In the paper this procedure is further developed to non-linear systems of and Van der Pol types taking into account some second order non-linear terms. It is shown that the effect of this non-linearity can be detected by the procedure proposed while it cannot be investigated by using the classical first prder stochastic averaging method (SAM). | Vietnam Journal of Mechanics, NCNST of Vietnam T. XX, 1998, No 4 (1- 10) ON THE INFLUENCE OF SECOND ORDER NON-LINEARITY ON RANDOM VIBRATIONS NGUYEN DoNG ANH, NGUYEN Due T!NH Institute of Mechanics, Hanoi 1. Introduction For many years the stochastic averaging method (SAM) has been a very useful tool for investigating non-linear random vibration systems, see . [1-5]. However, the effect of some non-linear terms cannot be investigated by using the classical first order SAM. The procedure for obtaining higher approximate solutions for the Fokker-Planck (FP) equation was developed in [6, 7] and then applied to Van der Pol oscillator under white noise excitation [8]. In the paper this procedure is further developed to non-linear systems of and Van der Pol types taking into account some second order non-linear terms. It is shown that the effect of this non-linearity can be detected by the procedure proposed while it cannot be investigated by using the classical first prder stochastic averaging method (SAM). 2. SAM of coefficients in FP equation Consider a. single- degree- of freedom system whose motion equation takes the form () where w, u are positive constants, c: is a positive small parameter, h and h are functions in x and :i;. The random excitation E(t) is a Gaussian white noise process with unit intensity. Using the following change of variables x { :i; = acos [ ] [ ] w-=-c:k;,k;·ltjJ 8) - · +-+r1-+r28a 8 8a 8!p 8a + (k 2 _ ak12 _ 8k22) 8¢> 8a 8 + (8¢>)2] + 2k12( 8a 2 2 82 ,p 8a ( 8 ) +k22 [ 8'P2 + . 8 8a8. 8) 8a 8!p 2] } 0"2 k 12 = - - sm . 2o(a,1(a,'P) + c: 2¢>2(a,o w-=0 81] w~~ = [k;,k;j]t[o,1,¢>2] 2 () () () where the operators [k;,k;j]l[c;ba,c;bi], [k;,k;j]l[c;bo,4>t,4>2] are defined as [k;, k;i l l [t) + k ( 8 t + 8c;bo 84>1) 12 + 2 11 8a2 + aa aa 8a8 a,P 0 8,. 0 (a) must be chosen from the condition for the function e(a)+ •14>lo(a), l1'>10(a, cp)J+• 2 [4>2o(a) =.P22(a, u(a,cp)] 0 () 3. Application In order
