Control of Non-linear vibrations using three-to-one internal resonances

Abstract

A weakly nonlinear vibration absorber is used to suppress the primary resonance vibrations of a single-degree-of-freedom weakly nonlinear oscillator subjected to periodic excitation. The linearized natural frequency (low frequency mode) of the nonlinear absorber is tuned to be approximately one-third of the linearized natural frequency (high frequency mode) of the primary nonlinear oscillator. The cubically nonlinear coupling of stiffness establishes the terms that develop three-to-one internal resonances. The low frequency mode required for the absorber can be achieved by a light-weight mass nonlinear attachment with small values of linear and nonlinear stiffness of coupling. The method of multiple scales is used to obtain the averaged equations that determine the amplitudes and phases of the first-order approximate solutions. Numerical results are given to show the effectiveness of the nonlinear absorber for suppressing nonlinear vibrations of the primary nonlinear oscillator under primary resonance conditions.