Nonlinear Dynamics 11: 401-405, 1996. @ 1996 Kluwer Academic Publishers. Printed in the Netherlands.

Random Vibrations of an Isoehronous SDOF Bilinear System

M. D I M E N T B E R G Mechanical Engineering Department, Worcester Polytechnic Institute, Worcester; MA 01609, U.S.A.

(Received: 11 December 1995; accepted: 3 July 1996)

Abstract, A single-degree-of-freedom (SDOF) system with bilinear restoring force and damping characteristics is excited by a white-noise force. For the case of small damping the asymptotic method of quasi-conservative stochastic averaging is applied, which reduces the problem to that of a single Ito stochastic differential equation (SDE) for total energy of the response. A special case of the isochronous system is considered, where the static equilibrium position coincides with the slope discontinuity point of the system's characteristics, such as that of a moored body without slack and pretension in the mooring cable. For this case an analytical solution is obtained for autocorrelation function of the response energy. For the limiting case of a vibroimpact system the results are shown to agree with the exact solution.

Key words: Bilinear system, random vibrations, response autocorrelation, stochastic averaging.

Consider an SDOF system, governed by equation of motion

+ h ( x , k ) + f ( z ) = C(t),

f ( x ) = f~2_x, x < 0 ; h(x,:b) = 2oz_:b, x < 0 ;

f ( x ) = a 2 x , x > 0; h(x,:i:) = 2o~+:b, x > 0, (1)

where ~ (t) is a zero-mean Gaussian white noise with intensity D. As can be seen, the system is of a rather special type, with its static equilibrium position coinciding with the slope discontinuity point of both the restoring force and the damping characteristics. Still it may be an adequate model of a moored body, excited by random ocean waves, in the case where neither pretension nor slack is present in the mooring cable. The cable itself is assumed to be elastic, and for positive displacement its elastic and damping properties are combined with those due to fluid forces (buoyancy and hydrodynamic damping) leading to parameters with the subscript "plus". On the other hand, the cable becomes inactive for negative displacements, so that the parameters with the subscript "minus" correspond to fluid forces only. Thus, without any loss of generality, we may assume that ~2+ > f~_, with the possibility of infinite f~+ included, this limiting case corresponding to vibroimpact motions with inextensible mooring cable. The system (1) is also of interest for the general theory of random vibrations as a rare example of the case, where the autocorrelation function (and therefore the spectral density) of the nonlinear system's response energy can be obtained analytically.

The following analysis is made for the case of a lightly damped system with small excitation intensity, using the asymptotic method of quasiconservative stochastic averaging (QCSA) [ 1- 4]. Accordingly, the parameters o~+, o~_, D are assumed to be proportional to some small

402 M. Dimentberg

parameter ~ (so that the RHS of Equation (1) is proportional to v'7), and a new state variable - the system's total energy - is introduced as

A = ;b2/2 q- U(x), U(x) = i f(x) dx, (2)

where U(x) is clearly seen to be the system's potential energy. For the bilinear system (1)

U(x) = f~2x2/2, x < O,

U(x) = a2x2/2, x > 0. (3)

Consider first the corresponding conservative, or undamped and unexcited system, as obtained from (1) in case g = 0. Then its total energy A is constant, and the natural half-period T1/2 (A) can be obtained by resolving Equation (2) for 2 and integrating over the coordinate x between its extreme values:

x +

f dx Tw2(A) = v/2[A - U(x)] ' (4) Z - -

where x_, x+ are the smallest and largest roots, respectively, of the equation U(x) = A. (Full period is obtained if the same integral with the reversed direction of integration is added, completing passage along closed-loop phase-plane trajectory with a given value of A.) Substituting expressions (3) into Equation (4) yields

x+ = V ~ / f } + , x_ = - v ~ / f L , T1/2 = 7c/f~, ft -1 = (1/2)(~+ 1 + f~-l). (5)

Equation (5) shows that the system (1) is quasi-isochronous, with its conservative counterpart having energy-independent natural period of oscillations.

The analysis of random response of the system (1) is based on introducing the total energy as a new state variable, together with x. Differentiating Equation (2) with respect to time and using Equation (1), we may reduce the latter to an equivalent pair of first-order equations (for

> 0 )

= -~h(x, dc) + :b~(t) = -~/2[A - U(x)] h(x, V/2[A - U(x)]) + d2[A - U(x)] ~(t),

= V/2[A - U(x)]. (6) 0~

As long as the first Equation (6) contains a small parameter in its RHS, the corresponding state variable A(t) is slowly varying. According to the theorem, rigorously proved by Khas'minskii [4], with e -+ 0 this slow state variable converges in the weak sense to a diffusional Markov process. The latter is governed by the Ito SDE, as obtained by averaging the drift and diffusion coefficients of the Ito equivalent of the first Equation (6) over "fast" state variable x [1-4]:

]k = -Q1/2(A)/Tw2(A ) + D/2 + v/DSw2(A)/T1/2(A) r/(t). (7)

Here ~(t) is a zero-mean Gaussian white noise of unit intensity and

x + x +

Qi/2(A) = i h {x, V/2[A- U(x)]} am, $1/7 = S V/2[A- U(x)] am, (8) 02-- 9~--

Random Vibrations of an Isochronous SDOF Bilinear System 403

where integration in the formulae for Q and S is made within a half of the natural period (which is possible due to symmetry in the absence of hysteretic-type nonlinearities). The second term in the RHS of Equation (7) may be identified as a Wong-Zakai diffusional correction to the drift coefficient [2, 3]. This reduction of the second-order problem to a single first-order SDE (7) for a slowly varying state variable - response energy - is a major advantage of the asymptotic QSCA analysis. It leads to an explicit analytical expression for stationary probability density of the response energy of the general quasi-conservative second-order system [1-4]; first-passage problems become more more simple as well, with some analytical results being available for a mean exit time from a given domain [2, 3].

For the bilinear system (1), however, even more analytical results can be obtained, which are usually not available for nonlinear systems with random excitation - namely, for a correlation function of the energy. Thus, after substituting Equation (3) for U(z) into expressions (8) and performing integration the shortened SDE (7) becomes

= -2c~A + D/2 + ~ ~l(t), (9)

where the "equivalent", or averaged-over-the-period damping factor is

+

f~;l + (10)

It may be noted, that if the Ito SDE (9) is transformed to the Stratonovich SDE, the latter would still contain non-zero constant term D/4 (rather than D/2). Such a term is also present in "common" stochastic averaging for quasi-linear systems, where it clearly corresponds to the second-order approximation in the basic Krylov-Bogoliubov asymptotic averaging scheme.

Equation (9) is also valid in the "vibroimpact limit", when f~+ is infinite, provided that the corresponding impact damping ratio o~+/f~+ is finite. This finite limit can be easily related to the restitution factor r by considering undamped free vibrations of the system (1) within the domain of positive displacements between two consecutive crossings of the equilibrium state x = 0 (the first one with positive, and the second one with negative velocity). As long as the damping ratio is assumed to be small compared with unity, the resulting relation is found to be c~+/9,+ = (1 - r)/rr, up to second-order terms in the damping ratio. In this way impact damping can be accounted for, such as that due to elastic wave radiation into ground from the mooring cable.

As long as the drift coefficient in the SDE (9) is linear in A, the method of moments can be used to obtain mean value and standard deviation rnA, erA respectively of A(t), as well as autocorrelation function of its zero-mean part

K A A ( t ) = ( A ( t ) A ( t + -

where angular brackets denote direct unconditional expectation operator. The latter is applied first to both sides of the Ito SDE (9), resulting in the following deterministic equation

rhA = --2c~raa + D/2 (11)

with constant steady-state solution

mA = D/4oz. (12)

404 M. Dimentberg

Then the Ito SDE for squared energy V = A 2 is derived from SDE (9), by using Ito differenti- ation rule, and once again unconditional expectation operator is applied. Constant steady-state solution for the mean square energy is then found to be

(A 2) = DrnA/2(~ = 2rn~, (7 A ---- ((A 2) - m ~ ) 1/2 = D/4c~. (13)

The first-order differential equation for KAa (r) can be derived now by multiplying A(t) - rnA and the RHS of the SDE (9) at time instant t + r , r > 0. Then application of the expectation operator yields (here and in the following primes denote differentiating with respect to r )

[£1A : --20!KAA, KAA = Cry. (14)

Integrating this equation and using the symmetry property of autocorrelation function yields

KAA(T) = Cr~ exp(--2c~[~-[). (15)

It is of some interest to compare this result with the exact one, available for the limiting vibroimpact case (ft+ --+ ec). In this case, the motion within a positive part of each cycle is degenerated into a velocity jump, which will be assumed as perfectly elastic (r = 1). Equation (1), which is perfectly linear between impacts, or for x < 0, is supplemented then with the rebound condition

k(t, + O) = - k ( t , - 0), z( t , ) = 0. (16)

The analysis for this case can be based on the following transformation, as proposed originally by Zhuravlev [5] (see also [2, 6])

z=lzl=zsgnz, ~ = i s g n z ; s g n z = l f o r z > 0 , s g n z = - l f o r z < 0 . (17)

This transformation clearly makes the new variable z(t) continuous at impact instants t , , with the minus sign in the rebound condition (16) being changed to the plus. The transformed Equation (1) for motion between impacts (for x < 0) after multiplication by sgnz is reduced to

+ 2c~i + ~ 2 Z z ~ ( t ) " sgn z, c~ = c~_, f~ = ft_. (18)

The remaining nonlinearity due to the sgn z factor in Equation (1) is irrelevant, the RHS still being a zero-mean Gaussian white noise with the same intensity D. Therefore, the response z(t) is Gaussian, with a zero-mean and autocorrelation function [2]

2 2 = D/4o~f~2 =

p(t) = exp(-c~lr l ) [cosc~r + (c~/ft)sin~dlrl], toe = v / - ~ - a 2 , (19)

where crx is clearly seen to be the rms displacement of a linear SDOF system with natural frequency ~. Using relations (17), the total energy may now be expressed in terms of a new variable z(t) and its time derivative as

I = (1/2)(~ 2 + ft2z 2) = (ft2/2)(v 2 + z2), v = ~ / a . (20)

The autocorrelation function of the zero-mean part of energy can be obtained now as

KAA(~) = J ( ~ ) - J ( ~ ) , OQ

J(r) = (1/4) ffff( 2 + a2z2)( 2 + a22;2)p4(z,2;r,~,~.c) dz dz~- d~ d~-, (21)

- - O O

Random Vibrations of an Isochronous SDOF Bilinear System 405

where P4 is a Gaussianjoint probability density of z(t) and its first time derivative at two time instants with a shift ~-. It can be easily written in terms of the autocorrelation function (19) for the general case. For simplicity, however, we shall consider only specific time shifts, namely

m, = 2~n/cvd, ~ = 0 , 1 , . . . ;

p('c,) = exp(-c@-,I) , p'(~-,) = O, On(r,) = -f~2exp(-c@-,I) . (22)

Then the four-dimensional joint p.d.f, is reduced to a product of the two identical two- dimensional p.d.f.' s of displacement and normalized velocities:

1 Z 2 - 2pzz.r + z~] p4-~P2(Z, zr)p2(v,V.c), p2(Z, Zr)=27CCr2 ~ exp ~2--( i---p~ j . ( 2 3 )

Using expressions (22) and (23) in the integral (21) yields formula (15), as obtained originally by the approximate QCSA method.

The above results may be of use for response analysis of (linear) secondary structures of small mass, attached to the main one (with a bilinear restoring force) via "cascade" approximation, regarding z(t) as a base excitation for a secondary structure. In general, this procedure would certainly require knowledge of the autocorrelation function (or spectral density) of z (t), which is difficult to obtain analytically other than for the limiting vibroimpact case. However, for some cases approximate procedures can be derived, that require only data on the energy of excitation, particularly when the secondary structure's natural frequencies are much larger than ft_ and much smaller than ~Q+.

Acknowledgement

This work was sponsored by ONR-URI Grant No. N00014-93-1-0917. This support is grate- fully acknowledged.

References

1. Stratonovich, R. L., Topics in the Theory of Random Noise, Vol. 1, Gordon & Breach, New York, 1963. 2. Dimentberg, M. E, Statistical Dynamics of Nonlinear and Time-Varying Systems, Research Studies Press,

Taunton, England, 1988. 3. Lin, Y. K. and Cai, G. Q., Probabilistic Structural Dynamics, McGraw-Hill, New York, 1995. 4. Khasminskii, R. Z., 'On the behaviour of a conservative system with small friction and small random noise',

Prikladnaya Matematika i Mechanica (Applied Mathematics and Mechanics 28, 1964, 1126-1130 [in Russian]. 5. Zhuravlev, V. E, 'A method for analyzing vibration-impact systems by means of special functions', Mechanics

of Solids 11, 1976, 23-27. (English translation of the Russian Journal Mekhanika Tverdogo Teta.) 6. Dimentberg, M. E, 'Pseudolinear vibroimpact system: non-white random excitation', Nonlinear Dynamics 9,

1996, 327-332.