Nonlinear Dynamics 9: 327-332, 1996. (~) 1996 Kluwer Academic Publishers. Printed in the Netherlands.

Pseudolinear Vibroimpact Systems: Non-White Random Excitation

M. D IMENTBERG Mechanical Engineering Department, Worcester Polytechnic Institute, Worcester, MA 01609, U.S.A.

(Received: 16 June 1994; accepted: 19 January 1995)

Abstract. Response analyses of vibroimpact systems to random excitation are greatly facilitated by using certain piecewise-linear transformations of state variables, which reduce the impact-type nonlinearities (with velocity jumps) to nonlinearities of the "common" type - without velocity jumps. This reduction permitted to obtain certain exact and approximate asymptotic solutions for stationary probability densities of the response for random vibration problems with white-noise excitation. Moreover, if a linear system with a single barrier has its static equilibrium position exactly at the barrier, then the transformed equation of free vibration is found to be perfectly linear in case of the elastic impact. The transformed excitation term contains a signature-type nonlinearity, which is found to be of no importance in case of a white-noise random excitation. Thus, an exact solution for the response spectral density had been obtained previously for such a vibroimpact system, which may be called "pseudolinear", for the case of a white-noise excitation. This paper presents analysis of a lightly damped pseudolinear SDOF vibroimpact system under a non-white random excitation. Solution is based on Fourier series expansion of a signum function for narrow-band response. Formulae for mean square response are obtained for resonant case, where the (narrow- band) response is predominantly with frequencies, close to the system's natural frequency; and for non-resonant case, where frequencies of the narrow-band excitation dominate the response. The results obtained may be applied directly for studying response of moored bodies to ocean wave loading, and may also be used for establishing and verifying procedures for approximate analysis of general vibroimpact systems.

Key words: Vibroimpact systems, random vibrations, narrow-band random excitation.

Use of certain specific modulus-type piecewise-linear transformations of state variables for vibroimpact systems, as proposed in [1], has proved to be a major breakthrough in analysis of these systems. These transformations effectively reduce the system to one without velocity jumps - with "regular" nonlinearity(ies) only - provided the impact is elastic. Moreover, in case of a "slightly inelastic" impact, where value of the restitution factor is close to unity, an approximate analysis can be made by applying any one of the asymptotic methods of nonlinear mechanics - such as quasiconservative averaging - to the reduced system with only small velocity jumps.

These transformations were proved to be particularly useful for systems with random exci- tation, since traditional methods for vibroimpact systems' analysis were found to be generally inadequate for solving such random vibration problems. Thus, they were applied in [2] to SDOF vibroimpact systems with a white-noise random excitation. Using quasiconservative stochastic averaging (QCSA) approach for the transformed state variables the analytical solu- tions have been obtained for stationary probability density for the response energy. It had been shown also, how the available exact analytical solutions to the stationary Fokker-Planck equation for motion between the impacts can be used for the case of a vibroimpact system with perfectly elastic impacts [3].

328 M. Dimentberg

There exists a certain specific case also, where the transformed equation of motion is found to be linear, except for the excitation term (provided that the impact is elastic). This is the case of a linear SDOF system with a single barrier, situated exactly at the system's static equilibrium position. This vibroimpact system may be called "pseudolinear". The nonlinearity in the excitation term in the transformed system in this case is found to be of a signature type, and in case of a white-noise random excitation it does not affect the solution. This means that not only stationary probability density of the response can be obtained analytically for the pseudolinear system, but also other response characteristics, usually obtainable for linear systems only, such as autocorrelation function and/or spectral density of the response. The possibility for such an extended analysis has been mentioned in [3], and the analysis itself has been presented in [4]. It seems to represent the unique solution available for the spectral density of the nonlinear system's response to random excitation.

After solution for the transformed response varaible(s) is obtained, it may become nec- essary, in general, to return to the original variable(s). This represents, in fact, a classical problem in theory of random processes: to find certain statistical characteristics of a given nonlinear function of a random process with given properties. The procedures for such an analysis are well known, and they may be particularly easy to implement in case of a Gaussian transformed response; such an analysis has been made in [4] for the response spectral density. However, when only variance(s) and cross-correlation(s) of the response state variables (at the same time instants) are of interest, this step is not needed since these second-order moments of the transformed state variables are found to be the same as those of the original ones. Thus, whereas in general the transformation of state variables decomposes the dynamical nonlinear problem into a dynamical linear one and inertialess nonlinear one, the latter may sometimes be of a secondary interest only. This is particularly the case with a problem, considered in this paper.

It should be admitted, of course, that in general pseudolinear systems comprise only a small subclass of the vibroimpact systems. However, their study still seems to be worthwhile, mainly due to two reasons. Firstly, in some applications the vibroimpact systems in question may be pseudolinear indeed; this may be the case with moored bodies, where the shift of the static equilibrium position due to initial preload or slack of the mooring cable, if at all, may be negligible compared with the level of dynamic displacements due to ocean wave loading. Secondly, the possibilities for extended analytical studies of these systems should provide a good insight into behavior of general vibroimpact systems, which may be helpful in development of approximate analytical methods for the general case; the latter, once again, may be checked with the use of baseline rigorous solutions as obtained for the pseudolinear cases.

Thus, in this paper a SDOF system is considered for the case of a random excitation with an arbitrary spectral density. The transformed system is still nonlinear in this case because of the signature-type nonlinearity in the excitation term. The solution for the mean square response can be obtained for the case of a narrow-band response by using Fourier series expansion for signum function; cases of resonant response at the system's natural frequency and of non-resonant response due to a narrow-band excitation are considered separately.

Consider a common spring-mass-dashpot system, excited by a stationary random force rag(t), where m is the system's mass. Let a rigid barrier be installed at the static equilibrium

Pseudolinear Vibroimpact Systems 329

position of the mass, so that the equation of motion of the mass between its impact against the barrier may be written as

~1 + 2c~9 + f~2y = g(t), y > 0 (1)

where y(t) is the mass displacement from its equilibrium position (and the barrier). When y(t) becomes equal to zero, the impact against the barrier with a subsequent rebound into domain y > 0 takes place. The rebound is assumed to be a perfect one, so that the magnitude of the mass velocity is completely retained (the case of elastic impact). Therefore the impact condition may be written as

y+=-y_ , for y=O; ? )+=y( t ,+O) , y(t,)=O, (2)

where t. is clearly seen to be the instant of impact. Since t. are not given but rather are govemed by the equation of motion itself, the vibroimpact system (1), (2) is nonlinear indeed. We shall see however, that it can be reduced to the linear one except for the excitation term in the RHS of equation (1).

This is done by introducing a new state variable x(t) as

+1, x > 0

y = Ixl = z sgn x, $ = 5: sgn z; sgn x = 0, x = 0 . (3) -1, x < 0

Using this transformation in the impact condition (2) yields 2+ = k_, implying that the new state variable x(t) has a continuous time derivative at impacts, i.e. when x = 0. This implies also that ~)(t) = 2(t) sgn x, since (d/dx)(sgnx) = 0 when x # 0. Using now transformation of variables (3) in the equation of motion between impacts (1), multiplying the resulting equation for x(t) by sgn x and using the identity (sgn x) 2 = 1, we obtain finally the following transformed equation of motion, which may be solved disregarding the impact condition (2) which will be satisfied automatically

+ 2~2 + ~2X = g(t) sgn x. (4)

This equation contains nonlinearity in the RHS only, because of the factor sgn x wih the excitation force 9(t). Thus, free vibration problems for the system (1), (2) become perfectly linear. Moreover, in the case where 9(t) is a zero-mean white noise the nonlinear factor sgn x becomes irrelevant: it can be easily shown, that in this case the apparent excitation process f(x, t) = 9(t) sgn x in the reduced equation (4) is also zero-mean white noise with the same intensity factor. Therefore, in this case solution for autocorrelation function and/or spectral density of the transformed state variable x(t) can be obtained easily, whereas those of the original state variable y(t) can be obtained then by using "inertialess" relation (3) between x and y [4]. The system (1), (2) may, therefore, be called "pseudolinear": although it may not be completely reducible to a linear one, still the transformed equation of motion may permit to obtain analytial solutions for much larger varieties of response characteristics compared with common random vibration problems. And here we shall study the case of excitation with an arbitrary spectral density ~bgg(w), though certain restrictions will be imposed.

Consider steady-state second-order moments (x2), (~b2), (x:b) of the transformed response as governed by equation (4); in view of the relation (3) they are equal to the corresponding second-order moments of the original response, (y2), @2), (yy), respectively. Analytical

330 M. Dimentberg

solution to this problem is obtained in the following for the case, where x(t) is narrow-band, and therefore can be represented as

x(t) = A(t) sin 0, 0 = At + (t). (5)

Here A(t), (t) are slowly varying amplitude and phase of x(t) respectively, whereas mean frequency A will be specified later. Then we may use Fourier series expansion of the sgn x function in the RHS of equation (4):

(4 ) ( 1 1 ) sgn x = sgn (sin 0) = sin 0 + ~ sin 30 + g sin 50 +. - . . (6)

Autocorrelation function Kfy(t) of the transformed excitation f(x, t) = 9(t) sgn z can be related then to the original excitation g(t) by direct term-by-term multiplication of the series (6), as written for time instants t and t + r:

Kef(r) = (f(t)f(t + r)) (~) ( 1 1 ) = (g(t)g(t + r)) cosAr + ~ cos31r + ~ cos5Ar + ... ;

(g(t)g(t + r)) = Kga(r ). (7)

Here angular brackets denote probabilistic averaging (mathematical expectation), which lead to vanishing of all cross-product terms, as well as all non-stationary terms with

cos{(2k-1)[(t)+(t+r)]}, k =0,1 , . . .

Taking then Fourier transforms of both sides of relation (7), we obtain power spectral density (PSD) ~IY (w) of the apparent excitation for the transformed system f(t) in terms of the PSD aa(w) of g(t):

(4 ) [ lgPaa(W-3A) = - A) + + A) +

1 ~ 1 ~gg(w + 5A)+. . . ] (8) + ~ ~gg(w + 3A) + e~gg(w - 5A) + ~

First of all, it can be seen that if 9(t) is a white noise, so that sgg(w) = if0 = constant, then q~//(w) = if0 since [5]

oo 1 71-2

(2k - 1) 2 = -8-" (9) k=l

Thus, in this special case the transformed system is seen to be perfectly linear indeed. In the general case, however, mean square responses of the system (4) should be obtained by direct substitution of the series relation (8) into expressions

(X)

(x 2) = f --0(3

oo 49Sf(W ) dw

- -00

o2 ss( o) (co 2-- 7 4o 2w 2 "

(10)

As long as the expected frequency of the response is to be determined, two cases will be considered here.

Pseudolinear Vibroimpact Systems 3 31

(i) Resonant case, A = f~ - components with frequencies close to the system's natural frequency dominate in the response. This case may be realized for low-damped systems (a

332 M. Dimentberg

in case f2 ~ 2u) of the series (13) should be withdrawn and complete direct (e.g. numerical) integration according to (10) should be performed for the corresponding term of the series (8).

Acknowledgment

This work has been sponsored by ONR-URI Grant No. N00014-93-1-0917. This support is gratefully acknowledged.

References

1. Zhuravlev, V.E, 'A method for analyzing vibration-impact systems by means of special functions', Mech. Solids 11, 1976, 23-27.

2. Dimentberg, M. and Menyailov, A., 'Response of a single-mass vibroimpact system to a white-noise random excitation', ZAMM 59, 1979, 709-716.

3. Dimentberg, M., Statistical Dynamics of Nonlinear and Time-Varying Systems, Research Studies Press, Taunton, England, 1988.

4. Dimentberg, M., Hou, Z., and Noori, M., 'Spectral density of a nonlinear SDOF system's response to a white- noise random excitation: A unique case of an exact solution', International Journal of Non-Linear Mechanics, to be published.

5. Gradhstein, I. S. and Ryzhik, I. M., Tables of Integrals, Series and Products, Academic Press, New York, 1980.