torsional vibrations of a non-linear driving system with cardan shafts

Journal of Sotmd and Vibration (1973) 26 (4), 533-550

TORSIONAL VIBRATIONS OF A NON-LINEAR DRIVING SYSTEM

WITH CARDAN SHAFTS

J. ZAHRADKA

CKD Prague, Research blstitute for Diesel Locomotives, Ceskomoravska 205, Praha 9, Czechoslorakia

(Receired 30 October 1972)

This paper deals with the analysis of torsional vibrations in a system incorporating cardan shafts. The solution is carried out for a driving system having a single degree of freedom, in which two Hooke's joints are in general angular misalignment. The resulting governing equation of motion is a non-linear, non-homogeneous, second-order, ordinary differential equation with periodic coefficients. Its solution is investigated in the region of main, subharmonie, and subultraharmonic resonance by means of the approximate Van der Pol's method of slowly-varying coefficients. In particular, a solution is obtained for a particular driving system on an ICL 1905 digital computer, and a number of the amplitude resonance curves are obtained. In view of the fact that a number of solutions exist for one tuning coefficient of a system, the domains of attraction leading to the different resonant solutions are identified after the corresponding singular points are classified. The widths of these regions depend on two main factors: the damping of the system and the angular misalignment of the cardan shaft.

1. INTRODUCTION

The problem of a driving system with cardan shafts is rather broad; thus, it may be solved by a number of different methods and.various interesting phenomena occur in these systems. In this paper, the problem is analysed from the point of view of the existence of torsional vibrations which occur in the cardan shaft. This problem has, of course, been discussed in a number of papers [1-5], but, in all these works, as well as in an earlier paper by the author [6], the system is linearized by a different method. The analysis of oscillatory motion in these linearized systems usually leads to an equation of the Mathieu type. It follows from Floquet's theory that, under certain conditions, violent torsional oscillations can occur. In such cases, the amplitudes of quasiharmonic oscillations increase linearly or exponentially with time [7]. It is known from experimental investigations of such systems with cardan shafts [8, 9] that the growth of torsional amplitudes is considerable but that the amplitude will stabilize at a certain constant value. This fact is caused by the presence of non-linearities which are not taken into account in linearized systems. Never- theless, the results for linearized systems can be assumed in certain cases to be sufficiently correct. In particular, in the region of the main resonance of a relatively strongly damped system, where, by the virtue of the damping, the interval of instability is very narrow, the system can be considered as a stable one and linearized theory can be applied.

The determination of further resonance regions (the domains of so-called subharmonic and subultraharmonic vibration which could be important in the operation of composite systems) cannot be obtained by means of linearized theory. These resonance regions occur in ranges which are integer or fractional multiples of the speed range for main resonance [I0] and it is necessary to remark that their occurrence in systems with Hooke's joints has already been investigated experimentally [11, 12], and also, for a system with one joint,

533

534 J. ZAHRADKA

the problem has been analysed theoretically [13-]. For this reason the problem of concern in this paper is that of the non-linearities which occur in a system having cardan shafts.

2. DESCRIPTION OF THE SYSTEM

The equation of motion for the given system has been derived in detail in a previous paper 114-1. It is necessary to assume several important preconditions under which the driving system can be reduced to the torsional system shown in Figure 1. This system consists of three torsionally flexible shafts of torsional stiffness cl, c2 and c3, connected by two perfectly spherical Hooke's joints. The directions of the shafting axes, in general, can be described by the angles cq and ct2; in this analysis, all the shafts are assumed to be coplanar. By means of the kinematic properties of Hooke's joints, a load of moment inertia O is put into oscillation on the end of the output shaft. The rotation of the load is taken to be opposed by a viscous damping of magnitude equal to k times the angular velocity.

453

- - a / ¢

-" ¢ o t ~ $ t

Figure 1. Line diagram of one-degree-of-freedom cardan shaft system.

If the twist in the input shaft is denoted by x, then the equation of motion has the form

+ (/~xx + b2~ + ~zx2 + ~4xg +/~5 x2 +/~622 + b72x 2 +/~sx222) Sin 2oJt

= P cos 20~t + ~ sin 209t + ~ , (I)

where ~c is the relative damping parameter, I2 o is the natural frequency of the linearized system and ~i,/~i, P, O., and/~ are constants dependent on kinematic arrangement, torsional stiffness, viscous damping parameter, moment inertia of mass on the output shaft and input angular velocity. The dots denote differentiation with respect to time, t.

NON-LINEAR VIBRATIONS IN CARDAN SHAFTS 535

For the analytical solution, it is preferable to express equation (I) in dimensionless form. Thus, introducing the new time variable

"r = I2ot, the coefficient of tuning

the static angular displacement of the linearized system

Xo

and the relative amplitude p = X / X 0 ,

one obtains the following equation of dimensionless form:

v" + Do' + v + (atv + a2v' + a3v" + a+vv' + aso 2 + a6v'v 2 + aTv"v 2 + asvv '2) cos 2tl~

+ (blv + bzv' + b3vv' + b4vv" + bsv 2 + b6v '2 + bTv'v 2 + bsv2v '2) sin 2113

= !12 sin 2tlr + ~lP cos 2~lr + llr. (2)

The resulting equation of motion (2) is a non-linear, non-homogeneous, second-order, ordinary differential equation with periodic coefficients.

3. SOLUTION OF THE GOVERNING EQUATION

This equation of motion contains, in its homogeneous part, parameters which are periodically variable in the time, z. This variation of the parameters results in the occurrence of parametric excitation and leads to parametric resonance in systems of this class. In view of this fact, it is necessary to consider solving this equation for the regions of parametric resonance, as well as for the regions of the main resonance and subharmonie resonances (given by the right-hand side of the equation). In some cases, parametric resonances occur in regions of non-linear resonances. The manifestations of the two phenomena--the subharmonic resonance and the parametric resonance--are different, however. In the former, the resonance oscillation is produced directly by the external periodic excitation, but, in the latter, it is due to the periodic variation of parameters in the system. In the solution of the governing equation (2), parametric resonances of the second and first order will be considered. These regions correspond to those of the main resonance and of the subharmonic resonance of order 1/2.

3.1. ANALYTICAL SOLUTION BY THE APPROXIMATE VAN DER POL'S blETIIOD 1N TIIE REGION

OF MAIN RESONANCE

In this case, the region of main resonance is identical with that of second-order para= metric resonance, For this range the solution of equation (2) is sought, according to reference [15], in the form

v = X + a cos 2tl~ + b sin 2,lr, (3)

where X, a, b are slowly varying functions of time, and for stationary solutions are constants. In practical calculations, it is more convenient to assume a solution in the form

v = X + Ae 2t~* + Be -2t~=, (4)

where A and B are, for stationary solutions, complex conjugate constants. When one substitutes the assumed solution (3), and its corresponding differentiations and powers,

536 J. ZAttRADKA

into equation (2), neglecting second-order derivatives of slowly varying functions X, a and b, or possibly X, A and B, and further using the method of harmonic balance, one obtains the following equations for transient vibrations:

a' = ~ [2Dlla - (1 - 4112) b - ( b t X + . . . . ) + !12"1,

b' = ~ [2Dqb + (1 - 4112) a + (a tX + . . . ) - tlp],

0 = X + (½aaa + ½atb + . . . ) + ( ½ b j a - ½biib + . . . ) - tlr. (5)

Equation (5) may be written in the form

a' = f 1 ( a , b , X ) ,

b' = f z ( a , b , X ) ,

0 = f 3 ( a , b , X ) .

The steady solution is then given by the relations

f l ( a , b , X ) = O,

f2 (a, b, X) = O,

t

fa(a, b, X) = O.

In practical calculations, by using the solution in the form (4), the following equations are obtained for the steady state:

f , ( A , B , X ) = O,

f 2 ( A , B , X ) = O,

f3 (A, B, X) = 0, (6)

where, as already mentioned, the functions A and B are, in the steady state, complex conjugate constants. Introducing their components

A = Y - i Z ,

B = Y + i Z ,

and using those relations in equations (6), one obtains a system of three non-linear algebraic equations for the unknowns X, Y and Z:

a,(x, Y, z ) = o ,

a~(x, Y, Z) = 0 ,

a3(X, r, z) = 0 . (7)

The solution of equations (7) can be obtained by the iterative Newton-Raphson method on the digital computer. It is first assumed that the functions (7) possess, in the interval of


solution that is sought, partial derivatives with respect to X, Y and Z and that Jaeobi's

J =

matrix,

has a determinant

BGt dG1 3Gt 3X 3Y OZ

3G2 3G2 3G2 3X 3Y 3Z '

IOG3 OG 3 0 G 3 3X 3Y 3Z

(8)

where Rv = ~ + b 2 ,

OF,

Rv=2v/-Y-X+ Z 2, and, at the same time,

v = X + R v c o s ( 2 q r - WR),

Z kv R = arctg-~.

The solution was computed for several alternatives of the driving system in Figure 1, where particnlar, realistic values were taken from experimental investigations I-9]. Results for several configurations, which were obtained by means of an ICL 1905 digital computer,

IJI # o.

Then, after the determination of suitable initial approximations for the roots,

Xo, Yo, Zo,

which can be obtained by solving the corresponding linear parts of equations (7) or determined by estimation, it is possible to expand the functions G~, G2 and G3, in the neighborhood of the points Xo, Yo and Zo, in Taylor's series.

For the function G~ the first approximation will be

( 0 G t ) [ Ax ( o o q at, + o ax(Xo, Yo, Zo) + k ~ ] o ' + \ OY Jo ~ oAZ' = ,

and, written in matrix form, the nth approximation will be

. . . . . . . Xn+t = J - I (X, Y, Z) , . G(X, Y, Z)n,

where JL~(X, Y, Z), means the inverse matrix of Jaeobi's matrix (8). After the determination of the difference AX,+I it is possible to obtain the expression

for the (n + 1)th approximation. It follows that

It is thus possible to limit the number of iterations by introduction of an appropriate relative error between the unknowns X, Y and Z in the nth and (n + 1)th approximations, respectively. As the first approximation, one can consider either the solutions of the linear parts of equations (7) or arbitrary initial conditions. The steady statesolution (3) can be simply converted into the following form:

538 j . ZAHRADKA

a re p lo t t ed f o r the r e g i o n o f m a i n r e s o n a n c e in F i g u r e s 2, 3, 4 and 5. These c o n f i g u r a t i o n s

- " " °

150 50

~ 5 5 - .

o / ~, " ,7 \

I

1 I I I ! ! 1 0 I I I I I I " ) '47 0 .50 -O.53 ) '47 0-50 0-53

-,,,/.no Figure 2 Figtre 3

Figures 2 and 3. Relative torsional amplitude R~ in the region of main resonance. Figure 2: ~t, ---- 5 °, ct: = 10% k ----- 8 kpcms. AA' , stable foci; FF' , saddle points; - - - , linear part. Figure 3: cti = 5 °, =, = 15 °, k = 8 kpcms. AA', stable fool; DD', unstable foci; DEF' , D'EF, saddle points; - - - , linear part.

I I l I

30

~ 20

5

,10

'I/ %..

01 1 I I I I I 0 47 0 50

4 0 [ I I 1 I t I 1

/ / \ io l - / ~ .

I ./~nc\ I

) 5 3 0-47 0 5 0 0.53

~/-w/.(2 o Figure 4 Figure 5

Figures 4 and 5. Relative torsional amplitude R° in the region of main resonance. Figure 4: ~q = 5 °, ~2 ---- 20 °, k = 8 kpcms. AA', stable foci; DD' , unstable nodes; DEF' , D'EF, saddle

• points; - - - , linear part. Figure 5: cq = 5 °, ct2 = 20 °, k = 2 kpcms. ABC", CBA', stable foei; CC', saddle points; DD' , unstable

nodes; DEF' , D'EF, saddle points; - - - , linear part.


differed in the angular settings of the Hooke's joints, ~1 and ~2, and in the magnitudes of the viscous damping, k.

From the above-mentioned graphs of relative torsional amplitudes in the region of main resonance, the following conclusions can be drawn.

(a) In the neighborhood concerned, that of the main resonance, several solutions of the differential equation exist for a certain value of the coefficient of tuning, tl, given by the corresponding branches of the calculated resonance curves.

(b) The difference between the solutions of the governing equation (2) in the bottom branch and in its linear part is rather small for the systems which are more strongly damped (see Figures 2, 3 and 4). For systems of this type having the same magnitudes of the viscous damping coefficient but with different angles of shafting, cz I and ~2, the values of the resonance curves are nearly the same. It is possible to draw a conclusion from these two facts: namely, that there exists a certain function of the difference between the angles ~1 and ~2 which is, for strongly damped systems, an approximate limit for the absolute torsional amplitude. This function is the pre-determined static angular displacement of the linearized system, Xo:

Xo = Isin 2 ul cos ,~ - sin s us cos "2J •

With increasing magnitude of this constant, Xo, the difference between the solution of the non-linear equation and that of its linear part also increases. In the same way, with increasing Xo the upper branch of the resonance curves gets nearer to the bottom one.

(c) The difference between the solution of the non-linear equation (2) in the bottom branch and that of its linear part is already evident for systems which are only slightly damped. It is evident from the graph of relative torsional amplitudes in Figure 5 that, due to non-linearities, the non-linear system oscillates with a smaller amplitude in the bottom branch than does the corresponding linearized system. On the other hand, the upper branches of the resonance curve come close to the bottom one which in form approaches a loop.

3.2. ANALYTICAL SOLUTION BY THE APPROXIMATE VAN DER POL'S METHOD IN THE REGION

OF SUBHARMONIC RESONANCE

Subharmonic resonance of order ½ is characterized by the property that the oscillation in this region consists of two components with different frequencies. The first of them is the frequency of forced vibration given by the right-hand side of equation (2) and the other frequency is half of the first one. Under certain initial conditions, the subharmonic oscillation of half frequency can dominate and the corresponding amplitude can reach substantial values. Then, for subharmonic resonance one can establish that

O)subh ~-- 2 Q 0 ,

where f2o is the natural frequency of the linearized system. As the governing equation of motion (2) contains periodic variable coefficients and as it is known from the theory of linear differential equations of this type that the unstable region of the first order (of greatest importance of all for Mathieu's equation) is in the neighborhood of the value

a~ = 2f2 o ,

therefore the region of subharmonie resonance of order ½ is congruent with the parametric resonance in the unstable region of the first order.

As only the steady state solution is of concern here, the approximate solution is sought

540 J. ZAtlRADKA

in the form v = X + Ae 21~' + Be -2t~' + Ce ~' + De -t~' , (9)

where, in the steady state, A, B, C and D are complex conjugate constants so that

A = Y - i Z ,

B = Y + i Z ,

C = U - i V .

D = U + i V .

According to the Van der Pol 's method for the steady state solution (i.e., by comparison of coefficients at ~2~,, e~, and absolute terms), the following system of algebraic equations is obtained for the unknowns X, Y, Z, U and V:

n , ( x , Y, z , u, v ) = o,

H2(X , Y, Z, U, V) = O,

H3(X , Y, Z, U, V) = 0 ,

U4(X , Y~ Z, U, V) = 0 ,

Hs(X, Y, Z, V, V) = 0 . (10)

The solution of equations (10) can be accomplished by application of the iterative Newton-Raphson ' s method, on a digital computer. Results for one of the previously mentioned configurations, obtained by means of an ICL 1905 computer, are plotted in Figures 6 and 7.

D I I l I I I

I n ,

0 . 4

C

! B' I 1 .00

0 " 3

• 0 . 2

0 ' 1

A' 0

0 . 9 7 1 0 3

~ • w/./ '2

I I 1 | I 1

W-O

i

! I I t , ! 1 0 9 7 1 . 0 0 1 . 0 3

F i g u r e 6 F i g u r e 7

Figures 6 and 7. Relative torsional amplitude W (Figure 6) and R, (Figure 7) in the region of subharmonic resonance.

Figure 6: =t = 5 °, ~: = 20°, k = 2 kpcms. AB, B'A', stable loci; BCB', stable loci; BB', saddle points; DD', saddle points.

Figure 7. ~1 = 5, =, = 20 °, k = 2 kpcms.

The steady state notation [16], to the form

v = X + R v cos(2qr - 7JR) + W cos(t/r -- I/'w), where

R o = 2 x / c ~ + Z 2 ,

IV = 2x/U -2 + V 2

and Z 7~R = a rc tg -~ ,


solution (9) can be simply converted, by introducing appropriate

(11)

V kUw = arctg--u.

From the graphs of the relative torsional amplitudes in the region of subharmonic resonance one can see the rather small width of the interval where the oscillation having the half-frequency of the exciting frequency on the cardan shaft dominates (given by points BB' in Figure 6). The width of this interval depends on the coefficient of viscous damping, k. With increasing magnitude of k the interval gets narrower.

3.3. ANALYTICAL SOLUTION BY THE APPROXIMATE VAN DER POL'S METHOD IN THE REGION

OF SUBULTRAHARMONIC RESONANCE

In order to test the validity of the non-linear theory, experimental work was undertaken [12]. In this investigation violent oscillations also occurred too, even when the rotation speed lay in critical speed ranges whose locations were fractional functions of the corresponding natural frequency.

As again the steady state solution only is of interest, the approximate solution is sought in the form

v = X + Ae 2t~ + Be -2~' + Ce ~i~ + De -~i~, (12)

where A, B, C and D are, in the steady state, complex conjugate constants and

and at the same time

il ct = 2--~ (n ,N = 2 ,3 ,4 . . . . )

n N _ m # integer. N ' n

According to the Van der Pol's method for the steady state solution (i.e., by comparison of coefficients at e 2~, e ~ , and absolute terms) the following system of algebraic non-linear equations for the unknowns X, Y, Z, U and V is obtained:

HA(X, Y, Z, U, V) = O,

H2(X, Y, Z, U, V) = O,

H~(X, Y, Z, U, V) = O,

n4(x , Y, Z, U, V) = O,

n s ( x , Y, Z, U, V) = O. (13)

542 J. ZAtlRADKA

This system of algebraic equations (13) is similar to the system (10) for subharmonic resonance. They differ in some values of the coefficients of the unknowns X, Y, Z, U and V. The solution of equations (13) again can be obtained by applying the iterative Newton- Raphson's method on a digital computer.

4. STABILITY OF STATIONARY SOLUTION

It is evident from the graphs of the relative torsional amplitudes (Figures 2-7) that for one tuning coefficient tl there are several stationary solutions of the governing equation of motion (2). The initial relations for the determination of stability are equations (5). These equations are obtained by means of Van der Pol's method in the form

a' = f~(a ,b ,X) ,

b' = f2(a, b, X) ,

0 = f s (a ,b ,X ) .

The stationary solution of system (14) is given then by the corresponding singular points and these can be obtained by solving the following system of equations:

A(a,b,X)=O,

f2(a,b,X) = O,

f 3 ( a , b , X ) = O .

The classification of singular points undertaken earlier by Poincar6 and Bendixon can be used to investigate the small perturbations from the stationary state. Provided that a,, bs and X~ are stationary solutions of system (14) (i.e., coordinates of singular points), and at the same time c~, /~ and ~ are small perturbations from the stationary state, then the corresponding variational equations for the system (14) will be

~'=~\aa)s +p

?sq ?sq ff = v \ aa /~+ P \ ab ls

?sq O = " \ a a ] +13\ab]~

The characteristic equation for this system is, therefor,

[afq + ~ \ ~ ) ,

(osq

?sq + ~ \axle"

(aM ?M _ ;. ?M Oa ]s ~ ab /s \ OX /s

(asq ?sq ?sq & /~ k ab /~ \ ax/~

= 0

NON-LINEAR VIBRATIONS IN CARDAN StlAFTS 543

or

where

flk Oa ]~

[orq \ aa/,

~k ~-"

~k22 - ilk2 + ~'k = 0,

o~k = \ ~gX],'

(OZql (ozq (Olql

aa/~ kOb/, \aX/~ I

(azq i'a.cq (azq dais \ab l s \aX]

(arq (azq i'a.rq aa/ , k ab/s ax/,

(15)

Subscript s indicates that the values a, b and X are determined by the equations for the stationary solution (i.e., that they are coordinates of singular points in the phase plane (a, b) in the case of X = 0 and in the case of X ~ 0 are coordinates of singular points either on the surface f3 (a, b, X) = 0 or in the projection of this surface on the plane (a, b)).

For the actual classification of singular points a short program was prepared for the digital computer rOSE 25. When the partial derivatives in equation (15) have been calculated then, on the basis of the roots of the equation, the character of the corresponding singular points can be determined. This analysis showed that in this system the usually available stability criterion, that of the vertical tangent line, is of no use. This system differs from systems where singular points of focus or saddle type usually only alternate. It is evident, if the calculated singular points are arranged according to the magnitudes of their corresponding torsional amplitudes, that the axiom of regular exchange of stable and unstable points does not apply. The character of the stationary solution is marked for the calculated resonance curves in the legends to the illustrations: the curves are divided into stable and unstable sections by capital letters according to the legend.

The singular points in the region of subharmonic and subultraharmonic resonance can be classified in the same way. If one wishes to obtain the solution in these regions, including further harmonic components (terms (9) and (12)), one must assume that not all the coefficients are functions which are slowly varying with time at an equivalent rate. One must separate the coefficients which, during the transient process, will vary with time faster than the other ones and whose first derivates with respect to time cannot be therefore neglected. If one approximates the solution of the equation of motion in the form of five components with coefficients X, A, B, C and D (or X, a, b, c and d) and chooses the members whose frequencies are close to the natural frequency of the system 017] (i.e., coefficients c, d) as those varying more rapidly with time, a system of five equations is obtained where two equations are algebraic equations. This system is then again determined by two initial conditions and the stability of its steady solution can be found for small

544 s. ZAtlRADKA

disturbances from the corresponding variational equations by using the calculated roots of the characteristic equation.

For the actual classification of singular points a short program was prepared for an ICL 1905 digital computer. The character of the stationary solution is marked for the calculated resonance curve in Figure 6 in the region of subharmonic resonance of order ½ (the capital letters are explained in the legend).

5. THE DOMAINS OF ATTRACTION

After the classification of the calculated stationary solutions of the governing equation (2) (or (1)) one can approach at last the definition of the regions of initial conditions leading to the different solutions.

5.1. SOLUTION IN TIlE REGION OF MAIN RESONANCE

If one considers, for example, the case of the configuration whose resonance curve is plotted in Figure 5, it can be seen that there are six stationary solutions for the coefficient of tuning q = 0.501.

The stationary solution of equation (2) (or (l)) was obtained by means of the approximate Van der Pol's method. The relation between the initial conditions x(0) and 2(0) in equation (1) and the values of the calculated components a, b, and X from equation (3) (or X, Y and Z from terms (7)) is given by

o r

x(o) = xo(x + a),

2(0) = 2XoCob,

x(O) = xo(X + 2 r ) ,

2(0) = 4xowZ .

The definition of the regions of initial conditions leading to different solutions, so-called domains of attraction, can be demonstrated in the phase plane (x, 2). The boundary of these domains, the so-called separatrix, can be found by means of the stroboscopic method. Suppose now that for definite trajectories in the phase plane (x, 2) one follows the projection of the reference point, and records the projections at time intervals which a period (n/w) of the assumed solution apart: i.e., at t = 0, rr]oo, 2re]to . . . , as if the reference point were illuminated stroboscopically in those time intervals. The trajectories can be called the trajectories in the stroboscopic plane with coordinates x*, 2*. For transient motion one obtains a certain trajectory in the Stroboscopic plane and for the stationary state one obtains a singular point.

The actual procedure of obtaining the separatrix is based on the idea of realizing "negative" time in the solution of equation (1). The starting point of this solution is the saddle point, the coordinates of which can be determined by the approximate Van der Pol's method. One starts the solution in the neighborhood of the saddle point and, by recording points X*Olrc/o9), 2*01n/w) one obtains, in accordance with references [15], [18] and [19], the separatrix with adequate accuracy.

The solution was performed for one of the previously mentioned configurations (the resonance curve of which is plotted in Figure 5) by means of an analogue computer simulation program X3AB for the ICL 1905 digital computer (the block diagram is shown in Figure 8). For this case and the coefficient of tuning having the values, respectively, ! I = 0.480, 0.495 and 0.501, the domains of attraction are shown in Figures 9, 10 and 11.

NON-LINEAR VIBRA'i-IONS IN CARDAN SttAFTS 545

F r o m these graphs the fo l lowing conclus ions can be drawn.

(a) In the case o f the conf igura t ion o f F igure 9 (i.e., for the coefficient r / = 0.480) there a re two s ingular poin ts in the s t roboscop ic p lane :

1 s table focus,

1 saddle poin t .

The s t roboscop ic phase t ra jector ies s ta r ted f rom the ne ighbo rhood o f saddle po in t 2 in negat ive t ime (p lo t ted thickly), on bo th sides, fo rm the sought boundary - sepa ra t r ix , which splits the s t roboscop ic p lane in to two par ts . The doma in o f a t t rac t ion s i tuated to the left o f the separat r ix , where s table focus 1 lies too , leads in all cases to the s ta t ionary s table so lu t ion o f the tors ional ampl i t ude given by the bo t t om branch o f resonance curve A A ' in F igure 5. F o r init ial condi t ions x*, ~* ly ing to the r ight o f the separa t r ix the ampl i tude o f

;in

;OS

Figure 8. Block diagram for the solution of differential equation (1) (dashed blocks are used for the solution in negative time). The values of introduced constants are as follows.

Const. 1 = ~L Const. 9 = ~t Const. 17 = I2o 2 Const. 2 = ~2 Const. 10 = b2 Const. 18 = 2o9 Const. 3 ---- ~ Const. 11 = b3 Const. 19 = - - p

Const. 4 = h4 Const. 12 = ~ , Const. 20 = - - Const. 5 ---- ~5 Const. 13 = bs Const. 21 = Const. 6 ----- a6 Const. 14 = bs Const. 22 = t¢ Const. 7 ---- ~ Const. 15 = ~7 Const. 23 = I Const. 8 = ha Const. 16 = 5s Const. 24 = ~'(0)

Const. 25 = x(0)

546 J. Z A I t R A D K A

f J I I I I 40 i 40

30 30

TO 20

I0 I0

"~ o :~ o

-I0

-20

-30

-40 - 0 6 -0-4 - 0 2 0 0 2 0.4 0 6 0.8 - 0 6 - 0 4 - 0 2

X"

3

0.2 0 4 0.6 0 8 ,t" Figure 9 Figure I0

Figures 9 and I0. Domains of attraction for the solution of equation (I). Figure 9: coefficient of tuning tl = 0.480; Figure 10: t/ = 0"495.

transient motion increases up to the overflow of the digital computer.

(b) In the case of the configuration of Figure 10 (i.e., for the coefficient ~l = 0.495) there are four singular points in the stroboscopic plane:

I stable focus,

2 saddle points,

1 unstable node.

The stroboscopic phase trajectories started from the neighborhood of saddle points 3 and 4 in negative time (plotted thickly) in one sense point toward unstable node 2, where they have a common tangent line. If one solves the given differential equation in negative time then this unstable node changes into a stable one 1-20] and corresponding stroboscopic trajectories started from the neighborhood of saddle points are directed to this singular point. The other two branches of the separatrix, obtained again from the neighborhood of saddle points 3 and 4 but in an opposite sense, together with those obtained in the first sense divide the stroboscopic plane into two parts. The domain of attraction situated to the left of the separatrix, where stable focus 1 lies too, leads in all cases to the stationary stable solution of torsional amplitude given by the bottom branch of resonance curve AA' in Figure 5. For initial conditions x*, :~* lying to the right of the separatrix the amplitude of transient motion increases again up to the digital computer overflow.

(c) In the case of the configuration of Figure 11 (i.e., for the coefficient tl = 0'510) there are six singular points in the stroboscopic plane:

2 stable foci,

3 saddle points,

1 unstable node.

NON-LINEAR VIBRATIONS IN CRADAN SHAFTS 547

The stroboscopic phase trajectories started from the neighborhood of saddle points 5 and 6 in one sense in negative time (plotted thickly) form the sought boundary-separatrix pointing to node 4. In the same way the separatrix trajectories started from the neighborhood of saddle point 3 in both senses are directed to this unstable node 4 too. The other two branches of the separatrix, obtained again from the neighborhood of saddle points 5 and 6 but in an opposite sense, together with the above mentioned boundaries, divide the stroboscopic plane into three parts. The first of them (cross-hatched), where stable focus 2 lies, is limited by the separatrix started from saddle point 3 in both senses. This closed domain of attraction corresponds to the torsional amplitude given by the branch BC' (or BC) of the resonance curve in Figure 5. The second region, where stable focus 1 lies, is the domain situated to the left of the separatrix started from singular points 5 and 6 in both senses but reduced by the closed region corresponding to stable focus 2. This domain of attraction leads in all cases to the stationary stable solution of the torsional amplitude given by the bottom branch of resonance curve ABA' in Figure 5. For initial conditions x*, :t* situated to the right of the separatrix 546 the amplitude of transient motion increases again up to the digital computer overflow.

((1) The area of the domain leading to stationary stable solution in the bottom branch of resonance curves depends on two basic values. The first of them is the static angular displacement of a linearized system, Xo. When the magnitude of this constant increases, i.e., when the difference between the angles ut and c~2 increases, the domain of attraction for stationary stable solution decreases. The second constant influencing the area of the above mentioned domain of attraction is the magnitude of the damping coefficient, k. When the magnitude of this constant k decreases the domain of attraction for stationary stable solution decreases too.

5.2. SOLUTION IN THE REGION OF SUBttARMONIC RESONANCE

From graphs of relative torsional amplitudes in the region of subharmonic resonance of order ½ it is clear that for a certain tuning coefficient !/there are again more stationary solutions of the governing equation of motion (2) (or (1)). The stationary solution of these equations was obtained by means of the approximate Van der Pol's method. The relation between the initial conditions x(0), ~(0) in equation (1) and the values of the calculated components A, B, C, D and X from equation (9) (or X, Y, Z, U, V from terms (10)) is given by

x(O) = xo(X + 2Y + 2U),

2(0) = 2XoOg(2Z + V). (16)

The boundary of the domains of attraction-separatrix can again be found by means of the stroboscopic method, based on the idea of realizing negative time in the solution of equation (1). The starting point of this solution is the saddle point, the coordinates of which can be determined in the stroboscopic plane from equations (16).

The solution was carried out for one o f the previously mentioned configurations, the resonance curve of which is plotted in Figure 6. For this case and the tuning coefficient r /= 1.01 the domains of attraction are shown in Figure 12. From this graphin the region of subharmonic resonance the following conclusions can be drawn.

For the given case, i.e., for the coefficient q = 1.01, there are five singular points in the stroboscopic plane:

2 stable foci,

3 saddle points.

548 J. ZAIIRADKA

Stable foci 2 and 2' as well as saddle points 3 and 3' are in central symmetry with respect to the point of the stroboscopic plane given by the coordinates

x* = xo (X + 2Y),

5c* = 4XotOZ, as follows from equations (16).

4 0 I I J ' ' 5 0 1 - I , t I x . ' ~ ~ -

- 2 0

- 3 0 - 4 O -

- 4 0 I I I -50 I I - 0 6 -0.4 -0-2 0 02 0.4 06 08 -15 -10 - 0 5 0 0-5, 10 15

X" x °

Figure I1 Figure 12

F igures 11 and 12. D o m a i n s o f a t t r ac t i on f o r the so l u t i on o f equa t i on (1). F igu re 11 : coemc ien t o f t un ing I I = 0 . 5 0 1 ; F i g u r e 1 2 : t l = 1 " 0 1 0 .

The stroboscopic trajectories started from the neighborhood of saddle point 1 in negative time (plotted thickly), in both senses, form the sought separatrix which divides the stroboscopic plane into two basic parts. The importance of this boundary is rather secondary. The stationary solutions given by stable foci 2 and 2' are the same; only they are shifted by an angle z~. In these systems, where the trivial solution is unstable, all regions of initial conditions in the phase plane lead to stationary oscillations of the same amplitide. In this case this region is limited. The limitation is given by the branches of the separatrix started from the neighborhood of saddle points 3 and 3', in both senses. The domain of attraction between these separatrix branches corresponds to the stable subharmonic solution of the differential equation for the torsional amplitude given by the branch BCB' of the resonance curve in Figure 6. For initial conditions x*, :t* situated outside this region the amplitude of transient motion increases up to the digital computer overflow.

If one compares the areas of domains of attraction leading to stationary stable solution in the regions of main and subharmonic resonance (Figure 9 and Figure 12), one finds that the boundary behind which the amplitude of transient motion increases is, in the case of subharmonie resonance, far from the stable focus and the corresponding domain of attraction is larger then for the cases of main resonance.

6. CONCLUSION

This paper has been concerned with the analysis of torsional vibrations in a non-linear system with a single degree of freedom. The main attention has been given to the


determination of stability of Stationary solutions and to the definition of the regions of initial conditions leading to different solutions. After the calculation of these domains of attraction one can find the condition under which a sudden change of torsional amplitude in the system can occur owing to external impulses or disturbances. I f the magnitude of some disturbance reaches such a value that the vector of this disturbance with initial point situated in the corresponding singular stable point extends beyond the separatrix of the given domain of attraction, then it will be possible for that system to undergo a change, with some probability, of the amplitude of torsional vibration [21"1.

For strongly damped systems, when the upper branch of the resonance curve is far from the bottom one (which represents the stable stationary solution) then the stable solution with small amplitude is strongly resistant to disturbances from the stationary state. For slightly damped systems the domain of attraction for the stationary stable solution is, in comparison with that of strongly damped systems, smaller. Therefore, the stable stationary solution is less resistant to disturbances from the stationary state. In these systems such disturbances can arise that the character of vibrations changes to those of larger amplitude (cross-hatched region), or a transient motion with increasing amplitude can come into existence. Otherwise, from results in the region of subharmonic resonance, one can deduce that subharmonic oscillations are, for slightly damped systems, sufficiently resistant to disturbances from the stationary state.

The disturbances from the stationary state can arise in a cardan shaft driving system from a sudden change of the angles of shafting, cq and c~2. For this reason it would be useful to determine the regions of attraction for these systems (driving systems of locomotives, cars, etc.), since in actual operation the magnitudes of these angles of the shafting often change. The analysis enables us to determine when the oscillation at upper branches can be dangerous from the point of view of dynamical stress in the structure of the cardan shafts, with obvious consequences for safety of operation.

ACKNOWLEDGMENTS

The author is deeply grateful to Professor Dr B. Porter, Professor of Dynamics and Control, University of Salford, for his help in the completion of this paper and for providing copies of his own papers. Thanks are also due to Doe. Dr.Ing. A. Tondl, DrSc. and Ing. F. Turek CSc, from the Research Institute for Machine Design, Bechovice, near Prague, who greatly facilitated the author's study and especially for their co-operation and advice. The author would like to express his gratitude to Ing.M. Herold from CKD Prague, who programmed these problems for the ICL 1905 and ZUSE 25 digital computers.

REFERENCES

1. F. BOIIM 1957 Maschinenbau ltnd Warme Wirtschaft 6, 153-161. Oscillation loading of cardan shafts in locomotive bogies. (In German.)

2. A. DORNIO 1965 Rivisti di htgegneria 7, 661-672. Dynamical instability of machines caused by presence of couplings with double cardan joint. (In Italian.)

3. K. A. NIKULINA 1971 Mashinostro]enie 9, 118-124. Parametric excitation of torsional vibrations on cardan shafts. (In Russian.)

4. A. P. PAVLENKO 1971 Vestnik Vsesoyttznogo Nauchno-lssledoL'atel'skogo htstituta Zhelez- nodorozhnogo Transporta 2, 26-29. Determination of dynamical loading in cardan shafts of locomotives. (In Russian.)

5. B. PORTER 1961 Journal of Mechanical Engineering Science 3, 324--329. A theoretical analysis of the torsional oscillation of a system incorporating a Hooke's joint.

6. J. ZAHRADKA 1969 Strofnicky casopis 3, 288-301. A quasiharmonie oscillation of a drive system with cardan shafts. (In Czech.)

550 J. ZAHRADKA

7. J. ZAI~DKA 1972 Technicke zpravy CKD Praha 1, 48-54. Stability of oscillatory motion in dynamical systems. (In Czech.)

8. R. PROKYSEK 1968 Thesis CVUT Praha. Vibration exciter with a cardan joint. (In Czech.) 9. J. ZAHgADKA 1971 Technicke zpravy CKD Praha 4, 40-46. Dynamics of a cardan shaft drive

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oscillation of a system incorporating a Hooke's joint. 14. J. ZAHgADKA 1972 Strojnicky casopis 5, 404--421. Torsional vibration of a nonlinear system

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66-02013. Dynamics of nonlinear systems with peridoically variable parameters. (In Czech.) 16. CtI. HAYASm 1964 Nonlinear Oscillation itt Physical Systems. New York: McGraw-Hill Book

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19. A. TONDL 1970 Revue Roumaine des Sciences Techniques 6, 1255-1262. The effect of a sudden change of the system parameter on the stability of nonlinear resonance vibrations.

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21. A. TONDL 1971 btgenieur Archly 41, 61-72. The effect of random step changes of system parameters on the stability of steady vibration of nonlinear systems.

torsional vibrations of a non-linear driving system with cardan shafts

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