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Some problems on nonlinear oscillations

Thia paper presents our research on the interaction between no linear oscillations, Van-der-Pol's systems subjected to complicated excitations ancl quasilinear o::;cillations in systems with large static deflections [1-4]. It is well known that there is always an interaction of some kind between nonlinear oscillations, namely, between the forced, parametric and self-excited oscillators. | Vietnam Journal of Mechanics, VAST, Vol. 29 , No. 3 (2007), pp. 187 - 195 Special Issue Dedicated to the Memory of ProL Ng uyen Van Dao SOME PROBLEMS ON NONLINEAR OSCILLATIONS NGUYEN VAN DAO Vietnam National University, Hanoi 19 Le Thanh Tong, Hanoi , Vielnam (This paper has been pu,blishcd in: IUTA.!vl Symposium on Ilccent Development in Non-linea r Oscill ations of Mechanical Systems, Hanoi, 2000) 1. INTRODUCTION Thia paper presents our research on the interaction between no linear oscillations, Van-der-Pol 's systems subjected to complicated excitations ancl quasilinear o::;cillations in systems with large static deflections [1-4]. It is well known that there is always a n interaction of some kind between nonlinear oscillations, namely, between the forced, parametric and self-excited oscillators. Each of these oscillators demonstrates definite sustained oscillations, comprising one or a combination of several modes. The principal questions to be answered are: \Vhat will happen if these oscillators arc coupled in some manner? Docs a resultant nonlinear oscillation exist and is it stable? The satationary oscillat ions and their stability have been paid special attentions. 2. INTERACTION BETWEEN EXTERNAL AND PARAMETRIC EXCITATIONS In this section, we examine some quasilinear oscillating systems subjected to external and parametric excitations. \Ve restrict ourselves to a class of quasilinear systems with two excitations. The following systems have been considered: The system with external excitations in principal resonance and parametri c excitation of the first degree in subharmonic resonance of the order one-half: i + w2 x = E{b.x - hx - 1x 3 + 2pxcos2wt + ecos(wt +CT)}. (2.1) The system with interaction between an external excitation and a parametric excitation of the second degree, both in the principal resonance: x + w2x = E{b.x - hx - 1x3 + 2px 2 cos wt+ e cos(wt + 1) } . (2.2) The system with interaction between an external excitation in .