Nonlinear Dynamics of Self-excitation in Autoparametric Systems

Abstract

Various types of self-excited oscillators are implemented into an autoparametric system, and the study of the solutions, stabilities and bifurcations, shows very different results. First, we implement the Rayleigh type oscillator into a suitable autoparametric system. The bifurcation analysis of the solution gives the result that there exists a stable robust heteroclinic cycle as nontrivial solution of the system. Taking the detuning sigma is not zero (near resonance) results in symmetry breaking of the heteroclinic cycle; a long periodic solution occurs. Second, we replace the Rayleigh oscillator of the autoparametric system. With a dry friction oscillator characterised by a small parameter. We can determine the boundary value of parameters of the system for having a nonsmooth periodic solution. By using the software package SlideCont, sliding bifurcations of the nonsmooth periodic solution can be studied. A numerical simulation shows that a 3-dimensional nonsmooth invariant manifold existing in the (full) system can be obtained. In this thesis we also implement a relaxation oscillator of van der Pol type into an autoparametric system. The possibility of destabilising the undesirable vibrations due to the stable normal mode of the system is studied by choosing a suitable tuning and appropriate deformation of the slow manifold. In the case of normal mode vibration derived from a relaxation oscillation, we need low-frequency tuning of the attached oscillator. In this way it is possible to make the quenching effective. Conditions for suppression of self-excitation of a 3 degree-of-freedom system is also studied. We consider vibrating systems containing self-excitation and parametric excitation. In the analysis we show that full and partly suppression of the excitation are possible to achieve when some conditions are met. Surprisingly, an vautoparametricv phenomenon takes place in the normal form of a three-mass system in 1 : 2 : 3 resonance; partial decoupling of the system occurs. We conclude that nonlinear dynamics obtained from embedding a self-excited oscillator in a higher dimensional system is of practical interest and at the same time it is a rich source of interesting phenomena.