EXTERNAL PRIMARY RESONANCE OF SELF-EXCITED OSCILLATORS WITH 1 3 INTERNAL RESONANCE/

The forced response of a class of weakly non-linear oscillators with s elf-excited characteristics is investigated. The non-linearity is symm etric, the external forcing is harmonic and the essential dynamics are described by a two-degree-of-freedom oscillator, whose linear natural frequencies satisfy conditions of 1:3 internal resonance. Firstly, se ts of equations governing the slow time variation of the amplitudes an d phases of approximate solutions of the equations of motion are obtai ned by applying an asymptotic analytical method. For primary resonance of the first mode, only mixed-mode response is possible, since the se cond mode is always activated through the non-linearities. On the othe r hand, when conditions of primary resonance of the second mode are me t, single-mode response is also possible. In both cases, a methodology is developed which reduces the determination of constant solutions of the slow-flow equations to the solution of two coupled polynomial equ ations. The stability analysis of these solutions is also provided. Ne xt, numerical results are presented for an example practical system, i n the form of response diagrams. These results show the,effect of some system parameters on the existence and interaction of various branche s of constant solutions. Then, more numerical results are presented, o btained by direct integration of the slow-flow equations in forcing fr equency ranges where these equations possess no stable constant soluti on. The results demonstrate the existence of periodic and chaotic solu tions of these equations. (C) 1997 Academic Press Limited.