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Analysis of some nonlinear deterministic oscillators using extended averaged equation approach

The paper presents an application of extended averaged equation approach in investigating some nonlinear oscillation problems. The main idea of the method is briefly described and numerical simulations are carried out for some nonlinear oscillators. The results in analyzing oscillation systems with strong nonlinearity show advantages of the method. | Vietnam Journal of Mechanics, VAST , Vol. 27, No. 2 (2005), pp. 32 - 40 ANALYSIS OF SOME NONLINEAR DETERMINISTIC OSCILLATORS USING EXTENDED AVERAGED EQUATION APPROACH NINH QUANG HAI Hanoi Architectural University Abstract. The paper presents an application of extended averaged equation approach in investigating some nonlinear oscillation problems. The main idea of the method is briefly described and numerical simulations are carried out for some nonlinear oscillators. The results in analyzing oscillation systems with strong nonlinearity show advantages of the method . 1. INTRODUCTION The method of moment equation is well known for analysis of random nonlinear oscillation phenomena and gives also good approximate solutions for systems with strong nonlinearity [15-16] . One way of extension the method to deterministic oscillation systems was given in [17] . In this paper, an extended averaged equation for deterministic one degree-of-freedom systems is presented and then some nonlinear oscillations are investigated in detail. The numerical results give good approximate solutions for the systems with weak, and strong nonlinearity. 2. EXTENSION OF MOMENT EQUATION METHOD TO DETERMINISTIC NONLINEAR VIBRATIONS In order to describe briefly the main idea of the extended averaging approach which was presented in [17], one considers a oscillation of one-degree-of-freedom system governed by a nonlinear differential equation (2 .1) z+f(z , z) =O , where dots denote time differentiation, f(z, z) is a nonlinear function of z , z. At the same time, consider the corresponding linear equation x + k 2 x = 0. For an arbitrary differentiable function w(t, x, one gets . dw - dt 8w =- ot 8w . +- 8w . z + - . (-f(z , z)) oz oz (2.2) x, z) using equations (2 .1) and (2.2) , 8w . +- ox x Denote the averaging operator (Borgoliubov & Mitropolskii) J 8w + - . (ox k 2 x) . (2 .3) [1-3] as T . T1 = hm T-+oo (.)dt. 0 (2.4) Analyzing Some Nonlinear .