recent results in random vibrations of nonlinear mechanical systems

R. A. Ibrahim ASME Fellow

Department of Mechanical Engineering, Wayne State University,

Detroit, Ml 48202

Recent Results in Random Vibrations of Nonlinear Mechanical Systems The influence of random vibration on the design of mechanical components has been considered within the framework of the linear theoiy of small oscillations. However, in some important cases this theory is inadequate and fails to predict some complex response characteristics that have been observed experimentally and which can only be predicted by nonlinear analyses. This paper describes some recent developments in the theory of nonlinear random vibration based on Markov methods and related problems in the design of dynamical systems. Research efforts have been focused on stability/ bifurcation conditions, response statistics and reliability problems. Significant progress has been made in developing new analytical methods and conducting experimental testing. These developments have helped to resolve some controversies, and to enhance our understanding of difficult issues. Experimental and numerical simulations have revealed new phenomena that were not predicted analytically. These include on-off intermittency, snap-through phenomena, and the dependence of the response bandwidth on the excitation level. The main results of studying the responses of nonlinear single-and two-degree-of-freedom systems to random excitations obtained by the author and others are discussed in this paper.

1 Introduction The nonlinear random vibration of mechanical compo

nents subjected to severe environmental conditions is one of the serious and difficult problems facing designers and reliability engineers. This problem mainly deals with the stochastic stability/bifurcation, response statistics and reliability of mechanical systems. The theory of nonlinear random vibration is a combination of applied mechanics, probability theory, and stochastic differential equations. The role of applied mechanics is to develop analytical models which describe the system dynamics. These models are usually given in the form of partial or ordinary differential equations. Probability theory and stochastic calculus are among the useful tools usually employed to describe the excitation and response statistics and to estimate the stability conditions and reliability of systems subjected to a random environment. The stochastic stability analysis of an equilibrium position is usually carried out on the basis of a linearized approximation to the equations of motion. If the equilibrium position is unstable in a stochastic sense, the linearized equations do not provide a unique bounded solution. On the other hand, if the system's inherent nonhnearities are included in the mathematical modeling, the solution trajectories, which emanate from an unstable equilibrium often end up in bounded limit cycles. Moreover, the nonlinear modeling also allows the designer to predict a wide range of complex response characteristics such

as multiple solutions, jump phenomenon, internal resonance, on-off intermittency, and chaotic motion. These phenomena have a direct effect on the reliability and safe operation of mechanical equipment. Accordingly, the designer has to estimate the reliability of nonlinear systems subjected to Gauss-ian/nonGaussian random excitations. In this case the engineer has to deal with both the catastrophic type and the fatigue type failures. The former is related to the distribution of extreme values of the system response, and the latter is related to the crossing rates at different levels of the system response. The issues of reliability and fatigue are beyond the scope of this paper; however, the readers may consult Bog-danoff and Kozin (1985), Bolotin (1989), Sobczyk and Spencer (1992), and the recent review by Clarkson (1994).

The system nonlinearity can be the result of several factors including the geometry, boundary conditions, and material characteristics. The geometry and boundary conditions can result in inertia and stiffness nonhnearities. For example, in a clamped-clamped beam the nonlinearity is governed by mid-plane stretching, while for a cantilever beam the nonlinearity can include inertia and curvature depending on the mode in question. These continuous elastic elements are usually described by a partial differential equation of the form

£(w(z,£0(t),t)) =U(z,t) (1)

Contributed by the Design Engineering Division for publication in the Special 50th Anniversary Design Issue. Manuscript received Oct. 1995. Technical Editor: D. J. Inman.

together with the appropriate boundary conditions, where £ is a linear/nonlinear operator, w(z,t) is the deflection which is a function of the space coordinate z and the time t, f0O) is

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a parametric excitation to the system, and U(z,t) is a random field excitation. To develop methods for evaluating the nonlinear response to a stochastic excitation, an approximate solution of the nonlinear partial differential equation can be obtained by expanding w(z,t) in terms of a finite number of eigenfunctions

>(z,0 = L 4>j(z)qj(t) (2)

where the eigenfunctions <$>jiz) are the linear free-vibration modes, and q-(t) are the corresponding generalized coordinates. The random excitation field U(z,t) can also be expanded in terms of 4>j(z)

u(z,t) = E </v(zH(0 (3)

where £y is the generalized force associated with mode ;'. Introducing relations (2) and (3) into Eq. (1) and applying

Galerkin's method (Kantorovich and Krylov, 1958) results in a set of nonlinear ordinary differential equations of the general form

{<?} + [ £ ( ' ) ] {?} + [ [ * ] + [ f o (0 ] ] {?}

where [C(t)] and [K] are the linear damping and stiffness matrix coefficients, £0 represents the parametric random component, and the vector *P includes all nonlinear terms due to material and structural nonlinearities. [K] is a constant matrix coefficient.

One should notice that expansions (2) and (3) converge when the number of terms is large enough to represent the dynamic characteristics of the structure. However, for nonlinear modal interaction between limited number of modes, e.g., two or three modes, where the excitation bandwidth is concentrated about one of these modes, the mathematical modeling may be developed only for these few modes. This approximation is only used for predicting and understanding complex nonlinear phenomena. If the excitation is wide band and the nonlinear interaction does not take place in complex structures, then the designer may use one of the finite element codes to estimate the response statistics.

In problems concerning structure failure, one of the most important questions is whether the response to a random perturbation remains bounded for all time. In the case of a parametric excitation, a more refined question is whether the equilibrium solutions, if they exist, are asymptotically stable. The method of stochastic Liapunov functions has been proven to be effective for stochastic ordinary differential equations (ODE's). Stochastic stability can be examined in terms of one of the stochastic modes of convergence such as mean-square stability and almost sure stability (see for example Ibrahim, 1985). Mathematicians (Arnold and Wihstutz, 1986) established a measure of the exponential growth of the response known as the Liapunov exponent. Unfortunately, Liapunov exponents are only estimated for linear stochastic differential equations.

2 Difficult Issues and Controversies The development of the theory of random vibration has

encountered a number of difficulties and controversies in its applications to engineering design problems. These difficulties include:

il) The analysis of nonlinear systems subjected to random excitation involves a number of difficult issues which require special treatment. When the random excitation is approximated by a white noise process, the rules of stochastic

calculus are not the same as the rules of ordinary calculus. For example, if the function Bit) is treated as an ordinary well behaved function, it obeys the following "ordinary" rules

dB2(t) = 2B(t)dB(t)

fbB(t)dB{t)=l-{B\b)-B\a)} (5)

On the other hand, if Bit) is a Brownian motion, which is characterized by independent increments and cannot be defined as a continuous analytic function, it will possess the following "stochastic" rules

dB2(t) = 2B(t)dB(t) + o-2dt

fabB(t)dB(t) = l-{B\b) - B2(a)} - \cr\b - a) (6)

where a2 is a constant positive parameter. The observed difference between the ordinary and

stochastic calculi is mainly due to the fact that the Brownian motion is not differentiable in the mean square sense, as it possesses continuous sample functions with unbounded variations, i.e.

lim E B(t + At) -B(t)

— (7)

Accordingly, the white noise Wit) is defined as the "formal" or the ordinary derivative of the Brownian motion process, i.e., Wit) = dBit)/dt. A comprehensive engineering treatment of the stochastic calculus supported by numerous examples has been documented by Di Paola (1993).

(2) The response of nonlinear systems to Gaussian excitation is non-Gaussian. This property results in problems of infinite coupling of response moment equations and in obtaining a closed-form solution of the probability density function. Closure schemes of the infinite coupled moment equations have been developed (see Ibrahim, 1985). However, in applying these closure techniques, precaution should be taken for preserving the moment properties and satisfying Schwartz's inequality.

(5) Different methods lead to different results for the same problem. For example, Ibrahim and Soundararajan (1983, 1985) employed three different methods to determine the stochastic stability of a liquid free surface in an upright circular container subjected to random vertical excitation. The three methods are Gaussian and non-Gaussian closure schemes, and Stratonovich stochastic averaging. The Gaussian-closure solution is found to be consistent with the condition of mean square asymptotic stability

D/2£ < 1 (8)

where 2D is the power spectral density of the excitation, and £ is the damping factor of the sloshing mode in question. On the other hand, the Stratonovich averaging solution agrees with the sample stability condition (stability with probability one)

D/2C, < 2 (9)

The non-Gaussian closure solution bifurcates at a critical excitation level where a jump in the mean-square response takes place. This critical level is defined within the range

1 < D/2C < 2 (10)

The exact point depends on the system parameters. The observed jump in the non-Gaussian closure solution is believed to be due to the nonlinearity introduced in the closure

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scheme, which is purely a mathematical artifice. Later, Sun and Hsu (1987) examined the validity of Gaussian and non-Gaussian closures by using a nonlinear system whose exact solution is known. They found that the Gaussian closure technique usually leads to the same general response curves as those of the exact solution, but has substantial errors in some cases. The non-Gaussian closure is found to be inapplicable and predicts erroneous behavior for the system in certain parameter ranges, including a faulty prediction of a jump in response as the excitation varies through a certain critical value. However, away from the bifurcation point defined by the critical excitation level, the non-Gaussian shows better nonlinear representation of the system response statistics.

(4) The Fokker-Planck-Kolmogorov (FPK) equations of two identical systems in which one is excited by a physical white noise and the other is excited by a mathematical white noise are different if the excitation appears at the next-to-the-highest derivative (Gray and Caughey, 1965).

(5) Almost sure stability (with probability one) does not guarantee stability of moments (Kozin and Sugimoto, 1976; Ariaratnam, 1981; Ibrahim et al., 1983, 1985).

(6) Numerical simulations and experimental results reveal new phenomena such as the widening effect associated with a shift of the response spectra as the excitation level increases (Mei and Wentz, 1982; Reinhall and Miles, 1989; Moyer, 1988; and Ibrahim et al., 1993) and on-off intermit-tency near the bifurcation point of parametrically excited systems (Ibrahim and Heinrich, 1988).

3 Methods of Analysis During the past thirty years a number of techniques have

been developed to investigate the response properties of nonlinear systems. These techniques include: (/) Markov methods based on the FPK equation or the Ito stochastic calculus, (ii) Gaussian and non-Gaussian closure schemes (Ibrahim, 1985), (Hi) stochastic averaging methods (Roberts and Spanos, 1990; and Red-Horse and Spanos, 1992), (iv) equivalent linearization methods developed originally by Caughey (1963) and by others (see Spanos, 1981), (v) perturbation techniques (Crandall, 1963), (vi) Volterra-Wiener functional expansion (Schetzen, 1980), (vii) finite element methods, and (viii) Monte Carlo simulation. These approaches have been applied to dynamic systems with various forms of nonlinearities. However, there can be no general rule about the suitability of any method for a particular nonlinear system. The first three methods have been extensively used in random parametric vibration problems and systems involving nonlinear coupling with internal resonance.

When the excitation is modeled by a white noise process, the response of the system constitutes a Markov process and the response transition probability density function is governed by the system FPK equation. The solution of the FPK equation has been obtained for a limited class of dynamical systems. In general, the derivation of the exact solution for the response probability density function of nonlinear systems is not a simple task. Ilin and Khasminskii (1964) and Kushner (1969) developed approximate techniques based on successive solutions of the system FPK equation. The FPK equation is split into a linear zero-order and a higher-order part containing the system nonlinearity. An iterative scheme is then established based on the fundamental solution. Caughey (1971, 86) obtained closed form stationary solutions of some special cases of first- and second-order systems. Caughey and Ma (1982, 83) constructed the stationaiy solution of a class of nonlinear oscillators subjected to white noise excitation. Under the assumptions that the system satisfies both Lipschitz and growth conditions a well behaved unique solution of the stationary FPK equation can be ob

tained. Additional necessary but not sufficient assumptions require that the potential energy and the system Hamiltonian possess continuous second order derivatives. Caughey and Ma also indicated that if the stationary density of a system is obtained, it may be possible to obtain the approximate non-stationary response by using perturbation techniques. The work of Caughey motivated others (Dimentberg, 1982; Yong and Lin, 1987; Lin and Cai, 1988; Cai and Lin, 1988; Soize, 1991; and Moshchuk and Sinitsyn, 1991) to derive the stationary solution of dynamical systems described by a set of Ito equations with invariant measure (e.g., probability density). Lin and Cai (1988) split the drift and diffusion coefficients, in the FPK equation, into the circularity and potential probability flows, respectively. Soize (1988) and Moshchuk and Sinitsyn (1991) represented the system by a combination of Hamiltonian and non-Hamiltonian parts, such that the dissipation energy is proportional to the excitation energy. Di Paola and Falsone (1993) analyzed the random response of nonlinear systems driven by non-Gaussian delta-correlated processes. Musculino (1993) has reviewed the main results of the response of linear and nonlinear structural systems subjected to Gaussian and non-Gaussian filtered excitations.

3.1 Exact Solution of the Fokker-Planck-Kolmogorov Equation, (i) There exists a stationary solution for the probability density function of nonlinear systems described by the second-order nonlinear differential equation:

d2q dq dU(q) dB(t) —:r + I 1 = c (11) dt2 dt dq dt v '

where q is the system response coordinate, H(q) is a nonlinear smooth function whose gradient gives the restoring force and f is the damping parameter. This type of nonlinearity is encountered in many structural elements such as clamped-clamped beams and suspended cables. The stationary solution of the FPK equation of system (11) is

where D = <x2/2f, a dot denotes differentiation with respect to time t, and C is the normalized constant defined by

1

/ exp{-£>"'n(g)}^

(ii) For nonlinear systems described by the Hamiltonian H(q, h), where q and h are the generalized displacements and momenta of the system, respectively, the equations of motion are

dH dH

where Fnc stands for nonconservative forces and W(t) is a white noise process. For simplicity Fnc is taken in the form:

dH Fn.c.= -c-^ + W(t) (15)

where c is the damping coefficient. It is also assumed that the excitation level is proportional to the damping force. Equations (14) can be written in terms of the Stratonovich differential equation

dl{{^}\^{f(q,h)}dt + [G]{dB} (16)

where the white noise is replaced by the formal derivative of the Brownian motion process B(t), i.e., W(t) = ]/2DdB(t)/dt,



y2D is the mean square root of the excitation such that B(t) has a unit variance, and

{/(q>h)} dH

[s]

dH

dh , dH

0 0 0 1

[G] = [0] [0] [0] [g]

The diffusion matrix [<x] takes the form

[0] [0] M [0] 2£»[5]

(17)

(18)

It is assumed that the stochastic differential Eqs. (16) possess a unique solution q, h which constitutes a diffusive Markov process. In this case the transition probability density function p(q, h, t) is governed by the FPK equation

dt ._, Kffq)p) Kffh)p)

dqt dh:

1 " " d2(p[o-]ji) + ̂ L E \ , J = 0 (19)

• y = i / = i dhjdhi

The solution of this equation is given by the Gibbs distribution:

Ps(H) =pae Dl (20)

where p0 is a positive real constant determined from the normalized condition

c0 0 r c

1 = / Po[e-DH}dH (21)

The Kth joint moment of response coordinates is

4 ?r • h^» OO .00

,*1

Xp0e-~5mq'p)dql

• A * 2 " )

.dqndhl...dhn (22)

where i£ = E ;̂- According to a theorem developed by / = i

Caughey and Ma (1982) the stationary solution (20) of the FPK equation is well-behaved and unique.

The method of stochastic averaging has been extended to treat nonlinear systems subjected to arbitrary colored Gaussian excitations, which are modeled as the output of multidimensional linear filters to white Gaussian noise. Recently, Roy (1994) has used the method of averaging based on a perturbation theoretic approach of the FPK equation. For nearly Hamiltonian systems perturbed by parametric excitations of uncorrected noises, Roy showed that the state probability density function is governed by a reduced equation which depends on the excitation parameters.

3.2 Closure Schemes. The previous subsection has shown that the stationary solution of the FPK equation exists only for limited class of nonlinear systems. Alternatively, one may generate a general first-order differential equation which describes the evolution of the response joint statistical moments. This equation can be generated from the FPK equation or the Ito stochastic calculus. Ibrahim (1985) has documented the application of the two methods by several examples. With reference to the FPK equation method the moment equation is derived by multiplying both sides of the

FPK equation by 3>(X) = X%'... X*" and integrating by parts over the entire space - °° < X < <», i.e.

lkx,k2,...,k

,<* « dp(X,t)

J —ca J — co dt • dX„

. 0 0 - C O

= - / . . . / *(X) £ ^ dXx... dX„ J — oo J — 00 ; _ i 1 = 1 dX;

. G O • 0 0

- / .../ *(x)E E Z J — oo J — 'n _• 1 • i

« " a2{P(x,t)*t])

i=\j=i

dX,... dX„ dXidXi '

(23)

where w*, ,^. . . ,^ = E[X^X^ ... X,f'<], £ [ . . . ] denotes expectation, /, and (Tij are the first and second moment increments evaluated from the system state equations as follows

ft(X,t) = lim —EiX^t + AO - X ( ( 0 ] \t^o b.t

or,j(X,t)= lim —£[{*,-(*+ Af) Ar~>o at L

-Xi{t)}{Xj{t + l^t)-Xj{t)}\ (24)

This process results in a first order linear differential equation of the system response. For linear systems this equation is generally consistent. However, for nonlinear systems, this equation constitutes an infinite coupled set of differential equations of the form

mki,k2,---,kn =rhK = MK(mK,mK+l,...) (25)

The higher order moments of order greater than K, (where K = kl + k2 + . . . +k„), must be replaced by moments of order K or less. This can be achieved by using one of the cumulant-neglect schemes. A cumulant is a statistical parameter whose first and second orders are equivalent to the mean and variance of the process, respectively. If higher-order cumulants vanish, then the process is Gaussian, which is completely described in terms of the mean and variance. However, if higher-order cumulants do not vanish, then their values provide a measure of the deviation of the process from being Gaussian. Higher-order cumulants are related to corresponding order and lower order moments. Thus if third- and fourth-order cumulants are set to zero, one can express third-and fourth-order moments in terms of second- and first-order moments, and the closure is said to be Gaussian. A first-order non-Gaussian closure is established if fifth- and sixth-order cumulants are set to zero, which implies that third- and fourth-order cumulants do not vanish, and fifth- and sixth-order moments are then expressed in terms of fourth- and lower-order moments. Cumulants up to eighth order are derived in terms of moments in Ibrahim (1985).

As mentioned in Section 2, the closure schemes solutions can lead to erroneous results specially in problems dealing with stochastic parametric stability. For nonlinear coupled oscillators the non-Gaussian closure scheme usually gives reliable results (Ibrahim, et al, 1990, 93; Di Paola and Falsone, 1993; and Musculino, 1993).

4 Applications

4.1 Single Mode Random Excitation. More than 80 percent of the published work on nonlinear random vibration deals with one mode excitation. This includes rods, beams, cables, simple pendulum, and liquid sloshing in moving containers, see Fig. 1. The random response of a clamped-clamped beam has been extensively examined within the framework of one mode excitation. Nonlinear systems de-



P < P c r

Y(t)

(a) clamped clamped beam, pre-buckling case (b) postbuckling case

(c) Cantiliver beam under filtered parametric excitation

(e) suspended cable under parametric excitation

(d) liquid container under parametric random excitation

Fig. 1 Schematic diagrams of simple nonlinear dynamical models which exhibit complex phenomena under random excitations

scribed by one mode excitation are usually modeled by the nonlinear differential equation

q + 2£a„q + colq + cV(q,q,q) = f (f) (26)

where £(r) is an external excitation and W may include nonlinear stiffness, damping, and inertia terms and e is a small parameter.

4.1.1 Systems with Nonlinear Stiffness, (i) Supported Beams: The case of nonlinear stiffness has been extensively investigated. The equation of motion of the clamped-clamped beam shown in Fig. 1(a) is similar to Eq. (11) and is known as Duffing's equation. The early study of single mode response under random excitation goes back to the work of Lyon et al., (1991). They considered f (r) in Eq. (26), or W{t) in Eq. (11), as a narrow band Gaussian process derived from a wide band excitation of a resonant filter centered at frequency w1

ft dt2 ~^2 + 2^/w i dt + a>U- W{t) (27)

where W(t) is a white noise. Their results revealed multi-valued response characteristics which have the same general appearance as those for sinusoidal forcing except that the peaks are much sharper. The same feature has been observed by Lennox and Kuak (1976). They used quasi-static method with small but finite bandwidth. However, their method does not allow one to investigate the influence of bandwidth. Fang and Dowell (1987) analyzed the response of a Duffing oscillator to a narrow band random excitation by numerical simulation. Their results showed that multi-valued mean square responses can occur for mono-level excitations having very narrow bandwidths. As the bandwidth increases,

the multi-valued responses give way to single-valued responses. Similar results were obtained using different approaches, such as, averaging method (Davies et al., 1988, 90, 92; Roberts and Spanos, 1986), stochastic linearization (Di-mentberg, 1971; Iyengar, 1989; Roberts and Spanos, 1990; Roberts, 1991), probabilistic linearization technique (Iyengar, 1992), numerical simulation (Davies et al., 1988, 90, 92; Iyengar, 1989, 92), and experimental testing (Lyon et al, 1961). An alternative approach has been proposed by Davies and Nandlall (1986). They modeled the excitation as the response of a lightly-damped second-order filter to white noise. Thus the four-dimensional FPK equation can be written for the response. An approximate, time-dependent solution based on a Gaussian closure scheme was obtained from the four-dimensional FPK equation. The result was smoothed time histories for the mean square displacement and velocity. When plotted in a phase plane, the solution showed two stable attractors and a saddle point for the case in which the excitation had very narrow bandwidth, in a certain frequency range. This behavior is related to the well-known jump phenomenon. As the bandwidth of the excitation was increased, the phase plane was reduced to a single stable attractor (sink), indicating the elimination of the jump phenomenon.

The FPK equation for a Duffing oscillator subjected to filtered excitation has been solved numerically by Kapitaniak (1985) using the path-integral method. He found the stationary probability density function of the response process to be bimodal. Davies and Liu (1990) investigated the same system via the averaging method and numerical simulation. For narrow bandwidth excitation, they showed that the probability density function has multi local maxima with random jumps. For wider bandwidth excitation the probability density



function exhibits only a single local maxima. A similar result was obtained by Iyengar (1992) using numerical simulation. Koliopulos and Bishop (1993) studied the response statistics of nonlinear oscillators subjected to narrow band random noise based on a quasi-harmonic assumption. They showed that the method can easily incorporate the possible jumps of the system via the input-response amplitude curve. This curve defines critical input values which establish the region of possible jumps. In this case the probability density of the response amplitude exhibits two peaks.

Iyengar (1986, 88) has presented a stochastic stability analysis of a nonlinear hardening-type oscillator subjected to narrow band random excitation. He showed that only one statistical moment is stable and, hence, the multi-valued predictions of the equivalent linearization analysis do not correspond to the real behavior of the system. Koliopulos and Langley (1993) improved the stability analysis for the same system. They showed that, when jumps between competing response states occur, the choice of an appropriate value for the kurtosis of the response is crucial for a reliable estimation of the local statistical moments and for a more accurate stochastic stability analysis than the equivalent linearization method. Furthermore, a modification is proposed which takes into account the influence of certain nonlinear terms in the variational equation for the case of strong nonlinearities.

(ii) Cables. Suspended cables are very rich in mathematics and dynamics. They received extensive deterministic studies. However, the random analyses of cables are limited and few. The response statistics of a linear string were analyzed by Lyon (1965), while the nonlinear treatment was considered by Caughey (1960) and Lyon (1960). Tagata (1978) examined the planar motion of a string with cubic nonlinear -ity excited by filtered white noise. He used a quasi-static averaging analysis together with the FPK equation and obtained conditions for existence of steady state responses. The analysis revealed multiple-valued solutions similar to those predicted by the deterministic nonlinear theory of the Duffing oscillator. The response probability density exhibits non-Gaussianity of large concave shape as estimated by digital simulation. Richard and Anand (1983) considered the planar response and stability of cables subjected to a narrow band random excitation. Tagata (1989) extended his previous work to investigate the concave shape generation mechanism. He indicated that the main reasons why the concave shape in the response probability density function is generated are the result of high frequency of some finite amplitudes which result in saturation phenomena and growth of the higher harmonic oscillations arising from the nonlinear stiffness. This is in addition to the jump phenomenon which takes place under narrow band excitation. Recently, Chang et al. (1994) analyzed the nonlinear coupling of the in-plane and out-of-plane motions of a suspended cable in the neighborhood of 2:1 internal resonance and under random loading. It was found that the bifurcation of the out-of-plane mode takes place at a critical excitation level. Above this level the two modes interact in a form of energy sharing. The response characteristics are governed by damping ratios and sag-span ratio which results in variable system stiffness. To the knowledge of the author, the random interaction of cable dynamics with the dynamics of surrounded fluids (air or water) has not been addressed in the literature.

4.1.2 Systems with Nonlinear Inertia. Another class of mechanical systems characterized by inertial nonlinearities includes partially filled containers, see Fig. 1(d), subjected to parametric random excitation (Ibrahim and Heinrich, 1988) or a cantilever beam, see Fig. 1(c), excited parametrically by a filtered white noise (and Ibrahim and Yoon, 1993). A

cantilever beam subjected to a filtered white noise whose center frequency is close to twice the natural frequency of the first mode may be described by the differential equation

, mb q"h + 2rCbq'b + rlqh - ea=-qfqb

mb

+ e2b^qh(qbql + q'l?)=0 (28) ?n„

q'} + 2£fq'f + qf = - jlD W" ( r ) (29)

where qb is the lateral deflection at the tip, r is the ratio of the first mode natural frequency to the filter center frequency, £b and £,• are beam and filter damping ratios, mb is the total mass of the beam, fnb = (33/140)mfc, a and b are constant parameters, and e is a small parameter. The last expression in the beam Eq. (28) represents the inertial non-linearity due to the axial drop of the beam while the coupled term with the filter acceleration is the parametric excitation term. Both Monte Carlo simulation and experimental testing revealed that in the neighborhood of a bifurcation point there exist two regions characterized by on-off intermittency. The following discussion is confined to the experimental observation near a bifurcation point.

Under a particular modal excitation, the excitation level is increased from a very low level at which the beam does not respond and remains dynamically stable even with a given manual perturbation. As the excitation level increases the beam exhibits three different response regimes depending on the excitation level. These are zero motion, partially developed motion, and fully developed motion. Over the excitation level defined by the first regime, the beam does not oscillate because the structural resistance forces overcome the input energy. The second regime is characterized by intermittent periods of motion followed by periods of no motion. The third is fully developed random motion which exhibits continuous random oscillations of the beam. Similar response features were observed by Ibrahim and Heinrich (1988) for the case of liquid sloshing under parametric random excitation. Figure 2 displays these three regimes in which the zero motion regime includes an uncertain region where the beam may or may not oscillate, and if oscillation occurs it is too small to be measured. The stochastic bifurcation of the beam response is not sharply defined due to the presence of an uncertain region adjacent to the partially developed regime. It is clear that the experimental results displayed in Fig. 2 reveal new regimes that were not predicted analytically. These regimes are the uncertain and the partially developed random motions.

The uncertain and partially developed regimes have features in common with what is known in fluid mechanics as "on-off intermittency." In fluid mechanics (Townsond, 1976), the term on-off intermittency describes a flow alternating between long periods of regular laminar flow, interrupted by periods of shedding vorticity. The mechanism of on-off intermittency is different from those reported by Pomeau and Manneville (1980) and Grebogi et al. (1982, 87). Piatt et al. (1993, 94) and Heagy et al. (1994) have reported the on-off intermittency in a class of one dimensional maps that are

• multiplicatively coupled to either random or chaotic signals. In this case the on-off intermittency is a result of time-dependent forcing of a bifurcation parameter through a bifurcation point. Here the "off" state is nearly constant and can remain so for very long periods of time. The "on" state is a burst, departing quickly from, and returning quickly to, the "off" state. Piatt et al. (1993) indicated that the on-off intermittency can be generated by systems having an unstable invariant (or quasi-invariant) manifold, within which is found a suitable attractor. The on-off intermittency due to a time



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Fig. 2(a) Dependence of mean square response on the mean square excitation level showing four different response regimes under the first mode parametric excitation

TIME. sac .

Fig. 2(b) Time history records of excitation and response showing the on-off intermittency regime

dependent forcing of a bifurcation parameter through a bifurcation point may be demonstrated by the simple example

dX

dt = -X3 /x(t)X (29a)

with bifurcation at fi(t) = 0. The time dependence of the bifurcation parameter /x(f)

can be replaced by chaotic forcing (e.g., generated from Lorenz's system, Hammel and Piatt, 1994). In this case one may write Eq. (27a) in the form

Fig. 3 Time history records of chaotic motion of Lorenz's type attractor y and the pitchfork model X exhibiting on-off intermittency

dX

dt -X3 - (y -b)X (29b)

where y is a chaotic process generated from the modified Lorenz equations

dt <7> dq

dt

dy_

dt + zy — evq,

dz

dt = -e(z + a(y-l)) (30)

where e, v, b, and a are constants. A typical t ime history records of both y and X are shown

in Fig. 3 which is taken from Hammel and Piatt (1994). It is seen that the X motion experiences on-off intermittency when the bifurcation parameter y is chaotic.

It is clear that the mechanism of the observed intermittency in the uncertain and partially developed regimes of the beam motion is the same as the one in the example above where the filtered signal from the shaping filter is t ime dependent . This signal appears as the bifurcation parameter q'j- in the equation of motion of the beam (28).

4.2 Multi-Mode Random Excitation. T h e case of multi-degree-of-freedom systems involves new problems in addition to those encountered in single-degree-of-freedom systems. The new problems mainly are due to the nonlinear modal interaction which is only significant if the natural frequencies, u>0 are commensurable, i.e., if they satisfy the internal resonance condition T,kiwi = 0, where the kt are integers. Unde r this condition, the mode which is directly excited interacts with other modes in the form of an energy exchange. This type of modal interaction is also referred to as autoparametr ic (Minorsky, 1962). In the absence of internal resonances, the response is dominated by only the directly excited modes. The energy sharing in the random vibration of nonlinearly coupled modes is believed to have been first addressed by Newland (1965). The random response of two degree-of-freedom systems with autoparametric coupling was examined by Ibrahim and Roberts (1976, 77), Schmidt (1977a, b), Ibrahim and Heo (1986), Soundararajan and Ibrahim (1988), Li and Ibrahim (1989), Ibrahim et al. (1990), and Nayfeh and Serhan (1991). These authors considered different nonlinear coupled systems such as coupled beams, liquid free surface sloshing interacting with an elastic support, shallow arches, suspended cables, and hinged-clamped beams. For systems governed by

228/ Vol. 117, JUNE 1995 Transactions of the AS ME


quadratic nonlinearities it was found that the Gaussian closure scheme leads to nonstationary response statistics while the non-Gaussian closure gives a stationary response to a white noise excitation. Systems with cubic nonlinearity exhibit complex response characteristics in the neighborhood of the condition of internal resonance. It was reported that unbounded response statistics take place at certain regions above and below the perfectly tuned internal resonance. For regions well remote from the exact internal resonance condition the system experienced linear response behavior. These results were qualitatively .verified experimentally by Roberts (1980), Ibrahim and Sullivan (1990), and Ibrahim et al. (1990). In some cases the experimental results revealed an uncertain region of excitation level over which the coupled mode may or may not interact with the directly excited mode. Ibrahim and Li (1988) and Li and Ibrahim (1990) considered the nonlinear interaction of a three degree-of-freedom structural system subjected to a wide-band random excitation. The nonlinearity of this system resulted in different critical regions of internal resonance. The response statistics were predicted by using Gaussian and non-Gaussian closure schemes, and Monte Carlo simulation. While the non-Gaussian closure predicted multiple solutions in the neighborhood of an exact internal resonance, the Monte Carlo simulation yields only the branch of the stationary solution that corresponds to a zero set of initial conditions.

The next two subsections describe the dynamic behavior of clamped-clamped beams in the neighborhood of two different internal resonance conditions. The first case corresponds to pre-buckling condition while the second deals with the post-buckling state.

4.2.1 Clamped-Clamped Beam (Straight Beam Case). Nonlinear continuous systems such as beams and plates subjected to random excitation, can exhibit new phenomena not found in discrete nonlinear systems. Ibrahim et al. (1993) studied the stochastic bifurcation of a clamped-clamped beam under wide-band random excitation in the neighborhood of the internal resonance condition <u3 = <w, + 2&>2, which is mainly governed by the value of pre-applied axial static load, see Fig. 1(a). The equations of motion of the first three modes of the beam are

Yt + 2^53,^ + 7b\Yx + CXY3 + C2Y13 + C3YXY2 - C4YX

2Y3

+ C5YXY2 - C6Y2Y3 - C7Y3

3 = CsYb(t) (32a)

Y2 + 2«T2532Y2 + ai2Y2 + C9Y23 + CWY2Y3

2 + CnY2Y2

+ CuYxY2Y3 = 0 (32b)

Y3 + 2£3w3Y3 + a>2Y3 + CA3YX + CUY? - CX5YXY2

+ CUY2Y3 - C17YXY2 + CISY2Y3 + C19Y3

3 = C20Yb(t) (32c)

where Yi is the displacement of mode i, C, and S3,, are constants which depend on the beam geometric and dynamic properties and static pre-applied in-plane load P. The double sign indicates whether the axial static force is tension or compression, respectively. It is seen that the first and third modes are coupled linearly, while the three equations are coupled nonlinearly through nonlinear terms. The system possesses internal resonance of combination type co3 = cox + 2 w2 when a = P/P0 = 0.78, where PQ is the Euler buckling load. All <w,- depend on the parameter a which is taken as the control parameter. The external excitation Yb is assumed to be Gaussian wide-band process whose correlation time is much smaller than any characteristic period of oscillation of the system. In this case the response can be approximated by

3.50

o.o 0.5

Fig. 4 Bifurcation diagram showing three regions: I- two mode interaction, II- three mode interaction, and III numerical instability

a Markov state vector which is established through the following coordinate transformation:

{Y„Y;,Y2,Yi,Y3,Yi} = {XX,X2,X3,X4,X5,X6} = X (33)

Let mK(a) denote the K\h moment of the vector X as / -> °3, i.e.

mK(a)= \imE\X1(t,a)X2(t,a)...Xn(t,a)\K, K>\

(34)

If the moment mK bifurcates for a = ac but converges to a non-zero limit for a > ac, one says that the system bifurcates in Kih joint moment. Figure 4 shows a bifurcation diagram on the <j2-a plane. The critical value of a which separates the two-mode response (region I) from the three-mode response (region II) is the lower boundary of the hatched area. The bifurcation diagram given in Fig. 4 includes another region (III) where the non-Gaussian solution is numerically unstable.

The experimental results revealed some new phenomena not predicted analytically. For example, the power spectral density functions of the shaker and beam are shown in Fig. 5. It is seen that there is a significant shift of the resonance peaks as the excitation level increases. Another important feature is that the response spectra become wider as the excitation level increases. This feature implies that the stiffness nonlinearity enhances the degree of randomness of the response signals. The estimated probability density functions, from numerical simulation, displayed significant deviation from normality. Furthermore, the response revealed non-zero mean under zero-mean excitation. This non-zero mean was attributed to the fact that the straight beam is under initial axial load.

4.2.2 Clamped-Clamped Beam (Post-Buckling Case). When the axial static load exceeds the Euler buckling load, i.e., P > Pcr, where Pcr is the Euler buckling load, see Fig. lib), the motion of the beam is governed by the nonlinear partial differential equation

d2U(x,t) d\U(x,t) + U0(x)) m ; + EI dtl dx*

+P d2(U(x,t) + U0(x)) EA d2{U(x,t) + U0(x))

dxl 2L

f {d(U(x,t) + U0(x))/dx)2dx -m-

dx1

d2Y(t) (35)



104 156 Hz

Fig. 5 Estimated spectra of excitation and beam response from experimental measurements for three excitation levels: (Sy)m a x = 0.02 (mv)2 / rad / s, (Sy)m a x = 0.01 (mv)2 / rad / s ,

(Sy)m a x = 0.005 ( m v ) 2 / r a d / s , the corresponding response spectra show three peaks corresponding to the first three modes. The peaks experience a shift as the excitation level increases associated with a widening effect in their bandwidths.

where U(x, t) is the deflection at position x and time t measured from U0(x); E is Young's modulus of the beam; / is the area moment of inertia in bending; and A is the cross-sectional area of the beam. The static axial load P exceeds the Euler buckling load Pcr, i.e.:

P„ + f 2 /. Jn

EA rL{ dU0(x)

dX dx (36)

The deflection of the beam is written as an expansion in terms of the first n modes:

U0(x) + U(x,t) = c V 0 U , ( x ) + £ g,-(0*,-(*) I (37)

where V0 is the initial deflection of the beam at x = L/2; C = I/Oj(L/2); cjf = qt/h, the q,(t) are the generalized coordinates; and the <&,(x) are the linear free-vibration modes of a clamped beam.

Applying Galerkin's method to Eq. (32) for the first two modes and introducing a linear viscous damping, one obtains the nondimensional equations motion (Lee and Ibrahim, 1994)

Q'[ + 2£xrQ\ + r2Q{ + er2(l.5Q2 + 1.8046}!)

+ C2r2( l .804G,Gf + 0.5G?3) = ~f„Z(r)

Qi + 2(2Q'2 + Q2

+ er2(3J4Q!Q2) + e2 r2 ( l . 8687Q2Q2 + 6.74L?3) = 0

(38a)

where the excitation Z(T) is generated from a the second order linear shaping filter

EtQ,2]

0.000 0.005 0.010 0.015 0.020

E[Q23]

0.000 0.020

Fig. 6 Dependence of the second mode bifurcation on the damping ratio of the linear shaping filter, f , = 0.05 A A Gaussian solut ion, f , = 0.1O O Gaussian solution, and • • non-Gaussian solution, (, = 0.2 • a Gaussian and m m non-Gaussian

Z" + 2£frfZ' + rfZ = e2W(r) (38b)

where T = <o2t, r = o>i/co2, e = b/L, rf = o)f/a)2, fa = 1.32/K0) col = 720.0756£/ / (mL 4 ) , a>2 = co2{VQ/h)2, co2 = 2.76089w^, Z(T) = LY(t)/bco2 = -LoiJY(j)/bm\, and G?,(r) = L/bqiU), i = 1,2 and a prime denotes differentiation with respect to T. Z(T) is the response process of the linear filter (38i>) whose bandwidth and center frequency are controlled by the parameters &• and rf. The quadratic nonlinear terms in Eqs. (38a) are due to the curvature resulting from the static deflection, whereas the cubic nonlinear terms are due to mid-plane stretching. The first two natural frequencies give rise to the one-to-one (r = 1) internal resonance condition at V0/h = 1.66. The departure from the exact internal resonance is expressed by the internal detuning parameter A, defined by r = 1 + e2A. The stochastic bifurcation of the second mode depends to a great extent on the excitation level and system parameters, such as system damping ratios, and the internal detuning parameter A. The bifurcation behavior also depends on the damping ratio of the linear shaping filter equation which is related to the bandwidth of the filtered excitation. Figure 6 shows that the second mode bifurcation takes place at different excitation levels due to different damping ratios of the linear shaping filter. If the damping ratio of the linear shaping filter increases, a higher excitation level is required for the second mode bifurcation. For very small filter damping ratio, such as (£y = 0.05, the non-Gaussian closure was found to encounter numerical instability.

The results also demonstrate that in the neighborhood of exact internal detuning A = 0 the energy is transferred from the externally excited mode to the second mode. It is found that modal interactions take place for a wider range of the internal detuning parameter for the non-Gaussian closure solution than for the Gaussian solution. The mean-square responses are found to be not symmetric about the exact internal resonance condition due to the presence of a cubic nonlinearity. For the same damping ratios, the Gaussian closure solution yields a second-mode bifurcation at a higher excitation level than the level due to the non-Gaussian closure. At relatively low excitation level the response is uni-modal and the first mode mean square is governed by a



I L i l J .1 i ' iV .Li '

1500 2000 T

2500

Fig. 7(a) Sample of time history records of excitation and response according to Monte Carlo simulation for f, = f2 = 005, r = 0.99, and <r2 = 60,000

Fig. 7(b) Sample of time history records as measured experimentally for the shaker Y and beam response U showing the snap-through case, for excitation level exceeding the critical level

linear relationship with the excitation level. As the system damping ratio increases, the region of two mode interaction shrinks.

There is a critical excitation level below which the beam oscillates about one potential well (one of the buckled positions). Above this critical level, the beam oscillates between the two wells where snap-through takes place as shown in Fig. 7(a) as predicted by Monte Carlo simulation. Experimentally, however, for a long test duration, it was difficult to identify the excitation level due to the inevitable uncertainty of the boundary conditions. At relatively high excitation level the snap-through phenomenon was observed as shown in the

time history records of Fig. Kb). The response signal U is found to fluctuate between the two potential wells in an irregular manner.

5 Conclusions Nonlinear random vibration must be considered in the

design stage of mechanical systems. Several complex characteristics owe their origin to nonlinear effects. The theory of nonlinear random vibration is limited in predicting these characteristics. Numerical simulation as well as experimental tests are important in exploring other phenomena such as snap-through, on-off intermittency and stochastic chaos. These phenomena have been revealed in single and two-mode excitations. The on-off intermittency takes place in systems with multiplicative excitation or with coupled systems possessing internal-resonance conditions. Under a filtered white-noise excitation, the nonlinear response and stochastic bifurcation of a clamped-clamped beam have been studied in the neighborhood of a 1:1 internal resonance. The mean-square responses show that the bifurcation of the second-mode response depends on the excitation level, system damping ratios, internal detuning parameter, and the damping ratio of the filter equation. The damping ratio of the filter equation is related to the bandwidth of the filtered excitation. The response bifurcation is examined at different excitation levels, detuning ratios, and damping ratios of the linear-filter equation. The range of two-mode interactions predicted by non-Gaussian closures is wider than that predicted by Gaussian closure. Monte Carlo simulation is found to be in good agreement with the analytical results in predicting the second mode bifurcation boundary. The effect of the damping ratio of the linear filter on the response statistics is found to reduce the interaction between the two modes.

Acknowledgment This research is supported by a grant from the National

Science Foundation under grant number MSS-9203733 and by additional funds from the Institute for Manufacturing Research at Wayne State University. The author also would like to thank his former students Drs. Soundararajan, Heo, Li, Yoon, Afaneh, and Lee; and Messrs. Heinrich and Evans for their contributions of the results presented in this paper. The author would like to thank the reviewers for their valuable comments.

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recent results in random vibrations of nonlinear mechanical systems

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