.2, 2) he can easily determine the position of thefirst point 1' of the second sequence 1', 2', ... By repeating the solution he can obtainas many points for fitting the trajectory as he desires.

Fig. 7.2, 2

Compare now the stroboscopic method with the approximate solution effected, forexample, by means of the van der Pol method; in the application of the latter, thesolution is sought in the form

y=acoswt +bsin wt (7.2,4)

where, for a non-stationary solution, the coefficients a, b are slowly varying functionsof time and investigations concerning stability in the large are carried out using thephase portrait analysis in the (a, b) phase plane (see Section 4.4). The more closely thetrajectories in the (a, b) phase plane approach those in the stroboscopic phase plane

(y)*), the better the approximation (7.2, 4) satisfies the solution of (7.2, 1)


and the more perfect is the fulfilment of the assumption that a(t), b(t) are slowlyvarying functions of time, for then the relations

(y)* = a , (y)* = cob (7.2, 5)

are satisfied to a higher degree. The relations (7.2, 5) were also the reason why theplane ((y/co) *, (y) *) was used in place of the plane ((y) *, (y) *) in most examples pre-sented here. Since the solution y(t) is not identical with the approximate solution(7.2, 4) because it generally contains higher harmonic components (also in the case ofsteady solutions), the positions of the singular points in the (a, b) plane are not iden-tical with those in the ((ylco)*, (y)*) plane; consequently, the phase portrait obtainedin the ((y/co)*, (y) *) for t = 0 can no longer be made applicable to the initial conditionsat t = to by mere turning. For these delayed initial conditions (or for excitationP cos (cot ± qq)) - if an exact procedure is in order - the problem must be solvedseparately. For the Duffing equation the effect is analyzed in a paper by FIALA andTONDL (1974). The differences between the two procedures (the van der Pol and thestroboscopic method) grow larger with increasing non-linearity of the system beingexamined; qualitatively, however, the methods are in close agreement even forcomparatively strong non-linearities.

(b) Method of television scanning

The method is based on repeated solution of the equation of motion (or of a set of suchequations if the system is governed by more than two initial conditions) for initialconditions which - except for one which is being varied - are kept constant. In thespecified interval of values the variable initial condition is changed in steps. Afterrunning through the whole interval of values, the other initial condition is changedby a certain value and the process of step-wise changing of the original variablecondition is repeated. In this way the analyst scans a plane whose coordinates arethe values of the two initial conditions - hence the term the television (TV) scanningmethod. Each transient solution is examined by means of a suitable criterion (whichwill be outlined at the end of this section) to determine whether or not it leads toa particular steady solution. TV scanning also actuates and controls the motion of thegraph plotter stylus. Logical circuits controlled by the criterion cause the stylusto dip whenever the transient solution converges to the steady one. The result of thisprocess can be, for example, a hatched area representing the domain of attraction ofa particular steady solution. If a system is determined by more than two initial condi-tions, the domain of attraction can be represented by plane sections or three-dimen-sional surfaces. Fast repetitive procedure has been found expedient when an analoguecomputer is used in the solution. From the point of view of practical solving and pre-sentation of results, the application of TV scanning is not restricted to systems withone degree of freedom excited externally or parametrically; the method has beenfound equally suitable for analyses of self-excited systems with several degrees offreedom.

(c) Method of circular scanning

This method differs from TV scanning by an auxiliary subroutine which is providedto generate a function whose reference point moves continuously in a circle. Thecircle, called the circle of disturbances, is centred at a definite point, for example,a singular point. At regular time intervals the values of the function (the coordinates

7.2. Methods for disturbances in the initial conditions 206

of the reference point on the circle) are introduced to the problem being solved in fastrepetition as the initial conditions. After a circle has been completed, the radius ischanged automatically by AR (Fig. 7.2, 3). The analyst can maintain either a constantangular velocity of rotation on the individual circles (and equal numbers of pointscorresponding to the initial conditions of the various solutions in fast repetition oneach circle) or a constant peripheral velocity of motion on the individual circles (andobtain a number of points whose spacing is kept constant and which grows larger

(v)"

Fig. 7.2, 3

with increasing radius). Using a suitable criterion and the memory circuits he thendecides whether or not a solution leads to the given steady solution. The circularscanning also actuates the stylus of the graph plotter whose motion is coordinatelycontrolled by the circuits which determine whether or not a transient solution leadsto the given steady solution.

The domains of attraction of a particular steady solution are obtained in a waysimilar to that used in TV scanning. The method of circular scanning, however, wasproposed primarily to facilitate solution of the second problem, i.e. investigation ofstability of a particular steady, locally stable solution against disturbances in initialconditions which are not fully determined. This problem will be dealt with in thenext section.

Returning now to the question of the criterion for determining whether or not atransient solution leads to one or the other steady solution: consider the so-calledintegral criterion and its application to the following example. A steady solutionwith a large amplitude should be distinguished from that with a small amplitude.Let the time interval T" used for the criterion application be bounded by times t1

and t2, and denote by x(t) the velocity of a chosen coordinate which describes the motionof the system being examined. The integral monitored on the computer is

tS t,

J = f [x(t)]2 dt or J = f Ix(t)I dt . (7.2, 6)t, t,

Denote by J1 the value of J for the large-amplitude steady solution and by J2 that forthe small-amplitude (or non-oscillatory) solution. It is found that

J1,J2.Choosing a suitable value JO from the interval J2 < JO < J1 the transient responseleads to the large-amplitude solution for J > JO and to the small-amplitude one forJ < J0. A detailed discussion of the various criteria may be found in the monographsby ToIDL (1970 a, 1970 b, 1973 a). Questions of analogue computer programming forthe application of the methods and criteria outlined above are dealt with in a mono-graph by FIALA (1976).


7.3. Investigation of stability in the large for not-fullydetermined disturbances

This section will discuss the principle of solution effected by means of the circularscanning method. Consider first a one-degree-of-freedom system for which two stablesteady solutions (one resonant, the other non-resonant) exist for a specified value of Co.In the stroboscopic phase plane they are represented by the stable foci 1 and 3,respectively (Fig. 7.3, 1). The separatrix s is formed by two stroboscopic trajectoriesleading to the saddle point 2; from this point issue two trajectories terminating atpoints 1 and 2.

Fig. 7.3, 1

Consider the system to be in a definite steady state, for example, resonant vibration,to which corresponds focus 1 in the stroboscopic phase plane. The points of interest arethe disturbance in the initial conditions from the steady state which will lead to thereturn to the original steady solution, and the disturbance which will lead to a qualita.tive change, i.e. to the transition to non-resonant vibration represented by focus 3.

In the coordinates (y)*, (v)* a disturbance from the steady state is defined by

A(y)* = y(0) - Yi

A(v)* = 1 y(0) - vlw

where yl, v1 are the coordinates of the singular point 1. For L(y)* = 0(v)* = 0, i.e.for y(0) = yl, y(0) = (ov1 no transient motion will arise and the system will immediatelyvibrate in resonant vibration. For fully determined disturbances in time t = n(2-x/w)(n = 0, 1, 2, ...) the question can be answered directly by consulting the separatrixdiagram shown in Fig. 7.3, 1.

So much for fully determined disturbances. Consider now the effect of disturbancesfrom the steady state, which are no longer fully determined but contain a stochasticelement introduced in their definition. Choose the initial conditions at time t = 0such that it will make the distance between the starting point and the singular point(point 1 in the case being considered) in the phase plane equal to a given value, R. Thismeans that the starting point is chosen quite freely, and so represents the stochasticelement because it may lie anywhere on the circle with radius R centred at point 1(Fig. 7.3, 2).

7.3. Investigation of stability for not-fully determined disturbances 207

Fig. 7.3, 2

Definition 7.1. A disturbance from the steady state represented in the stroboscopic phaseplane (or Hayashi's plane) 1) by singular point 1 with coordinates y1, v, is defined by thedistance R between the starting point with coordinates y(0), v(0) . (v(0) = 1/(0 y(0)) andthe singular point 1. The initial conditions are defined by any point of the circle describedby the equation

R2 = [y(0) - y,]2 + [v(0) - v1]2 (7.3, 1)

and termed the circle of disturbances.

The position of the starting point on the circle with radius R is, therefore, stochasticand it is assumed that the starting point has equal probability of occurring anywhereon the circle. Denoting by Al the length of the are of the circle of disturbances, whichlies inside the domain of attraction relating to the singular point 1, the relation

Alp(R) = 27rR

(7.3, 2a)

defines the probability that a solution with a disturbance in the initial conditions suchthat the starting point can lie anywhere on the circle with radius R centered at point Iwill again lead to the steady solution represented by the singular point 1. If n (a suffi-ciently large number) solutions, starting from points uniformly distributed on thecircle of disturbances, are examined and k solutions are found to lead to the steadysolution being investigated, the probability p(R) is also given by the relation

kP(R) _ -n

(7.3, 2b)

This equation makes it possible to determine the probability p(R) directly (by meansof an analogue or a digital computer).

1) The term Hayashi's plane (used for reason of briefness) denotes the phase plane of thesystem of two first-order differential equations, which is obtained by application of thevan der Pol transformation (the (a, b) phase plane in Section 4.4). It was used by TONDLin his monograph (1973) because Professor Hayashi was the first who employed thephase portraits in this plane.


Definition 7.11. Probability on the circle of disturbances, p(R), is the probability thatthe disturbed solution at time t = 0 will lead to the given steady state; in the stroboscopicphase plane (Hayashi's plane) a disturbance is defined by any point on the circle ofdisturbances.

Knowledge of p(R) for a single value of R does not provide sufficient information.If, on the other hand, one knows the probabilities on the circles of disturbances ofdifferent radii R, i.e. the function p(R) (Fig. 7.3, 3), one can gain a clear and compre-hensive idea of the effect of disturbances from the steady state being examined. Con-sulting the diagram, we see that up to R' the probability p(R) is equal to one, that isto say, so long as R < R' arbitrary disturbances can cause no change of the steadystate. The circle with radius R' is the largest circle which can be inscribed inside therespective domain of attraction from a particular singular point in the stroboscopicphase plane.

p(R)

0.5

0

Fig. 7.3, 3

R'

As mentioned above, all information so far presented relates to disturbances at timet`= 0. Upson dividing the period 2n/a) into n equal time intervals and determiningpx(R) for the beginning of each interval, the mean value of the probabilities on thecircle of disturbances is found to be defined by the formula

_ n

p(R) _ - E pa(R) . (7.3, 3)n x=1

This mean value makes it possible to eliminate the dependence on time.

Definition 7.III. The probability on the circle of disturbances, p(R), obtained from(7.3, 3) is the mean value of the probabilities on the circles of disturbances established forvarious instants uniformly distributed over the interval of a vibration period. It thusrepresents the probability on the circle of disturbances with which a solution with a distur-bance defined on the circle of disturbances will lead to the steady solution being examinedat any instant.

The probability on the circle of disturbances can also be employed for determiningdisturbances defined in different ways (see the already quoted monograph by ToNDL(1973 a)).

Since in practical cases the occurrence of small disturbances is much more frequentthan that of large disturbances, and the magnitude of a disturbance can, for the mostpart, be restricted to a finite value, we can choose the density of occurrence of thedisturbances and express it also in terms of the circle of disturbances. To that endintroduce the function of the probability density of disturbances on the circle of


disturbances, /(R); the function can be bounded (see Fig. 7.3, 4) by the condition thatthe maximum disturbance will not extend beyond the circle of disturbances withradius R0.

f(R)

Fig. 7.3, 4Ro R

The function f(R) satisfies the relation

f,f(R)dR = 1

f(R) 0 for

0 f (R) = 0 for

R < Ro,

R>R0.(7.3, 4)

If function f(R) is applied to the probability function on the circle of disturbancesp(R) obtained as indicated above, we can, given the function of the probability densityof the occurrence of the disturbances on the circle of disturbances, establish directlythe probability with which the steady solution being examined will be attained by theeffect of any disturbance at time t = 0.

Definition 7.IV. If the specified function of the density of distribution of the occurrenceof disturbances on the various circles of disturbances satisfies (7.3, 4), the probability Pthat any disturbance from the steady state being examined will lead to that state can bedetermined from the equation

no

P = _f f(R) p(R) dR0

(7.3, 5a)

when the function p(R) is known and the disturbances arise at time t = 0, or from theequation

Ro

P = f f(R) p(R) dR0

(7.3, 5b)

when the function p(R) is known and the disturbances arise at any time.

In this case the numerical value of the probability represents a measure of estimatingthe resistance of the steady state in question to disturbances.

To facilitate solution, it is possible to choose for the function I (R) the linear function

f(R) = R0(1Ro/

(7.3, 6)

and for R0 a value which is common to a whole class of systems, for example, a multipleof the static deflection corresponding to the amplitude of the excitation force. Thisenables the analyst to compare resistances of certain solutions of various systems or tooptimalize a particular parameter.14 Schmidt/Tondl


As mentioned in the preceding section, solutions to problems of the sort discussedare expediently obtained by means of the circular scanning method, which makes itpossible to determine not only the domains of attraction but the function p(R) as well.The graph plotter stylus, which is controlled by circular scanning and the logiccircuits deciding whether or not a transient solution leads to the given one, moves ata uniform rate and draws a vertical line whenever the solution in fast repetition con-verges to the steady solution. In the segments of the circle of disturbances in whichthe transient solutions fail to lead to the steady one, the stylus is stopped. If the circleof disturbances lies whole in the domain of attraction being examined, the stylusdraws the entire prescribed length and the probability p(R) = 1. After completinga circle, the stylus is automatically reset by R and the process is repeated. The resultingdiagram contains a series of vertical lines whose lengths represent, to an appropriatescale, the values of the probability p(R) ; by fitting a curve to the end points of theline segments we obtain the dependence p(R) (Fig. 7.3, 5).

1

P(R)

I-AR R

Fig. 7.3, 5

Consider now systems with several degrees of freedom, for example, a system excitedby a periodic force with a period T = 27s/w whose stable steady solutions are charac-terized by singular points in the n-th dimensional stroboscopic phase (or the n-dimen-sional Hayashi's) space. Just as in the case of one-degree-of-freedom systems thesolution, although approximate, is independent of time. Proceeding in the wayoutlined for one-degree-of-freedom systems examine the effect of the disturbancesfrom the given steady state replacing, logically, the circle of disturbances by thesphere of disturbance of radius R centered at a particular singular point.

Definition 7.V. I/ yoi' Yo2' , yon are the coordinates of the given singular point,yl(0), y2(0), ... , y.(0) the coordinates of the starting point, and Xk = yk(0) - yok (k = 1,2, ... , n) in the stroboscopic phase (Hayashi's) space, the sphere of disturbances is definedby the equation

nxk=R2

k=1(7.3, 7)

If the analyst plans to use a computer for a practical solution, the points on thissphere must be replaced by a finite number of points. Since it is usually much easierto make such a replacement for carves than for surfaces, the analyst should proceedin two steps. First, the spherical surface is replaced by an exactly defined number ofcurves distributed as uniformly as possible; second, the curves are replaced by a finitenumber of points. Although this double replacement tends to make the procedureslightly less accurate, it enables the analyst quickly to form a qualitative opinionof the situation. The simplest substitute curves seem to be the principal circles of the


sphere of disturbances defined by the quations

xk+xJ=R2, (k,j=1,2,...,n;k=xs=O (s= 1,2,...,n;s4k,s=j).

(7.3, 8)

These equations represent the curves of intersection of the coordinate planes (Xk, x,)and the sphere of disturbances. The points on the sphere surface are thus replaced by

points lying on the principal circles whose number is N = (2). This system of circlesis termed the substitute sphere of disturbances. \\

Definition 7.VI. The substitute sphere of disturbances. The substitute sphere of disturb-ances is the set of all starting points for a disturbed solution that lie on the principalcircles of the sphere of disturbances defined by (7.3, 8).

Assume that a starting point has the same probability of occurring at any point ofthe substitute sphere of disturbances.

2 that a disturbedDenote by p$(R) = pkf(R) s = 1, 2, N = ())the probability

solution will lead to the given steady one where the disturbance is described by any ofthe points of the principal circle of disturbances. The mean value of all probabilitiesps(R) is defined by the equation

Y

P(R) = 1 E p8(R)N 8=1

(7.3, 9)

and represents the probability that a disturbed solution will lead to the steady solutionwhere the disturbance is described by any of the points of the substitute sphere ofdisturbances. To simplify the discussion the same notation is used for the functionp(R) as for the probability on the circle of disturbances of one-degree-of-freedomsystems.

Definition 7.VIII. The probability on the substitute sphere of disturbances p(R) isthe probability that a disturbed solution at time t = 0 (at any time when using Hayashi'sspace) will lead to the steady solution being examined. The starting point of the disturbedsolution may lie at any point of the substitute sphere of disturbances. The value of p(R)is the mean of the probabilities on all principal circles of the sphere of disturbances.

If the analyst is using the stroboscopic phase space and wishes to eliminate thetime from his considerations, he can - just as in the case of one-degree-of-freedom sys-tems - divide the period of steady vibration into a number of equal parts, deter-mine the functions ps(R) for the beginnings of the individual intervals and by estab-lishing the mean values ps(R) obtain the probability

1 vP(R) _ - E ps(R) . (7.3, 10)

N8=1

This is the probability that a disturbed solution at any time will lead to the steadysolution where the disturbance is defined by the substitute sphere of disturbances.

Note: Although the use of Hayashi's space is more straightforward and eliminatesthe time dependence directly, the stroboscopic phase space approach is more rigorous.14


A procedure similar to that just outlined can be used to determine the probability Pthat any disturbed solution will lead to the given steady solution if one knows thevariation of function p(R) (at time t = 0) or of function p(R) (at any time) and thefunction of the density of distribution of the probability of occurrence of the disturb-ances, I (R). The resulting relation is formally analogous to (7.3, 5).

7.4. Examples of investigations concerning stability in the largefor disturbances in the initial conditions

Example I. For the purpose of comparison with the approximate procedure outlinedin Section 4.4, an analysis will first be presented of the Duffing system described byequation (4.4, 1). Figs. 7.4, 1 and 7.4, 2 show the phase portraits in the stroboscopic

Fig. 7.4, 1

Fig. 7.4, 2

7.4. Examples for disturbances in the initial conditions 213

phase plane obtained for x = 0.05 and y = 10-2 (i.e. the same as those used in Section4.4) and j = 1.3. The first portrait relates to the initial time a = 0, the second toz = 7r/21); both show the effect of time of application of the disturbance. Unlike thoseobtained by application of the approximate method of Section 4.4, the two diagramsdo not become one when turned through an angle of 90° relative to one another.Fig. 7.4, 3 shows the phase portrait obtained for q = 1.5 and r = 0. To facilitate com-parison the separatrix established by application of the approximate method is shownin dotted line. It can be seen that although the qualitative agreement is close, theresults differ quantitatively. The difference grows larger with increasing value of thenon-linearity y.

Fig. 7.4, 3

Example II. Consider a one-mass, two degrees-of-freedom (two-dimensional) systemor whose motion is described by the differential equationsexcited by a circulating vect

mx + xlx + Cax + Nxy2 = mEw2 cos wt ,

my + x2y + C2y + y2xy = mEw2 sin wt(7.4, 1)

where xa, x2, cl, c2, fl, y2, E are positive constants. Assume that the system is tuned tothe internal resonance for which the ratio of c1/c2 is close to 4. Introducing the relativedeflection u = x/E, v = y/E, the notation col = c1/m, w2 = c2/m (the partial frequenciesof the linearized system without damping) and the substitution coat = r, the equationsof motion can be written in the dimensionless form

zv" + D2v' + Ka + youv = fl2 sin

where,q = w/w1, D1 = x,/ma), D2 = x2/mwa, K = w2/wi, fl = N1E2/CU Ye = c2/Ca 72--/C2'If --- 1, application of the van der Pol transformation (the solution is sought in

the form

u" + Du' + u + Nouv2 = y12 cos ?-C ,

(7.4, 2)J}

u=acos)Ji+bsingt,(7.4,3)

V=ccos2qz+dsin2vz


where the coefficients a, b, c, d are assumed to be slowly varying functions of time(constants for a steady solution)) enables equations (7.4, 2) to be converted to thefollowing system of first-order" equations:

11a' = - 2 D1a +--

Ll - Iq 2 + 2 l3o(c2 + d2)J b

b'2

D1b - I 1- 712 + 12

/90(C2 + d2) I a + 2

C2 D2c + l 2 Yobc + V - 2 ri -

2' 7,,a] dl

= - D2d - -f - K(1)2

--I--2

Yoa I c 7r] J 1

1'/.4. 4)

In a definite interval of ri, system (7.4, 4) can have the semi-trivial solution a z 0,b + 0, c = d = 0 as well as two stable and two unstable non-trivial solutions. Ineach of these pairs, the solutions differ by the phase angle rather than by the magni-tude of the amplitude. Since a detailed analysis of the results is presented in a mono-graph by Tot wL (1970 a), only the information necessary for further discussion willbe given below.

The results relating to stationary vibration in the case when K = 0.3, D1 = D20.1, yo = 3 10-2 and N0 = 0.8 .10-2 are reviewed in the diagrams included in

Fig. 7.4, 4. The diagrams show the dependence of the amplitudes rl = (a2 + b2)112 andr2 = (c2 d2)1/2 on q (ro denotes the semi-trivial solution) and in the planes (a, b) and(c, d) the corresponding values for various q. The results obtained for the stable solu-tions are drawn in heavy lines, those for the unstable solutions in light lines.

The figure also includes the results of an analysis of the domains of attraction ob-tained by means of the method of TV scanning for fully determined disturbances andri = 1.15. The results refer to disturbances in the initial conditions of the solutiondefined by the stable singular point with coordinates al, bi, ci and dl. The disturbancesare marked a = a(0) - al, # = b(0) -- bi, y = c(0) - ci and 6 = d(0) - dl. Since thespace of the initial conditions is four dimensional, plane sections are taken of theseparatrix surfaces with always two of the coordinates a, i, y, 6 made equal to zero(for example, c(0) = ci, d(0) = d.). The diagrams in Fig. 7.4, 4 relate to the case ofc = d = 0 when the singular point belongs to the semi-trivial solution. In the dia-grams of r1(n), r2(rj) and in the planes (a, b), (c, d), the points corresponding to thissolution are marked by numeral 1. The cross-hatched areas represent the regions of theinitial conditions which lead to the semi-trivial solutions, the hatched areas thosewhich lead to the non-trivial solutions (in the diagrams of r1(rl), r2(,q) and in the planes(a, b), (c, d) the non-trivial solutions which, as is recalled, are two differing only as

to the phase angle, are marked by points 3 a and 3 b). Since\2

6, there ought to be

six plane sections. As, however, all the initial conditions in the (a, fl) plane lead to thesemi-trivial solution, this plane is omitted. The diagrams shown in Fig. 7.4, 5 aresimilar to those just discussed except that the singular point 3 a corresponds to oneof the non-trivial stable solutions.

The results obtained in determination of the function p(R) by means of the methodof circular scanning are shown in Figs. 7.4, 6 and 7.4, 7. The first figure (correspondingto Fig. 7.4, 4) belongs to the case when the singular point represents the semi-trivial

7.4. Examples for disturbances in the initial conditions

0 1.5 0.5 1 1.5

0

9

MEN

c

0\

vI---=_22222222222-.

vor`vnnr w

j20

rl= 1.15

20 d

d

20

m

Mown.411

NEEN-11 IN

i0

216

Fig. 7.4, 4


Fig. 7.4; 5


pip (R)

0.5

0 5 10 R 15

lR)Pad

0.5

0

PO' (R)

0.5

0

5

5 10 R 15Fig. 7.4, 6

5

10 R 15

10 R 15

p, 6(R)

0.5

0 5 10 R 15

p (R)0.5

0 5 10 R 15Fig. 7.4, 7

0

paa(R)

0.5

0

p,5(R)0.5

0

0

5

5

5

5

10

10 R 15

10 R 15

10 R 15

10

10 R 15

1

P(P)

0.5

0 5 10 R 15

1

p(R)

0.5

0 5 10 R 15

solution and the probabilities indicate that with which the disturbed solution willconverge to the semi-trivial one. The auxiliary diagrams (drawn to a smaller scale)show the probabilities on the principal circles, pas(R), pdC(R), ... , used to obtain thecourse of the function p(R). For a linear function of the density of distribution of theprobability of occurrence of the disturbances on the various substitute spheres, definedby (7.2, 6) and Ro = 15, the probability is P = 0.64.

R 15


Fig. 7.4, 7 (corresponding to Fig. 7.4, 5) shows the results obtained in the case whenthe singular point represents one of the two stable non-trivial steady solutions. Sincetheir amplitudes are the same, the two solutions are not distinguished one from anotherin the diagrams; the probabilities shown express that with which the disturbed solu-tion will lead to the stable non-trivial one (i.e. irrespective of the fact that thereexist two such solutions differing only as to the phase angle). For I (R) according to(7.3, 6) and Ro = 15, P = 0.99.

As the results imply, the semi-trivial solution is less resistant to disturbances thana non-trivial solution. Comparing the results obtained by means of the two proceduresused in the analysis, it is seen that the method of the p(R) function, by virtue of itssimplicity and clarity, provides data which are easy to apply particularly when promptand comprehensive information is desired.

Example III. The example just discussed involved the case of special tuning (tointernal resonance) when the equations of motion were converted by means of thevan der Pol transformation to an approximate system of four first-order differentialequations. This procedure, although it failed to solve the equations of motion directly,avoided the need to examine the effect of the instant of disturbances. In the present

rm2w2cosc t

Fig. 7.4, 8 Fig. 7.4, 9

example, the equations of motion will be solved directly by means of the method ofcircular scanning. Consider a two-degree-of-freedom system with two equal masses;the spring connecting them has a broken-line characteristic (Fig. 7.4, 8). The lowermass rests against a linear spring whose rigidity is the same as that of the linearportion of the characteristic of the upper spring for small deflections from the equili-brium state. Two identical linear viscous dampers with a damping coefficient x areattached to the two masses (Fig. 7.4, 9). The two masses are acted on by constantforces, Ql and Q2, the lower one is in addition excited by a harmonic force m&w2 cos (Ot.Denoting by yl, y2 the deflections of the upper and lower mass, respectively, the motionof the system is described by the differential equations

myi+x(y1 -y2)+F(y1 -y2) =Qi,(7.4,5)

}my2-x(yi-y2)-1'(yi-y2)+xy2+cy2=Q2+mEw2cosCot


where

219

E(y1 -- y2) = c(y1 - y2) for Iyi -- y2l < a, (a> 0) ,

= c(y1 - y2) + c1(y1 -y2-a) for yi - y2 > a ,=c(y1-y2)+c1(yl-y2+a) for y1-y2<-a.

Introducing the notation

c =w2 xD, Q1- Q2- = yi, (7.4,6)o, =ql, -q2,

m main C C Wo

the relative coordinates

y2-= x1, =x2y1

E E

and the transformation

(7.4, 7)

coot = z (7.4, 8)

equations (7.4, 5), after rearrangement, take the form

xl+D(xl --x2)+ (xl -x2)[1 +9'(x1 -x2)1=q1x2 - D(xl - x2) + Dx2 - (x1 - x2) [1 + 9)(x1 - x2)] + x2 = q 2 + 7J2 Cos 1]t}

(7.4, 9)where

9'(x1-x2)=0 for Ix1 -x2I <a (a>0)

= Cl (1 - a1 )for I xl - x2I >

aC E Ixl - x21 E

Figs. 7.4, 10 to 7.4, 12 show the amplitude-frequency characteristics obtained byanalogue computer solutions for the relative deflections x1, x2 and their difference

1

10

-10

ncx(x1)

0.5

min(x1)

1.0 1.5 2.0

Fig. 7.4, 10


Fig. 7.4, 11

Fig. 7.4, 12

xl - x2 in the case when ql = 4, q2 = -6, a/e = 10, cl/c = 2, D = 0.05. The lowerresonance is seen to have a linear character. The non-linearity of the upper springdoes not manifest itself except in higher resonance.

Fig. 7.4, 13 shows the trajectories of the steady solutions (both resonant and non-resonant) in the partial phase planes (xl, xi/q) and (x2, x2/q) for a relative excitationfrequency = 1.86. The points I, II, III and IV mark the position of the referencepoint at time z = 0, 2 7c/,q, n/r and 2 it/n. The coordinates of these points define the


Fig. 7.4, 13

initial conditions for which the steady solution, at the specified times are obtaineddirectly, without transient motions. They determine the position of the singularpoints in the stroboscopic phase plane. The method of circular scanning makes itpossible to obtain not only the function p(R) on the various principal circles of thespheres of disturbances but also, by means of a second plotter, the domains of attrac-tion in the various plane sections of the space of the initial conditions. As in ExampleII, two initial conditions (corresponding to two coordinates of the singular point) arekept constant, and two are varied in accordance with the circular scanning. Thesevariable coordinates of the singular point describe the centre of the circle of circularscanning. Since six combinations of the principal circles are possible for four initialconditions (hence the six diagrams of the function p(R)), there exist six combinationsof the plane sections of the separatrix surface. The solution of the example was carriedout for q = 1.86 and four initial times, i.e. = 0, 2 7c/ri, 7r/,q, 2 7r/,q. Figs. 7.4, 14 to7.4, 17 refer to the resonant solution and show the domains of attraction at theseinitial times for disturbances in the solution's neighbourhood. The regions of theinitial conditions leading to the non-resonant solution are shown in hatching. Fig.


z=o

Fig. 7.4, 14


x'(0)ii

20

Fig. 7.4, 15

224 of stability in the large7 Investigation

Fig. 7.4, 16


10 20 xl(O)

Fig. 7.4, 17

15 Schmidt/Tond I


1

P."(R)

0.5

0

1

p ,x2

-x1

(R)

0.5

0

I I:

Nk.

10 R 20

7

10 R 20

Fig. 7.4, 18

-e =0

1 x211.

Tl

3 X2 rl

1

px1l ,x (R)2

0.5

1

P.,,.x(R)2

0.5

0

0.5

0

10 R 20

10 R 20

10 R 20

1

pp, (R)

2

0.5

0

I

\

10 R 20

II;

1

\. \

10 R 20


,e=0

Fig. 7.4, 19

15


10 x2(0)

Fig. 7.4, 20


Fig. 7.4, 21


37t271

Fig. 7.4, 22

Px, X (R)11 1

0.5

1

Px x (R)2 1

0.5


I

1 I

I

\

10

px;x2 (R) PXZx2 (R)

0.5 0.5

R 20 0

.1i

\ \ ?

1

P , , (R)x11x2

0.5

10 R 20

0 10 R 20 0 10 R 20

1

Fig. 7.4, 23

-e =0 P(RI

0 10 R 20

0.5

11

0 10 R 20

231


7.4, 18 shows the corresponding functions p(R) on the various principal circles of thespheres of disturbances established directly by means of an analogue computer solu-tion, as well as a diagram of the function p(R) whose points correspond - accordingto definition - to the mean of all values for a particular R.

Figs. 7.4, 19 to 7.4, 23 present similar diagrams for the non-resonant solution.Comparing the functions p(R) on the principal circles of disturbances at various times,it is found that the functions at r = 0 and 7c/ii, or at z = a 7c/r) and 2 7c/,q differ onlyslightly from one another; the difference between p(R) at times a = 0, 7c/q and thoseat 2 7C/,q, s 7c/,q is more substantial.

A comparison of the results of the resonant steady solution with those of the non-resonant one allows the analyst to draw the following conclusions:

At , = 1.86, R' (maximum R for which p(R) = 1) is larger for the non-resonantsolution; however, once this R' has been exceeded, the function p(R) for the resonantsolution decreases much more rapidly. Consequently, the region of disturbances fromthe steady state defined by (7.4, 5), in which all solutions lead to the steady solutionbeing examined, is larger for the non-resonant solution. If, however, the region ofpossible disturbances from the steady state is enlarged, the function p(R) of the reso-nant solution decreases more slowly than that of the non-resonant one. These findingsare also confirmed by a comparison of the values of the probability P for specifiedfunctions of the probability density /(R). For /(R) according to (7.3, 6) (triangulardistribution) and Ro = 20,

Pres -> res = 0.878 , 1'nonres -+ nonres = 0.885.

For rectangular distribution with function /(R) defined by1

f(R)= Ra

0

for

for

R<Ru

R> Ro(7.4, 10)

and Ro = 20,

Pres -+ res = 0.834 , Pnonres -+ nonres = 0.668

The results of a detailed analysis of the effect of the coefficient of tuning rl on theresistance to disturbances from the steady resonant solution (see a monograph byTONDL (1973a) or (1979c)) may be summed up as follows:

For two-degrees-of-freedom systems in which the coefficient of tuning 77 is varied sothat the system approaches the resonant peak and thus also the limit of the intervalof two steady locally stable solutions, the volume of the domain of attraction of theresonant solution grows smaller less rapidly than it does for one-degree-of-freedomsystems. The same can be said about the probability that disturbances from thesteady state will not lead to a transition to the non-resonant solution. In this respecta larger number of degrees of freedom has a favourable effect if the primary require-ments put on the system are good resistance to disturbances from steady resonantvibration and large amplitudes (as, for example, in the case of vibratory conveyors).

7.5. Investigation of stability in the large for other typesof disturbances

The disturbances whose effect will now be examined are not determined by the initialconditions; they are of the type described in Section 7.1.

7.5. Other types of disturbances 233

Depending on whether the steady state in question is non-oscillatory (equilibriumposition) or oscillatory (periodic vibration) the methods of examination can be dividedinto two groups.

In the first case the system is subjected to a disturbance of a certain type; theinitial conditions at time t = 0 (at the beginning of the disturbance application)correspond to the steady state. If a transient solution converges to this steady state,the parameters of the disturbance lie in the domain of attraction of the steady statebeing examined and can be varied, for example, by means of the method of TVscanning. Let the disturbance be characterized by two parameters, for example A(the pulse amplitude) and TO (the time of the disturbance duration) and let theseparameters vary within the intervals

Al<A<A2, T01<TO <T02.

One parameter, for example A, is kept constant, the other is varied in steps of AT0through the whole interval (T01, T02). Next, A is varied in steps of AA and the processis repeated. A transient solution in fast repetition is obtained for each value of theparameters and, by means of a suitable criterion, an estimate is made whether or notthe transient process converges to the steady solution. The logical circuit which appliesthe criterion controls the dip and lift of the graph plotter stylus; its vertical and hori-zontal displacements are regulated by the circuits which effect the gradual changesof the disturbance parameters. The result of the procedure is the hatched area of thedomain of attraction of the steady state in the (A, T0) plane.

In the second case (the steady state being examined is oscillatory -periodic vibration)the procedure has several variants whose choice is controlled by the instant of thedisturbance application. If the domain of attraction is to be examined for a definitetime instant, the analysis proceeds as in the former case. At the instant the disturbanceis applied the system is subjected to initial conditions which realize the steady statewithout a transient solution.

Consider now the case when the beginning of the disturbance application is stocha-stic rather than deterministic. Let the frequency of the steady oscillatory solution beco (the vibration period T = 2rc/w). Assume that the probability of occurrence of theinstant of the disturbance application is the same throughout the whole vibrationperiod T of the steady solution (jT < t < (j + 1) T (j = 1, 2, ...). Divide the period Tinto N equal time intervals and apply N times the disturbance characterized by theparameters A, TO at the beginning of each interval. The probability that the disturb-ance applied at any time will lead to the steady state is defined by the relation

p(A,T0)=N (7.5,1)

where k is the number of transient solutions which converge to the steady state. Inpractice the subsequent procedure depends on the type of the stationary vibration,that is, whether it is excited externally or parametrically, or whether it is self-excited.

In the first case the disturbances are applied successively with a delay (r/N) T(r = 1, 2, ... , N). The logical circuit which applies the criterion of convergence of thetransient solution controls the motion of the graph plotter stylus in one direction(for example, vertical). If a transient solution leads to the steady state, the stylusmoves 1/N of the total pre-set range. After examination of N transient processes, thestylus marks out a segment k/N long of the total range corresponding to the probabili-ty 1. Following completion of N such solutions, one of the parameters, for example A,


is changed by AA and the process is repeated until the whole interval of A is traversed.If the other parameter is also varied in steps, a set of p(A) diagrams is obtained forvarious values of To. On the basis of these diagrams the analyst sets up the axono-metric representation of the function p(A, To).

In the second case - investigation of stability of stationary self-excited vibration -the procedure is more complicated. An analogue solution can proceed as follows: Onepart of the computer (system I) performs a slow solution of the equations of motionfor parameters corresponding to the state without a disturbance. The other part of theanalogue (system II) is programmed to solve the equations of motion in fast repetition.This part - complete with the logic circuits - carries out the actual solution of theproblem ; system I serves only as a source of the necessary initial conditions which arespecified so as to lead to the solution whose stability is being examined without tran-sient processes. Let T1 be the period of the steady solution for system I. Divide thisperiod into n intervals and use the deflections and velocities obtained after a Ti/ntime interval for the initial conditions of the fast repetition solution in system II.Since the solution in system II proceeds at a very fast rate,

T1 = NT2 (7.5, 2)

where T2 is the period of the steady solution in system II, and N is a large integer,N ' n; N/n is a large integer. The instant at which the initial conditions from system Iare applied to system II represents the time at which system II is subjected to a dis-turbance of To duration. The time in the course of which system II carries out thesolution, is (1/n)/T1; it is divided into three intervals, viz.

TI=To+T'+T"(7.5,3)

n

To is the time of the disturbance duration, T' is the time sufficient for stabilizing thetransient response, and T" the time during which a suitable criterion (for example,integral) is applied to the logic circuits. The time interval T' must be a multiple of T2such that during its course the transient vibration becomes stabilized. In most casesthis is achieved with a factor of 15 to 30. This explains why special stress is laid onthe requirement that N/n should be a large integer.

The logic and control circuits of the part of the analogue which handles system IIare the same as those used in previous examples.

As in Section 7.3, given the function of the probability density of the disturbances,/(A, To) (for disturbances of the type of a force pulse - Fig. 7.1, 1 or a step change ofa parameter lasting a definite time - Fig. 7.1, 2b) or /(Q1, QII) (for disturbances ofthe type of a step change of a parameter - Fig. 7.1, 2 a), one can determine the prob-ability that random disturbances of a certain type will lead to a given solution.Functions /(A, To) or /(QI, QII) satisfy the relations

Co 00 Co 00

f f /(A, To) dA dTo = 1 , f .f /(Q1, Q1I) dQ1 dQ11; (7.5, 4)-00 -00 -00 -00

they can be defined in a restricted interval, that is, be non-zero in finite intervals ofthe values of A, To or Q1, QI1. The probability P is then defined by the equation

00 00

P = f f p(A, To) /(A, To) dA dTo (7.5, 5a)-00 -00

7.6. Other applications of the results 235

or by the equation00 00

P = f f p(Q1, Q11) f(Q1, Q11) dQ1 dQ11.-00 -00

(7.5, 5b)

The reader interested in additional details of the method outlined above is referredto a special monograph by TorrDL (1973 a).

7.6. Other applications of the results

Determination of the probability P (for disturbances in the initial conditions or forother types of disturbances) for a specified density of distribution by means of thefunction /(R) in relation to one of the parameters of the system makes it possible toestablish the dependence of P on that parameter. The function P(17) (,q is the variableparameter, for example, the excitation frequency) can then be used in solution ofother problems, for example, that of comparing the resistance of various steadysolutions in the whole interval of the excitation force, in which more than one steadylocally stable solutions obtain.

By way of example, consider that two such solutions, one resonant (I), the othernon-resonant (II) exist in the interval (%, %). Denoting by

P1(n) where ri, < 71 < i72

the probability with which a disturbed solution at the specified function of the densityof distribution of the disturbances from the steady solution I will again lead to thatsolution, and by

P11(77) where fh < 17 <q2

the probability with which a disturbed solution at the same function of the densityof distribution of the disturbances from the steady solution II will again lead to thatsolution, the analyst can use the values

n, a1 (' 1

P1 = P1W dI7 , Pu = P11(n) dq (7.6, 1)772 X71 772 - TIi

,h ,li

as a measure of comparison of the global resistance of the two steady solutions todisturbances from the steady state. In practical applications, the integrals of (7.6, 1)can be replaced by summations.

Expressions (7.6, 1) can also be used in evaluation of the resistance to disturbancesof various systems. It is, of course, necessary to arrange the differential equations ofmotion of the systems being compared to the same, preferably the dimensionless formand to operate with the same function of the density of distribution AR) or I (A, To)or possibly f (Q1, Q11) and the same type of disturbances when determining P('q).

Optimization of a parameter of a system, for example, tuning, also belongs to theclass of problems being discussed. The system is required to possess, in the highestpossible degree, the desired property (for example, a very large amplitude of vibrationof a particular mass) in combination with maximum P(?j), maximum probability thatno random disturbance will cause transition to another steady state. If 71 is the para-meter to be optimalized, the analyst should know the dependence P(71) as well as


the dependence A(q) which expresses the relation between the desired property ofthe system and the parameter rI. The optimum value of n can be obtained from theconditions that the function

0(iI) _ [P(n)]ki [A(n)]"` (7.6, 2)

should be at its maximum. Exponents k, n express the weights of the respective func-tions; by their means the analyst can lay the stress on the requirement of max P(,q)or max A(rI). If A(,q) is to be very low, the exponent n must be negative.

7.7. Examples

Example I. Consider a one-degree-of-freedom system whose parameter undergoesa sudden change (Fig. 7.1, 2a). The examination is to show whether or not this effectresults in a qualitative change of the steady solution (for example, the change of theresonant to the non-resonant solution). The mass m rests against a spring with abroken-line characteristic of the restoring force F(y) and is excited by a harmonicforce with an amplitude Q. The mass is also acted on by a constant force Q0; the damp-ing is linear viscous, with the coefficient of proportionality X.

The motion of the system is described by the differential equation

my + -y* + F(y) = Q0 + Q cos wt (7.7, 1)

wherecy

F(y) = cy + cl(y - a)cy+ci(y+a)


for jyj a (a'> 0)

for y > a

for y<-a.

%Q=D, o=q, yoQ ux (7.7,2)

and the transformation Qt = z, equation (7.7, 1) can be rearranged to the dimension-less form

x" + Dx' + x[1 + 99(x)] = q + cos ilr (7.7, 3)

where

?'I99(x)J0 for

Ix, < a-,YO

c 1 a 1for

1xI> a

c yo Ix1 YO

The solution was obtained for the following values of the parameters:

D=0.05, c1=2, a=10,c yo

'n = 1.15. (7.7, 4)

The prestress was varied (by a sudden change) within the range

-2<q1<6, -2<q11<6 (7.7, 5)

7.7. Examples 237

By way of preliminary information, Figs. 7.7, 1 to 7.7, 4 show the amplitude-fre-quency characteristics for various values of prestress q. It is readily seen that

[max (x)]4 = -[min (x)]_Q (7.7, 6)

The interval of n in which two stable steady solutions (one resonant, the other non-resonant) exist is found to grow narrower with increasing q. In the range of q conside-red here the value ri = 1.15 always lies in the region of the two-valued stable steadysolution.

x

15

10

5

0

-5

-10

-15

Fig. 7.7, 1

Additional information may be obtained by consulting Figs. 7.7, 5 and 7.7, 6, whichshow the diagrams of the domains of attraction for various values of q and z = 0.The region of the initial conditions which lead to the resonant solution can be seen togrow larger with increasing (in negative as well as positive sense) prestress.

The next figures show the diagrams of the function p(q1, qII) indicating the prob-ability that the resonant (Fig. 7.7, 7) and the non-resonant (Fig. 7.7, 8) steadyvibration will lead again to resonant (non-resonant) steady solution in consequenceof a step change of the prestress (from qI to q11) at any time. Since F(-y) = -F(y),i.e. the nonlinear characteristic is symmetric,

p(-q1, -qn) = p(q1, qn)Using the simplest possible function of the probability density of occurrence

/(qj, qII) = (q2 - q1)2for qI, qII inside the interval (q q2) ,

(7.7, 7)

(7.7, 8)

0 for qI, qII outside the interval (q1, q2) ,


0.50 0.75 \ 1.00 1.25 1.50 71

Fig. 7.7, 2

Fig. 7.7, 3

7.7. Examples 239

that is, assigning the same weight to any step change for ql < qI < q2, qi < q11 < q2,we obtain the following values for the probability P:

Pres - res = 0.858 , Pnonres --> nonres = 0.965

The conclusion which can be drawn on the basis of these results is as follows:In systems described by (7.7, 3) and having parameters as indicated above, the

probability of step changes of prestress which satisfy the assumptions made inSection 7.1, of changing resonant vibration to non-resonant and vice versa, is verylow. Consequently, such systems are highly resistant to step changes of prestress. Inthe event that a qualitative change of steady vibration does occur, it is very difficultto return the system to the original state just by application of additional changes ofprestress.

Example II. This example shows the effect of a change of a parameter applied fora definite time interval only (Fig. 7.1, 2b). The system to be analyzed is a verysimple model of that in which relative dry friction is the cause of self-excitation. Arigid body of weight m rests on a continuous conveyor belt moving at velocity v(Fig. 7.7, 9). The body is bound by a linear spring of rigidity c and its motion iscontrolled by means of an absolute damper with damping of the dry friction character(idealized by Coulomb friction). Since the dry-friction absolute damper has a stabi-lizing effect, a stable equilibrium position always exists. (Rather than by a singlevalue of the deflection coordinate, this position is described by an interval of values -for details see a monograph by TONDL (1970b)). However, steady locally stableself-excited vibration is apt to exist at a certain velocity v of the belt motion andrelative dry friction between the body of mass m and the belt; in the phase plane it is


Fig. 7.7, 5

represented by a stable limit cycle. In the case being considered there exists, in addi-tion, an unstable limit cycle which surrounds the steady equilibrium position andforms the separatrix; the separatrix marks out the region of the initial conditionswhich lead to the equilibrium position from that of the initial conditions which leadto steady self-excited vibration.

Analyze now the effect of a sudden disturbance characterized by a step change ofa parameter, which is in action for a certain time. Examine the effect of a suddenchange of the magnitude of the dry friction force at the point of contact of the bodyresting on the moving belt. This change can be caused, for example, by minuteparticles of a material which landed on the belt, was dragged by the belt's motionbetween the contact surfaces of the body mass in, remained there for some time andwas removed from the contact. surfaces by the belt's motion.

To simplify the analysis assume that only the magnitude of the friction force andnot the character of the dry friction undergoes a change in this process. The change

7.7. Examples

Fig. 7.7, 6

241

can be expressed by means of a coefficient, a multiple of the original value. Theequation of motion (not considering the step change of the friction force) is

mi-mgfT(v-z)+0sgnx=0 (7.7,9)

where g is the acceleration of gravity, f the coefficient of dry friction between bodyand belt, p(v - i) the function of the dependence of the relative friction, and 0 thecoefficient of Coulomb's friction of the absolute damper.


V= Q=Ym a=.and the time transformation Qt = r, we obtain the following equation

x"+x-F(V -x')+Ssgnx' =0 (7.7,10)16 Schmidt/Tondl

(0'9'9)8TL

2F3

L

a2aaj

aqucS4ttiqu4a;o

uopsBt3sanuI

'LUZ

;

7.7. Examples 243

The function F(V - x') is assumed to have the simple form (the decreasing slope ofthe dependence is important for self-excitation effects)

F(V - x') = fosgn (V - x') - /1(V - x') (7.7, 11)

where fo = 1, fl = 0.1 for the case being considered.Taking the step change of the friction force into consideration, the equation of

motion becomes

Fig. 7.7, 9

x"+x-[1+K(z)]F(V-x')+Ssgnx'=0 (7.7,12)

where

K(z) =J0 for r<1ri, 'r>V2

K = const for r1 < < z2 .

Assume that K can take values only in the interval -1 < K < 1 and that F(V)' 6;consequently, for K = 0, vibration represented by a stable cycle in the phase planeexists in addition to the stable equilibrium position.

Examine now the effect of a step change of the relative dry friction on the equili-brium position. For purposes of information, Fig. 7.7, 10 shows the records of thelimit cycles in the phase plane for various values of 6 and V = 4. The larger (stable)limit cycle is seen to surround the unstable one.

Fig. 7.7, 11 shows the boundaries of the domain of attraction for the non-oscillatorysolution (equilibrium position) in the case of V = 4 and S = 0.02 and 0.03 in a dia-gram having coordinates K, To/T where T = 2 is the period of self-excited vibration.The boundary K(To/T) can be seen to have local minima in the neighbourhood ofTo/T = n + 1/2 and maxima in the neighbourhood of To/T = n (n = 0, 1, 2, ...).The resistance to disturbances of the type specified increases very rapidly withincreasing S. In the often-quoted monograph by TONDL (1973 a), which discusses thisand other cases in greater detail, the diagrams shown in Fig. 7.7, 11 are additionallyprocessed by specifying the density of the probability of occurrence of disturbancesin the form of rectangular areas; points lying inside these areas represent the valuesof the disturbance parameters having the same probability of occurrence.

Assume now that prior to a change of a parameter, the system vibrated in steadyself-excited vibration. The results obtained in terms of the probability functionp(K, To/T) are shown in Fig. 7.7, 12 (3 = 0.2) and Fig. 7.7, 13 (6 = 0.3) (V = 4 inboth cases). Unlike the non-oscillatory solution, the resistance of the steady self-excited vibration to disturbances decreases with increasing b. For small values of 6,the probability function p(K, To/T) was equal to one in the whole range of K, To/T.As the results suggest, a disturbance caused by a sudden increase of the force of relativedry friction lasting a certain time cannot result in transition to the steady equilibriumposition. Such transition can be achieved for larger values of S by application of a dis-turbance caused by a sudden decrease of the force of relative dry friction lasting16


-2.5 2.5

x

Fig. 7.7, 10

5=0.06

x

7.7. Examples 245

5 =0.02

K

0.5

-0.5

1 2 3 4T 5

ITFig. 7.7, 11

Fig. 7.7, 12

1

K

0.5

0

-0.5

1

6=0.03

2 3 4

TYT

5

a certain time. For larger values of S and longer time of duration (larger T°1T) a compa-ratively small decrease of the relative dry friction can result in transition to the steadyequilibrium position. The effect of the duration of disturbance, T°, is also of interest.Compared with the case when the system is in equilibrium position prior to the appli-cation of the disturbance, it is less marked and decreases comparatively rapidly withincreasing T/To; consequently, for larger values of 6, the function p(K, T°/T) afterthree to five periods (T°1T = 3 to 5) becomes largely independent of To/T and takesthe value of unity or zero. For p = 1 and p = 0 and increasing To/T, the edges of the


Fig. 7.7, 13

body representing the function p(K, To/T) have the shape of a rapidly attenuatingdamped natural vibration. They are shifted half a period, that is, To/T = 1/2, againstone another. The upper edge of the body (for p = 1) corresponds somewhat to thediagram shown in Fig. 7.7, 11; the lower edge is displaced roughly by To/T = 1/2.

The findings revealed by simultaneous evaluation of the resistance of two solutions,one non-oscillatory, the other oscillatory can be summed up as follows: For smallvalues of 6, the change of the non-oscillatory to the oscillatory solution caused bydisturbances of the type discussed is apt to take place more readily than that of theoscillatory to the non-oscillatory solution. Analyses such as that presented above canexplain the irreversibility of certain phenomena occurring in practice.

Additional examples concerning the effect of disturbances of the type of forcepulses will be analyzed in the chapters which follow.

8. Analysis of some excited systems

8.1. Duffing system with a softening characteristic

Consider a Duffing system which differs from the conventional model (Section 4.4) bya negative coefficient of the cubic term. An analysis of this system is interesting fromphysical as well as methodological aspects. It will be shown later in the section thatsome methods of analysis fail to provide a complete answer to the problems of stability(both in the small and in the large) of the steady solutions. This fact was pointed outin two papers by TONDL (1976c).

The system being examined is described by the equation (written in dimensionlessform and including the shift of the time origin)

y"+xy'+y-yy3=cos(nt+92) (8.1,1)

where x, y are positive constants, and (p is the phase shift angle. If the steady solutionis sought in the form

y=Acosqr (8.1,2)

use of the harmonic balance method leads to the following equations for determining Aand 92:

A(1 - ayA2-r2)=cos99(8-1,3)

Axri = sin 97 .

The dependence A(rk) is obtained by help of the inverse function n(A) which is readilydetermined from the equation

71 4 - 2 1 - 4 yA2 - 2x2 712 + 1 - 4 yA2 - A2 = 0. (8.1, 4)

The dependence 92(,q) is described by the equation

tan 9' = xrn(1 - ; yA2 - rig)-1

The backbone curve is defined by the equation

2)112.A = 2 (1 -. (8 1 6)s , . ,

V3y

this equation implies the necessity of satisfying the inequality rl < 1. The limit envelopehas the form of the rectangular hyperbola

AL = 1 . (8-1,7)x'n

248 8. Analysis of some excited systems

The following three cases can occur:

(a) The limit envelope and the backbone curve intersect at two points.(b) The limit envelope touches the backbone curve.(c) The limit envelope and the backbone curve do not intersect.

The case (b) constitutes the boundary between the cases (a) and (c). The curve A(ri)of case (a) consists of two branches. The lower branch has the usual form of resonancecurve of a system with a softening characteristic for which it is typical that A at anypoint is smaller than a particular value. The lower branch is chatacterized by that Aat any point is always greater than this particular value (Fig. 8.1, 1).

A

A5

AL

Fig. 8.1, 1?l

A

As

AL

Fig. 8.1, 2n

The curve A(,q) of case (c) also consists of two branches, one lying to the right, theother to the left of the backbone curve (Fig. 8.1, 2).

The same curves are obtained when using the van der Pol or the Krylov and Bogo-ljubov method. Stability of the solution is established by application of the rule ofvertical tangents (see Chapter 2). This procedure, however, fails to provide a compre-hensive picture of the solution stability.

If y = A cos TIT + x, where x is a variation of the variable y, is substituted for y in(8.1, 1) the equation in variations takes the form

x"+xx'+x- zyA2(1+cos2,nr)x=0. (8.1,8)

This is the Mathieu equation. The solution on the boundary of the region of first-order instability may be approximated by

x=uCOS71-C+vsinlT. (8.1,9)

Substituting (8.1, 9) in (8.1, 8) and comparing the coefficients of cos 17-C and sin'qTleads to a system of homogeneous equations in u and v. As the condition of non-triviality of the solution implies,

A = 2 12(1 _ n2) + [(1 - ,72)2 _3x 1y2]1/2)1/2 (8.1, 10)3 Vy

This establishes the boundary of the region of first-order instability of the steadysolution (8.1, 2), which corresponds to the approximate boundary of the region of first-

8.1. Duffing system with a softening characteristic 249

order instability of (8.1, 8). It should, of course, be noted that other instability regionsare also likely to play a role. In the case being examined a major influence is exercisedby the region of zero-order instability (the solution on the boundary of which is theMathieu equation of zero order) which comes into consideration only for negative valuesof the constant term of the coefficient of x in (8.1, 8). The boundary can be approxima-tely determined from the condition

i - Z yA2 > 0 . (8.1, 11)

In the chart of the regions of instability of the Mathieu equation this approximationcorresponds to the replacement of the boundary of the region of zero-order instability bya vertical straight line passing through the origin. The same condition is obtainedwhen checking the stability of the solution of (8.1, 8) for the average values of thecoefficients. The characteristic equation thus arrived at takes the form

A2+xA+1 - yA2=0.As subsequent examples will show, the boundary implied by (8.1, 10) yields the

same result as the rule of vertical tangents. However, the boundary implied by thecondition (8.1, 11) cannot be obtained when using the van der Pol or the Krylov andBogoljubov method. Figs. 8.1, 3 to 8.1, 5 show A as a function of ri for y = 0.01 andx = 0.2 (Fig. 8.1, 3), x = 0.175 (Fig. 8.1, 4) and x = 0.15 (Fig. 8.1, 5). The stablesolutions are drawn in heavy solid lines, the unstable ones in dashed lines, and thevarious regions of instability (obtained using (8.1, 10) and (8.1, 11) are shown cross-hatched or dotted. Fig. 8.1, 3 also shows the backbone curve (the dot-and-dash line)and the limit envelope (the light solid line). The arrows in Figs. 8.1, 4 and 8.1, 5indicate the transient (jump) phenomenon arising as the excitation frequency is slowlyincreased or decreased. As Fig. 8.1, 5 reveals, if the excitation frequency is decreased

15

10

0 0.5 1

Fig. 8.1, 3


15, .........................

.............................A I'. .. .'. ..

10

0 0.5

Fig. 8.1, 4

1

rL

1.5

:1:

1

A

10

Fig. 8.1, 5

0.5 1

It1.5

at a definite value of q, the steady solution with resonant amplitude loses stability anddivergent vibration is initiated.

These results were checked and confirmed by analogue solutions. Fig. 8.1, 6 showsthe extreme deflection [y] as a function of tj. The curve drawn in light line correspondsto the case shown in Fig. 8.1, 4, that shown in heavy line, to the case in Fig. 8.1, 5.


10

[y]8

0

Fig. 8.1, 6

divergent vibration

0.5 1

In the investigation of stability in the large which follows the approach using thevan der Pol (or the Krylov and Bogoljubov) method will first be shown to yield incor-rect results in the case being examined. Since they are identical with (4.4, 2) exceptfor the minus sign of the coefficient y, the corresponding transformed differentialequations will not be repeated here. The results obtained in investigations of the do-mains of attraction fot y = 0.01 and x = 0.175 are shown in Fig. 8.1, 7 (iq = 0.5)and Fig. 8.1, 8 (71 = 0.8). In the first case, the resonant solution corresponds to a pointof the upper branch, in the second, to a point of the lower branch. As a stability analysis(see condition (8.1, 11)) and an analogue solution reveal, the resonant solution of thefirst case is unstable rather than stable as implied by Fig. 8.1, 7.

-15 -10 5

a10 15

Fig. 8.1, 7


15

b10

5

0

-5

-10

FS

SPFS4

Fig. 8.1, 8

10

-10 -5 0 5 10- a 15

The domains of attraction expressed in terms of the coordinates of the initial condi-tions were obtained by a direct analogue solution of (8.1, 1) in the course of which theinitial conditions y(0), y'(0) were varied in the interval (-15, 15) by means of the TVscanning method. In some cases the equation was solved for the excitation cos rlt aswell as for the alternatives -sin qz, -cos 97T and sin riz, applying the initial conditionsat time z = 0. Depending on the initial conditions, the transient solutions could con-verge to the following steady solutions: non-resonant, resonant and divergent. Fig.8.1, 9 shows the domains of attraction obtained for y == 0.01, x = 0.175 and variousvalues of q (the cos ri's excitation). Fig. 8.1, 10 shows the domains of attraction foridentical values of y and x, for il = 0.82 and four types of excitation.

In another examination the system was subjected to a pulse force disturbance appliedin combination with the harmonic excitation. This disturbance had the form of adecaying oscillatory pulse (see Fig. 7.1, 1) generated by means of an analogue circuitdescribed by the equation

u'+2DQu'+Q2u=0. (8.1, 12)

If the initial conditions u(0) = U, u'(0) = 0 are applied to this circuit at time z = 0,the disturbance pulse is P(a) = -u'(r)/S2. It is typical of this circuit that for constant Uthe value of P(z)max does not vary with S2. The value of the coefficient D is chosen soas to make Pmax = 2 U = Po (Po is the amplitude of the disturbance pulse). Owingto the effect of D, the interval between r = 0 and the instant of the first zero crossingof P(z) is slightly longer than To/2 where T. = 27r/S2. Theoretically, the length of thispulse duration is unlimited. The characteristic parameters which are varied, are P0(Po = Pmax) and To/T where 27r/T is the natural frequency of the linearized systemwithout damping. Since a linear variation of To is preferred, Q must be varied accordingto a rectangular hyperbola relation. The value of To/T was varied in the interval

8.1. Duffing system with a softening characteristic

15

0

-5

-10

-15 -10 -5 0 5 10 151y(0)

77.0 .

vibrations:

non-resonantI

resonant divergent

0.75..

263

Fig. 8.1, 9

254

15

10

5

0

-5

-10

-15

8. Analysis of some excited systems

0.2cos r)z; .

vibrations:

non - resonant

Fig. 8.1, 1001I

sin riz

resonant 0 divergent

(0.4, 8), and the value of PO in the interval (0,5). To show the effect of the instant ofthe pulse application, the problem was solved for the following four alternatives ofharmonic excitation: cos rat, -sin fit, -cos rit and sin fit. The corresponding diagramswere obtained for y = 0.01, x = 0.175 and n = 0.75. The domains of attraction arerepresented in plane diagrams having the coordinates Po, T0/T. Fig. 8.1, 11 shows theresults obtained for the case of the system vibrating in non-resonant vibration beforethe application of the pulse. Fig. 8.1, 12 depicts the case of resonant vibration. Thenon-resonant vibration is seen to be much more resistant to disturbing pulses than theresonant vibration. The effect of the phase shift between the excitation and the instantof the disturbance application is more distinct in the latter case.

The most important results obtained in this section can be summarized as follows:

For systems having a softening characteristic of the restoring force the classicalmethods for establishing the domains of attraction, which use approximate first-orderdifferential equations based on the assumption of slowly varying amplitudes (or a


Tl= 0.75

0 4

5

A

0 4

vibrations;0 non-resonant

cos rlt

T°/T

T°/T

III

8

8

1resonant

0

0

0

-sinrtt

4 8

'-T°/T

4

sin r1t

T./T

divergent

Fig. 8.1, 11

8

slowly varying amplitude and phase), do not yield results that are quantitatively oreven qualitatively correct. Neither do they disclose the existence of divergent vibra-

tion.The domains of attraction expressed in terms of the coordinates of the initial condi-

tions differ substantially from those obtained in investigations relating to the resistanceof a particular steady locally stable solution to disturbances of a specified type.

Three domains of attraction, that is, that of resonant, of non-resonant and of diver-gent vibration, can be obtained when a decaying oscillatory pulse is applied to theDuffing system having a softening characteristic. Under its effect the steady resonant


TI= 0.75

cos T1

0 4

T,/T

-cos'r1t

A

0 4

vibrations :

non resonant

Fig. 8.1, 12

To/T

I

8

A

8 0

resonantI 04

divergent

sin 'qt

To/T8

vibration is apt to change more readily to the divergent than to the non-resonant

vibration.

8.2. Some special cases of kinematic (inertial) excitation

This section is devoted to an analysis of systems I and II considered in Section 1.2(Fig. 1.2, 6). Equation (1.2, 5) which describes the motion of system I is not only non-homogeneous but - as in the case of parametric excitation - also has periodically

8.2. Some special cases of kinematic (inertial) excitation 257

variable coefficients. It is woth noting that parametric excitation can be linear as wellas non-linear. Introducing the dimensionless deflection y = x/a and carrying out thetime transformation

Sgt=T (8.2,1)

where Q2 = 2k/m, (1.2, 5) takes the dimensionless form

y" + 28y' + y + eiy3 + e2(y -- cos ri-r)3 = cos pp - 671 sin rib (8.2, 2)

where_ x _ w yla2 y2a2

mil, 77

S2, E1

mQ2, 82

ms22

The following alternatives are studied:

(a) El=e2=E;(b) el -El -E' combinations of a softening and a hardening spring(C) E1=-2E2=-e

The fundamental analysis is made by means of the harmonic balance method; tofacilitate the formal calculation, the time shift O/r7 is introduced where 0 is the phaseshift between response and excitation.

For alternative (a), (8.2, 2) becomes

y" + 26y' + y + e{y3 + [y - cos (77t + )]3}= z cos (77Z + ) - 877 sin (77z + 0) ; (8.2, 3)

the stationary solution of (8.2, 3) can be approximated by

y = A cos 77i . (8.2, 4)

Application of the harmonic balance method leads to the following equations forobtaining A and 0:

A[1 - 772 + a E1 + A2) + ; e(cos 20 - 3A cos 0)]_ (2 -E ; s) cos 0 - 877 sin 0 , (8.2, 5)

A[2677 - ; e(A sinO - sin 2i)] _ (a + 3 e) sin 0 + b77 cos 0 . (8.2, 6

In the calculation, (8.2, 5) and (8.2, 6) are used for determining the function 77(O) forvarious values of A ; the points of intersection of the curves of the two sets yield thevalues of 77 and 0 corresponding to the gradually varied A. Equations (8.2, 5) and(8.2, 6) rearranged for the !purpose of this calculation have the form

2\1 +

3EICos0

271

- b sA71

+ \ 2 4A - 1 - 4 e [2(1 + A) + cos 20

-3Acos0]=0, (8.2,7)

b(2Asin 0s 0) 2 + 4 E(1 -}- A2 - 2A cos 0) . (8.2, 8)

Fig. 8.2, 1 shows the calculated relation between A and 77 for b = 0.05 and severalvalues of e; Fig. 8.2, 2 shows the corresponding functions O(77). It can be seen that the

17 Schmidt/Tondl


results obtained for alternative (a) are not qualitatively different from those establishedfor the model shown in Fig. 1.2, 5.

10

A

5

0.50

r =-0.01 -0.005 0 0.005 0.01

i 1 1

i 1

Fig. 8.2, 1

1 I 1.5

E =-0.01

90

00.5

Fig. 8.2, 2

=0 \\F 0.01

1

Different results, however, are obtained for alternative (b) (i.e. for elFor this alternative the rearranged equations of motion takes the form

y" + 25y' + e{3y2 cos (p + 0) + 2 y[1 + cos 2(71a + O)]}

= (a + ; e) cos (pi + 0) - on sin 0) + 4 e cos 3(ni + 0) . (8.2, 9)

Proceeding as before, the equations for obtaining A and 0 become

I1+3elcos071

2-5s 0 + 2 A -1 - 4e(2+cos20-3Acos0)=0,(8.2, 10)

sin 0 1 + 3 e(1 + A2 - 2A cos 0) . (8.2, 11)5(2A - cos 0) 2 4

The curves A(ri) and O(rb) for 5 = 0.05 are shown in Figs. 8.2, 3 and 8.2, 4. Theunstable solutions are drawn in dashed lines assuming the applicability of the rule ofvertical tangents.

To illustrate the effect of e, Fig. 8.2, 5 shows the axonometric view of the responsecurve A(rj) (without indicating the stability of the solution). The assumption concerningthe solution stability will be checked for correctness in a later analysis.

For the case of e > 0, that is, excitation via a hardening spring: they differ qualita-tively from both those of (a). As will be shown later, there exists no stationary solutionfor larger values of e > 0 in a certain range of iq - 1, and the transient solutions repre-sent divergent vibration for any initial conditions. For smaller values of e > 0, thereexist two domains of the initial conditions: one for which all solutions converge to astable stationary solution with a finite vibration amplitude, and another (lying outsidethe former) which leads to divergent vibration.

8.2. Some special cases of kinematic (inertial) excitation

20

A

15

10

5

0

l1 0.0061 Y j I1

I

I I

j i II

I I1

0.01III i I~ 0.01

I I

I

I I

I I

0.075 I1 I

I III

0.006

0.005

0

-0.005

-0.001j0.5

Fig. 8.2, 3'2 1.5

2,59

These results were confirmed both by an analysis using the van der Pol method, andanalogue solutions. The solution to (8.2, 2) is now sought in the form

y=ucos71x+vsin77t (8.2,12)

where u and v are slowly varying functions of time for a transient solution. Applicationof the well-known procedure leads to the following equations

u' = I l -817(2u - 1) + I1 - 2 - 3 s(2u - 1)1 v1,J\

V,

= \ 1 /2 - (1 - n2) u + 4 e[u(u - 1) + 1 + v2]} .

J

1(8.2.13)

The singular points of the system represent stationary solutions which are identicalwith those obtained by means of the harmonic balance method recalling that therelations

u=Acos0, v = A sin Papply. The roots of the equation

au au-au av

av ay

au av

=0

(8.2, 14)

(8.2, 15)

17


180°

4)

90

0'°0.5

e =-0,051800

4)

90

0L--- _,1 1.5 0.5 1

1800

4)1800

4)

90 90

0 0

C =-0.01

s=0.01

i

d(A=201

1 1.50.5 1 1.5 0.5

where U(u, v) and V(u, v) are the right-hand sides of (8.2, 13) for the arguments ofthe coordinates of the singular points, define the stability and type of a singular point.Substitution in (8.2, 15) results in

[21(6 + 2)12 - (a Ev)2 + [1 - 12 - 4 e(2u - 1)]2 = 0 .

As this equation implies, satisfying the inequality

11 - 12 - 4 8(2u- 1)]2 - (ev)2 +(26n)2 > 0 (8.2, 16)a

is the conditition of stability for a singular point (assuming that 6 > 0).If the condition

[1 -12 a s(2u - 1)]2 < (2 -ev)2 (8.2, 17)

E=0.0075

p(A=20)

i

6(A=20)


Fig. 8.2, 5

is satisfied, the singular point is a saddle or a node; if it is not satisfied, the singularpoint is a focus. Since, in the case of equality, condition (8.2, 16) is a boundary betweensaddles and nodes, the singular point - if conditions (8.2, 16) and (8.2, 17) are satisfied- is a stable node. If condition (8.2,17) is not satisfied, the singular point is a stablefocus because inequality (8.2, 16) is always satisfied. Fig. 8.2, 6 shows the boundariesof the domain of stability, the types and positions of the singular points in the(u, v)-plane for the case corresponding to Figs. 8.2,3 and 8.2, 4 for e = 0.0075; (a) rep-resents the case for 91 = 0.95, (b) that for j = 1.05, and (c) that for 71 = 1, e = 0.006.

A solution of (8.2, 13) yields the phase portraits which provide information about thesolution stability for any initial conditions, and, if several singular points exist, alsothe domains of attraction - in the case being discussed, the domain of attraction ofthe stable singular point and the domain of divergent vibration. Some examples ofthe phase portraits are shown in Fig. 8.2, 7: (a) to (c) are drawn for e = 0.0075 and71 = 0.95, 1 and 1.05; (d) for e = 0.006 and q = 1 (8 = 0.05 in all cases).

The separatrix (drawn in heavy full line) is formed by two trajectories entering thesaddle point (SP) and divides the domain of attraction of the stable focus (Fs) or thestable node (Ns) from the domain of divergent vibration. Since only divergent vibrationexists for the whole range of values of u and v corresponding to the case shown inFig. 8.2, 7 c, the earlier conclusions are fully confirmed.

Fig. 8.2, 8 shows the maximum deflection ([y]) as a function of frequency q for thecase of 6 = 0.05 and e = 0.0075 (in the vicinity of 71 = 1, the vibration becomes diver-


-20a)

20

0 U

SP0

ONS

20 -20b)

-20 0 u 20

c)

Fig. 8.2, 6

0 U 20

gent) and of e = 0.006 obtained in an analogue solution of (8.2, 3). Fig. 8.2, 9 showsvibration records for the case of s = 0.0075 and two different values of ri. As the firstrecord reveals, at 71 = 1 the vibration becomes divergent even for zero initial conditions.The other two records correspond to initial conditions close to the separatrix.

For alternative (c) (r = -62/2 = -e) the A(71) curves corresponding to 6 = 0.05and several values of a are shown in Fig. 8.2, 10. As e is increased, the maximum valuesof A grow larger at a fast rate, and after a certain positive value of a has been exceeded,the resonance peak is no longer well defined. In fact, as A grows larger, the two branchesof the A(i) curve move ever further apart. Starting from a certain value of 71, twostationary stable solutions, one resonant, the other non-resonant, exist for every 11 > 1.

Applying the procedure and notation adopted for system I, the equation of motion(1.2, 6) which describes system II can be converted (for the alternative el = -e2 = -e)to the following dimensionless form :

J + 26y' + y + a e[(4y2 + 1) sin 99 sin i7T - 2y sin 29sin 21)x]

= cos cp (cos i7z - 281 sin rir) -- a e sin 3q sin 31lT. (8.2, 18)


The case when c = 0 (the motions of the suspension points of the two springs are inphase) will not be discussed here because it can readily be dealt with by introducingthe relative deflection or, alternatively, by solving (8.2, 18) which, for q? = 0, turns outto be a simple, non-linear non-homogeneous differential equation. If the motions of thesuspension points are in opposition (T = 7c/2, sin (p = 1, cos 4p = 0), there exist onlya very small external excitation with frequencies q and 3,q and a linear and a non-linear parametric excitation with frequency q; of these, the latter has the greatesteffect.

200.95

V

15F = 0.0075

10SP

5

U

-15 -10 -5 5 10 15

-5

-10

-15

a)

-10

-15

I 1 1 -20

b)


20

v =1.05

15

SP10

s 5

-15 -0 5 5 10 15 u

10

15

20

v n=11 5 S E=0.006,

10'

F.

5

5

-15 10 5 5 10 15 u

-5

-10

-15

-20

c)

0,6 1 1.4 1l

Fig. 8.2, 8


[y]

20

10

-10

-20

11

11=t05

y(O) = 14

y(0) = 0

Fig. 8.2, 9

[y]

20

y(0) = 0

j2(0) = 0

Applying the procedure adopted for system I, the equations for obtaining A and 0turn out to be

226C sin i C cos 3 ((3A 1 )ai 2A 17 + A - 1 -

4e sin S +

A- 4SC cos J

=0, (8.2, 19)

C sin 0 + ; E[S(A2 + 1) cos - 2SCA cos 20]} [26(A - C cos (P)]-i

(8.2, 20)

where S = sin ip, C = cos q'.


20

A

15

10

5

0

/

/

-0.005

0.004

0.0025

=0

0.00251-0.005

0.5

Fig. 8.2, 10

10

5

25 =0.075

10I

1.5

C = 0.01

2b=0.1

2

20

15

10

5

01 1 1 0,

0.5 1 'rt 1.5 as

Fig. 8.2, 11 Fig. 8.2, 121 Tt 1.5


The results obtained for the alternative q' _ 7r/4 are presented in Figs. 8.2, 11 and8.2, 12. The first figure shows the A(i?) curves for e = 0.01, the second for e = -0.01and two values of 26. The dependences O(q) corresponding to these resonance curvesare shown in Figs. 8.2, 13 and 8.2, 14. The results obtained for the alternative T = 7r/2,two different values of 28 and s = ±0.01 are presented in Figs. 8.2, 15 and 8.2, 16 (theA(q) curves are the same and the 0(ij) are merely shifted for the two values of T).The A(97) curve has two branches. One branch corresponds to the stable solution, withan amplitude so small as practically to coincide with the q axis in the diagram. Theother branch, which corresponds to the unstable solution (this is always a saddle pointfor any 71 in the (u, v) plane), is a curve whose minimum value of A is obtained for rJslightly less than 1.

cP = n/4

e=0.0125 = 0.075 -

0.1 -

(A=20)

3nJ2

(A=20)

<p =n/4

e = 0.01

25 = 0.075 -0.1

(A=20)

x/2

(A=20)

_N121 00.5

Fig. 8.2, 137Z 1.5 0.5 1 n 1.5

Fig. 8.2, 14

As in the case of system I, the solution of (8.2, 18) can be sought in the form (8.2, 12);for q = 7r/2 this leads to the equations

1U, =I

[-28riu + (1 - 712) v + 4 e(1 + u2 + 3v2)1,2,q)

V, -\2 /

L_2- (1 - ) u - 4 uvl.\/jj (8.2, 21)

It can be shown that for q 1, there always exist two singular points, one being astable focus or node (for n = 1), the other a saddle. Fig. 8.2, 17 shows the phase por-traits obtained for e = 0.01, 26 = 0.075 and three different values of 77 (0.9, 1, 1.1).The separatrix divides the (u, v) phase plane into two domains of attraction - thedomain of the stable singular point and the domain of divergent vibration. The vibra-tion records obtained by solving (8.2, 18) for c = ir/2, e = 0.01 and 26 = 0.075 (Fig.8.2, 18) confirm the results arrived at earlier. (Fig. 8.2, 18a represents the case of17 = 0.9, Fig. 8.2, 18b that of 71 = 1 and different initial conditions y(O) (indicated inthe diagrams; y(0) = 0 in all cases).


20

A

15

10

5

0.5

Fig. 8.2, 15

0

1 I 8=0.01' 25=0. 1

1I II1 1 1

1

I

I

250.075

1

To give a general idea of the effect of the phase shift qp, Fig. 8.2, 19 shows the A (ij)curves for e = 0.01, 26 = 0.075 and three different values of 4p. As mentioned earlier,the case of q = 0 is a commonly occurring external excitation corresponding to alter-native (a) of system I; the case of 99 = n/4 corresponds to alternative (b) of system I.For 49 = rc/2 when the external excitation is very small and a trivial solution is prac-tically the only stable steady solution, divergent vibration is also possible for n - 1.


20

NS

-20

a)

-20c)

Fig. 8.2, 17

0

0

u

u

'n =1

SP

20 -20 0 u 20b)

20

This means that for a definite value of n (71 - 1) there exist two domains of initialconditions - one leading to small, the other to divergent vibration.

The main results obtained in this section can be summed up as follows:

In special cases the response to kinematic (inertial) excitation differs substantially,both quantitatively and qualitatively, from the response to an excitation force actingon the mass or from that to a conventional kinematic excitation. Special cases of thissort are encountered, for example, in systems whose mass is mounted between two non-linear springs, one softening, the other hardening. When the kinematic excitationderives from the motion of the end of one of the springs, the special case occurs whenthe spring involved is the hardening one. When the suspension points of both springsperform harmonic motions, the special case arises as the two motions become shiftedin phase. The difference in the response grows particularly striking in systems havinga combination of non-linear springs for which a similar system under an externalexcitation acting directly on its mass would appear linear because the non-linearities ofthe two springs compensate one another. It should be stressed that the qualitativedifference may in fact become so great that no stable steady solution exists in a parti-cular interval of the excitation frequency and all unsteady solutions are divergent.


71=0.9 n=1y(0)=15 y(0)=12 y(0)=11

20

Y

0f

20

Y

0

20

AA

15

10

5

-20I

-20

b)

Fig. 8.2, 18

1

II

Ire=0.012b=0.075

1

Y n l2= \Sp=n14

ik

0

W =x /2

0.5

Fig. 8.2, 191.5

Although a domain of initial conditions leading to a stable steady solution with a finiteamplitude exists in some cases, the initial conditions outside this domain result indivergent vibration.

8.3. Parametric vibration of a mine cage

One of the examples mentioned in Section 1.2 in connection with parametric excitationreferred to a mine cage in which the stiffness corresponding to the restoring force fora lateral deflection of the cage varies periodically at a constant travel speed of the cage.The cage is carried along guide bars by means of cage guides spaced distance 1 apart;in the case being considered, to < l where to is the spacing of the bar supports as well asthe length of the guide bar. As Fig. 8.3, 1 (drawn horizontally for ease of clarity)shows, the restoring force acting on the cage guide as the guide bar deforms is definedby both the stiffness of the supports and the stiffness of the bar and depends on theposition of the cage guide, i.e. on the distance between the cage guide and the guide

1-10

1

x

10

i V

0

k=kp[1+f(,,)t)

W=Z 2x0

Fig. 8.3, 1

8.3. Parametric vibration of a mine cage 271

support. Assuming a constant speed of travel of the cage, v, and linear springs, thestiffness at the point at which the cage guide rests against the guide bar is defined by

k = ko[1 + f(wt)] (8.3, 1)

where w = i 2,r, ko is the mean stiffness and f (wt) is a periodic function with period0

2ic/w. Fig. 8.3, 2 shows the stiffness ratio k/ks (ks is the stiffness midway of the span)as a function of the cage guide position. The curves are drawn for several values ofthe kT/kso ratio; hT is the stiffness of the guide bar support and kso the stiffness ofthe guide bar on absolutely stiff supports midway of the span. The optimum value ofthe kT/kso ratio is 0.5. This value, however, does not correspond to the kT/kso ratio ofactual structures.

kks

0.5

Fig. 8.3, 2- M. X/10

1

To simplify subsequent considerations, replace f ((ot) by a harmonic function. Thestiffness of the restoring force which is acting on the first cage guide then becomes

kl = ko(1 + Iu cos wt) (8.3, 2)

and that of the restoring force acting on the next cage guide

k2 = ko[1 +,u cos (wt - V)] (8.3, 3)

where

w=2rc 1 - 10

to

In the derivation of the equations of motion of the cage it is assumed that the cagetravels at a uniform speed v in guides without clearance and that the centroid of thecage lies at its centre.

The cage has a mass m and a moment of inertia about the axis passing through thecentroid normal to the axis of travel I = mr2. The motion of the cage can be describedeither by a lateral deflection y and an angular deflection 0 (Fig. 8.3, 3) or by deflectionsyl and y2 of points 1 and 2. Assuming 0 to be small, these coordinates are related asfollows :

yi= 1sin0+y= 10+ya

1(0+u)(8.3,4)

y2= a l8-y= +where u = 2y/l.


Fig. 8.3, 3

The potential energy (ignoring the part of the potential energy which is impartedto the cage in the gravity field at angular deflection 0, i.e. V' _ mgl(1 - cos 0)) isdefined by

2V =4

l2ko{[2 +,u cos cot +,u cos((ot - s')] (02 + u2)

+ 2y[cos cot - cos (c)t - v')] Vu} . (8.3, 5)

The kinetic energy of the cage is

2

2T = r2m12 my2 = mI.

-I- u2 (8.3, 6)

If damping is not considered, use of the Lagrange equations of the second kind resultsin the following equations of motion

+ Q { [1 + ,u[cos cot + cos (cut - V)D 6 + 2 µ[cos not - cos (cut - ip)] u} = 0 ,zi + S22{ 11 + . 4u[cos cot + cos (cut - OR u + 21 ,u[cos cot - cos ((ot - 0] 6) = 0

(8.3, 7)

where S21 = (l/r)Vko/2m, 522 = 1/2ko/m are the mean natural frequencies of torsionaland lateral vibrations, respectively.

The instability intervals of the first kind and first order are described (in the firstapproximation) by the relations

2Q8 - a ,ua88 < co < 2Q8 + 2 ua88 (s = 1, 2) ; (8-3,8)

the instability interval of the second kind and first order is defined by the relation

1'Q3' Q2I - 2 it I ±a12a21 < w < I Q1 D2I -21- P I ±a12a21

where

(8.3, 9)

au=a22= 2Y2(1+cosa12=a21= 21/2(1 -cosy).2 2Since a12a21= a12 = a21 > 0, only the plus sign has a meaning in inequalities (8.3, 9).

For V = 0 + 2n, i.e. l/lo = n (n = 1, 2, ...), all = a22 == 1, an = a21 = 0, the intervalof the second kind does not exist.

For = (1 + 2n) 7s, i.e. l/lo = a n (n = 1, 2, ...), all = a22 = 0, a12 = a21 = 1,that is, the intervals of the first kind disappear.

On the basis of these findings one can draw the important conclusion that only onekind of instability interval can be quenched by altering the pitch of the cage guides.

The width of all instability intervals can be affected by changing the value of the


coefficient It, that is, by reducing the variability of the stiffness of the restoring forces.This can be done by mounting the cage guides elastically on, for example, flexiblerubber elements having a stiffness k2. For the arrangement shown schematically inFig. 8.3, 4 the total stiffness K is then described by

1 _ 1 + 1 _ k2 + k°(1 + ,u cos cot)

K k2 k°(1 + p cos cot) k°k2(1 + It cos wt)

lia7

Fig. 8.3, 4

777P71

k=k°(1+ucosc')t )

Assuming It < 1, one obtains from the above equation the approximate relation

K = k2k°(1 u cos wt) k2k°(1 + k0 ,u cos wt\ (8.3, 10)

k2+k0(1+,ucoswt) k2+k° k2+k° /which implies that the effect of the stiffness variability is the smaller, the lesser is theratio k2/(k2 + k°). In other words, the use of soft-mounted cage guides is advantageous.

It has been assumed so far that the cage moves in its guides without clearance. Toestimate, at least qualitatively, the effect of the cage-guides clearance, consider asimplified case when a12 = a21 = 0 and system (8.3, 7) decomposes into twoindependentdifferential equations of the same type. This means that the lateral and the torsionalvibrations are not bound one to another. Either equation can be rearranged (using thetime transformation Q,t = x and the notation q = (o/Qg, w = 1 /0° or w = u/u° where0° and u° denote the values of the coordinates for which the clearance is taken up) tothe dimensionless form

where

w"+ f(w-1)(1+,ucosqz)=0 (8.3,11)

0 for IwI < 1

f(wv-1)= w-1 for w>1w + 1 for w < -1

Note that this is a case of non-linear parametric exittation. To obtain a limited steadysolution for any q, assume the damping to be linear and progressive and write thedifferential equation of motion in the form

w"+(b+bw2)w'+f(w-1) (1 +It cosp)=0 (8.3,12)

where b is the coefficient of linear damping and S that of the progressive one. Both band 8 are assumed to be small compared to unity. Approximate the solution at mainparametric resonance by

w=Acos(2qr+q) (8.3,13)

18 Schmidt/Tondl


where A, q' are coefficients slowly varying in time for unsteady solutions. Accordingly,(8.3, 12) can be converted to a set of two first-order differential equations

A' _ (1) [-

2

n(b+ 46A2)+ 2IuQ sin 2991 A,

=\ 2n)2+Q(1-}

,ucos292/J17 7_

where

1

for A>1 ,0 for

The steady solution is obtained from the equations

,a n(b + ; 3A2) = a fuQ(A) sin 2T

(2 X7)2 - Q(A) u Q(A) cos 2p . }

Eliminating (p leads to the equation

Q=I cos1

A

- AA2 1

J=-

( tan 1 VA2 - 1 - AA21 /

A<1.

(8.3, 14)

(8.3, 15)

(4 2)1/2 = Q - 2 (b + 4 SA2)2 14 1u2Q2 - Q(b 4 6A2)2 + 4 (b + 4 oA2)4J1/2

(8.3, 16)

from which q(A) and thus also A(q) can be determined. From the equation

\b + I bA2) ?tan 299 - - 2

()-Qor the equation

(b + 46A2

tan T2

(T-2n) -(1

2 P)Q

(8.3, 17)

one can establish the relationship 92(n). The backbone curve is described by the equa-tion

n = 2 VQ(A5)

and the limit envelope by the equation

,uQ(AL)27

b+46AL

(8.3, 18)

(8.3, 19)

As shown by TONDL (1976b), the system being analyzed belongs to the class ofsystems to which applies the rule of vertical tangents, that is, the points of the reso-nance curve at which the tangents are vertical form the boundaries between stable


and unstable solutions. The author has also shown that the effect of non-linear para-metric excitation causes subharmonic resonances of order 1/N (N = 2, 3, ...) to existbesides parametric resonances of order N.

Fig. 8.3, 5 shows the resonance curve (A (,q)), the backbone curve (A 5(n)) and the limitenvelope (AL(q)) drawn for the case of b = 0, 6 = 0.01 and It = 0.4. In contrast withthe case of linear parametric excitation when the equilibrium position is unstable ina certain interval of the excitation frequency, the equilibrium position is found to be

10

A

A

0- 1

Fig. 8.3, 5

2 3

stable over the whole range of q; however, in the intervals in which parametric reso-nance occurs, the equilibrium position is only locally stable. The parametric resonanceshown in the figure is larger than that which occurs in the case of linear parametricexcitation and extends, theoretically, from zero value of the excitation frequency.Fig. 8.3 6 shows the results of analogue computations: the curve of the extreme deflec-tion [w] vs. the relative excitation frequency, and the vibration records obtained foril = 1.8 (A, B) and q = 3.2 (C) rae added to complete the information. It can clearly beseen that not only the main resonance of the first order (marked 1) but also the sub-harmonic resonances of order 1 /2 and 1 /3 (marked 1/2, 1/3) are present. The domainsof attraction shown in Fig. 8.3, 7 (for q = 1.4) and in Fig. 8.3, 8 (for q = 1.8) wereobtained by solving (8.3, 14).

As the theoretical analysis has revealed, only one type of parametric resonance canbe quenched by altering the pitch of the cage guides. A more effective means is aflexible mounting of the guides having a slight prestress which eliminates the unfavour-able effect of clearance, that is, the non-linear parametric excitation and in turn, thepossible occurrence of subharmonic resonances. Such an arrangement makes it possibleto raise the speed of cage travel.

The above study of the vibration pattern of a mine cage was made not just as anacademic excursion into the field of systems with non-linear parametric excitation;1S

R

m

0

B

amipp iiiiamI II H IIIllllllll iIIl01uUgIUIIIIIgmtllIIIIlI r

Fig. 8.3, 6

rl

,8 2

0

0

OR

VA

MA'

0

Fig. 8.3, 8

0

3 3,2

45

0135°

90 1135°'P. cp


rather it was to assist in finding the cause of failure of an actual structure. The collapsewas due to the effect of severe torsional vibrations of the cage which caused everysecond guide bar to break alternately on the two sides of the guide structure (Fig. 8.3, 9).

Fig. 8.3, 9

9. Quenching of self-excited vibration

9.1. Basic considerations and methods of solution

Since self-excited vibration impairs reliable operation and endangers the safety ofmiscellaneous machinery and structures, its suppression constitutes one of the majortasks of vibration engineering. An ideal and very expedient means to this end is theremoval of the source of self-excitation. However, this so-called active method is notapplicable in all cases. The systems in which it fails include those where self-excitationis an inherent characteristic of the technological process (for example, the cuttingforces in machine tools) or is inherent in the function of the device (for example, thehydrodynamic forces in journal bearings). Sometimes, as in the case of self-excitedoscillations - galloping - of high-voltage transmission lines, application of the activemethod is not feasible because of economic or operational reasons. In cases of thesesorts resort must be made to passive methods, that is, to paralyzing the destabilizingeffect of negative damping, which is obtained in the equations of motion when express-ing the action of forces producing self-excitation, by an increase in the level of positivedamping. This chapter deals exclusively with the passive methods of quenching self-excited vibration.

The practicability and efficacy of the various means used in connection with thesemethods will be examined using systems which belong to the class represented in itssimplest form by the van der Pol oscillator. Only systems with a finite number ofdegrees of freedom will be considered.

Let the portion of the system being examined, whose masses are acted on by self.excitation, be called the basic self-excited subsystem, or briefly, the basic system. Con-sider the self-excitation to have the classical form of that of the Van der Pol oscillatorthat is, the mass is acted on by damping having a negative linear viscous componentas well as a positive progressive component, which can be expressed as the product ofthe square of the corresponding deflection and its velocity. The study that followsconcerns two means of suppressing self-excited vibration, i.e. absorbers and resilientfoundations with damping or, more exactly, absorber and foundation subsystems.

Absorbers, a common and popular means of suppressing forced vibrations, are notfrequently used in self-excited systems. An analysis of their action in those systemswas published only recently. MANsouR (1972) was the first author to explain the basicproperties of absorbers as applied to a two-mass, two-degrees-of-freedom system (thesubsystem of the Van der Pol oscillator attached to the subsystem of an absorber). Ashe pointed out, an absorber is not always capable of compensating the self-excitationeffect represented by negative linear damping, and there exists an optimum value ofthe absorber's damping coefficient. These results were later extended by other authors,for example, TONDL (1975 a, b; 1976d; 1977), HAGIEDORN (1978), RowBOTTOM (1981).

9.1. Basic considerations and methods of solution 279

The principle of mounting on resilient or sprung foundations has been analyzed fordiscrete systems whose various masses perform only a lateral motion, as well as for rotorsystems. The literature relating to the first group of systems includes studies by TONDL(1975 a, b), (1976 d) and a comprehensive treatise including the action of absorberspublished in a book by TONDL (1980b). Of the fairly ample literature on the secondgroup, mention should be made of the work of TONDL (1965,1970c, 1971 a, 1980a), andKELZON and YAKOVLEV (1971 a, b).

Systems of the van der Pol type belong to the class of systems with soft self-excita-tion (provided that the action of forces (for example, friction) causing a qualitativechange in a system with hard self-excitation is not taken into consideration). Thepossibility of the occurrence of self-excited vibration is ascertained by means of ananalysis of the equilibrium position stability, that is, an analysis of the linearizedequations of motion written in terms of the coordinates of the deflections from theequilibrium position.

The conventional method of investigating the equilibrium position stability, whichinvolves an analysis of the roots of the characteristic equation using the Routh-Hur-witz criterion, has two disadvantages: it provides no more than an answer to thequestion of whether or not the system being examined is stable, and its calculationsgrow ever more time-consuming as the systems become more complicated. Consequent.ly, it is not capable of indiating either the natural mode with respect to which the systemis unstable or the frequency of the resulting self-excited vibration. Below is a descriptionof methods which are free of these disadvantages if certain assumptions are satisfied.It must be assumed that none of the roots of the characteristic equation is real andpositive (the characteristic equations of a vast majority of mechanical discrete systemswith n degrees of freedom has n complex roots). The exposition also refers to a methodof determining the dependence of the amplitudes of single-frequency vibration on aparticular parameter of the system. Although the method is approximate, it is moreexact than the conventional procedures because the solution is not approximated bythe natural modes of the abbreviated (undamped linearized) system.

The two methods to be described are both based on the assumption concerning theroots of the characteristic equation. For values of the parameters lying on the boundaryof the equilibrium position stability, one of these roots becomes imaginary.

The first method - named the boundary values method - is suitable for dealingwith fairly simple systems (having only a few degrees of freedom). It involves findinga set of values of two parameters of the system (for example, the coefficient of negativedamping P and that of positive linear damping x), lying on the stability boundary,where it holds for a root of the characteristic equation that A = iQ (Q is real and repre-sents the frequency of self-excited vibration initiated on this boundary. Substitutingfor A in the characteristic equation must cause the real and the imaginary parts of theequation to become zero. The two equations thus obtained, viz.

F1(Q;x,fl)=0, F2(Q;x,j9)=0 (9.1,1)

are polynomials in 9 and their coefficients are functions of parameters j9, X. The calcula-tions can generally proceed as follows: the value of one of the parameters (for example x)is varied in steps and the corresponding values of 9 and fl are obtained from the twoequations (9.1, 1). The two-mass systems to be analyzed by means of this method arecharacterized by the fact that one of (9.1,1) can be written in the form

Fl(, Q; K) = 0 (9.1, 2)

280 9. Quenching of self-excited vibration

where

K=x/3 (9.1, 3)

and the other in the form

F2(S2; K, x) = 0 . (9.1, 4)

For the two-mass systems being solved here, (9.1, 2) is quadratic in 9. This factpermits ready determination of the 9 - K dependence as well as the correspondingvalues of x ((9.1, 4)) and j9 ((9.1, 3)). Using a calculator with a graph plotter the curvefl(x) is drawn directly by increasing the parameter K in steps and calculating the cor-responding 92, j9, x in each step. Since two values of 92 exist for two-mass systems, wethus obtain a set of two curves, each representing the boundary of the stability regionfor one natural frequency. The boundary of the region of parameters, which is commonto both stability regions, is identical with that obtained by using the Routh-Hurwitzcriterion. It represents the region of the equilibrium position stability with respect toboth natural frequencies.

The second method has several features reminiscent of the procedure of determiningthe amplitude of single-frequency self-excited vibration as a function of a particularparameter of the system, for example, the tuning coefficient Q. Consider a system withN degrees of freedom whose motion is described in terms of the coordinates of the de-flections from the equilibrium position Xk (k = 1, 2, ... , N). The system is capable ofvibrating in n (n < N) modes of single-frequency vibration of frequencies Sts (s = 1,2) ... , n). The single-frequency solution can be approximated by the form

x1 = A cos Sgt,

x1=A1cosQt+B1sinQt, (j=2,3,...,N).By establishing the dependence of A on the system parameter Q whose optimum valueis to be found, one obtains the curves A (Q) corresponding to the various frequencies Qs.These curves can be either simply continuous or consist of branches. As shown inFig. 9.1, 1, they can feature either:

(a) an interval of Q for which no real solution A 4 0 exists, or(b) separate values, rather than an interval, of Q for which A = 0, or, finally,(c) a whole range of Q for which A + 0.

An investigation of the boundaries of the equilibrium position stability is aimed atdetermining the limit values Q = Q+ for which lim A = 0.

Q-Q'If the solution of (9.1, 5) is expressed in terms of the relative quantities

xkyk = A

(k = 1, 2, ... , N)

the solution of the parameters on the stability boundary may take the form

yl = cos Sgt,

y1 = a1 cos Sgt + b1 sin Qt (j = 2, 3, ... , N)(9.1, 6)

where a1 = A1/A, b1 = B1/A are coefficients which are generally different from zero forA -> 0. Based on the assumptions put forward above, substitution of (9.1, 6) in thelinearized equations of motion leads to 2N non-homogeneous algebraic equations eon-

9.1. Basic considerations and methods of solution

Fig. 9.1, 1Q

Fig. 9.1, 2

281

taming the unknowns a1, b1, Q and Q. These equations are linear with respect to thecoefficients a1 and b1. In most cases, parameter Q is not present in all the equations.Accordingly, the equations can be rearranged, for example, so as to contain the absorbertuning coefficient Q (its square at most) in only two of them. Coefficients a1, b1 as func-tions of 92 which are then readily obtained from the 2(N - 1) equations are substitutedin the remaining two equations containing Q, which enable Q to be determined asa function of 92(QI(Q), QII(92)). The points of intersection of these two curves give thevalues of Q and Q corresponding to the stability boundary (Fig. 9.1, 2).

By ascertaining the boundary values of Q for various values of another parameter,say x, we obtain curves x(Q), the so-called boundary curves in the (x, Q) plane. Thesecurves con;.,ect the values of parameters Q, x at which the amplitudes of self-excitedvibration converge to zero, that is, only a part of the boundary curves forms the bound-ary of the region of stability of the equilibrium position. This is the procedure used fordetermining the points corresponding to cases (a) and (b) shown in Fig. 9.1, 1. A tho-rough analysis of the boundary curves is, therefore, necessary. Further details concern-ing this subject may be found in a book by TONDL (1980b).

The method of boundary curves can be extended to systems governed by differential.equations of dependent variables zk (k = 1, 2,... , n) which are two-dimensional vec-tors of the deflections from the equilibrium position whenever the following assump-tions are satisfied :

(a) The differential equations are homogeneous; this means, for example, in the caseof rotor systems, that the rotors are fully balanced.

(b) All forces (elastic, damping, etc.) are expressed in terms of central symmetricfields - the absolute value of a force vector undergoes no change if the absolutevalues of the deflection and velocity vectors are held constant. Thus, for example, thevector of the restoring force is defined by the relation (denoting by z the deflectionvector)

P = - [fi(Izi) + '/2(IzI)] z .

In case of rotor systems, this means that the axis of rotation is vertical (or the effectof gravitation can be neglected) and all the elements such as bearings, glands, etc..are symmetrical about the axis of rotation.


(c) The roots of the characteristic equation have the same properties as those ofthe former systems.

The solution is arrived at as before, except that in rotor systems it is the angularvelocity of the rotor which is chosen as the step-wise varying parameter. On the bound-ary the solution has the form

zi = exp (iQt) , z1 = (A1 + iB1) exp (iQt) , (j = 2, 3, ... , n) . (9.1, 7)

We shall now outline the procedure used to determine the dependence of the ampli-tudes of single-frequency vibration on a particular parameter of the system (denoted Q).In this case it is necessary to proceed from the complete (rather than the linearized)equations of motion. Approximating the steady solution by (9.1, 5) and comparingthe coefficients of cos Qt and sin Qt one obtains a system of non-linear algebraicequations whose difficult solution can be avoided by applying the procedure by meansof which the unknown values of A1, B1, Q and Q are sought for a given value of X(the algebraic equations thus become non-homogeneous). By a suitable choice of thecoordinate xl one can either make the system linear with respect to A1, B1 or, as willbe shown by way of examples, determine these quantities successively from linearequations.

For systems described by vector equations the steady solution can be approximatedby

zl=Aexp(Wr), z1= (A 1-}-iB1)exp(iQz), (j=2,3,...,n). (9.1,8)

9.2. Two-mass systems with two degrees of freedom

The analysis presented below is concerned with three basic two-mass, two-degrees-of-freedom systems whose basic self-excited subsystem (basic system for short) is aVan der Pol oscillator described by the van der Pol equation. To curtail the exposition,the differential equations are presented already rearranged by means of the intro-duction of dimensionless coefficients of the linear terms. The time transformation isin all cases related to the natural frequency of the abbreviated (undamped) basicsystem.

The following schematic notation is used to simplify the description:

(a) Scheme in Fig. 9.2, 1 a denotes positive linear damping.(b) Scheme in Fig. 9.2, 1 b denotes a negative coefficient of the linear term of damping

having a positive progressive component-self-excitation of the Van der Pol type.(c) Scheme in Fig. 9.2, 1 c denotes the positive Coulomb dry friction.

The lateral deflections of masses Mk are denoted by xk and the stiffnesses of springsby ck.

G I(a) (b) (c)

Fig. 9.2, 1

9.2. Two-mass systems with two degrees of freedom 283

System I. The basic system having mass m2 and a spring of stiffness c2 is attachedto an absorber system (ml, cl)-Fig. 9.2, 2.

Fig. 9.2, 2 Fig. 9.2, 3

The system is described by the equations

xi + xQ(xi - x2) + 0 sgn (xl - x2) + Q2(xi - x2) = 0

X2 - ( - 8x2) x2 - P[xQ(xi - x2) + 0 sgn (xj -- x2) (9.2, 1)

+Q2(xl -x2)]+ x2=O 1

where it = ml/m2 is the ratio between the absorber mass and the basic system mass,

Q =V

cl/ml is the coefficient of absorber tuning, fi is the coefficient of negativeC2/m2

linear damping, 8 is the coefficient of the component of progressive positive damping,x is the coefficient of positive linear damping of the absorber, and 0 is the coefficientof dry friction. By introducing the relative deflection

x=x1 -x2equations (9.2, 1) may be given the form

x" + x2 + xQx' + 0 sgn x' + Q2x = 0 ,

X2 - (j9 - 8x2) x2 - Ia(xQx' + 0 sgnx' {Q2x) + x2 = 0 1

(9.2, 1 a)

or the form

x"+x2+xQx'+0sgnx'+Q2x=0,(9.2, 1 b)

,t) x2 + Px, - (N - 6x2) x2+x2 = 0-1+System H. The basic system (ml, cl) is mounted on a foundation subsystem (m2, CO -

Fig. 9.2, 3. The corresponding differential equations of motion can be rearranged togive the form

xl -x2=04 - M(xl - x2) + xgx2 + 0 sgn x2 + 82x2 = 0

(9.2, 2)

where M = ml/m2 is the ratio between the masses of the basic system and the founda-

/m2 is the coefficient of foundation tuning, x is the coefficienttion subsystem, q = O/Mi


of linear positive damping of the foundation mass motion, 0 is the coefficient of dryfriction of the foundation mass motion, and j9 and 6 are the coefficients of negativeand positive damping of the basic system.

System III. This system differs from System II only in that its self-excitation isprovided by the relative motion of the two masses (Fig. 9.2, 4). The notation of thecoefficients M, q is the same as in System II. The rearranged differential equations ofmotion have the form

xl - 0 - 6(x1 - x2)2 (xl - x2) + x1 - x2 = 0 ,

4 - M{ -[19 - 6(x1 - x2)2] (xl - x2) + x1 - x2) + xgx2+ 6 sgn x2 + 82x2 = 0

Fig. 9.2, 4

or, on substituting x = x1 - x2, the form

x" +x2 - (fi-8x2)x' +x=0,(I + M) x2 + Mx" + xgx2 + t sgn x2 + g2x2 = 0 .

(9.2, 3)

(9.2, 3 a)

Since dry friction is ignored (0 = 0) in the investigation of the equilibrium positionstability, all three systems belong to the class with soft self-excitation.

Consider first System I. The limit values of 9, x on the stability boundary can beestablished by means of the first method of Section 9.1, that is by substituting thesolution xk = Xk exp (iQr) in the linearized differential equations of motion. Carryingout this substitution in (9.2, 1 a) we obtain the characteristic equation which - recall-ing the considerations of the foregoing section - we can write as two equations, thatis,

S24 - [1 + (I + rL) Q2 - xfQ] 92 + Q2 = 0 ,(9.2, 4)

[x(I +,u)Q-N]Q2-xQ+PQ2=0Introducing the parameter

K=xfl (9.2,5)

leads to the equation

(92)1,2 = 2 [I + (I + u) Q2 - KQ] ± { 4 ([I + (I + it) Q2 - KQ]2 - Q2)1/2(9.2, 6)

9.2. Two-mass systems with two degrees of freedom 28,5

as well as to the equation

x = {K(Q2 - Q2)IQ[Q2(l +,u) - 1])1/2. (9.2, 7)

The calculation of the limit values of parameters x and ,3 can be convenientlycarried out as follows :

(a) Frequency Q corresponding to given values of x, # is obtained from (9.2, 6) forvarious values of parameter K (which is varied step-wise on the computer).

(b) The values of S2 are substituted in (9.2, 7), which yields the correspcnding valuesof X.

(c) fi is obtained from (9.2, 5).

Since (9.2, 6) has two roots, two values of Q are obtained for each K. The resultingtwo curves fl(x) represent the boundary of the equilibrium position stability correspond-ing, respectively, to the lower and the higher natural frequency of the system. Bothcurves start from the origin of the coordinates of the (x, j) plane. The area close to thex axis is the region of stability corresponding to the lower or the higher natural fre-quency. Only those values of x, fl which lie in both regions form the region of stabilityof the equilibrium position.

Curves fl(x) were obtained for various values of the tuning coefficient Q and theratio of masses u. They are presented in comprehensive diagrams, one of which isshown as an illustrative example. Curves j3(x) shown in Fig. 9.2, 5 a correspond to thelower frequency S2, those in Fig. 9.2, 5b to the higher Q (both were obtained for,u = 0.1). Fig. 9.2, 6 shows the various regions of the equilibrium position stability

0.25 0.25

P

0.2

0.15 0.15

1.05

0.1 0.1

0,05 1.2 0.05

0 0.2 0.4 0.6 0.8 0

x -

a) b)Fig. 9.2, 5

0.2 0.4 0.6 0.8 1

X ------ b

for Q = 1; the blank area represents the region of the equilibrium position stability.For small values of x the boundary curve fl(x) is close to a straight line passing throughthe origin. With increasing x the slope of the curve decreases, and the curve reachesits maximum. For a further increase of x, the curve becomes a decreasing function.The dash-line curve in Fig. 9.2, 6 represents the boundary of the equilibrium positionstability obtained by approximation effected by means of the diagonal terms of thequasinormal system (see, for example, a monograph by TON DL (1970b)). For largervalues of x, the two curves are seen to differ not only quantitatively but qualitatively


0.25

0.2

0.15

0.1

0.05

M0Fig. 9.2, 6

region of instability withrespect to lower natural frequency

region of instability withrespect to higher natural frequency

region of equilibrium positionstability

as well. MartsouR (1972) was the first author to point out the strange character of the#(x) curve forming the boundary of the equilibrium position stability (for Q = 1),which has its maximum at a particular value of x (Fig. 9.2, 6). mag increases withincreasing u and reaches the extreme at large values of x. For large values of It(,u > 0.3) curve fl(x) has the character of an increasing function in the interval 0 < x

1; with increasing x the curve slope (dfl/dx) grows smaller and the curve reachesits maximum for x larger than one. A larger value of Au can hardly be achieved inpractical applications of absorbers.

0.15

0.1

0.05

Fig. 9.2, 7Q-- .

X

6


0.25

0.2

0.15

0.1

0.05

0

Fig. 9.2, 8

0.6

287

Figs. 9.2, 7 and 9.2, 8 show the results of the examination drawn in comprehensivediagrams. The diagrams of the boundary values of the equilibrium position stabilitydrawn in coordinates of the coefficient j3 and the tuning coefficient Q for variousvalues of the damping coefficient x (the first diagram corresponds to ,u = 0.05, thesecond to It = 0.1) reveal the following: The optimum value of the tuning coefficientis less than 1, that is the optimum natural frequency of the absorber subsystem aloneis slightly lower than the natural frequency of the basic system and the differencegrows larger with increasing mass ratio 4u. There exists an optimum value of thedamping coefficient x. The maximum of the boundary value fi increases with increas-ing ,u.

As can be seen in the diagram of Fig. 9.2, 5, the stability region corresponding to thelower natural frequency grows larger with diminishing Q; the opposite applies to thestability region corresponding to the higher natural frequency. The most favourablecase as regards the region of stability of the equilibrium position occurs when theregions corresponding to the two ranges of natural frequencies overlap. Since thecharacter of the boundary curves is then the same, overlapping in general occurs when-ever the slopes of the two curves coincide. Fig. 9.2, 9 shows the curves of the optimumvalues of Q as a function of the mass ratio It obtained on the basis of this criterion(heavy line; the light line refers to linear material damping - see ToNDL (1980b)).

I

Q

0.50 1 2

Fig. 9.2, 9


Since the procedure applied to Systems II and III is the same as that for System I,only its results are presented below.

The 9(x) curves corresponding to the higher natural frequency have same characteras those of System I. The j9(x) curves corresponding to the lower natural frequencydiffer from those of System I in that with increasing tuning coefficient Q the stabilityregion grows larger, reaches its maximum and then starts to diminish. As the massratio M is increased, the regions of stability of the equilibrium position reach thismaximum at higher values of the foundation tuning coefficient q and their area growssmaller. For low values of M the 19(x) curves bounding the region of the equilibriumposition stability resemble those of System I. The diagram shown in Fig. 9.2, 10plotting the optimum values of the fundation tuning coefficient q as a function ofthe mass ratio M was evaluated in similar fashion to the case of System I. It can beseen that unlike the case shown in Fig. 9.2, 9 the function is not a decreasing oneover the whole range of M but has a minimum at M = 0.7. Since in practice therange of M of System II is comparatively broad, the boundaries of the region ofstability of the equilibrium position are drawn in axonometric view in the (19, q, M)space for x = 0.2 (Fig. 9.2, 11) and x = 0.4 (Fig. 9.2, 12). As the diagrams reveal,with increasing M the maximum value of j9 and consequently the ability to ensurecompensation of the self-excitation effect, decrease even for optimum tuning. On the

Fig. 9.2, 10

0

M

q2

Fig. 9.2, 11


0.15

0.1

0.05

0

Fig. 9.2, 12

1

q2

other hand, as M is increased, the system becomes less sensitive to accurate tuningof the foundation.

For System III, the fl(x) curves corresponding to both frequencies have the samefeatures as those for System I, the only difference being the fact that q > 1 for theoptimum value of the tuning coefficient and that the optimum value of q increaseswith increasing M (Fig. 9.2, 13). The regions of stability of the equilibrium position areshown in axonometric view in Fig. 9.2, 14 (x = 0.2) and Fig. 9.2, 15) (x = 0.4). Theircharacter is similar to that of the corresponding regions of System II.

1.5

1

0 1 2Al

Fig. 9.2, 13

19 Schmidt/Tondl


0.25

0.2

0.15

0.1

0.05

Fig. 9.2, 14

0.1

0.05

0 1

Fig. 9.2, 15

q

2

q

0.

The stability of the equilibrium position will now be examined using the methodof the boundary value curves. The method proceeds from the linearized differentialequations of motion and the assumption that .0 = 0. For System I the solution

x2 = cos QT ,(9.2, 8)

x = a cosQ-r +bsinQa


is substituted in (9.2, 1 b) and the following equations are obtained to determinea, b, 9, Q :

P - (1 + 4U) 92]a=92

b= P

a 1/2(9.2, 9)

Q1 = S2 (1 - a2+ b2)

Q1I=Da2+b2

For System II and System III, the problem is solved in a similar manner.The results obtained by means of this procedure are presented in Fig. 9.2, 16 which

shows an example of the boundary value curve for fl = 0.2 and,u= 0.1. The equilibriumposition is stable only for the values lying inside the loop. The coordinates of the centreof the loop can be taken for optimum values of the parameters x and Q. Fig. 9.2, 17shows the boundary value curves for ,u = 0.05 and three values of f4. As suggested bythe diagram, no value of x can stabilize the equilibrium position for j = 0.3. Sets ofthe boundary value curves were arranged into comprehensive diagrams, showing,for example, the effect of j9 when ,u _ 0.1 (Fig. 9.2, 18) and the effect of ,u when# = 0.1 (Fig. 9.2, 19). An increasing j3 or a decreasing µ causes the region of stabilityof the equilibrium position to diminish substantially.

The results obtained for System II are shown in Fig. 9.2, 20 to Fig. 9.2, 22, thefirst two figures indicating the boundary value curves for M = 0.3 and M = 1(j = 0.1 in both cases), the third containing the comprehensive diagrams ((a) - frontview, (b) - rear view; the cut-out portion of the cube represents the region of stabilityof the equilibrium position). For small values of M stabilization can be achieved forlow values of x in a narrow interval of q; for large values of M, x must be increasedbut the system is less sensitive to accurate tuning.

0.6

ae-

0.4

0.2

0

Fig. 9.2, 16Q 2

19*


0.6

05.X

0.4

0.3

0.2

0.1

0

Fig. 9.2, 17

0.6

x

0.4

0.2

Fig. 9.2, 18

i

n\

Q

Q -U.I

0.15

0.1

0.2

0.25


0.6

x

0.4

0.2

Fig. 9.2, 19

0.6

0.4

0.2

0

Fig. 9.2, 20

0

q

0.6

0.4

0.2

0.05

2 0

Fig. 9.2, 211 q 2

The results obtained for System III are shown in Figs. 9.2, 23 and 9.2, 24; the firstfigure presents the boundary value curves for M = 0.3 and = 0.1, the second (axo-nometric view) reveals the effect of the mass ratio M (for 8 = 0.1).

The procedure for calculating the amplitudes of single-frequency self-excitedvibration as a function of a system's parameter will now be explained using System Ias an example. Substituting the approximate solution

x=acos92e+bsinQt,)(9.2,10)

x2=XECOSS2t J}


0.4

0.2

0.6

0.2

0

2

a)

Fig. 9.2, 22


0.6

x

0.4

0.2

0 1

Fig. 9.2, 23

0.6

x

0.4

0.2

0

q 2

AU

1

9

0.30.1

2

Fig. 9.2, 24


in (9.2, 1 b) we obtain, for a given X2, the following equations

a = X2[l - (1 +,U) S22]

JUQ2

_ 2l

48X2/

b=X2 ,Y

aX2 1/2

QI=\l +a2+b2/

/ 2bX2 4 z

QII = a2 + b27c

(a2 + b2)112 ) .xQ

(9.2, 11)

The calculation carried out on a Hewlett-Packard 9830 A calculator proceeds asfollows: Using (9.2, 11) one obtains functions QI(S2) and Q11(Q) (Q is varied in steps)for several values of X2, and from the points of intersection of these curves, the corres-ponding values of Q. In this way the relationships X2(Q) and S2(Q) are established.

Some results obtained by means of this procedure for System I and System IIare shown below. Fig. 9.2, 25 shows functions X2(Q) and Q(Q) for various values of x

0.5

0.3 x

1 Qom 2


0.2

0

1.2

2

0.8

Xz

1 r

V 0.12

Q -

0.164-

0. 2

0.164

0.1

0

Fig. 9.2, 261 Q 2

0.164

x

297

and fi = 6 = 0.2, 4u = 0.1. Fig. 9.2, 26 shows the two curves for 6 = 0.2, fi = 0.3 andµ = 0.05. In Fig. 9.2, 25, the boundary value curve x(Q) has a loop form, in Fig.9.2, 26, a peak form. In the latter case, the absorber is not capable of quenching self-excited vibration for any. tuning and damping coefficient.

The effect of dry friction is evaluated using the results obtained for System II.In all cases discussed, j3 = 8 = x = 0.2. Fig. 9.2, 27 shows the curve X(q) for the massratio M = 0.3, Fig. 9.2, 28 for M = 1; the coefficient of dry friction a = 0.05 and0 = 0.1 in both figures. As the diagrams suggest, no stable equilibrium position existsunder the effect of dry friction; however, there exists an interval of the tuning coef-ficient in which only small-amplitude vibrations are present.

Many analogue solutions were carried out to check and complement the theoreticalanalyses. One of them, for example, was concerned with the dependence of extremedeflections (denoted by [x1], [x2], etc., for the purpose of differentiation) on the tun-ing coefficient Q (System I) or q (System II and System III).: all the diagrams,x = fl = 6 = 0.2. Fig. 9.2, 29 shows [x1], [x2] drawn as functions of Q (System I) for,u = 0.1 and 0 = 0.2. Dry friction can be seen to cause the interval of stable equili-brium position to be replaced by an interval of Q slightly broader than that correspond-ing to the case of 0 = 0, in which small-amplitude vibration occurs. Fig. 9.2, 30drawn for ,u = 0.1 shows the effect of a varying dry friction coefficient. As 0 is in-creased, not only the interval but also the amplitude of the small-amplitude vibra-


2

1

-&=0.050.1

I j 1

0

Fig. 9.2, 27

0

2

1

0.5

x2

0 0.5

Fig. 9.2, 29

q

1

1.5

1.5

q

Q -

Q2

2

Fig. 9.2, 30

2

0

0.1

,Y= 0.05 k'/

\

X

1

2 0

Fig. 9.2, 28

Q

I


tion grows larger. Fig. 9.2, 31 shows the effect of the ratio between the absorber andthe basic system masses, It, in the case of 15 = 0.1. An increasing It causes the intervalof occurrence of small-amplitude vibration to broaden. The small-amplitude vibrationdoes not always represent the only stable steady solution; in some cases there existintervals of Q with two stable steady solutions, one represented by small-amplitude,the other by large-amplitude vibration.

Fig. 9.2, 31

An example of the results obtained for System II for M = 0.3 is shown in Fig. 9.2, 32.As seen in the diagram, dry friction has a highly favourable effect, even in the casewhen for V = 0 no interval of q exists in which the equilibrium position is stable.A similar result is obtained for System III having the same value of M (Fig. 9.2, 33).

The next example relates to an investigation of stability in the large of System II.Fig. 9.2, 32 showed the case when two locally stable steady solutions existed for cer-tain intervals of the tuning coefficient q. The investigation that follows concernsthe resistance of the small-amplitude vibration to disturbances in the form of a sinusoi-dal pulse (Chapter 7). Let the disturbance function (a sinusoidal pulse) be describedby the equation

Asin2Tc r for

P(r) =0

TO

for

r< T2

?'(9.2, 12)

or >

9 -yFig. 9.2, 32


0

0 0.5

Fig. 9.2, 33

1 1.5

q -o

The values fi = 6 = x = 0.2 are common to all the alternatives considered in thediscussion. Since the vibration, particularly that in the coordinate x2, is small, thissteady state can, approximately, be regarded as non-oscillatory and the effect of theinstant of the pulse action can be ignored. Fig. 9.2, 34 shows an example of the domainof attraction in the (A, To/Tl) plane (T1 = 27c is the period of the natural vibration ofthe basic self-excited subsystem) for M = 0.5, q = 0.5 and 0 = 0.015, obtained bymeans of the automatic process on analogue computer using the method describedin Chapter 7. The values of the disturbance parameters lying in the cross-hatchedarea lead to large-amplitude vibration. Diagrams of this kind drawn for various valuesof 09 can be arranged in an axonometric pattern. Those corresponding to the case ofM = 0.3 are shown in Fig. 9.2, 35 (q = 0.5), Fig. 9.2, 36 (q = 0.75) and Fig. 9.2, 37(q = 1). All diagrams clearly show the influence of the tuning coefficient q on theparameters A, To/Tl. The closer the value of q is to unity, the more resistant is thesteady state being examined to disturbing pulses. For a certain value of 0, the bound-ary function A(T0/T1) has two distinct local minima.

0.8

0.6

0.4

0.2

0 1 2 3

T°/Tt

M = 0.5

q = 0.5

zY = 0.015

Fig. 9.2, 34

0•

4,

\


t

0.02

0.01

0

Fig. 9.2, 37

The main results of the foregoing discussion can be summed up as follows:

For system with a low ratio between the masses of the absorber and the basic sys-tem, 4u, absorbers were shown to have an optimum value of damping. In some casesof intensive self-excitation an absorber may not be able to ensure stabilization of theequilibrium position. The efficiency of absorbers grows rapidly with increasing U.However, an adequately high ,u is hard to achieve in most systems, for in actualstructures the mass of absorbers is usually comparatively low. Consequently, the ab-sorber is either highly sensitive to correct tuning or is not effective enough in suppres-sing intensive self-excitation.

In cases of this sort, resilient and damped foundation mounting (System II andSystem III) having readily attainable favourable mass ratios M (0.3 < M < 1) canbe used to advantage. Resiliently mounted systems are less sensitive to accurate tun-ing and respond effectively to application of dry friction. In correctly tuned systems,dry friction in combination with linear damping can, in fact, effectively reduce theamplitudes of self-excited vibration even in cases in which the action of linear dampingalone has been unsuccessful.

9.3. Chain systems with several masses

The systems which will be considered first have three masses with a one-mass basicself-excited system. The aim of the analysis presented below is to verify the possibilityof combining two of their means of quenching, that is, the absorber with the founda-tion subsystem. System IV (Fig. 9.3, 1) is a combination of Systems I and II. Intro-ducing the relative deflection v = xl - x2 leads to the following differential equationsof motion (after rearrangement to the dimensionless form in the coefficients of the

9.3. Chain systems with several masses 303

m3

Fig. 9.3, 111111211711111111111111

Fig. 9.3, 2

linear terms) :

v" + x2 +x_1Qv' + 0l sgn v' -{- Q2v = 0 ,

(1 +It)

x2_t

Iiiv ' - (I9 - ax2) x2 - x2 - x3 = 0 ,

x3 + x3gx3 - r53 sgn x3 + 82x3 - M(x2 - x3) = 0

where

Q2 = Cl/ml

c2/m22 C-3/M.3

ml m2q = ,u=-, M=c2/m2 m-2 m3

(9.3, 1)

System V (Fig. 9.3, 2) is a combination of Systems I and III, with the absorberconnected to the foundation mass. Writing

x1 = x2 - u , x3 - x2 = V, x2 = x

we obtain the differential equations

U11+X"-}-u-(fl-8u2)u1 =0X1, + M(u + x") + ,u(v " + X") + q2x -f- x2gx + P2 sgn x' = 0 , (9.3, 2)

V11 + X" + Q2v + x3Qv + t93 sgn v' = 0where

q2 = C2/m2

cl/mlQ2 =

C3/m3

c1/ml

m1M=-,m2

Since the procedure adopted for the solution was already outlined in the precedingsection, only some results obtained are presented here (for further details refer toToND7. (1980b)). The examples of the boundary value curves for System IV drawnin the (Q, q) plane, that is, in the coordinates of the tuning coefficients of the absorberand the foundation, are shown first. All the alternatives have the following parametersin common: x1 = fl = 0.2, ,u = 0.1, 0, = 03 = 0. Fig. 9.3, 3a shows the boundaryvalue curves for x3 = 0.2, M = 0.3 (for better clarity, Fig. 9.3, 3b shows the curvesof Q corresponding to the boundaries). Fig. 9.3, 4 shows the boundary value curvesfor x3 = 0.15, M = 0.3, and Fig. 9.3, 5 those for x3 = 0.2, M = 0.5. The domains ofthe parameter values in which the equilibrium position is unstable with respect to thelower or the higher natural frequency of the system, are done in hatching. Two sepa-


0

a)

1.5

0.5 1.0 1.5 Q 2.0

N

1.0

0.5L0

b)

Fig. 9.3, 3

0.5 1.0 1.5 Q 2.0

rate stability domains of the equilibrium position exist for the first example shown inFig. 9.3, 3. For the second and third example, owing to the effect of either decreaseddamping or increased mass ratio M, the lower region disappears, although it is broaderin the first example than the stability domain for larger values of q. It is thereforeadvantageous to provide a heavy enough damping of the foundation mass as well asa large enough foundation mass. Fig. 9.3, 6 is an axonometric view of the stabilitydomain of the equilibrium position for the case of ,u = 0.1, M = 0.3, fl = x3 = 0.2drawn in the coordinates Q, q (the tuning coefficients) and xI (the absorber dampingcoefficient).

In three-mass systems, the effect of dry friction is similar to that observed in two.mass systems. To demonstrate it by way of an example, Fig. 9.3, 7 shows a diagramplotting the amplitude of the foundation mass vibration X3 as a function of the absor-ber tuning coefficient Q for xl = x2 = 0.2, It = 0.1, M = 0.3, q = 0.9 and threevalues of the dry friction coefficients of the motion of the absorber and the foundation


0

Fig. 9.3, 4

0

Fig. 5.3, 5

0.5

0.5

In

In 2.0

mass subsystems, 0, and ?9'3. As the diagram suggests, the action of dry friction of theabsorber relative motion alone (01 = 0.05, 03 = 0) causes the interval of Q in whichthe equilibrium position is stable when dry friction is not present to be replaced bya somewhat broader interval of Q in which small-amplitude vibration occurs. Theeffect of dry friction of the foundation mass motion (01 = 0, 03 = 0.05) is greaterand as a result only small-amplitude vibration exists in the interval investigated,0 < Q < 1.5. These analytical results are confirmed in full by analogue computations.If the foundation mass is properly tuned, application of dry friction of the foundationmass motion can bring about a very efficient reduction of the self-excited vibrationamplitude in a broad interval of values of the absorber tuning coefficient.20 Schmidt/Tondi

306 9- Quenching of self-excited vibration

1.2

a)

0.6

of 1

0.4 I

----------1.2

1.0

0.8 q

0.6

0.2

Q

-4-A41j I

0.50

b)

Fig. 9.3, 6

1.0 1.5 2.0

0.4

Examples of the results of an analysis of System V which follow are presented indiagrams all drawn for j9 = x2 = 0.2 and ACC = 0.1. Figs. 9.3, 8 (M = 0.3) and 9.3, 9(M = 1) show the regions of stability of the equilibrium position in the coordinates Q,q, x2 (the absorber damping coefficient). Although the diagrams recording the effectof dry friction are on the whole similar to those obtained for System IV, they showan interesting anomalous effect of dry friction of the foundation mass motion. How-


X3

0.5

171 Y3

(a) 0 0

(b) 0.05 0

(c) 0 0.05

(a)

(a)

(b) L(b)t

(c)

(b)

(c)

0Fig. 9.3, 7

0.5 1.0 Q 1.5

Fig. 9.3, 8

1.4

307

ever, this phenomenon, not noted in other cases, was present only for M = 0.3 andonly within a certain interval of the foundation tuning coefficient q. Fig. 9.3, 10shows the relation between the amplitude of the relative deflection u = xl - x2and the tuning coefficient Q for q = 1.3, ,u = 0.1, M = 0.3, j3 = 6 = x2 = x3 = 0.2and for the friction coefficients of the foundation mass motion and absorber relativemotion 62, 03listed in the figure. For the alternatives (b) and (d) (02 = 0.05 and 0.06),the vibration amplitude reaches its maximum in the region in which the tuning isoptimal for other cases. As Fig. 9.3, 11 representing the alternative having q = 1.3suggests, the presence of this anomaly is confirmed by analogue computations. It isalso evident, to a lesser extent, in cases when q = 1 and 1.5.20


0.6

x3

0.4

0.2

Fig. 9.3, 9

Systems containing a basic subsystem with several masses are treated next. Sys-tem VI (Fig. 9.3, 12) and System VII (Fig. 9.3, 13) consist of a two-mass self-excit-ed basic subsystem with an absorber subsystem attached to either the upper (Sys-tem VI) or the lower mass (System VII) of the basic system. It is assumed that thebasic system of both VI and VII is acted on by self-excitation as well as positivedamping (produced, for example, by material damping of the elastic elements of thebasic system). An analysis was made of these systems, aimed at finding the most satis-factory position of the absorber. Examples solved in the analysis encompassed a rangeof the most usual engineering systems having the upper mass smaller than the lowerone. The important results of the analysis can be summarized as follows.

If the basic system is capable of vibrating in two modes, a single absorber attachedto either of the masses makes it possible, within a certain interval of values of thetuning coefficient in the neighbourhood of the natural frequency of the basic system,to quench only one mode of vibration. If the intervals do not overlap (as is usuallythe case), application of a single absorber cannot ensure complete quenching of self-excited vibration (Fig. 9.3, 14). The first vibration mode (of the lower frequency) iseffectively quenched by an absorber attached to the upper mass of the basic system(System VI), the second, by an absorber attached to the lower mass (System VII).The intensity of self-excitation in the first or second vibration mode is affected by thelevel of positive damping of the motion of the various masses of the basic system.Reduction of the positive damping of the lower mass brings about an increase in theeffect of self-excitation, particularly in the second vibration mode.

System VIII (Fig. 9.3, 15) consists of a three-mass basic system (masses, connectingsprings, relative positive damping and self-excitation are the same for all three mas-ses: m2=m3=m4=m; C2=C3=C9=C;%2 =x3=1L4=x) and an absorber sub-


U

1

1

Q

0.95

0.9

0.85

'(CO

(cl

0 0

(b); b 0.05 0

c 0 0.05(d)\ d 0.06 0

(d))

(a)

(b)

0 1 2 Q 3

(b)

I Id)Il

(a)

(b)

(c)

(a)

1 2 Q 3

Fig. 9.3, 10

309

system attached to the upper mass of the basic system. Fig. 9.3, 16 shows the bound-ary value curves in the (x1, Q) plane for u = 0.05 and several values of the coefficientof positive damping of the basic system x. Fig. 9.3, 17 shows the boundary valuecurves for x = 0.1 and several values of the absorber mass ratio ,u = mllm. j9 = 0.2in both diagrams. As the diagrams imply, a region of xl, Q in which the equilibriumposition is stable, is obtained for fairly large values of ,u. In case of high intensityself-excitation and light positive damping in the basic system, a low-mass absorberis not capable of stabilizing the equilibrium position.

As an analysis of System IX (Fig. 9.3, 18) reveals (TortDL (1980)) the equilibriumposition of self-excited basic systems (System IX has a two-mass one) which canvibrate in two modes, can be stabilized by application of two sufficiently large andunequally tuned absorbers.


2.0

1.5

1.0

0.5

01.5

1.0

0.5

0`-

1.0

1.5

0

1.5

1.0

4f

0.5

00.25 0.5

e2=0Y3= 0.05

I1.0 1.5

?Y2= 0.05-93 = 0

y

Q 2.0

1.15

1.3

1.5

0.25 0.5 1.0 1.5 Q 2.0

Fig. 9.3, 11 (IC denotes that different non-trivial initial conditions were applied)


oilI

0 u7777,,"ZI11211111 77//

Fig. 9.3, 12 Fig. 9.3, 13

0.52

0 0.5 I0 Q 1.5

Fig. 9.3, 14

111111111112111 11// /77-=Fig. 9.3, 15


0.6

X,

0.4

0.2

0 0.5 1.0 1.5

Fig. 9.3, 16

0.4

0.2

li

II

I \II

1 n \.

0 0.5

0.3

0.15/0.05 AL

1.0 1.5Q --

Fig. 9.3, 17

9.4. Example of a rotor system 313

/7777777777Fig. 9.3, 18

9.4. Example of a rotor system

Consider a system consisting of a rigid, ideally balanced rotor supported in air pres-surized bearings and rotating about a rigid but elastically mounted vertical axis(Fig. 9.4, 1 a) hinged in the plane of bearing 1. This arrangement corresponds, approxi-mately, to the mounting of the axis in a rubber ring (shown schematicallyin Fig. 9.4, 1 b), i.e. to the case of rigidity in the radial direction being several timesgreater than that against the angular deflections of the axis. The model representsthe spindle of a cotton-yarn spinning machine. Its motion is defined by three vectorsof the plane deflections, %, %, za. Let m denote the rotor mass, I the equatorial mo-ment of inertia about the axis perpendicular to the axis of rotation passing throughthe centroid T, and I. the polar moment of inertia about the axis of rotation. Let mobe the mass of the elastically mounted axis reduced to the plane of the second bea-ring. The rigidity of the axis mounting co is reduced to the same plane. Mounting by

0) b)

Fig. 9.4, 1


means of a rubber ring also represents damping of material character which can beexpressed as

-cok !;Izo

Izol

To simplify, write the equations of motion for a rigid rotor mounted in two supportsof rigidity c1, c2, respectively; in Fig. 9.4, 2, 1 denotes the distance between the planesof the supports, and h, l2 describe the position of the centroid T. Writing

l2 llal = a2 = (9.4, 1)

and denoting by w the angular velocity of rotor rotation, and by z1, z2 the vectorsof the deflections in the planes of mounting, one can describe the system shown inFig. 9.4, 2 by the following equations of motion (see TONDL (1973c) :

Az1+bz2-iwE(zl-z2)+wizl=0,bil + Bz2 - is E(z2 - zl) + w2; = 0 }

where

0,2 j, b=ala2- I, B=a2-}- j,M12

ml2

M12

I Iw I 10 2 Cl 2 C2E y' ' (01 , w2m12 I ml2 m m

Introducing the time transformation

wlt=zmakes it possible to give the equations the form

Azi +bz2 -ivE(zi -z2) +zl=0,bzi + Bz2 - ivE(z2 - zi) + p2z2 = 0

where

COv= -,w1

p2 =c2

.Cl

E

(9.4, 2)

(9.4, 3)

(9.4, 4)

Before writing the equations of motion, one should put forward the assumptionsconcerning the expression of the bearing forces. Since the primary aim of the analysisis to examine the flexible mounting of the axis and its essential effect on the limit ofinitiation of self-excited vibration (on the equilibrium stability in case of an ideally


balanced rotor), a very simple expression of the bearing forces will be used. Assumethe bearings to be identical, both of the hybrid type, and the resulting vector of thebearing forces to be expressed in terms of two independent components, one represent-ing the aerostatic, the other the aerodynamic contribution. In linear form, the aero-static component can be described by the product of rigidity c and the vector of thedeflection (this component has the character of a central force). The second componentrepresents the aerodynamic force which arises owing to the effect of shaft rotationand viscosity of the gas. For an incompressible medium the aerodynamic componentin its simplest form (see TONPL (1974a)) is described by (for the deflection vector z)

h(2z - i(oz) . (9.4, 5)

For a compressible medium and small amplitudes of the whirling motion, MARSH(1965) deduced the following definition of the vector of the aerodynamic forces

(Pe - iPy,) exp (i99) . (9.4, 6)

where (p is the position angle; the radial component Pe and the tangential componentP. are defined by

2cp2Lr e +2= 7P (9 4 7

/w)2,cpa l 1e A2(1 . , )

P, = 7CpaLrleA(1 - 2g7/w)

(9.4, 8)1 + A2(1 - 2ry/(o)2

where pa is the ambient pressure, L is the length of the bearing, r1 is the shaft radius,e = Izl/8 is the relative eccentricity of the shaft centre from the bearing centre, 6 isthe radial clearance in the bearing, and A is the compressibility number given by

A = 182 (9.4, 9)

where,q is the coefficient of dynamic viscosity of the gas. The quantities e, 99 and z arerelated as follows:

z = e8 exp (iqq) . (9.4, 10)

For an incompressible medium the equations of motion of the system shown inFig. 9.4, 1 can be rearranged to take the form

Azi + bz2 - ivE(zi - z2) + zl + K(2zi - ivz1) = 0 ,

bzi + Bz2 - ivE(z2 - zj) + z2 - zo + K[2(z2 - zo) - iv(z2 - zo)] = 0 ,

zp - M{z2 - zo + K[2(z2 - zo) - iv(--2 - zo)]) + q2 (__1+ k IzOI IzI 0

0

whereK = h ", M = - a,

mw1 moq2 = c0/mo

c/m

(9.4, 11)

K is the coefficient of the aerodynamic component, M the ratio between the masses,and q the tuning coefficient whose optimum value is to be established.

If the compressibility of the medium is taken into account, expedient introductionof the relative deflection

w=z2-zo


and some rearrangement changes the equations of motion to

Azi + bz2 - ivE(z, - zz) + --1[I + K(pe - ip,)] = 0 ,

bz,'+ Bz2 - ivE(z2 - z;) + w[1 + K(pe - iip,)] = 0,

z2 - w" -{- g2L(z2 - w) -{- k 1z2 - wlz2 - w'

- Mw[1 + K(pe - ip9,)] = 01z2-w'I

where(9.4, 12)

K - 7spaLri Ao(y - 2T')2mSwi

Pe 1 + !lo(v - 2q )2

11o(v -p = 1+ Ao(v - 2g9')z

Ao - 162 .

As a first step, find the boundary values v8 as functions of the coefficient of tuning qfor the system described by (9.4, 12). Substituting in (9.4, 12) the solution

zl = exp (WT), z2 = (Z + iW) exp (W7c) , zo = (U + iV) exp(iQv)

(9.4, 13)

leads to four algebraic equations determining Z, W, U and V as functions of Q, that is,

(1-F) KO (X+A-0') (Y -KoX)Z

_W

_' U

_V

__

H H A A

where(9.4, 14)

Ko=K(2S2-v), F=S22A-vQE, G=S22B-vS2E,H=S22b-vQE, A=1+Ko,X = -H - GZ+Z - KOW, Y = -GW+W+KOZ

and two equations for the function q(Q)

q = (U+ kV)-i{Q2U+M[Z- U -KO(W- V)]},g2 = (V + kU)-1 { Q2V -}- M[W - V + A

For different values of v these equations yield two systems of curves, gi(S2; v) andg1I(S2; v) the points of intersection of which describe the corresponding boundaryvalues of v8 as functions of q. The examples presented below were solved using aHewlett-Packard 9830 A calculator with a graph plotter.

Several alternatives were examined to establish the effect of the centroid position((a) : al = -0.25, (b) : al = 0, (c) : al = 0.5, (d) : al = 1, see Fig. 9.4, 3), the massratio M, of I/ml2 and the gyroscopic action. Some of the results obtained are presentedin diagrams showing the boundary value curves v8(q) (solid lines) and the curvesvK(q) (dashed lines). The minimum value, vsmin, is indicated by heavy solid line. Thenumbers in circles give the critical frequencies or the limit frequencies of initiation ofself-excited vibration corresponding to the first, second, ... critical speed. In all thecases examined, the values of the parameters were:

I/ml2=1, M=3, y=0.2, k=0.1, K=0.2

9.4 Example of a rotor system

rotor

Fig. 9.4, 3

317


The diagrams in Figs. 9.4, 4 to 9.4., 7 describe the studied effect for four alternatives,(a) to (d); to provide further information, they are supplemented by records of thecorresponding dependences of the self-excitation frequency Q (q) (Fig. 9.4, 8). As thediagrams suggest, an elastic mounting of the axis with damping can raise the lowestlimit of initiation of self-excited vibration well above that obtained in case of a rigidmounting (q = oo). The value of V$min corresponding to this arrangement is slightlylower than vSmin for the highest value of q shown on the right-hand side of the dia-gram. The increase of v8min produced by the effect of the elastic mounting of the axisdepends on the position of the centroid. For the parameters given above, alternative(a) shows the greatest (several times the original) increase of all.The analysis also revealsthat the maximum value of v8 min grows larger with increasing mass ratio M, increasingcoefficient of damping k and decreasing I/m12. The higher the values of M and I/m12,

6

VS

yK

4

2

6

PS

yK

4

2

ys

ysmin

vK ---

1 2

0

Fig. 9.4, 6

0.5 1 1.5 q2

Z77vs

ysminP.___

1

0 0.5 I 1.5 4 2

Fig. 9.4, 7

9.4. Example of a rotor system

2.5

2

2

1.5

1

0.5

(d)

(0)

(b)

(d) (c)

(d)

=(d)(b)

i , (a)

0

Fig. 9.4, 8

0.5 1 1.5 q 2

319

the more remote is the optimum position of the centroid from the centre of the spanbetween the bearings.

The results of an analysis involving a compressible medium are shown in Fig. 9.4, 9.The diagram of the single example reported here (for additional examples refer toToNnL (1980 a)) is drawn for y, k, M, I/ml2 used in the preceding analysis, for K = Ao= 1 and alternative (b) (al = 0, a2 = 1).

If compressibility of the medium is considered, one can obtain both the limit ofinitiation and the limit of extinction of self-excited vibration of a definite mode(TONDL (1974 a)). As can be seen in the diagram, this is what happened in the examplebeing discussed. For greater clarity, different kinds of hatching are used for the variousinstability regions, and encircled numerals for the corresponding vibration modes.To facilitate a comparison, the dependence of the critical frequency vx on q (for thedetermination of vk, see ToNnL (1980a)) and the correspondence with the differentvibration modes are shown in dashed lines. The instability region belonging to thelowest natural frequency is seen to be divided in two parts. If the rotor speed ofa system (having a given q) is raised, then - as long as q < 1.18 - the equilibriumcondition becomes unstable and self-excited vibration starts to arise from a certain

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(assuming, as before, an ideally balanced rotor) as follows:

Az' + bz2 - ivE(zj - z2) + zl[l + K(pe - ipq,)] = 0 ,

bzi + Bz2 - ivE(z2 - zi) + w[1 + K(pe - ip,)] = 0 ,

z2 -w"+42 z2-w+kIz2-wl

z2-w,Iz2-w (9.4, 16)

-M2v[l + K(pe - ipy,)] +,u(z2 - w" + u") = 0 ,

z2 -w"+u" +Q2(u+kolul lUl)=0

u' 1

where p = m3/mo is the ratio between the reduced masses of absorber and axis,

ko is the damping coefficient of the absorber, Q = (r'3/m3)1/2 is the tuning coefficientc/m

of the absorber, c3 is the stiffness and m3 the reduced mass of the absorber spring.Substituting in (9.4, 16) the solution

z,=exp(iQT), z2= (Z+iY)exp(i-Qv),(9.4, 17)

w = (W + iV) exp (iQx) , u = (U + iL) exp (i - Sgt)

yields the equations for subsequent determination of Z, Y, W, V as functions of Q,viz.

Z = (1 + KPe - F)/H, Y = -KP9,/H , (9.4, 18)

W = d [(1 + KPe) (H + GZ) -

V = d [(1 + KPe) GY + KP9,(H + GZ)]

where

F = Q2A - vS2E, G = S22B - vQ2E, H = Q2b + vQE,

A = (1 + KPe)2 + (P,K) 2,

_ Ao(v - 252)2P P - Ao(v - 2Q)P,

1 + A2(v - 252)2 ' 1 + AO - 2Q)2

and the equations for obtaining U and L:

U={-(1+,u)S22(Z-W)+92[(Z-W)-k(Y- V)]-M[W(1 + KPe) + VKP,,]}/,uS22,

L={-(1+E)S22(Y-V)+42[(1'-V)+k(Z-W)]-M[V(1 + KPe) - WKP,]}/uQ2.

(9.4, 19)

(9.4, 20)

For v changing step by step, the corresponding boundary values of Q (the pointsof intersection of curves QI(S2) and Q11(S2)) are obtained from the following equations:

Ql = Q[(Z - W + U)I (U - k0L)]112 , 1(9.4,21)

1/2Ql1 = Q[(Y - V + L)/(L + k0U)]21 Schmidt/Tondl

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10. Vibration systems with narrow-bandrandom excitation

10.1. Application of the quasi-static method

In what follows we investigate the vibrations of the system

y" + Ay = lady - by' - dy2y' - ey3 - g(T) . y (10.1, 1)

where now the parametric excitation g(z) is assumed to be a stationary narrow-bandrandom process with zero mean value,

g(r) = y(z) cos [2nz + rft)] = y(r) cos ip(r) (10.1, 2)

with the slowly varying excitation amplitude y = y(r) and phase ri(r). Further, a isa frequency variation, b the coefficient of a linear and d of a non-linear damping and ethat of a non-linear restoring force, while dashes denote derivatives with respect tothe dimensionless time T.

By analogy with the evaluation of amplitude formulae in case of deterministicexcitation, our aim is now to determine the statistical distribution of the vibrationamplitudes, the most important characteristic of a random vibration, as dependenton the system parameters.

The quasi-static method used by STRAToNovicu (1961) and LENNox and KUAK(1976) for problems of forced vibrations and by Baxter (1971) for parametricallyexcited vibrations of bars involves the following procedure.

Corresponding to the averaging method (Section 2.5), we assume the main parametricresonance case

A=n2and introduce the transformation

y = a cos 92, y' = -nasinq;, 9) =nr+t (10.1, 3)

where a = a(r) and 0 = o(z) are slowly varying functions. Thus we get the equationsin standard form

a' _ - 1 [2n2a - g(r)] a sin 92 cos 99 - ba sin2 99

- da3 sin2 99 COS2 qJ + 1 ea3 sin 97 cos3 qJ ,n

1[2n2a - g(z)] cos2 92 - b sin 99 cos 99

n

- da2 sin p cos3 92 + I eat cos4 92n

21*

324 10. Vibration systems with narrow-band random excitation

equivalent to (10.1, 1). Insertion of (10.1, 2) and trigonometric transformations give

a' = -naa sin 2T - 2 ba(1 - cos 2(p) - 8 da3(1 - cos 499)

+ 8n ea3(2 sin 2(p + sin 4q,) + 4n ya[sin (299 - y,) + sin (2q9 + y,)]

-na(1 + cos 2q,) - 2 b sin 2q, - 8 da2(2 sin 2(p + sin 4p)

+ 8n ea2(3 + 4 cos 299 + cos 499)

+ 4n y[2 cos y' + cos(2(p - y,) + cos(299 + y,)] .

According to higher approximations of the averaging method (compare BOGOLJUBOVand MITROPOL'SKIJ (1963)), the dependence on q,, and hence on the timer, of the quan-tities not containing y on the right-hand sides has to be iteratively elimated by meansof a suitable transformation

a=A+e(A,0), 0=0+6(A,0), 0=nr+0. (10.1,4)

In contrast to the expansion with respect to a single small quantity applied by STRA-TonTovicn (1961) and BAXTER (1971), a, y(r), and b are now assumed to be in-dependent small quantities. The first iteration for e, 6 yields, when the small quanti-ties of lowest order of magnitude are taken into account,

A' + n 88 = -naA sin 2 - 1 bA(1 - cos 20) - .1 dA3(1 - cos 40)2 8

+ 8neA3(2 sin 2 O + sin 4 0) - 4

nyA [sin (20 -V) + sin (2 0 + yp) ] ,

0' + n -m(1 + cos 20) - 2 b sin 20 - g dA2(2 sin 20 + sin 40)

+ gneA2(3 + 4 cos 20 + cos 40)

y[2 cos y + cos (20 - y,) + cos (20 + V)].+ In

The correction functions e, S are determined in such a way that they cancel on theright-hand sides the dependence on 0 in the terms not containing y. Thus we are ledto the differential equations

A' = - 2 bA - 8 dA3 + 4n yA[sin (20 - y,) + sin (20 + y')] ,

0' = -na + 3 eA2 + 1 y[2 cos V + cos (20 - y,) + cos (20 + V)]8n 4n

(10.1, 5)

10.1. Quasi-static method

and for the correction functions

from

and

n -naA sin 20 + 2 bA cos 20 + 8 dA3 cos 40

+ 8n eA3(2 sin 20 + sin 40) ,

n -na cos 20 - 2 b sin 20 - 2 dA2(2 sin 20 + sin 40)

1+8n

eA2(4 cos 20 + cos 40)

which we get, with arbitrary integration functions E0(A), 8o(A),

= I aA cos 20 + 1 bA sin 20 + + 1E dA3 sin 402 4n 32n

- 1 eA3(4 cos 20 + cos 40) + E,(A)32n2

6 nbcos20 + -- dA2(4 cos 20 + cos40)= -2 a sin20+4

+1

eA2(8 sin 20 + sin 4$) + 30(A) .32n2

325

We choose for simplicity e0(A) = 80(A) = 0, as in STRATONOVICH (1961), p. 103 andBAXTER (1971), and in contrast to BoGOLauBOV and MITROPOL'sKIJ (1963) where thecorrection functions are chosen such that the solution

y=acosry=Acos0+(Ecos0-A3sin0)+...acquires, by the correction terms in parentheses, no additional terms to the basicharmonic,

2n

(E cos 0 - Ab sin 0)cos 0 dq5 = 0.sinf

For the original quantity y, a first approximation yields, according to (10.1, 3),(10.1, 4),

y = (A +e) cos ((P+6)

= A cos 0 + 1 aA cos 0 + 1 bA sin 0 + 1 dA3(2 sin 0 - sin 30)2 4n 32n

-1 eA3(6 cos 0 - cos 30).32n2

Following the quasi-static method, we assume the random functions change soslowly, that is, that the correlation time of the random excitation g(r) is so large incomparison with the relaxation time of the amplitude - which is of the order of


magnitude of (nb)-' - that so-called quasi-stable values of the amplitude and phasecome to be established for every value of y(-t) and q(z). Therefore we assume that in(10.1, 5) the left-hand side derivatives A', (9' vanish and that only the excitationterms with the arguments 20 - p, which vary slowly for small a, are taken intoconsideration.

The slowly varying amplitude y as well as the phase 21 of the excitation being assum-ed constant, (10.1, 5) yields, by squaring and adding, besides A = 0,

4 y2 = n2(b + adA2)2 + (2n2a - 4 eA2)2 (10.1, 6)

from which follows the amplitude frequency formula

2n2a = 4 eA2 ± V 4 y2 - n2(b + 4 dA2)2 . (10.1, 7)

Division of the two equations (10.1, 5) gives

cot (2i - V) =2na - 3 eA2

4n

b+4dA2

1.0.2. Application of the integral equation method.Probability densities

The quasi-static method is based on the periodic solutions of (10.1, 1) for constantvalues of the slowly varying functions y and il found in Section 10.1 by help of thesecond approximation of the averaging method. The integral equation method leadsin a very straightforward way to these periodic solutions. Equation (10.1, 1) is anexample of the general investigation in Chapter 5. For the following compare SCHMIDTand WENZEL (1984).

Assuming only a constant excitation amplitude and phase or, in other words, aharmonic excitation (10.1, 2) for an individual realization, the integral equationmethod leads to the amplitude equation (5.5, 1). This equation simplifies to (10.1, 7)if we neglect the second order excitation terms which allow an evaluation of the secondresonance.

Real vibration amplitudes exist only if the amplitude-dependent threshold condi-tion

y > 2n(b + dA2) (102, 1)aholds. If the excitation amplitude is small enough to realize the sign of equality in(10.2, 1), then (10.1, 7) simplifies to the equation

3a = - eA2

8n2

for the backbone or skeleton curve, the geometric locus of all maximum values

Amax 2 1 rY b)- d 2n J

of the response curves for different excitation amplitudes y (the dashed line in Fig.10.2, 1).

10.2. Integral equation method. Probability densities 327

The amplitude values AV in the boundary points of the interval in which the reso-nance curves exist are determined by the condition

da

dA

for vertical points of the resonance curves. If e > 0, the lower boundary point yields

a=aV= - I11-4

1 2-n2b2,2n2 y

whereas the upper one leads to

1 y V9e2 + n2d2 3eb\ 2 4 ( 3eyla°_2n2 (2

nd- d Av = d b

2nv9e2 { n2d2/.

(10.2, 2)

Vice versa, if e < 0, the lower boundary point gives

1 y 9e2 + n2d2 3eb

a° 2n2 (2 nd+ d

the upper one

1 1av=- y2 - n2b2r2n2 4

Because A, > 0 holds,

y2 > n2b2(9e2 + n2d2)

4=

9e2

A, =0.

2 - 4 3eyAV - d (-b -2n V9e2 + n2d2

(10.2, 3)

is the condition for the resonance curves to exist beyond the linear resonance interval

y2 - n2b21 1

2- n2b2y '2n2 4 /

that is, for there to be an overhang of the resonance curves (Fig. 10.2, 1).Now as before the threshold condition (10.2, 1) holds. For the relatively small

excitation amplitudes

n2b2 < y2 < n2b2(9e2 + n2d2)4 9e2

the resonance curves remain in the linear resonance interval, no overhang exists.Insertion of av into formulae (10.2, 2), (10.2, 3) for AV yields the equation

2

a = a =8n2

f

24eeAV 2 + - (10.2, 4)

for a curve which reveals how the vertical boundary points of the resonance curvedepend on a and which we denote as vertical curve. Fig. 10.2, 1 gives an example for


-0.03 -0.02 -0.01

Fig. 10.2, 1

0.01 0.02 0.03 0.04 a

the group of resonance curves belonging to different values of y, for the (dashed)backbone curve and the (dashed-dotted) vertical curve where n = 1, e = 1, b = 0.04and d = 2, a very large value in order to make manifest the difference betweenvertical and backbone curve, which vanishes for d ---* 0.

For every a, the corresponding value Av is the smallest of all amplitude valuesbelonging to the stable upper branches of the group of resonance curves.

The amplitude formula (10.1, 6) can, because of (10.2, 4), be written in the form

P(b2 2 2 2

4 y2

- n2k(b + 4dA2l +

4 e(AV - A2) 12 Av +n3 dl

(10.2, b)e

which reveals the dependence between the excitation amplitude y, the resonanceamplitude A and the minimum Ao of A. The derivative dy2/d(A2) shows that y2monotonously depends on A2.

Our aim is to determine the probability density w(A) of the response amplitude A.To this end the probability density v(y) of the excitation amplitude must either beknown (say, by experimental investigations) or has to be determined from the knownprobability density u(g) of the excitation process g(r). Examples and general methodsof evaluation for the relationship between the probability densities v(y) and u(g)have been given by KROrA6 (1972). We assume the rather general case of a Weibulldistribution of the excitation amplitude, admitting additionally of a threshold value F,as the most comprehensive form to handle easily by analytical formulae :

(Y r kk k 1 20'

V(Y) (y -T) - e (10.2, 6)

Here a is the scale parameter and k the form parameter. With this often used distribu-tion various forms of probability densities can be described as Fig. 10.2, 2 shows for

1/P2.In particular, for k = 2 and F = 0 the Weibull probability density simplifies to

the Rayleigh probability density

_ Y,

v(Y) = a e 2a' (10.2, 7)Or


1

0.5

0 r+1

Fig. 10.2, 2

v(r)

0.6

0.2

dX 0 26 30 1-

Fig. 10.2, 3

-36 -26 -6

r+2

6 26 36 Y

r+3 a

of the excitation amplitudes, which corresponds with the Gaussian probability densityg4

U(g) = 1 e 2a°

V2n6

of the excitation process (Fig. 10.2, 3).The probability density of the response amplitudes associated with strips v(y) dy of

excitation amplitudes (Fig. 10.2, 3) can be obtained by the transformation

v(y) dy = dV(y) = dW(A) = w(A) dA (10.2, 8)

with the probability distributions V of the excitation amplitude and W of the responseamplitude, if y is a monotonic function of A (compare for instance STRAToNovicH(1961)). This condition holds for the problem at hand.

For the Weibull probability density of the excitation amplitudes (10.2, 6), thetransformation (10.2, 8) leads for the response amplitudes to the probability distribu-tion

AA)-P

W(A) _ -e V-2a (10.2, 9)

We first assume e > 0.


In case

>6d

we have to substitute (10.2, 5) for y(A) in (10.2, 9). Differentiation corresponding to(10.2, 8) yields the probability density of the response amplitudes

x-i-

_F k

w(A) = k (KK)

(9e2 + n2d2) A(A2 - A2) e V2a for A > A,(V21)

k 4K

where

2 2 2 2 2

K = 2 n2 (b+ 4 dA2) +[ + 12 Ao + n3eb]y

In case

< bd

6e

(10.2, 10)

(10.1, 6) has to be inserted in (10.2, 9). By differentiating we get the probability den-sity of the response amplitudes

2k ( )/' 1(&a-r k

w(A) = 6)k KaKa

[n2(dbJ

A - 6ea) + 4 (n2d2 + 9e2)A2J

e-1 °

where(10.2, 11)

Ka = 2 Vn2(b + a dA2)2 (2n2a - 4 eA2)2 .

Now assume e < 0. Then (10.1, 6) and the probability density formula (10.2, 11)

are valid for a > 6 . If A < - , (10.2, 5) has to be used, and the probability densityis (10.2, 10).

In the special case of a Rayleigh probability density (10.2, 7) of the excitationamplitudes, formula (10.2, 10) simplifies to

9e2 + n2d2 - 2L

w(A) = A(A2 - A2) e °1 for A > Av2a2

with2 2 3

L = n2 (b + 4 dA22

+ 3 e(A2 + A2) + 2 A2 +n2 2

whereas formula (10.2, 11) becomes

2La

w(A) _2

A [fl2(bd - 6ea) + 4 (9e2 + n2d2) A2] e

where

(10.2, 12)

(10.2, 13)

2 3 2

La = n2 (b + I dA2) + (2n2x -4

eA2) .


The probability density (10.2, 12) disappears if A tends to infinity and if A tendsfrom above to the minimum value A, of the vertical curve or to zero. As amplitudesbelow the vertical curve do not appear, w(A) = 0 holds for A < Av.

The probability densities (10.2, 12) respectively (10.2, 13) are drawn in Fig. 10.2, 4for the example a = 0.01, b = 0.002, e = 1, d = 0.2 and different values of the fre-

0.024

0.018

0.012

0.006

Fig. 10.2, 4


quency variation a, in Fig. 10.2, 5 for greater non-linear damping, d = 1. The verticalcurves (10.2, 4) are marked by dashed lines, they represent the lower limit for realresponse amplitudes. For increasing positive frequency variation a, this lower limitincreases, as too does the maximum of the probability density curves (which iscalled the most probable amplitude). The probability for positive response amplitudes,that is, the areal built by the probability density curves, decreases to zero if Jaj in-creases. This behaviour is accelerated by increasing non-linear damping, as comparisonof Figures 10.2, 4 and 10.2, 5 shows. Correspondingly, the probability that no reso-nance vibrations exist increases.

Fig. 10.2, 5

0.012

It is of interest to discuss the probability

W1 = W(A1 < A < oo)

that the response amplitude A is greater than a given level A1. Because of (10.2, 8),cc

2L

W, = f w(A) dA ==a_

°' A=A,Al

holds in the case of formula (10.2, 12) and

2La

W1 = eas

A=A,

in the case of formula (10.2, 13).Especially for Al = A, formula (10.2, 14) reads

_2 d.aallf g )(b Fj V)

(10.2, 14)

(10.2, 15)


and gives the probability of there being any positive response amplitudes at all.Correspondingly, the probability for vanishing response amplitudes is

1 - WV.

For the examples in Figures 10.2, 4 and 10.2, 5 and also for vanishing non-lineardamping, d = 0, formula (10.2, 20) gives the probability of positive response ampli-tudes drawn in Fig. 10.2, 6. The corresponding curves for a greater linear damping,b = 0.005, are given in Fig. 10.2, 7. The figure shows that this probability decreasesif linear or non-linear damping increases.

W,

1.0 d=0

The probability (10.2, 14) that the amplitudes are greater than a given amplitudelevel A, is sketched in Fig. 10.2, 8 for the same values as in Fig. 10.2, 5, for Ao = 0(that is, because of (10.2, 4), for a = 1/3000), as well as for A,, = 0.117 (that isa = 0.006) and Ao = 0.167 (that is a = 0.012).

Vice versa, (10.2, 14) yields the amplitude Al, amplitudes greater than which ariseonly with given probability p, as the (smallest positive) solution of the equation

n2 (b + I dAll2 14 e(AV - Ai) + 12e AV + n3 d12 = - 2 In p .


Examples of these amplitudes are given in Fig. 10.2, 8, at the points of intersection ofthe probability curves with the (dashed) line p = 0.1 and with the (dashed-dotted)line p=0.2.

Up till now we have investigated the first or main parametric resonance. We nowtake into consideration a general k-th parametric resonance

A = (kn)2.

The integral equation method leads (compare SCHMIDT (1975)) to the excitation partsof the iterative solutions

y2 r cos (k - 4) nz + s sin (k - 4) nzn4 [00 - (k - 2)2] [#k2 - (k - 4)2]

+rcos(k+4)nr+ssin(k+4)nt[k2 - (k + 2)2] [k2 - (k + 4)2]

1 1 r cos knr + s sin knz-[ k2-(k-2)2+k2-(k+ 2)2 k2

Insertion into the periodicity equations yields the amplitude frequency formula

where

2n2 Pk2 - (k - 2)2

rcos(k-2)n-e+ssin (k-2)n2y

+rcos(k+2)nz +ssin(k+2)nzlk2 - (k + 2)2

s 4y =

00 2

2n2a = 4 A2 + Gy2 + O(y4) ±vE

SvQvy2V - n2 (b -;4

dA2l (10.2, 16)

1 1 1

4n2 Ok2 - (k - 2)2 + k2 - (k + 2)21G=


and

G =1

v 2vnhv-2[v2 - (v - 2)2] [v2 - (v - 4)2] ... [v2 - (4 - v)2] [v2 - (2 - v)2]

with v - 1 factors in brackets in the denominator, especially

_ 1 _ 1 1Gl

202 16n2 '

G3 _512n4

The term O(y4) is not specified because it is not needed in what follows. Formula(10.2, 16) includes (10.1, 7) for k = 1. The threshold condition, generalizing (10.2, 1),is now

Gkyk > n (b +4

dA2l ,

and instead of (10.2, 5) the formula

/ 2 2 2 2 2Gky2k = n2I b dA2) + 4 e(Av - A2) + 12 Ao + n3

holds.Confining attention, for simplicity, to a Rayleigh probability density (10.2, 7) of

the excitation amplitudes, we getL1/k

W(A) _ -e2a=G2.1 k

and instead of (10.2, 12)

L1/k

2a2Gk'kw(A) _ (9e2 + n2d2) A(A2 - Av)e

g k62Gk1 kLl -1/k

In the same way formulae corresponding to (10.2, 11), (10.2, 13) can be found, whichcan be discussed quite analogously. It can be shown that the probability densitiesfor k > 1 already tend to zero for much smaller amplitudes than for k = 1.

11. Vibration systems with broad-bandrandom excitation

11.1. The amplitude probability density

In Chapter 10 we have assumed a narrow-band random excitation (10.1, 2) for whichthe bandwidth of the spectrum is much smaller than (nb)-1. In this chapter we considerthe opposite extreme, that the bandwidth of the excitation spectrum is much greaterthan (nb)-1. Here the apparatus of Marcov process theory is applicable (compare forinstance STRATONOVICH (1961), M. F. DIMENTBERG (1980)). Stratonovich assumesapproximately white noise excitation processes and thus gets results depending on theexcitation spectral density for special values. Following MrrROroL'sKls and KoLOMIEC(1976), WEDIG (1978), M. F. DIMENTBERG (1980) and other recent investigations, weconsider ideal white noise excitation processes, generalized derivations of Wienerprocesses, and are thus able to use the elegant mathematical tool of Ito equationcalculus, developed for instance in GICHMAN and SKOROCROD (1968, 1975), CHAS'MIN-sKIJ (1969) or ARNOLD (1973).

As a rather general equation of the type (5.1, 1), we investigate the vibration equa-tion

x" + ao2x = F

F = F(x, x', x", t) gj(t) x - bx' - dx2x' - ex3 - hx(x'2 + xx ")(11.1, 1)

where fi(t) is a white noise random process, f, g are the coefficients of forced and para-metric excitation respectively, b > 0 the coefficient of a linear and d > 0 that ofa non-linear damping, e the coefficient of a non-linear restoring force and h that ofa non-linear inertia force (compare, for the occurance of such forces, BOLOTIN (1956)and Chapter 12). The coefficients f, g and b are assumed small in the sense of theiterative procedure. For vanishing forced excitation, equations of this type havebeen investigated by STRATONOVICH (1961), NIKOLAENKO (1967), BAXTER (1971),SCHMIDT (1978), and DIMENTBERG, Is]Xov and MODEL (1981) and others; for vanishingparametric excitation see KtHNLENZ (1979). Simultaneous forced and parametricexcitation has already been considered by DIMENTBERG and GORBUNOV (1975) andMODEL (1978 a, b). For the following results compare SCHMIDT (1981 a).

Corresponding to the investigation of narrow-band excitations in Section 10.1,we may introduce by

x=acosp, x'_ -wasincpthe amplitude a > 0 and the phase

92 =a>t+t9

11.1. The amplitude probability density 337

as random functions of time, which leads to the standard form

-wa' = F sin 92 , -wa8' = F cos 99 .

We understand these equations as physical or Stratonovich equations, compare forinstance ARNOLD (1973). If they are written in the form

dyi = mi dt + ni i = 1, 2

with yl = a, y2 = 0 and d$ = fi(t) dt, the corresponding Ito equations are2

dyi = (mi + /ti) dt + ni d$, Pi = 1 Ydni

n1 .2 J=1 dy1

The additional Ito terms can be found to be

2

,u1 =4(02a

(1 + cos 299) -4(02

(3 cos 99 + cos 399)

2a

+16(02

2

142 = -__

sin 2p + (sin q) + sin 3q7) - 8w2 (2 sin 299 + sin 4q,) .

Thus the Ito equations can be written, after trigonometric transformations, in theform

dO=rdt+8d$ (11.1,2)

where

p=

q =

r =

8 =

ba ba da3 da3 eat e3 sin 4V

2+

2cos 24p -

8+

8cos 4q9 +

4wsin. 2p + a a3

- w a3sin 4(p +

4(t)/2 /2

2a + 4(02acos 2q9 - - -COST - 4/2 cos 3qi

+3g2a

+g2a

cos Zp16(02 40)2

- L sin (p + 2ual sin 299 ,

b da2 da2 3ea2 ea2 ea2-2

sin 299 -4

sin 297 -8

sin 499 +8uo

+2w

cos 2q9 + 8w cos 4T

2 2 2 /2sin

tgsin Q,

- (04 - w2 a cos 2q - 4a cos 4p -2w2a2

2g + 2w2a

2 ,+1-a sin 399 -4922sin2g9 - g92

2sin4q

-w

cos c +2w

+ 2w cos 299

The expressions in p and r depending on / and g correspond to the additional Ito terms.In F, in the sense of the iteration method x" is replaced by -(02a cos q).22 Schmidt/Tondl

338 11. Vibration systems with broad-band random excitation

As the basis for iteratively solving this equation, introduce, corresponding withSection 10.1, a transformation of amplitude and phase

a=A+E(A,0), cp=0+B(A,0) (11.1,3)

with small correction functions E and a which have to be suitably chosen. For thetransformed phases, let

0=cot+0, thatis, 0=p+8(A,0). (11.1,4)

Our goal is successively to determine the stochastic differentials for the randomprocesses A and O, written in the form

dA=Pdt+Qd , dp=Rdt+Sd Q. (11.1,5)

The stochastic differential of the process e. = E(A, 0) is, following of Ito's formula(ARNOLD (1973)),

as as a2E a2E I a2EdE =

aA+ ao dO +

2 aA2Q2 + aA aO QS + 2a02 S2 dt.

Therefore (11.1, 3) yields with (11. 1, 4), (11.1, 5), (11.1, 2),

as as I 2

dA -w a aAP

TOR

2 aA2Q2

a2E 1 a2E

aA aO QS - 2 a2s S2 -{- p(A -{- e, 0 + 6) dt

-I- L- aAQ - as S -{- q(A -}- E, 0 + 6) d$. (11.1, 6)

Analogously we getJJ

as _ as as 1 a2sP -R --a0 aA a0 2 aA22 2

aA QS- 2a22S2+r(A+E,0 +6) dt

a(P

+I-aAQ-abS+s(A+E, 0+6)Id$

With the notation (11.1, 5) the Fokker Planck Kolmogorov equation for the two-dimensional probability density w(A, O; t) of amplitude and phase reads

aw a(Pw)+

a(Rw) - 1 a2(Q2w) a2(QSw) 1 a2(S2w)

at + aA a0 2 aA2 + aA a0 + 2 ape

as is shown, for instance, in MrrROroL'sKIJ and KOLOMIEC (1976). We confine ourselvesto the important stationary case aw/at = 0. But even then no closed solutions of thisequation are known. Therefore we use the averaging method for iteratively evaluatingthe solution of (11.1, 7). For discussion on, and generalization of, the averaging methodcompare EBELING and ENGEL-HERBERT (1982) and EBELING, HERZEL, RICHERT, andSCHIMANSKY-GEIER (1986).

In the first approximation, which has already been evaluated by DIMENTBERG andGoRBuNov (1975) and MODELL (1978) for / = 0, the averaging method without cor-rection functions is used. In other words, for the solution of (11.1, 7) in P, R, Q2, QS,


S2 consider only the terms already found in (11.1, 2) and from these only the oneswhich do not oscillate, that is, which do not depend on 0 and which we can thereforeget by averaging with respect to 0. Correspondingly (STRATON0vICH (1961)), we canassume also that the probability density is a function of the amplitude only. Thus(11.1, 7) simplifies to

d=0dA

where

and

T= Pw -2 dA (Q2w)

__ _ 1392 - b 1 _ d 3P Pl4(02A + 1-6-0 2 A 8 A'

{2 2A2Q2=Q = 2+ g 82

2w w

Assuming W --> 0, w -> 0 for A --> oo, and therefore that ! - 0, we arrive at, ifwe use the abbreviation

22P-adA

J = Q2

the probability density

w=Cef'dAFor vanishing forced excitation, 1 = 0, comes

8w'b w0d1- s - fw = CA 9 e

and for vanishing parametric excitation, g = 0,_wm d'- A.

w = CA e 7' 8f'

In case g + 0 a partial fraction decomposition yields

4/2 + (g2 - 8(t)2b) A2 - 2w2dA4J =

therefore

A(4/2 + g2A 2) I

412 + (g2 - 8(t)2b + 8(t)2d/2 A22w2d g2

92 A + A(4/2 + g2A2)

2w2d 1- d12 AA+8w2 b--92 A 92 412 + 92A2

2 2 {292A2)f 9

J dA = - d A2 + In A - 4- (b 2 g/ ) In 1 -} 2

\\ J 4

(11.1, 8)

(11.1, 9)

(11.1, 10)

(11.1, 11)

(11.1, 12)

(11.1, 13)

22

340

and

with

11. Vibration systems with broad-band random excitation

4w'b 4w'df

4/2) 9° + 9` - w=da daw = CA (A2 + s e 9

g

4w'b 4w'df

(4/2g1

9'-

9'C-

2J C.

(11.1, 14)

The constants C respectively C have to be determined from the normalization condi-tion

00

f w dA = 1 . (11.1, 15)0

Formula (11.1, 10) is not included in (11.1, 14) for g 0, in connection with thesubtraction and addition of terms in (11.1, 12) - the first one respectively the lastone on the right-hand side - which tend to infinity for g --> 0. But if we heuristicallydevelop in (11.1, 13)

2A2)In l+g4f2

into a power series we get for

g2A2< 1

4/2

the formulaw'bAS-w3 Id-b9' 1d' 1g4'+1 9'A'-+-...

w = CA e 'f' ) 2 4f' 3 (4f')' 4 (4f')' (11.1, 16)

which now does contain (11.1, 10) for g -> 0. An exact verification of (11.1, 16) has tostart from (11.1, 11), to develop

1 _ 1 1

4/2 + g2A2 4f2 g2A21 +

412

into a power series and to integrate term by term.The formulae thus found reveal how the excitation coefficients f, g as well as the

damping coefficients b, d influence the probability density. Non-linear damping, aswell as linear damping for vanishing parametric excitation, cause the exponentialdecrease of the probability density and hence the existence of a positive normalizationconstant. But the influence of restoring and inertia force non-linearities with thecoefficients e and It respectively remains undetermined.

In order to determine this influence, we apply in what follows the second approxi-mation of the averaging method. For the evaluation of the second approximationP = P1 + P2, first determine in the expression for P, that is, in the first expressionof (11.1, 6) in brackets, the term of highest order of magnitude -co(ae/aO) such asto compensate the terms depending not on s, 6, but on 0 (which are omitted in first


approximation by averaging). Integration yields, up to an arbitrary function of theamplitude chosen for simplicity to be equal to zero (compare p. 325)

= bA 20 + dA3 sin 40 -eA3

cos 20 -eA3

E sin cos 404w 32w 8w2 32w2

+g2A sin 20 +

9z14sin 40.

8w3 64w3(11.1, 17)

Analogously, the counterbalance of the terms in R depending not on e, 5 but on 0by -w(a5/aO) leads to

5oU

cos

2 2

4 isin4

2

20 - 16 +4(t)3A2

- 19 cos 0 - - f9 cos 30 + 92 cos 20 + 92 cos 40.2w3A 6w3A 80 32w3

(11.1, 18)

The correction functions s and 5 produce numerous further terms in P not dependenton $. If we insert in the first expression of (11.1, 6) in brackets the first approximationof P, R, Q, S and transform the last expression as

(A + E)3 cos 4(0 + 5) = A3 cos 40 + 3A2e cos 40 - 4A35 sin 40,

we get for the additional terms of second approximation:

as as 1 a2E 2 82e 1 a2E 2p2aA

P1 - 2O 2 aA2Qi aA 60 Q1S1 2 a02

S

- be - b e cos 20 - bA5 sin 20 - M A2e

+ MA2e cos 40

2 2 8 8

dA36 sin 40 .+

3eA2 E sin 20 +eA3 6 cos 20 + 3eA2

a sin 402 4w 2w 8w

3

+ eA 6 cos 40 - 3whA2 s sin 40 - whA35 cos 402w 4

/2 z

E E cos 20 - 5 sin 204w2A22(02A4w2A2

3fg K 2

+ - 5 sin 0 + - 5 sin 30 +6w2

e + 4w2 E cos 2040 4(02

-92A6sin20+-g2Ecos40-92A6sin4o,

2w2 16w2 4w2

where again only the averaged terms (that is, the terms which do not depend on 0)are taken into consideration. The terms now neglected by averaging are of a higher


order of magnitude with respect to the small parameters than the terms neglected infirst approximation. Using (11.1, 17), (11.1, 18), we can show that

_ hbA3 deA5 9C/2A h f 2A 25eg2A3 9hg2A3P2

8 + 32032(04 + 16(02 128w4 + 64w2

In the same way, (11.1, 6) leads to the additional expression of second order

as as 64Q2 aAQ1

851 - 6 cos 0 + - sin 20 + g- b cos 20

from which we get, by using (11.1, 17), (11.1, 18) and averaging,

Q2 = Q2 2+ 2Q1Q2 + Q2

where

2Q1Q2 { Q2 = - 16w4 8w2 64w4 +32w2.

When we insert P = P1 + P2 and Q2, the integrand (11.1, 8) reads

-lA6 - MA4 - NA2 + GA(rA4 + SA2 + G)

with the abbreviationsj2 2d 2 g2

M= +m, N=b- +n, +$82w2 2

where small letters indicate terms of second order :

_l

de _m hb 5eg2

+5hg2 = 3ef2

n3e/2

16w2' 4 64w4 32(02 16w 8w2

_r

5eg2 hg2

+3e/2s _ hf2

64(04 32(02' 16(04 8w2

(r, 8 not to be confused with the coefficients of the Ito equations (11.1, 2)).If we consider a non-vanishing parametric excitation, g = 0, decompose into partial

fractions, set (neglecting, as above, terms of higher order)

1 _ 1 rA4

rA4 + SA2 + G SA2 + G (SA2 + G)2

and assume, because of the first approximation, S > 0, we come to

2TA 2ZAJ = -2UA - 4VA3 + I + 2 'A2+S (A2+.)

Susing the abbreviations

UM Nr+Gl+G(M+r)r .v_lS - (M+r)r2S 2S2 83 4S2

_ I N G(M - r) G(G1 + 2Nr) 3G2(M + r) rT

2 2S + 282- 283 + 284

ZG2rNS+S2-G(M+r)2Sb

3e/2A2 h f 2A2 5eg2A4 hg2A4

11.2. Statistical properties of the vibrations 343

The probability density yields

r- T T - UA'-VA'--w=CA A 2 + 8 )) e

zA+G/S (11.1, 19)

This reduces to (11.1, 14) if the additional terms of second approximation writtenwith small letters are put equal to zero.

In case of vanishing forced excitation, f = 0, (11.1, 19) simplifies to the formula

2

w = CA2T+1 e-aA'-vA4 , 2T + 1 = -SN = 1 - 80)b

. (11.1, 20)y2

A comparison with (11.1, 9) shows that the second approximation leads to additionalterms only in the exponential expression.

All probability density formulae derived contain only the second power of the forcedand parametric excitation coefficients /, g, from which it follows that the signs of thesecoefficients do not influence the probability density.

11.2. Statistical properties of the vibrations

The normalization constant can be evaluated from (11.1, 15) by numerical integration.Analytical formulae can be found in the following two special cases.

First special case. Vanishing forced excitation, / = 0. By use of the integralsCO

f xv-1 e-ex-ax' dx = (2a)-'12 F(v) ep'/8a D-r (a> 0, v >0) (11.2, 1),J }/2a0

with the parabolic cylinder function D and the Gamma function I' and

00

f x°-1 a-ex dx = Q-°1'(v) (Q > 0, v > 0) (11.2, 2)0

(compare for instance GRAD§TESN and Ry2iK (1971), p. 351 and 331) we can find

1 -C

00

f A-N/Se-GA'-VA4 dA

0

1

I

1(2V)(Nl4S)-1/4I' (±. - N

e°18vDU

2 2 2S(N12S)-112

2 P

2U(N/28)-1/2 j'r

2 - 2S)for V = 0 , U> 0

for VSO,

(11.2, 3)

under the condition that the argument of the Gamma function is positive, in otherwords, that the threshold condition

g2 > 4co2b (11.2, 4)

for parametric excitation to exceed a certain damping threshold value holds.


Second special case. Non-vanishing forced excitation, but V = Z = 0, especially thefirst approximation. If we substitute

x=A2+S

and use the integral00

f'x°-le-exdx=P-T(v,Loz) (T >0,Lo >0), (11.2,5)

F(v, ,u) being the incomplete Gamma function (compare for instance GRAD§TEJN andRYkEK (1971), p. 331) we get

CO

1 = 2 e(GU)IS (xT a-ax dx = 2 ecau>Is U-T-1I'(T + 1, -), (11.2, 6)C J

ads

where the conditions G/S > 0 and U > 0 can be assumed valid because of the firstapproximation. By means of the formula

00 (-1)n zT+n+lT(T+1,z)=r(T+1)-nEOnI(T-fin

{-1)(11.2,7)

in GRAD6TEJN and RYZIK (1971), p. 955, we can substitute for the incomplete Gammafunction the (simple) Gamma function if T $ -1, -2, ... For f --> 0, formula (11.2, 6)simplifies because of (11.2, 7) to the second formula (11.2, 3).

The derivative of the probability density (11.1, 19) is

dAC (A2 + S)T

-2L1 - 2UA2 - 4VA4) (A2 -}-

S)

Z

-+- 2TA2 (A2 + G +-

2ZA2je- UA-- VA-- A=+c/s

J

(11.2, 8)

The condition for extreme values, the vanishing of the expression in brackets, is anequation of fourth degree in A2.

In the first special case of vanishing forced excitation, the threshold condition(11.2, 4) has to be fulfilled, otherwise the probability of positive amplitudes is zeroand no vibration exists. We have

w(0) = oo for g2 < 8w2b

and

w(0) = C for g2 = 8w2b ,

in both cases no extreme value of the probability density exists and it decreasesmonotonicly for A increasing (left curves in Figures 11.2, 3 and 11.2, 4). Correspond-ing to STRATONOVICH (1961), we speak of an "undevelopped vibration".

In contrast,

w(0) = 0 for g2 > 8w2b .


In this case a maximum of the probability density exists, that is, there is a mostprobable value A. which is determined, because of (11.2, 8), by

4VA2 = - U -{- U2 + 4 (1 - 8(02b 2b

/V for V > O ,

\ 9

A2 92 8w2b 1 g2 - 8w2b - (11.2, 9)2g2U 20)2d+4m+4(I- 8w2b\r

g2/I

for V=O, UFO.Here the condition U > 0 is valid because of the predominant expressions of firstapproximation. Since T < 0, the derivative of the probability density tends to infinityfor A - 0 so that the probability density already assumes great values for smallamplitudes (half developped vibration).

For non-vanishing forced excitation, (11.1, 19) always yields w(0) = 0 so that oneor two maxima of the probability density exist. The derivative of the probabilitydensity is finite, equal to

-C (G) e-(ZS)'' for A = 0

so that the probability density assumes great values only for greater amplitudes(fully developped vibration).

In the second special case V = Z = 0, exactly one maximum follows from (11.2, 8),.that is, a most probable amplitude A. is given by

2UA,2n= Y+1/Y2+2GUV S

if

Y=T- 2-GSis written as an abbreviation. For Y = 0, that is, in first approximation if

g2(g2 - 8w2b) = gw2d f2

then in particular

A 2 G 2 If1 - IgI g2 - 8(,)2bm = -2SU V-d 2w2d

where the arrow indicates reduction to first approximation. For

Y2 2GU

S I

we get by series expansion

A2 = G + U, if Y>O,2SIYI

0, if Y<0g2 - 8(02b

,4/2 if g2 > 80b+ 2(02dI92 - 8w2bI

10, if g2 < 8w2b ;


that is, for / --> 0 the most probable amplitude tends to zero if g2 < 8w2b and to thepositive value given by (11.2, 9) if g2 > 8w2b.

Examples for the probability density (11.1, 19) are given in Figures 11.2, 1 to11.2, 4 where w = 1, e = 4, h = 0 and b = 1/2000 have been chosen and the normali-zation constant has been evaluated by numerical integration. For / = 0.01 andg = 0.06 (fully developped vibration), Fig. 11.2, 1 shows how a greater non-lineardamping d diminishes the probability of greater amplitudes; Fig. 11.2, 2 gives ford = 0.2 the difference between first (dashed line) and second approximation (fullline) ; the second approximation also somewhat diminishes the probability of greateramplitudes. Fig. 11.2, 3 shows (for d = 1 and g = 0.06) how the probability of greateramplitudes increases with forced excitation (left curve / = 0, undevelopped vibration,

w

6

4

I

0.1

Fig. 11.2, 10.2 0.3 0.4. 4 0.1

Fig. 11.2, 2

W4

10

8

6

0.1 0.2

Fig. 11.2, 30.1

Fig. 11.2, 402

0.2

0.3

0.3

04

0.4 A

0.5 A


fully developped vibration for the other curves). An analogous influence of parametricexcitation reveals Fig. 11.2, 4 for d = 0.2, / = 0 and g = 0.06 (undevelopped vibra-tion), g = 0.08 respectively g = 0.1 (half developped vibrations).

The momentsm

mk = f Akw(A) dA, k = 1, 2, 3, ...0

of the amplitude can be evaluated by help of the probability density.ishing forc d excitation we get the formulaFor the first special case of van e

r(k+1-N)DNk+1G79T=7)U

2 2S s -, -

S)N 1(U

(2V)kl4l'(2 - 2 2S 2 V 21Tmk = rk+1N)

I\ 2 if V=0, U>0.Ukl2r

1

(2 2S)

The second moment or the mean square amplitude is because of r(x + 1) = xr(x)

(2 2S)Ds-2 \ 2V)if VSO,

( U2VDx 1$ = {a

V2 V2s

_2 VM2

S-N 1 g2-4co2b if V=O, U>02SU w2 d+4m+4(1 - 8w2b

2 1r9 /

from where follows the rms value j/m2.As forry=0

N k+ 1 k 4c02b -2 if b= 0, k=2,28 2 2

+ -92 -1 if b=0, k=0

and because of GRAn6TEJ1 and RYi1E (1971), p. 1081 and 1108,

D-1(z) = ez1141/2

[i

(11.2, 10)

(11.2, 11)

D-2(z) _ -ez2I'4V

z{z f1 - $ (-)] -

V- e-z'/2} (11.2, 12)

hold, in the case of vanishing linear damping and V > 0 the mean square amplitudecan be expressed only by means of the r function and the error function

z

f e-E' d .

Y 0

(11.2, 13)


Because the argument U/ 2 V of the parabolic cylinder function is of the order ofmagnitude 1/g, the asymptotic formula (compare MAGNus and OBERJETTINGER (1943)p. 92)

4z)(v-3)--F...1

D,.(z)~e_z,/4z' Cl-v((v -

for z )' l , z > jJJvj

can also be used for the evaluation of the moments.For V = 0, U > 0, formula (11.2, 10) immediately shows the influence of the

system parameters on the mean square amplitude. For example, the influence ofpredominant order of magnitude of the parametric excitation g and the linear dampingb appears in the numerator which is

equal to gz for b = 0 ,z gz

andequal tog2

for b =8(t)2

equal to zero for b = 94)z .

The dependence on non-linear restoring force e and non-linear damping d, say, re-presents the formula, written for h = 0, g2 = 5wzb:

mzb

(11.2, 14)d + 85e

64wz

Examples of the diminishing of the mean square amplitude by the non-linear restoringforce and the non-linear damping are given in Fig. 11.2, 5 for b = 0.01, CO = 1. Incase e = 10, b = 0.01d, the influence of e and by this of the second approximationcauses a diminishing of the mean square amplitude by about 12 per cent.

0.12

0.10

0.08

0.06

0.04

0.02

Fig. 11.2, 5

d=0.3

d=1

5 10 15 20 e

11.2. Statistical properties of the vibrations

The first moment or the amplitude mean value is for V = 0, U > 0

_r (3 4(02b)

2b)m1 _I9I

Vd+r(i_ 4b) 4m-f-4(11 \ l

from which there follows with (11.2, 10) the dispersion

D=m2-miof the amplitude:

g2r2 13 - 4g4(t)2b)

21 12 4 2b- w -D- 9

349

\w2 d + 4m + 4 (1 - f r f2 (1 - 42b

\\ g2 l \ g2

In the second special case V = Z = 0 and non-vanishing forced excitation we derive,by using the integral

00

(' v-a-1 a-v-1 se

x°-1(x + T)-I e-ex dx = T 2 Q 2 e2 r(v) W 1-v-a v-a (,ce)22 '

0

(e > O,v>0)(compare for instance GRAD TEJN and RYZJK (1971), p. 333) and G/S > 0, U > 0, theformula

Mk =G k/4 k

WT/2-k/4,T/2-f-1j2-{-k/411\\G(C

(SU) r(2 + 1) W GU1T/2, T/2+1/2 su

ment of these functions is in thewhere W,u,(z) are the Whittaker functions. The argfirst approximation

GU 4cw2df2

S g4

This can often be assumed as large in comparison with 1 so that the asymptotic devel-opment (MAGNUS and OBERHETTINGER (1943), p. 89)

W)2} ... [ ,2 - (,u - n + 2 )2]IV' - (du - 2 )2J IV' - (it 2

µv(2) eZ'/2 vu 1 En=1 nl zn

(11.2, 15)

can be used. The first two terms of this development lead to the approximative formula

m =r( + i) GU+ k

2

2ST

Uk/2 GU + ST

where the second summands in the second fraction and by this the second fractionas a whole stem from the second term of the asymptotic development (11.2, 15). In


particular, the mean square amplitude, using the expression for the first approxima-tion, yields

M2 ='1 GU + 2ST , g2 3df2 - 2g2b

2 U GU + ST awed 2df2 - g2b

A comparison with the opposite limiting case / = 0 of formula (11.2, 10) for, say,vanishing linear damping, b = 0, now (for predominant forced excitation f) resultsin one and a half times the value following from (11.2, 10) for f = 0.

The probability

W`=W(A1<A<oo)

of exceeding a given amplitude level At is00

1VI = f w(A) dA .

In the second special case V = Z = 0 (vanishing forced excitation, f = 0, included),especially for the first approximation we can find, by analogy to (11.2, 6), that

Wz = 2 e(GU)IS U-T-lr(T + 1, AI U -fGU)

.

Insertion of (11.2, 6) yields

r(T +1,Alu+GS )Wt = (11.2, 16)

r (T+ 1' GS )

In evaluating Wt, formula (11.2, 7) can be used.In two examples, (11.2, 16) simplifies significantly. For T = 0, that is in first

approximation for

d/2=bg2 (11.2,17)

because I'(1, z) = e-z the formula2

Wt = WIl = e'-d1 (11.2, 18)

holds. On the other hand, for T = -1/2, that is in first approximation for

df2 = (b_)/2802

because

the formula

WI=W12=1-0(1IAIU+GS

)

1 - (' 1GS1

holds.


A comparison of these two examples clarifies the influence of different system para-meters. For vanishing forced excitation, / = 0, the last formula simplifies to

W12 = 1 - 0 (A1 VU) ,

while the condition T = -1/2 simplifies in the first approximation to2

b 8w2, /=0.

0.25

Fig. 11.2, 6

0.5 0.75 1 1.25 1.5 A, 1U

(11.2, 19)

(11.2, 20)

The curves given by (11.2, 18) for T = 0 and by (11.2, 19) for T = -1/2, / = 0are drawn in Fig. 11.2, 6 in general form, without confinement to the first approxima-tion. In the first approximation we get

cw2dU= a ,9

(11.2, 21)

and the conditions for W12 are (11.2, 20), whereas the condition (11.2, 17) for W11 holds(say) for b = / = 0 or for

2

b 8w2,2 = 9

4

/

8w2d(11.2, 22)

In other words, W12 shows, in comparison with W11, the decrease of the probability ofexceeding Al if T = 0 changes to T = -1/2, in the first approximation if the lineardamping increases from zero to the value given by (11.2, 22) and if the forced excita-tion vanishes. On the other hand, W11 gives, in comparison with W12, the increase ofthe probability for exceeding Al if particularly in first approximation the forcedexcitation increases from zero to the value given by (11.2, 22) and

b= 92

8w2

holds.The influence of non-linear damping d manifests itself - in the first approximation

and for vanishing forced excitation, / = 0 - only in (11.2, 21), that is, in a modifica-


tion of the abscissa scale. For instance, if the non-linear damping increases to thep-fold value, U increases correspondingly, so that the probability of exceeding thediminished level

At

remains the same; or, in other words, for a fixed level At, the probability of exceedingdiminishes by the additional scale factor }lp. Fig. 11.2, 7 shows how the probabilityW11 (say, for / = b = 0) diminishes from the dashed line to the full one when thenon-linear damping assumes the four-fold value. Fig. 11.2, 8 gives the correspondingdiminution of the probability W12 (say, for / = 0, b = g2/8(02) when again the non-linear damping assumes the four-fold value.

Fig. 11.2, 7

WZT

1 lip

0.8

0.6

0.4

0.2

\\

2

0.25 0.5 0.75 1

Fig. 11.2, 81.25 1.5AifU'

11.3. Non-stationary, transition and two-dimensional probability density 353

In order to discuss the influence of the other non-linearities, say of non-linear inertiawith the coefficient h, by means of the formulae of the second approximation, wehave first to take into consideration the conditions V = Z = 0 of (11.2, 10). Thesecond condition leads, approximatively, to the condition (11.2, 11) for W11. Choosefor instance b = f = 0, so that instead of (11.2, 15) the formula

U-c"2d_ h

g2 2

holds. The condition V = 0 yields

3e = 2co2h .

Consequently, non-linear inertia, as well as non-linear damping, leads to an additionalscale factor. Fig. 11.2, 9 presents an example for a scale factor 0.9 which follows forinstance for non-linear inertia h = 9.5 and co = 1, d = 1/4, g = 0.1 and which causesan increase of the probability W11 from the values of the dashed line to those of thefull line.

W,

0.8

0.6

0.4

0.2

0.25 0.5 0.75 1 1.25 1.5 41VU

Fig. 11.2, 9

11.3. Non-stationary probability density, transition probabilitydensity and two-dimensional probability density

On the basis of the stationary probability densities thus found, we consider in thissection the dependence of the probability density on time, in other words, the firstexpression in the Fokker Planck Kolmogorov equation (11.1, 7). We use a methodintroduced by STRATONOVICH (1961), compare also Nikolaenko (1967).

Corresponding with Section 11.1, write the Fokker Planck Kolmogorov equationin the form

8w _ a(Pw) 1 82(Q2w) 8yjat 8A + 2 8A2 8A

23 Schmidt/Tondl


First, assume the probability density as a product,

w(A, t) = w(A) ry(t) .

Separation of variables yields

d(p

dt _ 1d(dPAw)

+ 2 d2 w)J = const .IL 2 _J

If we write -A for the constant, we get

99 = e-a(t-to)

as well as the differential equation

Hd2w+Idw+Kw=0dA2 dA

for w, using the abbreviations

H 2Q2,

2

dA- P ,

K2

dA2dP

-I 2

This differential equation can be simplified by writing

w(A) = E(A) v(A)

and introducing the variable

a=xA2,that is,

ddA = 2xA da ,

d2 d_ 2m + 4x2A2

d2

dA2 da da2

(11.3, 1)

(11.3, 2)

(11.3, 3)

(11.3, 4)

(11.3, 5)

with a constant x which we will fix later on. If we use the notation

v(A) = v(a) ,

we come to the differential equation

2

4x2A2HE da2 + 2x (2AH dA + AIE + HE)da

z{ HdA2-}-I+KEly=O. (11.3,6)


Choose E in such a way that the coefficient of dv/da vanishes,

2AH dA = -(AI + H) E, (11.3, 7)

from where follows, assuming AH + 0, the expression

E = ef (2s+2A) (11.3, 8)

for E. Substituting in (11.3, 6) the derivatives of E by (11.3, 7) and the formuladerived by differentiation of (11.3, 7), and considering the abbreviations (11.3, 4) and(11.3, 5), we get the differential equation

4a2d2a2+Ir=0(11.3,9)

using the abbreviation

A2P2

- - A2P2 A2P - 2A22dA-H = - + dA

44

Q Q2

Insertion of the formulae derived in Section 11.1 reveals the dependence of the expres-sion H on the amplitude,

- 3+ 1 G + (4A + N') A2 + 3M'A4 + 51A6 1

4 2 G+SA2+rA4 +4(G + SA2 + rA4)2

x [-G2 + 2G(28 + N') A2 + (2GM' - N'2 - 4N'S + 80r) A4- 2(M'N' + 2M'S - Gl + 4N'r) A6 - (M'2 + 2N'1 + 4S1 + 8M'r) A6

- 2(M' + 4r) 1A1° - 12A12]

using the former abbreviations and

M'd hb 25eg2 9hg2

4 4-

64w4 32w2 '

g2 f2 {2

N' = b -g22 + 961w4 glw2

differing from the corresponding expressions M, N in second respectively first approxi-mation.

If forced excitation does not vanish, we can, in connection with the predominanceof small amplitudes A, in general assume that

G > SA2 + rA4.

Developping the denominators of H into series leads to the expression

II+N' +S +2A A2 -N'2+952+ ION'S -8GM' -8Gr+8SAA4G 4G2

6G21 - GM'N' - 7GM'S - 7GN'r - 15GSr + N'2S + 7N'82 + 2S3 - 4GrA)+2G3

x A6 + O(A8) . (11.3, 10)

23'


If we neglect the amplitude terms of sixth and higher degree in (11.3, 10) and choosethe free constant x such that

2G V N' 2 + 9S2 + ION'S - 8GM' - 8Gr + 88A ,

we get the differential equation (11.3, 9) for v in form of the Whittaker equation

d2v4a2 - = (a2 - 4pa + 4q2 - 1) vdal

with

N' + S + 2A0

4Gxq = .

(11.3, 11)

Following the eigenfunction method of STRATONOVICH (1961), we use the eigenvalues

A=A,, (v=0,1,2,...)

and corresponding eigenfunctions

(v=0,1,2,...)

for given boundary conditions, in our case vanishing values for A = 0 and A , oo.The first eigenfunction for A0 = 0 is the stationary probability density

wo(A) = wgtet(A) (11.3, 12)

as comparison of (11.3, 2) with the stationary case of Section 11.1 shows.The unstationary probability density can be written in the form of the eigenfunction

development00

w(A, t) = Cowo(A) + E Cw,,(A) e-x"(' 6 (11.3, 13)v=1

generalizing the assumption (11.3, 1) of one product. Integration of (11.3, 3) leads,for Ay 4 0, to

Co

f w,,(A) dA = 0.0

Under the assumption that no two eigenfunctions have the same eigenvalue,

A., + A,, for a +v,we shall now establish an orthogonality relation. The differential equation (11.3, 3)reads

Aµw ,

aAPLWJ=A,w,


for two different eigenfunctions. Multiplication by w"/wo respectively w"lwo and inte-gration yields

('f w" a '[w,] dA = A,, J w

°

" dA ,

./(' ° (11.3, 14)

J J n a IF[w"] dA = A" W" W" dA .

0 J 0

The expression

J = f z aA Y7[woa] dA

f 2

(Q2woo) dA ,z aA (Pwa) dA - 2 z U-2

a and z being arbitrary functions, can be written in the form

(11.3, 15)

a as 1

J

a2J = za aA (Pwo) dA + z Pwo

aAdA - 2 za

aA2(QZwo)

(iA

z

zaA aA

(QZwo) dA - 2J

zQ2wo aA2 dA .

The first and the third expression cancel out because wo is the stationary solution. Inthe same way, the fourth expression can be written as double the second one withnegative sign. Thus we get

fpwo= - aA dA 2 zQ2wo aAz dA.

Integration by parts yields

('2 (zQ2wo) a dAJ aA (zPwo) or dA - 2

J'2

aA2

= f a P[woz]a dA

aA

if we assume

as azz-=a-aA aA

or

xWW[woa] = aWf [woz]

on the boundary. When we choose

a = w", z = w"

(11.3, 16)

(11.3, 15) and (11.3, 16) show that the left-hand sides of the two equations (11.3, 14)are equal. The right-hand sides give the equation

(A,,-A") J "dA=6,w"wWO


that is, the orthogonality of the eigenfunctions belonging to different eigenvalues withweight 1 /wo. Normalizing the eigenfunctions by

f w2I 'dA=1,

,1 wo

we can write the orthogonality relation in the form

fJMultiplication of the eigenfunction development (11.3, 13) by wµ/wo, putting t = to,

integrating and using the orthogonality relation gives

C = r w(A, to) w,(A) dA .J wo(A)

When the initial distribution is the Dirac delta function,

w(A, to) = b(A - A0) ,

then the eigenfunction development (11.3, 13) is the transition probability density

°O w ,(A) w(A0) -a;.(e-t')p,.(A, AO) _ I ve

v=o wo(Ao)

where Ao = 0. By means of the transition probability density and the initial distribu-tion, all distributions of finite order can be evaluated. In the case of a stationaryinitial distribution, the two-dimensional probability density

w.(A, A0) = pt,,_z(A, A0) wo(Ao)

satisfies the simple formula

CO

w.,(A, A0) _ wv(A) wv(Ao) e-lvl=I, r = t - to. (11.3, 17)v=0

The Whittaker equation (11.3, 11) which we have found here has for2q not equal to an integer

the system of linearly independent solutions

Mp,q(a) = aq+1/2 a-ale 1Fj(q - p + 2 , 2q + 1, a) ,

Mp. -q(a) = a-q+112 a-art 1F1(-q - p + 1, -2q -I- 1, a)

where 1F1 is given by

1F1(r,s,a)=1 }

r(r+1)...(r+v-1)a'v-l8(s+1)...(s+v-1)v!'

the so-called Pochhammer function or confluent hypergeometric function, which canbe expressed in the form

T(s) I'(1 - r) (8-1>1F, (r, 8, a) = f(8 - r) -r (a)


by generalized Laguerre functions. A system of solutions also linearly independentfor 2q equal to an integer is given by the Whittaker functions

W,,q(a) = I,( F(q

2q)

p)Mp,q(a) + r(z +2q) p)

Mp -q(A) .

We can transform the Whittaker equation (11.3, 11) by

v(a) = aq+1/2 e-,12 u(a)

to the differential equation

d2u du ( 11a)da+lp q- 2/u-0

which has (KAMKE (1959)), for natural numbers 2q + 1 and the boundary conditionsthat u is limited for a - 0 and not greater than a power of a for a -> oo, the eigen-values p - q - 1/2 = 0, 1, 2, ... and the eigenfunctions Ln2q)(a)

In our case

N'+ S+2A4Gx

q=0

the eigenvalue equation yields

An=(2n+1)Gx 2 -S n=0,1,2,...

or, after inserting the formulae found above,

An - \n + 2)452

(02 + 2w26(t)4 +

cotAn

+g-22 - 2 + O(f2r{'bf,Vb ,rlg3) .

(11.3, 18)

(11.3, 19)

The solutions are, in the approximation of the terms written explicitely in (11.3, 19),given by

2n2+2n+1 2 bAn

4w2 2

(±. n+ i 2452- 1 2df2+(4n3+8n2+5n+1)n942) ( 2) w2 16w4

We obtain real values An (and x because of (11.3, 18)) if forced excitation and non-linear damping are small enough for the radicand to be non-negative,

d f 2 < w2b2 + (4n3

2+ 1) w 1) n 94(11.3, 20)

4(2n

This inequality holds for every n if it holds for n = 0 in which case it reads

df2 < w2b2 .


At least one of the two values A for any integer n is positive if the inequality

d/2 < 4n(n + 1) w2b2 + 2n2 + 2n + 1 bg2 - 3n2 +3n + 1 4g(2n + 1)2 (2n + 1)2 4(2n + 1)2 owe

or the inequality (with this inequality also (11.3, 20) is met)

22co2b

g > 2n2+2n+1is valid. The last one also holds for every n is it holds for n = 0:

g2 > 2co2b .

In the case of vanishing parametric excitation, g = 0, (11.3, 19) yields

\\ W 2

For linear damping, d = 0,

(11.3, 21)

(11.3, 22)

Equation (11.3, 22) shows that A is proportional to n, (11.3, 21) shows approximatelythe same, from which it follows that higher terms in the development (11.3, 13)quickly decrease as t increases. In general, A > 0 holds so that the non-stationarysolutions tend to the stationary ones after some time.

If forced excitation vanishes, y = 0, the coefficient 17 of the transformed differentialequation (11.3, 9) reads, after developing the denominators into series and neglectingamplitude terms of sixth and higher degrees, in the form

- 382 - 2N'S - N'2 + 8SA + M'S2 - M'N'S + N'2r + 7N'Sr - 4SrA A24,82 2S3

-M'2S2 - 6M'82r - 2N'S21 + 4M'N'Sr + 6S3l + 6N'Sr2 - 3N'2r2 + 8Sr2A+

4f14 .

484

The choice of

3G 282 M'2S2 + 6M'S2r + 2N'S21 - 4M'N'Sr - 6831 - 6N'Sr2 +3N'2r2 - 8Sr2A

(where the radicand is positive because the first term is of highest order of magnitude)leads again to the Whittaker equation (11.3, 11), where now

2

= (21 - 2(02b 2 4c)2A

4 g2 - g2

and, by developing into series and neglecting terms of higher degree,

2(o2bp=1-2

+p,g

with8eb2 6co2hb2 5eb 3hb egg

pdg2 dg2 coed + d + 2(o4d


which yields the eigenvalue equation

402 = -n(n - 1) g2 - 4naw2b + [(2n - 1) g2 + 4cw2b] p' .

The condition 2 0 holds, for instance, for g2 7 b (linear damping small in compari-son with parametric excitation) and

eg2 > 2n(n - 1)aw'd2n-1

or for g2 > b, vanishing non-linear restoring forces (e = 0) and

b ?(n-1)

that is, always for n = 0 and n = 1, for

b> 2 if n=2,d 9h

and so on.Following (11.3, 8), (11.3, 4) and (11.1, 8), the function E can be given in a more

explicit form by

dAQa + 2AdAJ/dQ2pE=e

12P-dQ* dQa

- e2 I Qd4A) d9` a a

wsta tCAQ2

So we can derive explicit formulae for the transition probability density and by thisfor different probability densities of finite order, for instance in the case of prevailingforced excitation the formula

u'.,(A, AO) -x AAowstat(A) wstt(Ao) e-2 (A'+Ai)

C Q2(A) Q2(AO)

00

x L(°)(xA2) L;°)(xAo) e-IGI=Iv=0

for the two-dimensional probability density (11.3, 17) in form of a Laguerre functiondevelopment.

This way the Fokker Planck Kolmogorov equation can lead to explicit results onmulti-dimensional amplitude distributions in rather complex non-linear vibrationproblems.

12. Systems with autoparametric coupling

12.1. Basic properties

As an example of coupled vibrations in a system with two degrees of freedom system,we shall investigate systems with autoparametric coupling. In such systems a forcedexcitation of one mode acts, because of the non-linear connection between differentmodes, as a parametric excitation of another mode. We shall consider in particularthe autoparametric vibration absorber system investigated by HAxToN and BARR(1972) and IBRAHTh( and ROBERTS (1976, 1977); compare SCHMIDT and SCHULZ (1982).It consists of a main mass and a weightless beam with a concentrated end mass mountedon the main mass (Fig. 12.1, 1).

I/ v,C,

B7>O

Fig. 12.1, 1

Forced vertical vibrations of the main mass impose forced axial vibrations and, incase of a parametric resonance, additional lateral vibrations on the beam which reacton the main mass through inertial non-linearities to give a vibration absorber ef-fect.

The equations of motion of various systems with autoparametric coupling are ofthe form

xl x 21x1 = -B1 1 - Dlxlzl --tC,xi - Ex1(x2 + x1 1)+ Jx1x2 + K1xlx2 + L1(x2 + x2x2)

00

- E (Gj cos jcwt + Hf sin ja)t) xl ,j=1

00

x2 22x2 = (P1 cos j0)t + Q1 sin jwt) - B2x2 - D2x2x2j=1- C2x2 + Fx2 + K2x1x2 + L2(xl + x1 i1)

12.1. Basic properties 363

For the case of the autoparametric vibration absorber, these equations follow easilyfrom the Lagrangian formalism, where x1 is the lateral displacement of the absorbermass m, x2 the vertical displacement of the main mass M,

'Q2'' 02= 'u1 2M M+m

with the lateral spring stiffness v and the vertical spring stiffness ,u, with B2, Dii, Ctithe linear and non-linear damping and the non-linear restoring forces acting on the ab-sorber mass and the main mass respectively, E the coefficient of non-linear inertiaforces, K1, L2 the coefficients of coupling terms, Pq, Qq the Fourier coefficients of verti-cal forces, and setting

F=G,=HI=J=K2=L1=0. (12.1,1)

The dimensionless equations corresponding to (6.1, 8) are, with (12.1, 1),

yi + 11Y1 = Alai % - -Yi - DlY2

C2Yi - Ey1(yi + ylyl) K1yly'

a) w o)

1 00

Y2 +'2Y2 = 2Y, (P3 cos jr + Qp sin jr) + A2a2y2

w j=1

- B2y2 - B2 y2y2 -

2

y2+ L2(yl + ylyl )a) Lv W 2

In the non-resonance case neither t nor 22 is the square of an integer, and the firstapproximative solution turns out to be

1 P1cosjv+Q,sinjryno=0, Y210=-

w2 j=192 2 - j2

Now we assume a two-fold resonance

1 = n2, 22 = n2

with integers n, n2. Instead of (6.1, 7) now

CO -a)1an

aU -a)2 (t) -(")21 - ..(01 as(02

(12.1, 2)

holds with two fixed frequencies a 1, w2 in the neighbourhood of a). As we shall show,the case of a single resonance is included in the following analysis. The first approxi-mation is

yu=rcosn2+8sinn-r,

y21 = r2 cos n2'r + 82 sin n2Z + y21o

364 12. Systems with autoparametric coupling

This leads to the second approximation for y2

Y22 = Y21 - x2(r2 cos 762-C -1- 82 sin nT)

+ B2 (s2 cos n2z - r2 sin net) +4 - (r2 + 82) (82 cos n2z - r2 sin n2z)

2 2

+32n,2(t)

[s2(3r2 - s2) cos 3n2z + r2(3$2 - r2) sin 3n2z]2 2

-} 43 22 (r2 2 2+ s2) (r2 cos %T + s2 sin net)

- 382) cos 3n2r + s2(3r2 - s2) sin 3n,-c]+32 2('02 [r2(r22 2 2

+4n2

n2t$2nn2

[(r2 - 82) cos 2nt + 2r8 sin 2nt]n, 2

Using this, the second approximation of yl turns out to be

Y12 = Y11 - a1(r cos nz + s sin nc) + B1 (8 cos nr - r sin nr)nco

+ --(r2 + 82) (s cos n2 - r sin nr)

+ 32nvo[s(3r2 - 82) cos 3nt + r(3s2 - r2) sin 3n-c]

+ 4n212 (r2 + s2) (r cos n'c + s sin nr)

+ 32n2G)2[r(r2 - 3s2) cos 3n-c + s(3r2 - 82) sin 3ni ]

- 2 (r2 + s2) (r cos nt + s sin nz)

- 16 [r(r2 - 3s2) cos 3nr + s(3r2 - s2) sin 3nt]

K1 co j2

[(rPf -sQf) cos (n + j) i + (sPf + rQ1) sin (n + j) i

20J2 f 1W n2 - y2 n2 - (n + .2)2

-f- V2nn2- (n - j)2

(rPj + sQf) cos (n - j) v + (sPf - rQj) sin (n - j) cl

+ n2(1 - a2) K1 [(rr2 - ss2) cos (n + n2),r2[(n + n2)2 - n2]

+ (sr2 + rs2) sin (n + n2) -c]

+ n2(1 - a2) Kl[(rr2 + ss2) cos (n - n2) z

2[(n - n2)2 -19'22; n2]

+ (sr2 - rs2) sin (n - n.2) z]

12.1. Basic properties

+ 2[(n + n2)2K,

n2] [(rs2 + sr2) cos (n + n2) r

+ (ss2 - rr2) sin (n + n2) r]

+n2B2K1 [(rs2 - sr2) cos (n - n2) r

2[(n - n2)2 - 792fn2] co

+ (ss2 + rr2) sin (n - n2) r]2

2+ n2D2K1(r2 + 82)[(rs2 sr2) cos (n + n2) r

8[(n + n2 )2 - n2] C(o

+ (ss2 - rr2) sin (n +n2)] r

n2D2K1(r2 + s2)

+ 8[(n - n2)2 - 92nn2] w[(rs2 -sr2) cos (n - n2) r

n

+ (ss2 + rr2) sin (n - n2) r]

3C2K1(r2 + 82)

8[(n + n2)2 - n2] w2[(rr2 - ss2) cos (n + n2) r

+ (sr2 + rs2) sin (n + n2) rJ

3C2K7 (r2 + 82)

+ 8[(n - n2)2 - 02n n2] C2[(rr2 ss2) cos (n - n2) r

+ (sr2 - rs2) sin (n - n2) r]

366

9nDK+ 64[(n + 3n )2 1 n2] w

{ [(rs2(3r2 - s2) - sr2(382 - r2)] cos (n + 3n,) r- 2 2 2

+ [ss2(3r2 - s2) + rr2(3s2 - r2)] sin (n + 3n2)r}

) + sr2(3s2 - r2)]+ 64[(n - 3n )2 - t92n,n2] a {

[rs2(3r2 - s22 2

X cos (n - 3n2) r + [ss2(3r2 - s2)

- rr2(3s2 - r2)] sin (n - 3n2) r}

+64[(n + 3 2)21- n2] w2

{ [rr2(r2 -3 s2) - ss2(3r2 - s2)] cos (n + 3n2) r2 2 2

+ [8r2(r2 - 3822) + rs2(3r2 - s2)]

x sin (n + 3n2) z}

- 382) + ss2(3r2 - 82)]2 2+ 64[(n - 3n )22 - 9 ,n2] C02

{ [rr2(r2

X cos (n - 3n2) r + [sr2(r2 - 3s2i- r82(3r2 - 82)] sin (n - 3n2) r}

2n2K L2(r2 82)4n2 2 n2s) (r cos wr + s sin nr)n, 2

+ n2K-L2[r(r2 - 382) cos 3nr -I- s 3r2 - s2) sin 3nr

4(472 - 0,2-) ( ( ]


It is not easy to interpret these approximations, which include much informationabout the time dependence of the solutions. The periodicity equations (2.3, 3) stem-ming from them are, in vector form,

a\8/-bl(-r)+pl(-s) + ql(r

-k(r2(-8)+82(r)]+b[r2(r)+32(

s)]

} d lsz(s22 2- 3r2) (-s) + r2(r2 - 3s2)\ r )]

+ c I rz(3s2 - r2) ()+828_3r()]=Oand

a2 \82/ - b2 (-r2) (r22rs32) + (q2)0

with the abbreviations

1a =n2a - +1 1 ko2 2 4n2 - 0!"n2,

a2 = n22a2

n2E 2n4K1L2 A23C

3C2 A2-4W22,

bl = w (Bl +'1 A2), b2 . n02(- 2 +

RzA2)

2n2K1 P2nPI = (4n2 - #.n,n2) a,.,)2 P2 =lz

pn,

2n2K1 1 /1qi (4n2 - 0nnn2) w2

Q2n , q2 = (02Qn,

k=bn;(2n2-2)K1

1 = 62,=n2Lz

_ 2n 9C2K1c Stn,64cw2

where

A2=r2+32 e

b=bnn b2K1

2

2n 3nD2K13n,

320

2 2 2fl2 = r2 + s2

(12.1, 3)

(12.1, 4)

(12.1, 5)

are the squares of the partial amplitudes (more exactly: the squares of the resonanceparts of the first approximation of the partial amplitudes) in direction yl and y2respectively.


The equations (12.1, 4) are linear with respect to r2, s2 and their solution is

Br2 = a2l(r2 - s2) + 2b2lrs - a2p2 - b2g2Bs2 = 2a2lrs - b2l(r2 - $2) - a2g2 { b2p2 , 1

using the abbreviation

B=a22- b22Squaring the equations (12.1, 6), adding and dividing by B leads to

BA2 = 12A4 + p2 + q2 - 21P2(r2 - s2) - 4lg2rs .

(12.1, 6)

(12.1, 7)

The equations (12.1, 3) and (12.1, 4), complicated as they are, already show thecoupling between the vertical vibrations of the main mass, represented by r2, s2, A2,and the lateral absorber vibrations, represented by r, s, A. If l = 0, that is, if theresonance condition n2 = 2n does not hold, the vertical vibration is (in the approxi-mation at hand) not influenced by the lateral vibration, but it always influences thelateral vibration. The latter influence is given already by L2, n2 and the forced excita-tion terms Pen, Q2n in a1, pl, q1, but it is complicated by the linear r2, s2 terms in (12.1, 3),if k, b do not disappear, that is, if n2 = 2n and K1 + 0, and by the non-linear r2, s2terms, if c, d not disappear, that is, if 3n2 = 2n and K1, C2 respectively D2 are notzero.

The process of evaluating the first approximations also enables us to recognize thebehaviour of higher approximations. The coupling term K1y12y21 leads to terms contain-

ingcos (3n - n2) z which add to the periodicity equations (12.1, 3) not only forsin

n2 = 2n, but also for n2 = 4n. In detail, this term leads to

Y1 add =n2D1K1

n2 { [s(3r2 - $2) r2 + r(3s2 - r2) 82]64n[(3n - n2)- - #(sn-n2)2n2] w

X cos (3n - n2) r+ [r(3s2 - r2) r2 + 8(82 - 3r2) 82]X sin (3n - n2) z}

+n2 (C1 - 2n2w2E) K1

64n2[(3n - n2)2 - w2{ [r(r2 - 3s2) r2 + s(3r2 - s2) 821

X cos (3n - n2) r+ [s(3r2 - s2) r2 + r(382 - r2) s2]X sin (3n - n2) _t}

from which there follow the additional terms

- 62n nD1K1 fr2(8(3r2 - s2)\ +

s2(r(3$2 - r2))]

16w LL \r(3$2 - r2)/ 8(82 - 3r2)

+ 622n (2n2w2E - 0) K2

Lr

(r(r2 - 3s2)\

+s

(s(3r2 -

82no 16w2 2 s(3r2 - s2) 2 r(3s2 - r2)

- 15nnnD_K1

I r2s(3r2 - s2)\ +

s2(r(3s2 - r2)

4w r(r2 -3s2)/I

s(3r2 - $2)

+ a4n (2n2 0_)2E - Cl) K1 [r2 /r(r2 - 3e2)/

+ s(s(3r2 - $2)

no 4w2 ts(s2 - 3r2) 2 r(r2 - 3s2)/ 1

on the left-hand side of (12.1, 3).

(12.1, 8)


Insertion of (12.1, 6) into (12.1, 3), (12.1, 8) gives two coupled non-linear equationsfor r and s which can be solved numerically. A formula for the amplitude A onlyof the absorber vibrations can be found if we

a) exclude the resonances 3n2 = 2n and n2 = 4n, orb) restrict ourselves to the approximation (12.1, 3), (12.1, 4) and assume the non-

linear restoring and damping forces in vertical direction to vanish,

C2=D2=0, thatist c=d=0. (12.1,9)

In what follows, we assume one of these two restrictions as given. Then the insertionof (12.1, 6) into (12.1, 3) mentioned above yields

(Ba1 - a2klA2 + b2blA2)(r)

s

+ (Bpi + a2kg2 + a2bq2 + b2kq2 - b2bp2) (_:)

+ (Bq1 + a2kq2 - a2bg2 - b2kg2 - b2bq2) t sr

- (Bbl + a2blA2) + b2klA2) (-8) = 0, (12.1, 10)

twotwo equations which contain r, s, in contrast to (12.1, 4), besides the linear factorsin form of A2 only.

The determinant condition for a non-vanishing solution r, s reads

(Ba1 - a2klA2 + b2blA2)2 + (Bbl + a2blA2 + b2klA2)2

= (Bp1 + a2kg2 + a2bq2 + b2kq2 - b2bp2)2

+ (Bgl + a2kq2 - a2bg2 - b2kg2 - b2bg2)2 . (12.1, 11)

This is a biquadratic equation in A the coefficients of which are of fourth order inA2 in the case of restriction a) and independent of A2 in the case (12.1, 9) of the secondrestriction b), which we assume in what follows.

Abbreviating

a1=a-eA2, b1= f+gA2

instead of (12.1, 5), we get the solution of (12.1, 11)

[(Be + a2kl - b2bl)2 + (Bg + a2bl + b2kl)2] A2

= Ba(Be + a2kl - b2bl) - Bf(Bg + a2bl + b2kl)

f { -B2[/(Be + a2kl - b2bl) + a(Bg + a2bl + b2kl)]2

+ [(Be + a2kl - b2bl)2 + (Bg + a2bl + b2kl)2]

X [(Bpl + a2kg2 + a2bg2 + b2kq2 - b2bp2)2

+ (Bg1 + a2kq2 - a2bg2 - b2kg2 - b2bg2)2]}112 . (12.1, 12)

We shall first discus the case n2 + 2n (no internal resonance). Because b = k- l = 0,(12.1, 12) simplifies to the amplitude formula

(e2 g2) A2 = ae - f9 (e2 + g2) (p2 2+ q) - (et + ag)2 (12.1, 13)


which again shows that the motion of the main mass influences the absorber amplitu-des only by means of the values L2, P2n, Q2n, n2 in e and by pl, q1.

The equation A = 0 gives the resonance interval

at=±Vp2+qi-/2

which is real if linear damping is smaller than an excitation threshold value,

I<Vpi+gi.The resonance curves given by (12.1, 13) coalesce, that is, determine a maximumamplitude only for positive non-linear damping g.

Fig. 12.1, 2 gives resonance curves corresponding to (12.1, 13) for e = -0.65,a = al, / = 0.001, pl = 0.002, ql = 0 and different values of non-linear damping g,Fig. 12.1, 3 analogously for g = 0.1 and different values of forced excitation pl. Itshows that increasing non-linear damping diminishes, and forced excitation increases,the amplitudes.

Formula (12.1, 7) for the main mass amplitude now reads

BA22 22=p2+q2

or, in the original coefficients,

222(,)20,2 2 2 2 2n2fiJ (n22 + B2) A2 = Pnx + Qn, ,

-0.005

Fig. 12.1, 2

-0.02 -0.015

Fig. 12.1, 3

-0.002

-0.01

0.002 a,

-0.005

(12.1, 14)

0.002 0.004 al

24 Schmidt/Tondl


and is independent of the absorber motion. Fig. 12.1, 4 shows an example for

B = 9(9a2 + 16.10-6) , q2 = 0

and different values of pl. The resonance curves given by (12.1, 14) have the maxi-mum value

'/ 2 2A2 max = 1 Pn. + Qn.

for a2 = 0.

n2coB2

Fig. 12.1, 4

12.2. Internal resonance

(12.1, 15)

For internal resonance, n2 = 2n, the absorber equation (12.1, 12) can be simplifiedif we take into consideration only the terms of highest order of magnitude, in parti-cular if we omit the non-linear damping terms,

Bg C b2kl , (12.2, 1)

thus giving

klA2 = aa.2 - /b2 + (/k2(p2 + q2) - (42 + fa2)2 (12.2, 2)


n2u02K1L2A2 = 2n2aw2ala2 - B1B2 ± Y Ki(p2n + Q2 n) - n2w2(alB2 -;- 2a2B1)2 .

(12.2, 3)

The amplitude formulae (12.2, 2) respectively (12.2, 3) show, in the approximationat hand, no influence of non-linear damping g and non-linear restoring force e. Thecorresponding resonance curves nevertheless coalesce by means of the frequencyparameters al, a2 in the radicand. Fig. 12.2, 1 gives an example of the resonance curvescorresponding to (12.2, 3) for n = o) = 1, K1 = 3, L2 = 1/3, al = a2, B1 = 0.001,B2 = 0.004, Q2 = 0, and different values of P2.

More complicated is the determination of the main mass amplitudes because12.1, 7) contains not only A but also r2 - s2 and rs.

12.2. Internal resonance

-0.15 -0.1Fig. 12.2, 1

-0.05

If we abbreviate (12.1, 10) by

u(8r)+v( Sr)+\s/ +

v(-s)+ V (r ) - U(-r) = 0

that is

0.05 0.1

371

0.15 cc,i

(u±v)r=(U-V)s, (V+U)r=(v-u)s, (12.2,4)

we get by multiplying these equations with each other the equation

(u+v)(V+ U)r2 -(v-u)(U- V)82=0or, writing the left-hand side in the form xA2 + y(r2 - s2),

(uV + vU) (r2 -s2)= -(uU + vV) A2.

Multiplying (12.2, 4) by (u - v) r and (u + v) s respectively and adding yields

2(uV + vU) rs = (v2 - U2) A2.

This gives (12.1, 7) as an equation for A2 and A only:

(uV+vU)BA2=(UV+vU)(12A4+p2+g2)+ 2(uU + vV) 1p2A2 + 2(u2 - v2) 1g2A2.

Confining attention to the terns of highest order of magnitude

u = Ba - a2k1A2 , U = B/ + b2k1A2 ,

v = a2kp2 + b2kq2 , V = a2kq2 - b2kp2

24


and to a cosine-shaped forced excitation, q2 = 0 (without loss of generality, by anappropriate time shift analogous to (5.1, 12)), we get

[2a2b2klA2 + (a2f - b2a) B] kA2

= 2af1BA2 + (a2f - b2a) (kp2 - kl2A4) .

If forced excitation P2 does not vanish and is so great that the radicand in (12.2, 2)is positive, insertion of (12.2, 2) leads to

2 2- b2) ± 2a2b2 Y k2'p2 - (ab2 + fa2)2] k2A2L(ab2 + flat) (a2

[(ab2 + fa2) (a2 - b2) ± 2a2b2 1' k2p2 - (ab2 + fa2)2] (a2 + f2) ,

that is, to the surprisingly simple formula

k2A2 = a2 + f2


4n2w2K2A2 = n2W2a2 + Bi (12.2, 5)

This shows that, in contrast to the single resonance formula (12.1, 14), the main massamplitudes depend only (besides on w) on the absorber motion parameters n, K1,al, B1. For al = 0, the minimum amplitude

A = B1

2 2nw (K1

is assumed. Fig. 12.2, 2 shows an example of (12.2, 5) with n = 1, w = 1, Kl = 3,and Bl = 0.001.

To discuss the amplitude formulae (12.2, 3) and (12.2, 5), we shall introduce thedetuning

E = al - a2which, as the approximative equations (12.1, 2) show, does not depend on the variablefrequency co, but only on the fixed frequencies w1, w2:

w1

12.2. Internal resonance 373

Formula (12.2, 3) can be written in the form

2

n2w2K1L2A2 = 2n2w2 (a, -/

- 2 n2(t)2E2 - B1B2 -I- V- ,

with

(12.2, 6)

V = VKi(P2n + Q2n) - n2w2[(2B1 + B2) a1 - 2B1E]2 .

The resonance curves given by (12.2, 6) are approximately equidistant - in thevertical direction - to the vertical curves (compare Section 10.2.)

2

n2w2K1L2A2 = 2n2(,02 a1 - 2 -2 n2w2E2 - B1B2 (12.2, 7)

for which 1 = 0 and which are, because of KIL1 > 0, hyperbolae meeting the a1axis at

E2 B1 B2alf 2 4 + 2n2w2

(dashed curves in Fig. 12.2, 1). Fig. 12.2, 3 gives an example of (12.2, 6) correspond-ing to Fig. 12.2, 1 and P2 = 0.0003.

-0.15 -0.1

Fig. 12.2, 3-0.05 0.05 0.1 0.15 02 a

Differentiating (12.2, 6) with respect to a1 and putting dA/dal = 0 gives an equationof fourth degree in a1 as well as in e for the amplitude extreme values:

[4n2w2(2a1 - E)2 + (2B1 + B2)2] [(2B1 + B2) a1 - 2B1 ]2

= 4K1(P2n + Q2n) (2a1 - -)2.

Vertical tangents for the resonance curves occur for dal/dA = 0, that is, for V- = 0from which it follows that

(2B1 + B2) a1 = 2B]-+- 1 V P + Q.2.nw


In particular for vanishing detuning, s = 0, amplitude extreme values appear for

4

al = ±nw(2B1 + B2)

K2l(p22. + Q2n) - (B1 + -vertical tangents (coalescence of the resonance curves) for

2n (12.2, 8)K1 _al = f nw(2Bi + B2) Y '2n+ Q2

The maximum main mass amplitudes can be evaluated from (12.2, 5) (for Q2n = 0)with (12.2, 8) :

A2 P2n B16 2 + B21

2 max = [2nw(2B B2) 2B B , 4n2w2K21 + 2) 1( 1 + 2) 1

For non-vanishing detuning, one of these values is greater than the value

P2 B22 2n 1 nA2 max = 4n2w2(2B1 + B2)2 + 4n2w2K2

(12.2, 9)

for e = 0. A comparison with the maximum main mass amplitude (12.1, 15) forsingle, not internal resonance (for no absorber at all) and Q,, = 0 shows equality forB1 = 0 whereas the expression (12.1, 15) is for 2B1 = B2 quadruple, for B1 = B2ninefold and for B1 = 2B2 twenty-five-fold the value of the first expression in (12.2, 9).

Given the damping coefficient B2 and the excitation coefficient P2n of the main massthe absorber effect augments when the coupling coefficient K1 augments and ismaximal with respect to the absorber damping B1 for the minimum value A2 maxThe latter can be found by putting dA2 max/dB, = 0, that is, from the equation

B1(2B1 + B2)3 = 2K2IF2n

which simplifies to

6(28+1)3=2x (12.2,10)

with2 28=81 (>0), x= KB 2n (>0)

B2 2

The solution of (12.2, 10) is given in Fig. 12.2, 4.

-1 0 1 2 3 4 lgmFig. 12.2, 4

12.3. Narrow-band random excitation 376

If we do not exclude the influence of non-linear damping by (12.2, 1), we can findinstead of (12.2, 2)

(012 + 2b2klg + Bg2) A2 = (aa2 - fb2) kl - Bfg

2b2klg + Bg2) k2(p2 -I- 12) - [(ab2 + fat) kl + aBg]2 .

In the original coefficients, the condition jr = 0 for vertical tangents reads

[2n2K1L2(ociB2 + 2a2B1) +\2n2a2

+ B22

1a1Dj

J2

2 2

< L4n2KiL2 +2K1L2B2Di

(a2 B2 ) Dl (P2 2 12n +Q2n) ,

[ W2 4n2(i2 / c)-J

an equation of sixth order ina1, a2. For vanishing detuning e, it simplifies to an equationof third order in ai :

[2n2K1L2(2B1 + B2) (2n2a1 + B i D112ai

< C4n2KJL2 -{-2 2 22K1u2B2D1

+(2+4nw2)D2]w2(p2n2

An estimation for small values of the non-linear damping coefficient D1 shows thatin comparison with linear damping, D1 = 0, the interval (-al, al) of real amplitudesdiminishes if the excitation coefficients P2n, Q2n are not too small,

2Ki(P2n + Q2n) > (2B1 + B2)2 B1B2 .

In the case of non-linear damping, the formula for the main mass amplitudesbecomes very complicated and reveals, as formula (12.2, 5) for linear damping wouldalready do if we had considered a higher approximation, a dependence on the forcedexcitation.

It should be noted here that the at first sight surprising result, that the main massamplitudes do not depend on the forced excitation of the main mass, is not really sosurprising because the al interval for the two-mode vibration with the main massamplitude (12.2, 5) and by this the maximum amplitudes A2 do depend on the forcedexcitation.

12.3. Narrow-band random excitation

In what follows we shall assume, as in Chapter 10, that not a periodic excitation, buta stationary narrow-band random excitation with mean value zero

P(t) = z(t) cos [2n(ot + S(t)]

acts on the main mass. The excitation amplitude v(t) and the excitation phase 8(t)are slowly varying functions, the process P(t) is assumed Gaussian distributed, sothat the excitation amplitude is Rayleigh distributed, with probability density

n2

2a=v(n) = -e6

A generalisation to Weibull distributed excitation amplitudes is possible as in Chap-ter 10.


The probability density of the response amplitudes associated with strips w(n) doof excitation amplitudes can be found by the transformation

v(n) do = dV(n) = dW(A) = w(A) dA (12.3, 1)

using the fact that n is a monotonic function of A. The amplitude formula (12.2, 3)can now be written in the form

K1n2 = n2o 2(a1B2 + 20i2B1)2 + (n2w2K1L2A2 - 2n2(o2a1a2 + B1B2)2

so that (12.3, 1) yields

dW(A) = -e-9'with

99 = 1 n2w2(a B+ 2a B I)(n2a)2K L A2 - 2n2w2a a+ B B 2772K262

1 2 2 1) 1 2 1 2 1 2)J

By differentiation we get the probability density of the vibration amplitude

2n2w2L2Aw(A) = (n2602K1L2A2 - 2n2w2a1a2 + B1B2) a-I' (12.3, 2)

a2K1

or, because of (12.2, 7),

w(A) = 2n4 24L2 A(A2 - A2) a-9,. (12.3, 3)a2

The factor in parentheses vanishes on the vertical curve A = A,. Because amplitudesbelonging to parts of the resonance curve below the vertical curve do not appear,A > Ao, we have to assume v(A) = 0 for negative values of the above factor inparentheses as in Section 10.2, so that always w(A) > 0 holds. Correspondingly,the same expression in parentheses in dW(A) has to be omitted if it is negative.

The whole factor preceding a-9' in (12.3, 3) ensures that the probability densitydisappears if the amplitudes approach A,, > 0 or zero whereas e'9' causes the probabili-ty density to disappear for great amplitudes.

Formula (12.3, 2) for the probability density of the vibration amplitude enables usto evaluate the moments

00

mk = f Akw(A) dA , k = 1, 2, 3, ...0

of the amplitude. Abbreviating in what follows (12.3, 2) by

w(A) = c1A(c2A2 + c3) C,)' CCI

where c4 > 0 and the signs of c1 and c2 are chosen so that c3 > 0, we get, writing

A2 = x and !lc6 ,

2

the equation00

-C2c .x' c.,c,c.2Ink = C8 f xk/2 (C2x + C3) e dx

0


and, using formula (11.2, 1),

Mk = C2C6(2c2C4)-(kl4)-1 r 2 + 2) e(1I2Ac, D-(k/2)-2(c3 O C4)

+ cc6(2c2c4)r (-- + 1) e(1I2)c3c,D-(k,2)-1(c3 1/2c4)

with the parabolic cylinder function D and the Gamma function T. The seond momentof the amplitude yields

m2 = C6 c36 e(1/2)C c*D-2 (c3 2C4)

c 2C42 V2C42C 2

where (11.2, 12) and, because of the recurrence relation for parabolic cylinder func-tions (compare GRAD6TEJNand RYZIK (1971), p. 1080)

Dr+1(z) - zDv(z) + pDp-1(z) = 0

and (11.2, 11),

-(1I4)LD-3(z) = V g (1 + z2) e(1I4)z= f1-(.j;-)11

2 e

holds with the error function

V

IP(i) =-J

e` dt.0

The probability that the resonance amplitudes A are greater than a given levelAl is

W1=W(Aj<A<co)

From (12.3, 1) it follows thatCO

W1 = f w(A) dA < 1 .At

Because a2 = a1 - s, we get

W1 = expL

2K2a2 (n2w2[(2B1 + B2) a1 - 2B1s]2L 1

+ [n21V2K1L2Ai - 2n2w2a1(a1 - e) + B1B2]2}1 (12.3, 4),

for

and

n2w2K1L2A2I > 2n2w2a1(al - e) - B1B2

n201

2R2e[(2 B,+B,)a, -2B,s]'

W1=e 1 (12.3, 5).

for

n2C02K,L2Al < 2n2w2a1(a1 - e) - B1B2


that is, for

2n2cw2a1(a1 - e) > B1B2

and sufficiently small At, independently of Al.For At = 0, there follows the probability of there being non-vanishing amplitudes

at all

Wo = expL 2K262

{n2w2[(2B1 + B2) al - 2B1E]21

+ [2n2co2a1(al - e) - B1B2]2}1 (12.3, 6)

for

2n2ow2a1(a1 - e) < B1B2and

nzwz-2%2az

((2B,+B,) a, -2B,e1zWo=t 1

for

(12.3, 7)

2n2co2a1(a1 - e) ? B1B2

We shall discuss the probability Wt. Formula (12.3, 5) states that the maximumvalue Wt = 1 would appear for

(2B1 + B2) al = 2B1e

(but then (12.3, 5) does not hold because Al < 0 would follow) and that Wt mono-tonously diminishes if al removes from this value. In every case, Wt < 1 holds sothat the probability of vanishing amplitude A is not zero.

The condition

dWtOda

for extreme values of Wt leads, if we use formula (12.3, 4), to a cubic equation in al.In order to permit an analytical discussion, we confine ourselves to vanishing detun-ing, s = 0. In that case, the three extreme values are for al = alo = 0

- 1 (nswag LsAl +B, Ba)'W10 = e 2I2aa1 (12.3, 8)

and - for

1 /tai = 2a1M = K1L2A1n2u02

B1 4 B2

(only real and different from alo = 0, if

n2aO2KKL2A2 > Bi + 4 B2

holds) -(B,+Bz/2)1 (2nzw2S,LaA2-(B,-Bz12)x1

2%2azWtm = e 1

(12.3, 9)

(12.3, 10)


It can be shown that W10 < W1m because of (12.3, 8), so that Wt has a minimum W10for al = 0 and maxima W1m for al = ±alm. These maxima appear because the con-dition for the validity of (12.3, 4) holds for al = ±a,..

If the condition (12.3, 8) for maxima in f alm + 0 does not hold, W10 is the maxi-mum value of W1, as the factor

BI + ; B2 - n2w2KL2Ai

of a1 in (12.3, 4) shows.

Vice versa, the amplitude level A1i amplitudes above which have only the (per-haps small) probability W1, follows from (12.3, 10)

n2w2K1L2Al = - Klv21n!B-W2 + 2 (B1 - B2)2 , (12.3, 11)

(B1+.)if

n2w2KKL2Ai > B2 + 1 B2

and from (12.3, 8)

ifn2w2KiL2Aa = -2K1a2 in W1 - B1B2 , (12.3, 12)

n2w2KKL2Al 2 <B 2 + 4 B2.

If the equality sign in the last condition holds, the formulae coincide. They show thedependence of Al on the different parameters. Whereas (12.3, 12) causes a diminutionof Al for any increase of the damping coefficients B1 or B2, (12.3, 11) leads to a mini-mum Al for a certain damping ratio which follows from

B2 - 11 / B2 1\3 2K1a21n W1

t2B1 /J 2B1 + l Bi

AZ

8'i0{.

12(2BE

FI

8 0.8

6 0.6

4 0.4

2 0.2

-8 -6 -4 -2 0 2 4

Lg[-2K 262 In W,109'1

(12.3, 13)

Fig. 12.3, 1 Fig. 12.3, 2


for (say) B1 given. Fig. 12.3, 1 gives the solution of (12.3, 13). Examples of the depen-dence of At on B2/B1 represented by (12.3, 11) (full lines) and (12.3, 12) (dashed lines)are shown in Figures 12.3, 2 to 12.3, 4 for n2co2K1L2 = 1, K1262 = 10-6, B1 = 0.02and Wt = 0.0000454, Wt = 0.00674 and Wt = 0.0821 respectively. The dottedline marks the condition for either formula (12.3, 11) or formula (12.3, 12): on theleft-hand side of the dotted line the equation (12.3, 11) holds, given by the full line,whereas on the right-hand side (12.3, 12) is valid, given by the dashed line. ForWt = 0.0821, no real amplitude At results for B2/B1 > 5.59 because then the pro-bability of there being non-vanishing amplitudes at all is smaller than W1.

0.4

0.3

0.2

0.1

s1 2 3 4 5 6 B2/B 2 3 4 5 6 821,9

tFig. 12.3, 3

12.4. Broad-band random excitation

Fig. 12.3, 4

By analogy with Section 12.1, we shall investigate the system of two stochasticdifferential equations

x +S22xt=F1(t,xti,x,,,x,), i=1,2; v= {1, 2} (12.4,1)

where

F1 = -B1x, - Dxixi - Cxi - Ex1(x12 + xlxl )

+ Jxlx2 + K1x1x2 + L1(x2 + x2xa) - gi(t) x1and

F2=fi(t) - B2x2+Hxi+K2xix2+L2(xi +xlxi).Now 121 and Q2 are eigenfrequencies, dashes denote differentiation with respect totime t, and the white noise random excitation j(t) acts as a forced excitation with thecoefficient /(+0) in the second and eventually as a parametric excitation with the co-efficient g in the first equation.

In particular, for the autoparametric vibration absorber,

g=J=K2=L1=0holds, corresponding to (12.1, 1).

Introducing by

xt = at cos 99, , x, _ -Shat sin p,

12.4. Broad-band random excitation 381

partial amplitudes ai > 0, and phases

Ti = Dit + Pias random functions of time, we get the differential equations (12.4, 1) in standardform

-Dia; = F; sin 4pi , -Qjaa9i = Fj cos q,, . (12.4, 2)

In the expressions (12.4, 2) for Fi, we can replace - up to higher approximations,which are generally neglected - because of (12.5, 1)

X, = -Q;xi + bills(t) . (12.4, 3)

The second part on the right-hand side is taken into consideration only in the termK1x1x2 of highest order of magnitude, thus (12.4, 2) reads

-Q1ai = Q1a1(B1 + Dal cost 971) sin2 q)1 - Cai sin 991 cos3 9'1

QlEai(sin2 99, - Cos2 9)1) sin qI1 Cos q71

+ Ja1a2 sin (p1 Cos qJ1 cos 972 - Q2K1a1a2 sin q)1 cos q71 Cos q72

+ S22L1a2(sin2 922 '- cos2 992) sin 99,

- (g - K1f) sin 971 cos 971

-S22a2 = Q2B2a2 sin2 T2 + Hai cos2 q71 sin 972

- Q1K2a1a2 cos q91 sin T2 cos x22

tsin sin 22 ,

-Q1a1(9i = S21a1(B1 + Dal cos2 991) sin 991 cos 991 -Cai cos4 991

- Q12Eai(sin2 q71 - Cos2 91) cos2 921

+ Ja1a2 cos2 q)1 COS 992 - Sl2K1a1a2 cos2 q71 cos 922

t+ cos 4p1(sin 2 992 - Cos2 992) - (9 - K1f) a1S (t) cos2 921

-S12aA = Q2B2a2 sin 922 cos 992 + Hal cos2 9'1 cos 992

- ,Q2K2ata2 cos 991 Cos2 922

+ Q1L2a2(sin2 921 - cos2 991) cos 922 + fi(t) cos T2

We understand these equations as physical or Stratonovich ones. If we abbreviatethem in the form

dyk=mkdt+nkd ,where

and

yt=a,,,

k=1,2,3,4

Yi+2=01 (i=1,2)

d = j(t) dt,the corresponding Ito equations can be written

4

dyk = (Mk + ,ak) dt + nk dg S , ,uk = 1 Y, ank nj (12.4, 4)2 1=1 ay,


where the additional Ito terms are now

Kf 2afl1

_OS4 9)1 ' P2 = 2

522521C

(a - K J12

I2

2 2a2

,u2 =4522

(1 + cos 291) sin 2991

COS2 T2 ,

f2

-'a' 2 a2sin 2q92 .

2 2

After trigonometric transformations, the Ito equations (12.4, 4) can be written inthe form

dai=pidt+gid , dt'i=ridt+sid$ (12.4,5)where

B,ai 3 3

p1 = -Blal + - cos 24v1 -

Dal + Dal cos 4g91 + cal sin 29912 2 8 8 4521

3 Q as!;

Casin 4991 - 4 al sin 4q' + 4, a1a2 sin (2991 +

1 1

K Q2L1a2+

452a1a2 sin (2q'1 - 9,2) +

22

922 sin (T, + 2992)

1

22L1a2 3(g - K1/)2 a1 (g - K11)2 al-+

2S2,sin (Tj - 2992) +

16521 + 4522cos 2T,

+ (9 - K1f)2 a1cos 4Tp1

16521

B2a2 B2a2 Hat 521K2a1a2P2 = -

2+

2cos 2992 - 212

sin 992 +412

sin (991 + 2992)2 2

2- ,Q1K2a1a2sin (991 - 2992) +

Lalsin (2991 +.p2)

q1 =

q2 =

Lag

4122

;2 12-452

sin (29'1 - 992) + 4Q2a + 4i22acos 2992 ,

2 2 2 2 2

- (9 - K1f) a1 sin 2991 ,2121

- sin 9922

2 2 2 2R1

Dal sin 29'1 - Dal sin 4991 +3Ca1 + Ca

r, sin 2991 - cos 29212 4 8 8121 2121

Calcos 4991 -

521Ea1 - S21L ai Cos 2991 -S21Ea1

cos 49918121 4 2 4

+4Q

2- cos (2991 + 992) + 42 cos (2q'1 - 9)2) +2

2 cos 9921 1 1

2 2 2 2

+ 212

2cos (991 + 2992) +

212

2cos (T, - 2922)

1a1 1a1

(g - K1/)2 (g - Klf)2- - 4522 sin 2991 -8122

sin 4491 ,1 1

12.4. Broad-band random excitation 383

B2 HaQr2 = - - sin 2992 -

a

cos 99z +Sl2K2a1

cos 921 +242a1

cos (9,1 + 2492)zz

2

+'f12K2a1

cos (921 - 2992) + 4S2 cos (212 + TO4 za2

Lag f2+ cos (2991 - 992) - jQ2a2 sin 29)2

4Q2a2 2a2

K1181= 2 (1 +cos2TI),

1

82 = - f cos 992Q2a2

when we use abbreviations

K=S22K1-J, L=2QL2-H.Starting from the differential equations (12.4, 5), the corresponding Fokker Planck

Kolmogorov equation which determines the probability density can be evaluated.Because the latter differential equation depends in a rather complicated manner ona,, 0,, and t, we introduce, for an iterative solution, first a transformation of the partialamplitudes and phases with suitably chosen small correction functions ei and Si :

at = At + ei(A,,, 0v)

99i = 0i + bi(AV, P,.) .i = 1, 2 , (12.4, 6)

We shall write, for the transformed phases,

01 =Qit+et, (12.4,7)

that is,

O'i = ei + ai .

The aim is to determine iteratively, for the four stochastic processes Ai, Oi (i = 1, 2),the stochastic differentials, which we shall write in the form

dAi=Pidt+Qid$, dOi=R,dt+Sid$. (12.4,8)

Ito's differentiation formula (compare for instance ARNOLD (1973)) leads to thedifferentials

2 aeidAf + E -aEidei dOj + Qei dt=

8A1 j=1

of the stochastic processes e1(A,,, v = {1, 2}, if V denotes the operator of theadditional Ito terms,

12

2 a2 a2= 9jE 1(Q+s2)+Q1Q2aAaAaAa2 2

+ sls2 + E Qfska2

60 , a02 j, k=1 aA, a0k


Using this and equations (12.4, 7), (12.4, 8), (12.4, 5), we get from (12.4, 6)

2 aEi aei aEidAi =

-j=

40-1 + aA9P +

a01

Rf - Vsi + pi(A, + E,,, 0 + 8b) dt

r 2

as'aEi+L-a1(a Q,-i )]d $ .d (12.4,9)

Analogously, the equation

dOi = (-i=1

(01 aS602 + aSti

aAfP, + aaz

a0l R1/-Obi + ri(A F s,, qi I- 8v) dt

L J JJ

i r-El lab,Qx -I a Ss + si(A,, -f ,,, 0e + S.) d,

LL ` t

can be found.

12.5. Fokker Planck Kolmogorov equation

The Fokker Planck Kolmogorov equation of the probability density w(A, 19, t) is,for a system of differential equations of the form (12.4, 8),

aw + a(P1w) + a(P2w) + a(R1w) + a(R2w)

at aA1 aA2 a01 a02

_ 1 82(Q12w) 1 a2(Q2w) 1 a2(S2w) 1 a2(S2w)

2 aAi + 2 aA2 + 2 a0i + 2 a02

+ a2(Q1Q2w) + a2(Q1S1w) + a2(QIS2w) + a2(Q2S1w) + a2(Q2S2w) + a2(S1S2w)

aA1 aA2 aA1 a01 aA1 ae2 aA2 a01 aA2 ae2 9

(12.5, 1)

Even in the important stationary case

aw =0aat

which we assume in what follows, no closed-form solutions of this equations areknown. We shall use an approximation method (compare STEATONOVICH (1961)) tofind an iterative solution.

In the first approximation, in the sense of the averaging method, we consider in(12.5, 1) only the non-oscillating (that is, not depending on 01, 02) expressions andseek correspondingly also the probability density as a function of Al and A2 only(compare STRATONOVICH (1961)). For such a probability density, (12.5, 1) simplifies to

2 2 2 2 2

Tw + T1aw + T2 aw - Q1 a w + Q2 a W

+ Q,Q2a W

(12.5, 2)aAl aA2 2 aAi 2 5A2 aAlaA2


where

_ aPl aP2 aR1 aR2 1 a2Q2 _ 1 a2Q2TaA + aA + aO + aO 2 aA2 2 aA21 2 1 2 1 2

1 a2S1 1 a2S2 a2(Q1Q2) a2(QISl) a2(Q182)

2 a e - 2 aO2 - aA1 aA2 aA1 3A9 3A1 ao2

a2(Q281) a2(Q282) 82(S2S2)

aA2 80 aA2 802 aol ao2'

386

(12.5, 3)

Tl = P1 _ 8Qi a(QIQ2) _ a(QISS) - 8(Q182)

aAl aA2 aO1 AT2 = P2 _

8Q 2 a(Q1Q2) _ 8(Q281) _ a(Q2S2)

aA2 aA1 801 802

The correction functions ej, at in (12.4, 6) are not yet used (put equal to zero) infirst approximation. We assume for simplicity 92 > 01 and exclude an internalresonance 122 = 2Q1. Then we get

T_- B1+B2-3DA2+(g-Klf)2 f2

2 8 16111 4122A2 '

_ BiA1 DAi (g - K1f)2 AlTl2 8 16111 '

B2`92 /2T2

2 + 4D2A2 ' (12.5, 4)

= (g -K1f)2 A,

8121

2/2

Q2-2112

Q1Q2 = 0 .

In this approximation, the coefficients (12.5, 4) of the Fokker Planck Kolmogorovequation (12.5, 2) do not reveal the dependence on non-linear inertia and restoringforces.

We now use the transformation (12.4, 6) in order to find a higher-order solution ofthe Fokker Planck Kolmogorov equation. For simplicity assume

K2=L3=0which holds in particular for the autoparametric vibration absorber.

Choose the correction function E1 so that in the equations (12.4, 8), (12.4, 9) forP1 the terms of pl which are of greatest order of magnitude and independent of phaseand which contain neither the square of the excitation coefficients f, g nor et, 6,multiplied by other small parameters, are cancelled by the expressions

2 ae,- E .Q, -=1 aO,

25 Schmidt/Tondl


in (12.4, 9). We shall succeed in doing so by choosing

EB1A1 sin 20, + DAI sin 40

CAI cos 20, - CAI cos 401 4521 1 32521 1 8 521

I32 521

1

--- 61 cos 401 -452 /2121A2

cos (201 + 02)1(1 + S22)

K!AIA2cos (201 - 02).

491(291 - 522)

Correspondingly, cancel the terms of greatest order of magnitude in P2, R1 and R2which do not depend on phase by choosing

2

E2 =B222A

sin 202 -452 //252 ) cos (201 + 02)

2 2(1 2

}LA1

cos (201 - 02)4522(2121- 522)

2 2 2

S1 - BI cos 201 +DAI

cos 2O1 +DAI cos 401 + CAI sin 201

4521 8521 32521 4521

CA2 sin 40, - EAI sin 20EAI sin 4i

+ 32522 1 4 1 16 1

+4121(2A+ 522)

sin (201 + 02) +4521(25212 2)

sin (201 - 02)

sin 02+ 212

KA2

1122

and2

S2 =B2

cos 202 +LA

sin (2451 + 02)492 4122(2521- 122) A2

LAIsin 20+

492(291- 522) A2 ( 1 - z)

These correction functions have to be inserted in (12.4, 8), (12.4, 9). Thus we canfind, if we also exclude an internal resonance 122 = 4521, the additional terms of secondapproximation in P1 independent of phase

EB1AiP1 add

+=

832QCDA15

2 (12.5, 5)1

By (12.5, 5) we get for the coefficients (12.5, 4) of the Fokker Planck Kolmogorovequation (12.5, 2) - because the other terms are, as terms neglected in second appro-ximation, of power two in the excitation coefficients - the additional terms

T1 add = PI add (12.5, 6)

and

'aPI add _ 3EB1A1 5CDAi

dd = + (12.5 7)a8A1 8 32522

,

12.5. Fokker Planck Kolmogorov equation 387

If we consider instead of the system (12.4, 1) the corresponding system with onedegree of freedom for x1 only, which reads, by use of (12.4, 3),

xi S2ix1 = -B1x1 - Dxixl - Cxl - B(x12 + x1xl) - (g - K11) fi(t) x1

with the coefficient g - K11 of parametric excitation composed by the direct para-metric excitation and the autoparametric excitation in (12.4, 1), the probability densityof the amplitude is given by an ordinary differential equation, and thus in secondapproximation by (11.1, 20):

_ snial 1

w (A,) cA1 gf>' e-UA1-VA1I

U Q D 2Q EB1 + 5CB1 - E (12.5 8)(9 - K1/)2 (g - K1/)2 2(g - K1/)2 2

V =16(g

1

- Kif)2(3DC - 2S212DE - 5CEB1 + 2D2 2B1) .

By analogy with this form of solution, which enables us to separate the behaviourof the probability density for Al - 0 where the exponential factor tends to one fromthe behaviour for greater Al where the exponential factor predominates, we shalltry to solve the partial differential equation (12.5, 2) iteratively in the form

w(A1, A2) = cAiA2 e (12.5, 9)

choosing from the beginning coo = co = 0 by analogy with (12.5, 8).Abbreviating (12.5, 9) by

w = cAlA2eO

we find, for Al + 0, A2 + 0, that

aAl -cAiA2 eo

(A21 aol + aA

a2w i

,

(2 ao )2 - 2 820 ]aA2 - cAiA2 e A + aA A2 + aA2 '1 1 1 1 1

aA = cAiA2 ell (A2 } aA I

2 2

a2W

alpaA2 = cAiA2 eO L(A + aA 2 A2 + a1

2 2 2/ 2 2

(12.5, 10)

Consider exponents 0 up to the sixth power in A1, A2, assuming ckt = 0 for k + 1 > 7in (12.5, 7). Then

ao= 2c 0A c A 3c A2 2c .A A C12A2 4c As 3c A2A

M, 2 1+ 11 2+ 30 1+ 2 1 2+ 12 2+ 40 1+ 31 1 2

+ 2c22A1A2 +013`12 + 4c41`4iA2 + 3C32`4i`42 + 2c23`41A2

+ c14A2 + 6c60A1 + 5cb1A1A2 + 4c42AiA2 + 3c33AiA2

+ 2c24A1A2 +

and analogous expressions for aO/aA2 and the second derivatives will hold.25'


We have to insert all these expressions into (12.5, 10) and the resulting expressions,as well as (12.5, 4), (12.5, 6), (12.5, 5), (12.5, 7) into (12.5, 2), and divide by w. Acomparison of equal powers of A1, A2 leads, after a lengthy analysis, to equations foris is C.

We find for the lowest powers,

12 . (j-1) _0

A4522

and

/ColA24Q2

0

from which it follows because forced excitation does not vanish, that

j=1, cot=0.Using this, we find

A1

A2

and

C11=0,

1: 2 C02 = - 1 2 B1 - B2 + (1 - 22) (9

1

4!1: C21=0,A2

A1: c12=0,A2: c03 = 0 .

In the next step we derive from the coefficients of

As1

A2C31 = 0 ,

(12.5, 11)

iAi:S22

C22 =3 + - (EB1 - D) - B1c20 - (1 + i) (g 4Df)2 C20 (12.5,12)

2 1

A1A2: C13 = 0

A2,2. 4c04 = -

further that

At 4A2 C41=0,

2'Q2B2C02 2

/2C02

As:i n2C32 Bi { (3 } 2i)

(g8Dllf)21C-30 = 0

so that we can choose

2AlA2:

AA2,2A3.

2

C30 = C32 = 0 ,

C23 = 0 ,

C14 = 0 ,

Cob = 0 .

(12.5, 13)


Finally the coefficients yield

A15A2.

A4.1

C51 = 0 ,

D EB1 (9K11)2

C424+

4C20

4521

(5 + i) DC- 2B1c4o +

3251

21c20 +2(2+i)040)

A 3IA2: C33 =0,

AI22

4/2 -A2

2C24 = [iBi B2 - (3 -f- 2i - i2)

(g8Q2lf)2

c22 , (12.5, 15)

A3,1A2. C15=09/2 2124.A2

-+ -c = - B .) CC06 2 040222,51 Q2

It shows that the only eventually non-vanishing coefficients we obtain are j = 1,c02 given by (12.5, 11), C22 given by (12.5, 12), coo given by (12.5, 13), c42 given by(12.5, 14), c24 given by (12.5, 15), and c06 given by (12.5, 16), whereas i, c20, c40, c60remain undetermined. In order to secure the correspondence with the one-degree-of-freedom solution (11.1, 9), choose in first approximation

2

i = 1 - 8_1B12

(12.5, 17)(g - Klf)

and

= -cD2ID

20 ' "40(9 -K,/)2Consequently, (12.5, 11) simplifies to

C02 = -

2'Q2B2

389

(12.5, 14)

(12.5, 16)

(12.5, 18)

f2

so that (12.5, 13) reads

C04 = 0

and (12.5, 12) leads to

(3 + i) S22EB1C22 =

8J2

If we use the

second!

approximation (12.5, 8) of the one-degree-of-freedom solution,we have to choose (12.5, 17) and instead of (12.5, 18)

Q1D 2Q1EB1 5CB1 E

020

_(g - K1f)2 + (g - K1/)2 2(g - K1f)2

+ 2 , (12.5, 21)

c40 = - 1 (3DC -2 Q1DE - 5CEB1 + 2QlE2B1) . (12.5, 22)16(g-Klf)2


Using these formulae, we get from (12.5, 12)

C22(3 + i) S22 E// - K1/)2 II T 2D2EB 5CB , 12.5, 23( )22 -

16S21 f2lg 1 - 1]

from (12.5, 14)2

C42= -3 [2iDE - 5(2 + i) DC + 2E2(g -Klf)2112

5OC2Bi 2+ 3(3 - i) Sl1E2B1 - 5(3 - 2i) ECB1](g - Kif)2

+J

and from (12.5, 15)

S! 3+aC24

4f2B2

1 - 2 Bl C22

Formula (12.5, 16) yields

C06 = 0 .

The formulae thus found show that c02 and c20 (in (12.5, 21) because of the firstnegative term of highest order of magnitude) are negative. In formula (12.5, 22) forc40 the first two terms are of highest order of magnitude, they yield a negative coeffi-cient c40 if

3C > 2S21E . (12.5, 24)

The first approximation formula (12.5, 20) gives c22 > 0 for i > -3 whereas the secondapproximation formula (12.5, 23) leads to c22 < 0 if both (or neither) of the conditions

i> -3,E(g - K1/)2 + 2S22IEB1 > 5CB1

hold. If in (12.5, 24) the equality sign holds, so that coo 0, the second condition(12.5, 25) simplifies with E > 0 to i > -5, that is, the only condition for c22 < 0 isi> -3.

The highest order part of c42 is negative for

2iS21E > 5(2 i) C,

whereas c24 has the same sign as c22 if

(1 - i) B2 > (3 + i) B1 .

The order of magnitude of c20 and c40 is

D

(9 - K1/)2

whereas the order of c02 and c22 is B1, 2/f2, the order of c42 is D/f2 and the order of c24is B1, 2/14 if we assume 521, 2, E and C of order one and (g - K,/)2 and B1 of the sameorder.

We evaluate the coefficients for the example Q1 = 1, £22 = 2.5, C = E = K1 = 1,D = 0.1, B1 = 0.001, B2 = 0.004, g = 0 and different values of the forced excitation

12.6. Behvaiour of the solution .391

parameter /. The first approximation formulae (12.5, 17), (12.5, 18), (12.5, 19)'(12.5, 20) yield

f C20 C02 C22

0.07 -0.6327 -20.4082 -5.1020 0.3774 (12.5, 25)0.10 0.2000 -10.0000 -2.5000 0.25000.20 0.8000 - 2.5000 -0.6250 0.0742

The second approximation formulae (12.5, 21), (12.5, 22), (12.5, 23) and the ensuingformulae give

f C20 C40 C22 C42 C24

0.07 -20.0100 -1.2372 -0.3586 31.9125 -0.29160.10 -9.5500 -0.6063 -0.8750 20.3047 0.00000.20 -2.0125 -0.1516 -1.3730 5.6599 0.8045

(12.5, 26)

12.6. Behaviour of the solution

In order to evaluate the extreme values of the probability density, we have to deter-mine the points of intersection of the curves

aw _0

8A1

that is,

kck1AiA2 l + i = 0and

aw _00A2

that is,

1CJ, AkjA$ + 1 = 0 .

A numerical evaluation is possible in every case. For k + 1 < 6 we can find a cubicequation for All,

4C40C22A1 + 2(C22C20 + 2C40C02) Al + [2c20C02 + (i - 1) C221 A2 +'C02 = 0(12.6, 1)

and a corresponding formula

2 _ 1A2 (12.6, 2)2c22Ai + 2c02

Neglecting c40, (12.6, 1) yields the solution2

4C22C20`41 = -2c20C02 + (1 - 2) C22

± V4c20C02 + (1 - j)2C22 - 4(1 + 2) C22C20C02 . (12.6, 3)


If we further neglect c22, we get

A2 - - 2 A2 - - 11- - 22620 2602

(12.6, 4)

If forced excitation is so small that i < 0 (/ = 0.07 in the examples above), (12.6, 4)has no real solution A1, A2, that is, the probability density diminishes monotonicallyfrom w = oo for Al = 0. If, as for the other examples, i > 0, (12.6, 4) determines amaximum value of the probability density. Using in the first approximation thevalues (12.5, 25), in the second approximation (12.5, 26) and neglecting c22 by (12.6, 4)or taking 622 into consideration by (12.6, 3), (12.6, 2), we can find for f = 0.1

Al max A2 max

first approximation, without 622 0.1000 0.4472first approximation, with c22 0.1003 0.4474second approximation, without 622 0.1023 0.4472second approximation, with c22 0.1014 0.4464

and for f = 0.2

Al max A2 max

first approximation, without c2, 0.4000 0.8944first approximation, with 622 0.4049 0.9253second approximation, without 622 0.4458 0.8944second approximation, with 622 0.3756 0.7815

In the first approximation the coefficient 622 is positive and shifts the maximum togreater values of Al and A2, whereas in the second approximation 622 is negative andhas the opposite influence.

As in the first approximation c22 is positive, (12.6, 3), (12.6, 2) lead to a secondextreme value (a minimum) of the probability density:

Al min = 3.1544 , A2 min = 6.3182 for / = 0.1 ,

Al min = 2.8670, A2 min = 5.7470 for / = 0.2 .

For these values the probability density is practically zero, for instance for f = 0.1Wmin/Wmax is less than 10-41.

In the second approximation this minimum disappears because then c22 is negative.By means of (12.5, 25), (12.5, 26) we can determine how the ekl terms contribute to

the exponent in the probability density formula (12.5, 9) for Al = Al max, A2 = A2 max(in second approximation, with c22) :

Contribution of

J C20 C02 C22 C40 C42 624

0.1 -0.09817 -0.49825 -0.00179 -0.00006 0.00043 00.2 -0.28376 -0.38175 -0.11825 -0.00301 0.06873 0.04232

This shows that 620, 602 and in the second place 622, have the greatest influence on theexponent. The influence of 642 is greater than that of c40 and c24.

12.6. Behaviour of the solution 393

An especially strong influence of c42 is revealed in the example B1 = 3, B2 = 10,C = g = 0, E = 6, D = 0.1, B1 = 0.001, B2 = 0.004 and K1/ = 0.3, particularlyfor K1 = 1.5, f = 0.2 or K1 = 3, f = 0.1 or K1 = 6, f = 0.05. The formulae of secondapproximation and (12.6, 4) give

i = 0.2, c20=-5.8,

and

c40 = 7.05, C24=0

X1 J Cot C22 C42 Al max A2 max

1.5 0.05 -160 -576 -1578 0.1313 0.05593 0.1 - 40 -144 - 394.5 0.1313 0.11186 0.2 - 10 - 36 - 98.6 0.1313 0.2236

that is, the same ratio

c42 : co, = 9.8625 .

The general formulae of the approximate maximum amplitudes (12.6, 4) on thesystem parameters are found, by inserting (12.5, 17), (12.5, 21) and (12.5, 19), to be

(g - K,f)2 - 8QB1A2 max = 2S11D - 4S12EB1 5CB1 - E(g - Kl f )2 '

/2'42

2max = 2

2S12B2

These show that the parameters of the first equation (12.4, 1) and the forced excitation/ combined with K1 determine Al max whereas the parameters 812, f and B2 of the secondequation (12.4, 1) determine A2 max. Increasing eigenfrequency 812 and linear dampingB2 diminish A2 max. Increasing non-linear damping D and non-linear restoring force Cand in general also increasing eigenfrequency 821 and linear damping B1 diminishAl max. The amplitude A2 max is proportional to forced excitation /, whereas Al maxincreases with the combination g - K1 f of forced and parametric excitation.

Fig. 12.6, 1 gives the maximum amplitude Al mx as dependent on B1 for B1 = K1_ E = C = 1, D = 0.1, g = 0 and different values of f. The dependence of Al maxon f for different parametric excitation g and Bl = 0.001 is shown by the full curvesof Fig. 12.6, 2, the dashed line giving the corresponding maximum amplitudes A21IIaxfor B2= 2.5 and B2=0.004.

0.5

0.4

0.3

0.2

0.1

f= 0.2

0.0005 0.001 0.0015 0.002 B1

Fig. 12.6, 1

f0.1f1070.1 Q0s2 0.3 0.4 0.5 0.6 0.7

Fig. 12.6, 2

394 12. Systems with autoparametrie coupling

The above discussion of the probability density w(A1, A2) was possible withoutevaluating the integration constant c in this function. In order to determine quantita-tively the probility density we have to evaluate c by the normalization condition

00

ff wdA1dA2=1 (12.6,5)0

where w is of the form (12.5, 9). Using all non-vanishing coefficients evaluated aboveand the abbreviations

2 41P = -C02 - C22A1 - C42A1 '/ 2Q = -C24A1l (12.6, 6)f

as well as the integration formulae (11.2, 1), (11.2, 2), we can reduce (12.6, 5) to thesingle integral

CO

where

1 I P_ e8Q D_18Q UQ1 for Q=0,

2Pand D_1 given by (10.2, 8), (10.2, 10).

The integral (12.6, 7) can be evaluated numerically. If we assume

C201 c02 and c22 < 0 ,but

we get

I= J Ai e``A+`°°AJ(A1)

dA1C

0

J(A1) = f A2e-142-QA2

dA20

C42 = C24 = C40 = 0

1 1CO

A%, e°'ogidA1

C 2 C02 + C22A0

or, using the integration formula

J

x°-1 e-exdx = 'L"_1 e0 r(v) r(1 - v, Pr)

(12.6, 7)

(12.6, 8)

for ,u > 0 , v > 0 , larg zi < 7

(compare for instance GRADSTEJN and RY$nt (1971), p. 333) and putting x = A1,

1-i

C 4C22 (Cp2)

2 e r(1

2 i) r\1 2 - COC220

1

for QUO,

P>0

for i > -1 .

12.6. Behaviour of the solution

If additionally c22 = 0 holds, we find that

1'122/C = - - 1+i

4c02(-c20) 2

for i > -1 .

If we introduce instead of (12.6, 6) the abbreviations

P' _ -C20 - c22A2 - c24A22Q = -C40 - C42A2

the integration formula (11.2, 1) leads to the single integral00

1 eCo8A2 J'(A2) dA2

0

with00 Po

J'(A2) =.f AI, dA10

I 2(2Q) 4 es¢ D-l+b

2 \

1 -1+i pa P,

--1+i

( P F(

22 2)

3961

(12.6, 9)

for i> -1, Q'>0,

for i>-1, Q'=0, P'>0,

which has no advantage for the evaluation of c in comparison with (12.6, 7), but showsgenerally that, corresponding to the one-dimendional case of Chapter 10,

that is

(g - K1!)2 > 4Q B1

is the integrability condition for the probability density. If this condition does nothold, the probability of positive amplitudes is zero and no vibration arises.

Under the assumption

C201 C02 and C42 < Q r

but

C40 = C22 = C24 = 0

taking into consideration besides the second order coefficients the greatest coefficientc42, which corresponds better with the last example, (12.6, 7) reads

00A21_ 1 Ale`- I

c 2 J c02 + C42 AtdA1

0


The transformation

C42 2AixCo2

and the integration formula (compare GRADTEax and RYZIg (1971), p. 337)00

give

v 1 -gx

Jx e _ dx = 7sV,(20,0) (v > 0,

e > 0)1 { x2 sin 7cv

0

3-i l+i4C02 4 C42 4 sin

2(12.6, 10)

7tVi2 i (- X201 c4z, 0)VC

where V, are Lommel functions of two variables which can be represented by

V 1 + ( 2 , 0 ) = P S i 1(C) (12.6, 11)2 '-2

`` 2 J

with the asymptotic development (compare GRAD TEax and RYZ x (1971), p. 1000)

z+in-1(-1)°I'3

4-+i4--v I'1 4--

i4+ v

2 + O1e2-2n

a z v-0 211-,(4+4)1-,(4+4)

if i -1, -3, -5, ... (12.6, 12)

The probability that the amplitudes Al and A2 are greater than certain levels canbe readily determined by help of the two-dimensional probability density w(A1, A2).

The probability that Al is greater than a level a1, as dependent on A2, is00

W (a1, A2) = f w dA1 ,ai

while the corresponding probability that A2 exceeds a level a2 is00

W(Ai> a2) = f w dA2 .a'

The probability that a certain amplitude Al occurs, independently of A2, isCO

WA1) = f w dA2 ,0

the probability of a certain amplitude A2 independently of Al is00

W(A2) = f w dA1.0


By combination and use of (12.6, 5) we get the probability that Al is greater thana level al, independently of A2,

00o0 a, 00

Wj(al) = ffw dA1 dA2 = 1 - f f w dAl dA2a, 0 0 0

(Fig. 12.6, 3), the probability

A2

Fig. 12.6, 3at At

0000

0001,

W2(0'2) = f f w dA1 dA2 = 1- f f w dAl dA20 a3 0 0

that A2 exceeds a2, independently of Al (Fig. 12.6, 4), and the probability

AJ

a2

Fig. 12.6, 4At

W CG d, 00

W12(al, a2) = f f w dA1 dA2 + f f w dAl dA2a, 0 0 a,

a, a,= 1 - f f w dA1 dA2

00

that Al exceeds a level al and A2 exceeds a level a2 (Fig. 12.6, 5).

A2

a2

Fig. 12.6. 5al A,

We assume the probability density takes the form

e." + co. A2 + c4,Ai + c.,A4A2 + c..AiA2w=cAlA2e .

By analogy with (12.6, 8) we find thatco

(c.. + c..A2 + c..Al) a2rA c..A2 + ez,AiA2 + c.,A4A2 1 e

J2e

2

_- -2 cot + ez2A1 + ca2A1

(12.6, 13)

(12.6, 14)

(12.6, 15)

(12.6, 16)

as


if the expression in parentheses is negative, in particular that00

f A2 e`°'A2+`$Zg1A2+`49A1A2dA2/

1 4 (12.6, 17)J 2 (CO2 +

C22`412

+ C42A 1)0

The probability (12.6, 13) becomes, with (12.6, 17), if we neglect C40A4 in comparisonwith c20Ai,

a,

Ai c20A2

W1(al) = 1 +Cc 1

e2 4 dAl . (12.6, 18)

2 c02 + C22A 1 + c42A 10

This integration not being possible in closed form, we have to consider only one ex-pression in the denominator.

For not too great values a1, we can neglect (c22A2 + c42A2) A2 in comparison withc02A2 respectively with c20Ai and get, because of (12.2, 5),

1 +% \1

W1(a1)4c

C(-c20) 2 I' (1 2

2, -C20ai1

02

or, considering (12.6, 9),

+ 2

21

z C200C1

W1(a1) _ -

(C2

For numerical evaluation the development

00I'(v, fZ) = F(v) -I (-1)"`,uv+n (v

+ 0, -1, -2, ...)n-o nt (v + n)!

corresponding to (10.2, 4) can be used. The asymptotic representationfor great IM

v 1e-µ11+0GY-)]

(compare for instance GRAD§TEJN and RYZIK (1971), p. 956) yields

W1(a1) _

21 1

1_y1 + 2 1+0 \20a,)

(-C20) 2 al-ir 2

for

(12.6, 19)

(12.6, 20)

of F(v, u)

(12.6, 21)

IC201a1>1

(12.6, 22)

The probability W12(a1, a2) becomes, if we neglect the same terms and use (12.2, 2),(12.2, 5), (12.6, 17),

l1 + i 2\2

W12(a1, a2) = (1 - (12.6, 23)


Special cases of this formula are (12.6, 18) for a2 = oo and

W2(a2) = e0 2"2

for al = oo. Analogously to (12.6, 21), (12.6, 22), formula (12.6, 23) admits an asymp-totic representation for IC201 a i>> 1.

If we write (12.6, 18) in the form

00

C Ai eC22Ai

W1(al) _ - -2 C02 -F C22A 1 + c42A

dA1,

a,

we can, for greater values of %, neglect 602 + C22A2 in comparison with c42A4 andthus get instead of (12.6, 19)

/ C3-2 i-3

W1(x1)4C

( -620) 2 .I' ( 2-620ai) .

42

Using (12.6, 10) with (12.6, 11) and the first term of the development (12.6, 12) aswell as the asymptotic representation (12.6, 22) for I620I a2' 1, we find that

C02r1 -i\

/sin

\ 1

+i nl e°40"i

W1(al) - 2 ` 2/J- (12.6, 24)1-{

7LC42(-C20)2 a11-;

An example for the probability W1 is given in Fig. 12.6, 6 where, corresponding tothe examples p. 393, i = 0.2, c20 = -5.8 and C42 : Co2 = 9.8625. The dashed line isfound from formula (12.6, 22), the full line from formula (12.6, 24), taking intoconsideration 642. It shows that neglecting C42 yields probabilities of about the sameorder of magnitude. For values al < 0.5 the asymptotic formulae used do not hold.

1

Fig. 12.6, 6

The probability W2(a2) is sketched in Fig. 12.6, 7 wirh Q2 = 7 and different valuesof B2//2. It shows how this probability diminishes with increasing damping B2 respec-tively augments with increasing forced excitation / depending on the level a2.


W2(a2)

2 4 6 8 10 a'Fig. 12.6, 7

12.7. Application of computer algebra

The computer algebra methods sketched in Section 6.8 have been applied to the pro.blem at hand in order both to check and to generalize the analytical results found.The developed program consists, according to the analytical methods, of four steps.Firstly it transforms the differential equations into the standard form. Secondly itrealizes the iterative elimination of the fluctuating terms up to the approximationwanted. For this purpose the program expands amplitudes and phases into serieswith a small parameter. Thirdly the program realizes the averaging in the excitationterms. At last it provides the ansatzfunction chosen, inserts it into the Fokker Planckequation and determines, by means of coefficient comparison with respect to thepowers of the amplitudes, the set of coupled equations for the different coefficients ofthe ansatzfunction.

The computer algebra program not only verified the above formulae, it also realizedthe generalized ansatzfunctions

i In A1+ j In A,+ i1 In A, -In A,+ E cktA lAIk,1 even

w(A1,A2) = c e k+1+6

and

j2A2InAs+ E cktAiA2k, I even

w(A1,A2) = c e k+1+4

which take into consideration that the amplitudes Al and A2 are dependent also ifone of them is small. It showed that the additional terms disappear so that the ansatz(12.5, 9) is confirmed.

The program also gives the possibility to solve the problem in higher approximationor to solve even more complex problems where the analysis "by and" is not possibleor not suitable.

Other problems of coupled, especially of autoparametric vibrations as for instancevibrations of beam systems (frames) or vibrations of structures filled with liquid(containers) lead to similar equations of motion which can be investigated analogously(compare BARR (1969), BARR and McWHANNELL (1971), IBRAHIM and BARR (1975)).

Appendix

Method of obtaining phase trajectoriesin the phase plane

Consider a set of differential equations

dx= X (X' y)dt

dt = Y(x, y)

(A, 1)

and assume that X(x, y), Y(x, y) are functions of x and y such that they satisfy theconditions of a unique solution in the given space x, y, t. With the exception of thesingular points defined by the equations

X(x,y)=0, Y(x,y)=0 (A,2)

a single trajectory then passes through each point in the (x, y) plane. These assump-tions are, for example, always satisfied if X(x, y) and Y(x, y) are analytical functionsof x and y.

The calculation starts at a point of the phase plane whose coordinates are determinedby the initial conditions x(O) = xo, y(O) = yo (assuming that this point is not exactlya singular one); the slope of the trajectory passing through it is given by the equation

dy.

= N = Y(xw yo)/X (xm Yo)dx

(A, 1) implies

(A, 3)

d8 = [(dx)2 + (dy)2]1J2 = [X2 + Y2]1/2 dt . (A, 4)

The coordinates of the next point on the trajectory are approximately obtained fromthe coordinates of the point located on the tangent to the trajectory at a distanceds from the starting point (Fig. A, 1). They are described by the equations

xl = xo + dx = xo + ds sgn [X(xo, yo)]/(1 + N2)1/2 ,

yi = yo + dy = yo + ds N sgn [X(xo, yo)]I(1 + N2)112.(A, 5)

The calculations can be carried out using one of the following three procedures :

(1) A constant step along the trajectory, that is d8 = M = const.(2) A constant time step, that is dt = M = const. We can then substitute in (A, 5)

ds = [(dx)2 + (dy)2]112 = (X2 +- Y2)1/2 M . (A, 6)

(3) Combination of (1) and (2). Substituting in (A, 5)

d8 = (X2 + Y2)x lit (A, 7)

26 schmidt/Tondl

402 Appendix

Y'

dx

Fig. A, 1

dy

Yo

x

we obtain from (A, 4)dt = (X2 + Y2)%-1/2 M (A, 8)

where 0 < K < 0.5.

The disadvantage of procedure (1) (a constant step along the trajectory, that ismotion at a constant velocity) is its inability to provide clear information about theproximity of the singular point. In procedure (2) the proximity of the singular pointis readily discerned (the motion of the plotter is slowed down); however, the stepalong the trajectory ds is very small in the vicinity of, and fairly large at some distancefrom the singular point. The third procedure, if properly set up, does away with thedrawbacks while, at the same time, stressing the merits of the former two.

Below is a schematic program prepared for a Hewlett-Packard (type 9830 A) cal-culator. Since this calculator uses only capital letters, the following notation is intro-duced in the program:

x=A, y=B, dx=C, dy=D.Note: To define X(x, y) and Y(x, y) which are denoted by symbols in the schematic

program, it is necessary to substitute the appropriate expressions.

The procedure outlined below includes all three versions.10 SCALE ...20 S = f1 (+ for positive, - for negative time)30 M =40 DISP,,A=, B=";50 INPUT A, B60 PLOT A, B70 X = S * X(A, B)80 Y = S * Y(A, B)90 N = Y/X

I C = M * SGNX/(SQR(1 + N * N))100 2 C = M * SGNX * (SQR(X * X + Y * Y))/(SQR(1 + N * N))

3 C=M*SGNX*((X*X*Y*Y)tK)/(SQR(1 + N*N))110D=N*C120 A = A + C130 B = B + D140 GO TO 60150 END

(* denotes multiplication, t raising to a power)

The run is terminated by the command STOP.

Appendix 403

The example which follows shows a schematic program for obtaining the time de-velopment of a dependent variable, for example, x = x(t).

10 SCALE

20 S= +130 M =35 T = 040 DISP,,A=, B=";50 INPUT A, B60 PLOT A, B70 X=S*X(A,B)80 Y = S * Y(A, B)90 N = Y/X

1 C = M * SGNX * (SQR(X * X + Y * Y))/(SQR(1 + N * N))100 2 C = M * SGNX (SQR(X * X + Y Y))/(SQR(1 + N N))

3 C=M*SGNX *((X*X+Y*Y)TK)/(SQR(1 + N *N))110 D = N * C120 A = A + C130 B = B + D

1 T= T+ M/SQR(X * X+ Y* Y))1352T=T+M

3 T=T+M/((X*X+Y*Y)t(0.5-K)140 GO TO 60

150 END

The chief merit of the procedure just described is the simplicity of the program.In some cases, for example, when two stable singular points and a saddle point butno limit cycle exist in the phase plane, the procedure can be started from any pointand the trajectory will always tend to one of the stable singular points regardless ofthe size of the step. This circumstance has no qualitative effect; a similar statementcan hardly be made of other methods.

The size of the step must usually be reduced when the trajectories are very dense,for example, when the stable cycle lies close to the unstable one. In cases of this sort,reduction of the size of the step alone is frequently not enough and the progress ofthe solution must be checked by repeating the calculation at both positive and negativetime. Difficulties are sometimes encountered with slightly damped systems, especiallywhen the damping is non-linear so that virtually no damping exists in the neighbour-hood of the equilibrium position.

The method of solution of trajectories in the phase space in which the system isdefined by the equations

dx_dt

XJ(xl,...,xn) (Y=1, 2,...,n), (A, 9)

which satisfy similar conditions of unique solution in the (x1, ... , xn, t) space, resemblesthat described by (A, 1). The resulting relations similar to those obtained for (A, 1)are as follows:

n nds = [ (dx9)2)'I2 = dt (E X1)112 ,

9=1 9=1(A, 10)

26*

404 Appendix

xk=Nk, (N1=1;k=2,3,...,n),

x1

dxk = Nk dx1 ,

dx1 = dt sgn (X1) (I Xi)1,2/(1 + E Nk)1/2.j-1 ka2

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Index

abbreviated system 66, 68absolute stability 34absorber 283, 286, 297, 299, 302 - 304,

307 - 309, 320, 322absorber tuning coefficient 281, 283, 305,

321aeroelastic self-excitation 23air pressurized bearing 313amplitude 26, 42, 114

extreme values of 119, 141, 373forced 136maximum 116, 119, 173mean square 347, 350mean value 349minimum 173moments of 347, 376most probable 332, 345partial 158, 177, 366self-excitation 136

amplitude equation 115, 156, 190asymptotic stability 31, 34, 45, 78attraction, domains of 64, 81, 93, 95-97,

99, 111, 199, 204, 205, 207, 210, 221,232, 233, 237, 252, 254, 255, 261, 267,300

attractor 34, 90, 91, 199autoparametric coupling 362, 400autoparametric vibration 362, 380, 385,

400auxiliary curves 48averaging method 46, 338, 340

backbone curve 48-51, 55, 57-60, 62,63, 67, 68, 71-73, 117, 248, 274, 326

base circle 185basic system 282, 283, 299, 308

self-excited 282, 302beating 137bifurcation 41, 114, 154boundary curve 281, 322boundary value curve 290, 291, 293, 303,

316boundary values method 279broad-band random excitation 336, 380broken-line characteristic 15, 236

central symmetrical field 17centre 79, 82, 86centre-focus 89centre-node 89centre-saddle 89chain conveyor 22chaotic behaviour 34characteristic

broken-line 15, 236hardening spring 58load 100, 101restoring force 53, 55, 57, 76softening 14, 247, 248, 254, 255spring 63strongly non-linear 14symmetrical 14weakly non-linear 14

characteristic equation 27, 32, 67, 78,279, 282, 284

characteristic exponents 33circle of disturbances 204, 207-210circular frequency 112circular scanning method 199, 204, 206,

210, 218, 221combination resonance 28, 158, 165

difference type 159, 165gear 173parametric 28summed type 158, 163

combined excitation 25computer algebra 43, 196, 400computer-aided analysis 196constant prestress 112conveyor

chain 22

vibratory 232cotton-yarn spinning machine 313, 322

Coulomb friction 15, 282, 239coupled differential equations 154coupled vibrations 400coupling.

autoparametric 362, 400inertia 189strong 193, 195weak 188, 195

critical cases 34

416 Index

curveauxiliary 48backbone 48-51, 55, 57-60, 62, 63,

67, 68, 71-73, 117, 248, 274, 326boundary 281, 322boundary value 290, 291, 293, 303, 316resonance 57, 117skeleton 48vertical 327, 332

damping 53, 155, 393, 399destabilizing effect of 165identification of 48, 74internal 25linear 112, 119, 393material 25, 314non-linear 112, 124, 393viscous 52

decaying oscillatory pulse 252, 255Descartes' theorem 83, 88destabilizing effect of damping 165detuning 180, 372difference type combination resonance

159, 165dimensionless time 112, 155dispersion 349dissipation function 167, 185disturbance function 299disturbance pulse 252disturbances 200, 207, 233

circle of 204, 207-210, 232in the initial conditions 81, 201, 205,

206, 212probability density 208, 209, 234random 234, 235resistance to 235small 36, 39, 199sphere of 211, 232substitute sphere of 211

disturbing pulse 300divergent vibration 250, 255, 256, 258,

262, 267-270domains of attraction 64, 81, 93, 95-97,

99, 111, 199, 204, 205, 207, 210, 221,232, 233, 237, 252, 254, 255, 261, 267,300

dry friction 15, 59, 62, 63, 239-241, 243,245, 283, 284, 297, 302, 304, 305

Duffing system 35, 39, 45, 46, 57, 96, 97,204, 212, 247, 255

dynamic characteristic of the restoringforce 76

eccentricities 169eigenfrequency 189, 393elastic displacements 184equation

amplitude 115, 156, 190bifurcation 41, 154characteristic 27, 32, 67, 78, 279, 282, 284

coupled differential 154Fokker Planck Kolmogorov 338, 361,

384frequency amplitude 115generalized van der Pol 112higher-order differential 64higher-order solution of Fokker Planck

Kolmogorov 385in standard form 323integro-differential 41, 154

Ito 337, 382Lagrange's 167linear variational 31, 123Mathieu 19, 36, 248, 249periodicity 41, 114, 154, 156, 163, 194,

197, 366physical 381Stratonovich 337, 381van der Pol 112, 282variational 31, 67, 162Whittaker 356

error excitation 190excitation

broad-band random 336, 380combined 25error 190external 26, 77forced 112, 155, 336, 393, 399general periodic 112inertial 48, 112kinematic 19, 256, 269linear parametric 13, 112, 263, 275narrow-band random 323, 375non-vanishing forced 345parametric 22, 26, 61, 77, 155, 257, 270,

336, 343, 393stiffness 190stochastic parametric 13

external excitation 17, 26, 77external moment 185externally excited system 204

Floquet theorem 33, 44focus 78, 79, 81, 82, 84, 88, 91, 94, 261focus-node 89Fokker Planck Kolmogorov equation 338,

361, 384force pulse 200, 234forced amplitude 136forced excitation 112, 155, 336, 393, 399

non-vanishing 345FORMAC 196foundation subsystem 283foundation tuning coefficient 283, 288frequency

amplitude equation 115parameters 169pulling 138variation 42, 112, 155, 170, 180, 189,

194, 323

Index

friction 166Coulomb 15, 239, 282dry 15, 59, 62, 63, 239-241, 243, 245,

283, 284, 297, 302, 304, 305friction forces 25fully developped vibration 345

Gaussian probability density 329gear

combination resonance 173drive 166, 184spur 166torsional resonance 170

gearing down 191gearing up 191general periodic excitation 112generalized forces 167generalized van der Pol equation 112generalized van der Pol self-excitation 131global resistance to disturbances 235

half developped vibration 345hard self-excitation 29, 92, 112, 131, 150hardening spring 257, 258, 269

characteristic 58harmonic balance method 35, 49, 54, 65,

72, 257, 259harmonic resonance 146Hayashi's plane 207Hayashi's space 210, 211higher-order differential equation 64higher-order solution of Fokker Planck

Kolmogorov equation 385higher-order system 64hydroelastic self-excitation 23

identification of damping 48, 74inertia coupling 189inertial excitation 48, 112infinitesimal stability 31, 34instability 31, 34, 45

parametric 122instability interval 28, 272integral equation method 41, 43, 326integro-differential equation 41, 154internal damping 25internal resonance 28, 71, 178, 180, 184,

213, 218, 370isolated resonance curves 142Ito equation 337, 382

jump phenomenon 249

kinematic excitation 19, 256, 269kinetic energy 167, 185Krylov Bogoljubov method 46, 77, 199,

248, 249, 251

Lagrange's equations 167Lagrangian 167

417

Lienard method 91limit cycle 30, 80, 90-95, 99, 101, 111,

240semi-stable 80, 111

limit envelope 48, 50-52, 55-60, 62, 63,65, 67, 71, 72, 74-76, 247, 248, 274,275

linear damping 112, 119, 393linear parametric excitation 13, 112, 263,

275linear variational equation 31, 123Ljapunov's methods 34Ljapunov stability 31load characteristic 100, 101local stability 34locally stable 64, 199, 205, 239, 275locally stable self-excited vibration 239locally stable solution 64, 205

machine, cotton-yarn spinning 313, 322machine characteristic 100-103, 111main parametric resonance 28, 273main resonance 28, 35, 59material damping 25, 314Mathieu equation 19, 36, 248, 249maximum amplitude 116, 119, 173mean square amplitude 347, 350meshing tooth pairs 168method

averaging 46, 338, 340boundary values 279circular scanning 199, 204, 206, 210,

218, 221computer algebra 43, 196, 400harmonic balance 35, 49, 54, 65, 72,

257, 259integral equation 41, 43, 326Krylov Bogoljubov 46, 77, 199, 248,

249, 251Lienard 91Ljapunov's 34Monte Carlo 200quasi-static 323stroboscopic 201, 203, 204successive approximation 42, 155TV scanning 199, 204, 205, 214. 233, 252van der Pol 38, 77, 98, 199, 203, 204,

248, 249, 251, 259mine cage 22, 270minimum amplitude 173moment

external 185

of the amplitude 347, 376Monte Carlo method 200most probable amplitude 332, 345multi-frequency self-excited vibration 30multi-frequency solution 38multi-stage gear drive 166

418 Index

narrow-band random excitation 323, 375node 78, 79, 81-83, 88, 91, 261non-linear characteristic 14non-linear damping 112, 124, 393non-linear parametric excitation 13, 28,

112, 146, 172, 263, 273, 275non-linear restoring force 155, 393non-linear system 13non-resonance case 113, 156, 169, 189, 194non-stationary probability density 353non-symmetric restoring force 112non-vanishing forced excitation 345normal mode 68

one-frequency solution 135optimization of a parameter 235orbital stability 34oscillatory decaying pulse 200

parametric diminishing effect 120, 125,142

parametric excitation 22, 26, 61, 77, 155,257, 270, 336, 343, 393

linear 13, 112, 263, 275non-linear 13, 28, 112, 146, 172, 263,

273, 275stochastic 13

parametric instability 122parametric resonance 27, 28, 61- 62, 275

combination 28main 28, 273

parametric vibration 63, 138, 270parametrically excited system 59, 98, 204partial amplitude 158, 177, 366path of contact 166periodicity equation 41, 114, 154, 156,

163, 194, 197, 366phase plane 29, 77, 90, 92, 267, 401, 403

stroboscopic 202, 203, 206 - 208, 212,221

phase plane trajectory 77, 80phase portrait 81, 90, 91, 94, 96, 98, 99,

103, 104, 199, 203, 213, 261, 167phase relation 117, 120phase space 30

stroboscopic 210physical equations 381pitch radii 166Poincar6 mapping 202potential energy 167, 185practical stability 31prestress

constant 112static 169-171, 189

principal axes transformation 193principal circles 210, 211

of disturbances 232probability

of exceeding a given amplitude level 350on the circle of disturbances 208

on the substitute sphere of disturbances211

transition 358, 361probability density 326, 336, 376, 383, 384,

391Gaussian 329non-stationary 353of occurrence 237of the disturbances 234of the occurrence of the disturbances

208, 209two-dimensional 358, 361Weibull 328

program system ASB 196proportionality 12pull-in range 138

quasi-normal mode approximation, 70, 71quasi-static method 323

random disturbance 234, 235random excitation

broad-band 336, 380narrow-band 323, 375

relative tooth deflection 168resistance to disturbances 235resonance

combination gear 173combination parametric 28difference type combination 159, 165harmonic 146internal 28, 71, 178, 180, 184, 213, 218,

370main 28, 35, 59main parametric 28, 273parametric 27, 28, 61, 62, 275single 156, 189, 194subharmonic 13, 28, 29, 152, 275subsuperharmonic 28summed type combination 158, 163superharmonic 28torsional-lateral 173two-fold 157ultraharmonic 28

resonance case 41, 113resonance curve 57, 117

isolated 142restoring force

non-linear 155, 393non-symmetric 112symmetric 112

restoring force characteristic 53, 55, 57, 76rheo-linear systems 12rotor 25rotor system 29

saddle 78, 79, 81, 82, 84, 85, 88, 89, 91,94, 95, 102, 103, 203, 206, 261

saddle-focus 85, 88, 89secondary components 192

Index

self-excitation 13, 24, 25, 155, 239, 278,284, 288

aeroelastic 23

generalized van der Pol 112hard 29, 92, 112, 131, 150hydroelastic 23soft 29, 112, 279van der Pol 148, 150

self-excitation amplitude 136self-excited oscillation 111self-excited system 14, 17, 29, 77, 91, 129,

204basic 282, 302

self-excited vibration 101, 138, 234, 240,278, 322

multi-frequency 30single-frequency 30, 280, 293

semi-stable limit cycle 80, 111separatrix 90, 91, 104, 199, 203, 206, 240,

261, 267separatrix surface 214, 221several independent small parameters 43single resonance 156, 189, 194single-frequency self-excited vibration 30,

280, 293single-frequency solution 38singular point 30, 77 - 82, 101-103, 111,

203, 214, 221, 259-261, 267, 403stability of 39

sinusoidal pulse 299skeleton curve 48slowly varying functions 323small disturbances 36, 39, 199soft self-excitation 29, 112, 279softening characteristic 14, 247, 248, 254,

255softening spring 257, 269sphere of disturbances 211, 232spindle 313spring characteristic 63spring stiffness 166spur gear 166stability 122, 133, 151, 153, 162

absolute 34asymptotic 31, 34, 45, 78for small disturbances 39infinitesimal 31, 34in the large 31, 81, 199, 201, 206, 212,

232in the sense of Ljapunov 31local 34of singular points 39orbital 34practical 31

stability condition 165static characteristic of the restoring force

76static prestress 169- 171, 189stationary case 384

419

stiffness 35spring 166tooth 171, 185tooth-pair 170torsional 166

stiffness excitation 190stochastic parametric excitation 13strange attractor 34Stratonovich equations 337, 381stroboscopic method 201, 203, 204stroboscopic phase plane 202, 203, 206 to

208, 212, 221stroboscopic phase space 210strong coupling 193, 195strongly non-linear characteristic 14strongly non-linear systems 13subharmonic resonance 13, 28, 29, 152,

275substitute sphere of disturbances 211subsuperharmonic resonance 28subsystem

basic self-excited 282foundation 283

successive approximation method 42, 155summed type combination resonance 158,

163superharmonic resonance 28superposition 12surge 101

symmetric restoring force 112symmetrical characteristic 14synchronization 30, 138system

abbreviated 66, 68basic self-excited 282, 302Duffing 35, 39, 45, 46, 57, 96, 97, 204,

212, 247, 255externally excited 204higher-order 64parametrically excited 59, 98, 204program (ASB) 196rheo-linear 12self-excited 14, 17, 29, 77, 91, 129, 204strongly non-linear 13van der Pol 279weakly non-linear 13with many degrees of freedom 154, 166

threshold value 122, 133, 139

tooth deflection 168tooth error 166, 169-171, 184tooth stiffness 171, 185tooth pair 168tooth-pair stiffness 170torsional gear resonance 170torsional stiffness 166torsional vibrations 184torsional-lateral resonance 173transition probability 358, 361transmission ratio 185, 192

420 Index

TV scanning method 199, 204, 205, 214,233, 252

two-dimensional probability density 358,361

two-fold resonance 157two-frequency solution 137

ultraharmonic resonance 28undevelopped vibration 344

van der Pol equation 112, 282van der Pol method 38, 77, 98, 199, 203,

204, 248, 249, 251, 259van der Pol oscillator 24, 278, 282van der Pol self-excitation 148, 150van der Pol systems 279van der Pol transformation 213, 218variational equation 31, 67, 162

linear 31, 123vertical curve 327, 332vertical tangent 124vertical tangent rule 39, 40, 258, 274vibration absorber effect 362

vibrationautoparametric 362, 380, 385, 400coupled 400divergent 250, 255, 256, 255, 262,

267- 270fully developped 345

half developped 345locally stable self-excited 239multi-frequency self-excited 30parametric 63, 138, 270self-excited 101, 138, 234, 240, 278, 322single-frequency self-excited 30, 280,

293torsional 184undevelopped 344

vibratory conveyors 232

viscous damping 52

weak coupling 188, 195weakly non-linear characteric 14weakly non-linear systems 13Weibull probability density 328Whittaker equation 356

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