A new equivalent nonlinearization method for random vibrations of nonlinear systems

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A NEW EQUIVALENT NONLINEARIZATION METHOD F O R R A N D O M V I B R A T I O N S O F N O N L I N E A R SYSTEMS "

Lel Zhao Department of Architectural Engineering, Southwest Jiaotong University, Chengdu, Sichuen 610031, P. R. China QIu Chert Department of Engineering Mechanics, Southwest Jiaotong University

(Received 27 June 1995: accepted for print 8 November 1995) Introduction In the last two decades, responses stochastically excited nonlinear systems have been researched in deep-going and many methods of predicti~g the response of nonlinear systems to random excitations have been proposed. Because of various reasons, the FPK equation method can only be applied to linear systems and few simple nonlinear random systems. In general, many nonlinear systems are handled by some different approximate methods, and the equivalent ]inearization method has been used widely in engineering due to its simplicity and convenience [ I ] . The stochastic responses of ]inesr systems have been studied in great detail, and numerous analytic methods may be applied to treat both the stationary and nonatationary stochastic problems, so that responses of a nonlinear system may be obtained by means of a equivalent linear system. However, it must be noted that the resuhs determined by the equivalent linearization method are usually sm,ller than those of FPK equation method or Monte Carlo simulation, and the maximum error is on the low side about 20 percent ['2--3]. Hence, the error is obviously too big for structural systems with the higher accuracy. If the method were applied to structural reliability analysis, the results should have led to a tendency of dangerous even serious error. As the foregoing reasons, some equivalent nonlinearization methods and pad~ approximsnts are developed as an improvement to the linearization method [4-7"]. The concept of a statistical quadratization method is also introduced for determining the response statistics of random systerns ['8--107. The effectiveness of those methods has been demonstrated by many examples and is shown to give more accurate results than the equivalent linearization method. In this paper, the authors propose a new equivalent nonlinearizationmethod for random vibrations of nonlinear * This project is supported by National Natural ScienceFoundationof P. R. China

131

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L. ZHAO and O. CHEN

systems, i.e. , the given nonlinear system is substituted by a low-order nonlinear system with exact stationary solutions, and the solutions of the latter are taken as the approximate solutions of the former. The purpose of this paper is a contribution to the study of simple, feasible and accurate low-order nonlinear approach for treating problems of the stochastic response analysis of nonlinear systems under random excitations. The analytical and numerical examples show that the proposed method is simple in concepts, feasible in applications and accurate in results.

Proposed Equivalent Nonllneaflzation Method The equation of the motion for a nonlinear stochastic system can be written in the form X+g(X,X)=F(X) (1) where X, X and ~( are the random responses of the displacement ,the velocity and acceleration for the given system, respectively, g ( X , X) is a generalized nonlinear function of the system. F ( t ) is a random excitation. As a means of studying an approximate solution of Eq. (1), let us consider an auxiliary system that is described by a low-order nonlinear system. We assume that the approximate solution can be obtained from the auxiliary nonlinear system X + C j ( + K , X '/' = F ( t ) (2) where C, and K, are the equivalent damping and the equivalent stiffness of the auxiliary system, respectively. The error of low-order equivalent nonlinear, a random process, is expressed as { = g ( ~ , X ) - - C . ] ( - - K , X '/~ (3) which is the difference between the original system (1) and the auxiliary system (2). The magnitude of this difference will clearly be a function of unknown parameters C. and tL for the auxiliary system. An approximate solution of the original system (1) will be generated by selecting C. and K, in such a way that some measure of the difference t is minimized. As a minimization criterion on ¢, it will be required that the mean value of the scalar product iT{ be minimum. The necessary conditions for the criterion are a<~dt>

~

a

= o

~C, = u , 0K. (4) where the operator < • > denotes the statistical average or mathematical expectation of the appropriate variables. In order for the following discussion to be reasonably brief but still retain the essence of the proposed method, we assume that the excitation F ( t ) is stationary, is Gaussian and has a zero expectation, i. e. =0, < F ( t ) F ( t + r ) > = 2nSo~(r) (5) where So is the power spectral density of the noise, and 3(r) is the Dirac delta function. From a Fokker-Planck approach, the joint probability density of the stationary response for the auxiliary system (2) is given by p ( X , X ) = A e x p ( _ f f _ ~ , ( X' , 3 -

-,,o/,'

where A is a normalization constant. Therefore, the displacement X of the auxiliary system is non-Gaussian distributed and the velocity X is Gaussian distributed, but X and X are independent random variables. When we take X and ~ to be jointly stationary, we have < X 5>----0 (7) Eqs. (4) then can be reduced to

RANDOM VIBRATIONSiOF NONLINEAR SYSTEMS

C.=/,

K.=/

133

(8)

which will lead to the minimum for . Now, the original system (1) has been reduced to the solution of a rel ted auxiliary system (2) which may be solved by Fokker-Planck approach. The moments of various orders for random responges X and ~[ to the auxiliary system are obtained by

{

---- ( m - - l )

.==odd I

°

=-,i 31"[0,3(m+l)~/3I~C, I,

m even | (--1)'<0~

! ! (xSo/C.) -/'

101"~(1.3) ' ~

)

(9)

(-l)=>oj

where F( • ) denotes the Gamma (unction. Finally, the approximate solutions of the original system may be given by means of the proposed auxiliary system with exact stationary response, e. g. , the mean-equare responses are obtained from

1~(1.9) / 10~S0 ~ ,/s

=,,s,/c.,

(10)

However, Eqs. (8) are not explicit expressions for C. and IC, since the response statistics appearing on the right-hand aides depend on C, and IC. It is possible to see that an itecatlon solution procedure is generally required in the auxiliary system, which is also necessary to usual ]inearization techniques.

Accurac~ of the Proposed Method EXAMPLE I. System with Nonlinear Stiffness Let us consider a random system with cubic hardening stiffness described by ~(+~X-[-aXl = F (t) (11) By means of Fokker-Planck equation method, the exact solution of RMS response for the Even system (11) is [11]

~i.----- O. 6760("~) '"

(12)

Using the proposed method in this paper, the equivalent damping and the equivalent stiffness of the auxiliary nonlinear system (2) are given by c.=p,

v

['I'(1.9)( 10*o ~ l/'al'/' .

(is)

Then, the RMS response of the auxiliary system can be obtained directly, i. e.

(14) From the equivalent linesrlzation method, the RMS response may be given, i. e.

o'xL=0.5774(~) '"

(15)

The RMS response ~ of the given nonlinear system as a function of the excitation level is shown in Fig. 1. Results are given for the proposed nonlinearization, the usual linearization and the exact analysis. By comparison with the exact solution, the propoled nonlinearizationmethod gives a n even better solution. The error in the proposed solution is 6. 2 percent, much less than the error 14. 6 percent in lincarizatlon solution. EXAMPLE 2. System with Nonlinear Damping

134

L. ZHAO and Q. CHEN

25 20

j,~S j,,/:= i

15

~ .....

~.If>"

Exact analysis Proposed nonlinearization

-- - - - - U s u a l

linearization

I0

t; 0

So/ I

100

I

I

I

I

200

300

400

500

_

Fig. 1 RMS response of the system with nonlinear stiffness Let us consider a random system with energy-dependent nonlinearity described by X + ~ ( ~ ' +aX')~+aX---- F (t) (16) The exact solution of the RMS response o~ of the system is obtained by Caughey [12], i. e.

Using the proposed method, equivalent {actors o{ the auxiliary nonlinear system are given by 1,1,I 3C. C.-----(3~S0~( " ' 1 0 r ( l ' 9 ) r ( 1 . 7 ) ) / '" l-t271~.(I. 3) / ' K'----' I'(l" 3)

(

V(-iT--~.7)a/ ~ i ' 6 " ~ I

(18) The RMS response dx of the auxiliary system is obtained by

~z=o. 5878("s'/~'/ ' "

(19)

The result is slightly larger than that obtained by Caughey. The RMS response ~ of the linearization system is given by

(~s0/"

oK=0. 5000 N' ]

(20)

The results given by the proposed nonlinearization, the usual linearizationand exact analysis are plotted in Fig. 2, against the ratio o( the excitation level and the strength o{ the nonlinearity. The error in the proposed solution is - 4 . 2 percent, much less than the error 11.4 percent in the usual linesrization solution. EXAMPLE 3. System with Nonlinear Damping and Nonlinear Stiffness Let us consider a random system with nonlinear damping and nonlinear stiffness described by X + O ( I + ¢ , ) U ) X + a ( I + ¢ , X ' ) X = F (t) (21) The exact solution of the system cannot be obtained by Fokker-Pianck approach, but may be given by Monte Carlo simulation. The constants C. and K. of the proposed auxiliary nonlinear system are C.=R,~, K.=R,a (22) where

R, = (1+ ~/I+12~,So/~)/2, I'(1.3)( 311, ~,is, B'= ~ ~lOo~----):

R,=

[0.

5B,

(1+ v/1-+--c~B,)'1"

B_40F(1.9)P(1.7)

'-

-~?~3

~o

" R~

RANDOM VIBRATIONSOF NONLINEAR SYSTEMS

135

25

jj:

I / " I

20

15 I0 5

g ~/~ I

I

I

I

100

200

300

400

t~ 500

Fig. 2 RMS response of the system with nonlinear damping ( a = l . 0),legend as in Fig. 1 where el0 denotes the RMS response of degenerate linear system obtained from given system (21) when t~ and t, are equal to zero, respectively, hence

~o=~S./(~)

(23>

The RMS response ol of the proposed nonlinearization system is obtained by

~z----

B,

(24)

6(1+ ~+-'-Bt¢, ) At the same meaning, the RMS r~ponse ~ of the usual ]inearization system is obtained by o~L= 6-I ( -/I+ 12t,~0/R,- I)

(25)

For the case ¢,----¢,----2 ,~----0. 4 and a----1 ,the R M S responses ~ as a function of the excitation level are shown in Fig. 3. Results are given for the foregoing three methods. It is easily see that the result calculated by the proposed method compares more favourably with what obatined by Monte Carlo simulation than that given by the usual linearization.

Coaelmlon A low-order nonlinearizationmethod has been proposed and developed for the random vibration analysis of nonlinear systems. Although two approximate results accord very well with the exact analysis using Fokker-Planck approach or Monte Carlo simulation and evidently comparable in accuracy, the proposed new nonlinearization is more efficient procedure than the usual lincarization for all nonlinear random systems considered in this paper. The proposed approach has also the advantage of simple in concepts, reliable in accuracy and easy to apply. Especially, when the proposed method is applied to the random system with nonlinear damping, the RMS response of the auxiliary nonlinear system is slightly larger than the exact solution, which provides a safety insurance for engineering. Based on examples of three kinds of typical nonlinear systems, the accuracy of the proposed low-order nonllnearizationapproach is obviously superior to the usual fin-

136

L. ZHAO and Q. CHEN

a: 0.5

0,375 0.25

•/

f

. . . . . Proposed nonlinearization -- -- -- Usual linearization

O. 125

Fig. 3

l

l

[

0.2

0.4

0.6

I

0.8

I

S. -.._

1.0

R M S response of the system with nonlinear damping and nonlinear stiffness (~-- 0.4,ti = t,-- 2,a -- 1)

earization method, and appears to be well within the limits of practical engineering use{ulness•

References

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