Attenuation of primary resonance vibrations of a non-linear system using a non-linear vibration absorber

Abstract

In a single degree-of-freedom weakly non-linear oscillator subjected to a periodic external excitation, a small-amplitude excitation may produce relatively large-amplitude vibrations under primary resonance conditions, when the forcing frequency is in the neighbourhood of the linearised natural frequency of the non-linear oscillator. Jump and hysteresis phenomena that result from saddle-node bifurcations may occur in the steady-state forced response. A non-linear vibration absorber is thus used to suppress the primary resonance vibrations. The two linearised natural frequencies of the resultant system formed by the non-linear primary system and nonlinear absorber are not under any internal resonance conditions. The method of multiple scales is used to obtain the averaged equations that determine the amplitudes and phases of the first-order approximate solutions. Illustrative examples are given to show the effectiveness of the non-linear vibration absorber for suppressing non-linear vibrations of the forced oscillator under primary resonance conditions.