TAILIEUCHUNG - The influence of nonlinear terms in mechanical systems having two degrees of freedom
For many years the higher order stochastic averaging method has been widely used for investigating nonlinear systems subject to white and coloured noises to predict approximately the response of the systems. In the paper the method is further developed for two-degree-of-freedom systems subjected to white noise excitation. Application to Duffing oscillator is considered. | Vietnam Journal of Mechanics, VAST, Vol. 28, No. 3 (2006), pp. 155- 164 THE INFLUENCE OF NONLINEAR TERMS IN MECHANICAL SYSTEMS HAVING TWO DEGREES OF FREEDOM N GUYEN D ue TINH Mining Technical College, Quang ninh Abstract. For many years the higher order stochastic averaging method has been widely used for investigating nonlinear systems subj ect to white and coloured noises to predict approximately the response of the systems. In the paper the method is further developed for two-degree-of-freedom systems subjected to white noise excitation. Application to Duffing oscillator is considered . 1. INTRODUCTION It is well-known, the stochastic averaging method (SAM) is widely used in different problems of stochastic mechanics, such as vibration, stability and reliability problems (see . Mitropolskii et al, 1992; Red-Horse and Spanos, 1992; Zhu and Lin, 1994; Zhu et al, 1997) . However , the effect of some nonlinear terms cannot be investigated by using the classical first order SAM. In order to overcome this insufficiency t he different procedures to obtain approximate solutions have been developed for the nonlinear systems with one degree of freedom under white and coloured noise excitations (see . Anh, 1993; Anh and Tinh, 1995 ; Tinh, 1999) . In t he present paper this procedure is further developed for twodegree-of-freedom nonlinear systems subjected to white noise excitation. An application to Duffing system is considered and the effect of nonlinear terms can be detected in the approximate solutions of Fokker-Planck (FP) equation while it cannot be investigated by using t he classical first order SAM. 2. HIGHER SAM IN TWO-DEGREE-OF-FREEDOM SYSTEMS Consider t he motion equations of a mechanical system with two degrees of freedom = cfu (x1, x2, ±1, .±2) + c 2fi2( x1, x2, ±1, ±2) + Vfcr1~(t), i2 +wh2 = ch1(x1,x2 ,±1,±2) +c 2f22(xi,x2,.±1,±2) + Vfcr2~(t), i1 + wix1 () where w1, w2, cr 1, cr2 are positive constants and€ is a small positive .
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