THE METHOD OF EQUIVALENT LINEARIZATION FOR SYSTEMS SUBJECTED TO NON-STATIONARY RANDOM EXCITATION

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1 UNIVERSITY OF PITESTI SCIENTIFIC BULLETIN FACULTY OF MECHANICS AND TECHNOLOGY AUTOMOTIVE series, year XVII, no.1 ( 3 ) THE METHOD OF EQUIVALENT LINEARIZATION FOR SYSTEMS SUBJECTED TO NON-STATIONARY RANDOM EXCITATION 1 Marinică Stan *, 1 Petre Stan 1 University of Pitesti, Romania s: stan_mrn@yahoo.com, petre_stan_marian@yahoo.com KEYWORDS -random vibration, linearization, spectral density, standard deviation, response. ABSTRACT - The method of equivalent linearization is applied to the general problem of the response of non-linear discrete systems to non-stationary random excitation. Conditions for minimum equation difference are determined which do not depend explicitly on time but only on the instantaneous statistics of the response process. Using the equivalent linear parameters, a deterministic non-linear ordinary differential equation for the covariance function is derived. The theoretical analyses are verified by numerical results. An example is given of a damped Duffing oscillator subjected to modulated white noise. 1. SYSTEM MODEL To illustrate the procedure of equivalent linearization theory, let us consider the following oscillator with a nonlinear restoring force component. The ordinary differential equation of the motion can be written as:... 3 m() t + c() t + k() t + αk () t F() t (1) where m is the mass, c is the viscous damping coefficient, F(t) is the external excitation signal with zero mean and () t is the displacement response of the system. The reduced equation is... 3 () t p () t p () t p () t f() t + ξ + + α () whereξ is the critical damping factor, and p is the undamped natural frequency, for the linear system. As a next let us consider excitation described by subsequent correlation function ( ) R cos F τ De λτ βτ, (3) where parameters D>, λ>, β. Power spectral density function [1] of excitation we obtain from the relation: 1 SF( ) RF( τ) dτ (4) By substitution of the (3) in the (4) and integration we obtain Dλ + λ + β SF ( ) (5) π ( i) + λ( i) + λ + β or 89

2 S F Dλ + λ + β ( ) π λ + β + 4λ ( ). (6) Fig. 1. The power spectral density SF [ N s] of excitation for 1 1 D 5 N, λ 1 s, β 3 s Fig.. The power spectral density SF [ N s] of excitation for 1 1 D 5 N, λ 1 s, β 6,5 s. Power spectral density function of output we can obtain from the relation SF ( )/ m S ( ). (7) ( pe ) + 4ξ p So we obtain Dλ ( + λ + β ) S ( ). (8) πm { p + 3αpσ + 4ξ p } ( λ + β ) + 4λ 9

3 The displacement variance [] of the single-degree of freedom system under Gaussian white noise excitation can be expressed as, σ R() S( ) d. (9) Substitution of the (8) in the (9) and obtain Dλ ( + λ + β ) σ d mπ. (1) { p + 3αpσ + 4ξ p } ( λ + β ) + 4λ Integration [3,4] obtain + d π ( bh o 1+ hh 1 h3) d, (11) 3 ( i) + λ( i) + d ( i) + b( i) + b b( hh 1 h3 bo h1 d h3 ) where In this case where Using the notation 1 h b + λ, h b + λb + d, h λb + db. (1) h ( ξ p+ λ), h p + 4 λξp+ λ + β, h λp + ξp( λ + β ) 1 e 3 e b p p (1+ 3 ασ ), b ξ p. e 1 A (13) σ, (14) B A D p p p D p p λσ [1 α ( ξ + λ) 6 αλ ] + λ{ ( ξ + λ) + + λξp ξp+ λ + ξp+ λ λ + β λp ξp λ + β 8 ( ) ( )( ) ( )} B m p + p m p + p+ + p σ λξα 36 σ α{ λα[ 4 λξ ( λ β )]( ξ λ) + ( ξ + λ) [ + 4 λξ + ( λ + β )][ λ + ξ( λ + β )] + α[ λ + p p p p p p + pξλ λ + β + ξp+ λ λ + β + λαp + mσ pα ξp+ λ λp+ 3 3 ( ) ( ) ( )] } 1 { ( )[ + + p + p+ + p + p 4 5 ξλ ( β)][ 4 λξ ( λ β)] ξ αλ ( β) ξ λ β λ α α ξ λ λ β λα λξ λ β ξ λ + p( p+ ) [ p + 4 p+ ( + )][ p+ ( + )] + [p + } 3 + p ( + ) + ( p+ ) ( + )]} + 4 p m( p+ )[ p+ ( + )][ p + { αλ ( + + ) p p ( p+ ) ( + )} + p { [ p + 4 p+ ( + )]( p+ ) + ξ λ λξ λ β λ ξ λ β α λ ξλ λ β ξ λ λ β ξ λ λ ξ λ β λξ p+ ( λ + β )] 4 ξ p m( λ + β ) 4ξ pmλλ ( + β ) 4λmp pm( ξp λ) ( λ β ). l m p (15) (16) λξα (17) n p m p p p p p p p 4 36 α { λα[ + 4 λξ + ( λ + β )]( ξ + λ) + ( ξ + λ) [ + 4 λξ + ( λ β )][ λp+ ξ( λ + β )] + α[ p λ + pξλ( λ + β ) + ( ξp+ λ) ( λ + β )] + λαp } (18) 91

4 3 4 1 {{ αξ ( λ)[ λ ξλ ( β )][ 4 λξ ( λ β )] ξ αλ ( β ) ξ pλα( λ + β ) λ pα pα( ξ p+ λ) ( λ + β )} + p { λα[ p + 4 λξ p+ ( λ + β )]( ξp λ) p( ξp λ) [ p 4 λξp ( λ β )][ λp ξ( λ β )] α[p λ + pξλ( λ + β ) + ( ξp+ λ) ( λ + β )]}} r m p p+ p+ + p + p+ + p { s p m p p p p p ( ξ + λ)[ λ + ξ( λ + β )][ + 4 λξ + ( λ + β )] 4 ξ ( λ + β ) ξp λ( λ β ) 4λ p 4 p ( ξp λ) ( λ β ) Dλ[1 αp ( ξp λ) 6 αλp ] q Dλ{ p ( ξp+ λ) 8 λξp( ξp+ λ) ( ξp+ λ)( λ + β ) + λp + ξp( λ + β )} (1) obtain the equation lσ + nσ + rσ + sσ + q. () We can always find a way to decompose the nonlinear restoring force to one linear component plus a nonlinear component h( ) p ( + G( ) α), (3) where α is the nonlinear factor to control the type and degree of nonlinearity in the system. The idea of linearization is replacing the equation by a linear system:... () t + ξepe() t + pe () t f(), t (4) where p ξe ξ. (5) pe is the damping ratio of equivalent linearized system and p e is the natural frequency of the equivalent linearized system. To find an expression for p e, it is necessary to minimize the expected value of the difference between equations () and (4) in a least square sense. Now the difference is the difference between the nonlinear stiffness and linear stiffness terms, which is e h( ( t)) p e ( t). (6) The value of pe can be obtained by minimizing the expectation, of the square error: de{ e }. (7) dpe Substituting the equation (6) into equation (7) performing the necessary differentiation, the expression of p e can be obtained as: E{ G( )} pe p (1 + α ) p (1+ 3 ασ )., (8) σ where σ is the standard deviation of () t. This equation shows how the nonlinear component of the stiffness element affects the value of p e.. NUMERICAL RESULTS Consider in this example N Ns m kg k c m m m α Let us set the subsequent values of excitation parameters } (19) () 1, 36, 4, 3. (9) 9

5 Obtain: or σ D N s s 1 1 5, λ 1, β 3. (3) σ σ ,91 1 σ ,5 1 σ, + 34 (31) σ ,91 1 σ ,5 1 σ 16σ , (3) σ,5m. (33) Substituting the equation (33) into equation (8), obtain 1 pe p (1 + 3 ασ ) 7,6s. (34) In literature, very little attention has been paid to the frequency domain characteristics of nonlinear, dynamic systems excited by stochastic processes. It will be shown that this information can be of great value for the understanding of the system's stochastic behaviour. In the figures 1,, 3, 4 and 5, the power spectral density of the excitation, SF [ N s], is plotted for the different parameters D, λ, β. Figure 6 describes the harmonic peak with the same parameter values Fig. 3. The power spectral density D N s s 1 1 5, λ 1, β 3, S N s of excitation for F [ ] N Ns m kg k c m m m α 1, 36, 4, 3. 93

6 CONCLUSIONS Fig.4. The power spectral density N Ns m kg k c m m m α 1, 36, 4, 3. S [ m s] of response for The statistical linearization technique can also tackle a wide variety of problems and also provides approximate information on the frequency domain characteristics of the stochastic response. In this technique, a linear model, which optimally is the original, nonlinear system (in some statistical sense), is constructed. Due to the fact that response statistics of such a model can, in general, be evaluated analytically, statistical linearization is computationally very efficient. However, it only provides accurate approximation of the response statistics for weakly nonlinear systems. In this chapter, it is shown that the statistical linearization technique structurally underestimates the variance of the response of the piece-wise linear system (even for a moderate nonlinearity). This is dangerous when these estimates are used in failure criteria for practical systems. The cause for this underestimation of the variance can be found by comparing accurate, simulated frequency domain characteristics with those determined using the linear model. REFERENCES (1) Pandrea, N., Parlac, S., Mechanical vibrations, Pitesti University,. () Munteanu, M., Introduction to dinamics oscilation of a rigid body and of a rigid bodies sistems, Clusium, Cluj Napoca, (3) Zhao, L., Chen, Q., Equivalent linearization for nonlinear random vibration/ Probabilistic Engineering Mechanics, 9,

RESPONSE OF DUFFING OSCILLATOR UNDER NARROW-BAND RANDOM EXCITATION

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