Vibration Absorbers for a Class of Self-Excited Mechanical Systems

Abstract

An averaging methodology is employed in studying dynamics of a two-degree-of-freedom nonlinear oscillator. The main system is modeled as a van der Pol oscillator under harmonic forcing. The objective is to reduce its amplitude of oscillation near resonance, by attaching to it a damped vibration absorber with a Duffing spring. It is first shown that substantial reduction in the response amplitude can be achieved in this way. However, for some combinations of the parameters, the low-amplitude periodic motion of the system in the original resonance regime becomes unstable through a Hopf bifurcation of the averaged equations. Direct numerical integration shows that this gives rise to amplitude modulated or chaotic response of the oscillator, with much higher vibration amplitudes than the unstable periodic response, which coexists with these complex motions. Finally, it is shown that the present analysis can be employed in selecting the parameters in ways that exploit the significant practical advantages arising from the presence of the absorber, by predicting and avoiding these instabilities.