Design of a nonlinear vibration absorber using three-to-one internal resonances

• A nonlinear vibration absorber is proposed to attenuate primary resonance vibrations.
• The absorber is designed by establishing three-to-one internal resonances.
• The lower frequency mode corresponds to the vibration of nonlinear absorber.
• The absorber is realised by attaching a light-weight mass to the nonlinear oscillator.
• The absorber nonlinear stiffness is of opposite sign to that of primary oscillator.

A weakly nonlinear vibration absorber is designed to attenuate the primary resonance vibrations of a single-degree-of-freedom weakly nonlinear oscillator having cubic nonlinearity. The linearised natural frequency of the nonlinear absorber is tuned to be approximately one-third the linearised natural frequency of the primary nonlinear oscillator. The low frequency mode for the absorber is favourably considered based on the fact that the nonlinear absorber can be easily realised in practice by using a light-weight mass attachment with small values of linear and nonlinear stiffness of coupling. For a given primary nonlinear oscillator and absorber mass, implementation of three-to-one internal resonances requires the smallest value of the absorber linear stiffness among three options for utilising internal resonances to design nonlinear absorber. The method of multiple scales is used to obtain the averaged equations that determine the amplitudes and phases of the first-order approximate solutions to the vibrations of the primary nonlinear oscillator and nonlinear absorber. It is found that the absorber response may admit either forced vibration having the forcing frequency or a combination of forced vibration and free-oscillation term having one third the forcing frequency. The nonlinear absorber can effectively suppress the amplitude of primary resonance response and eliminate saddle-node bifurcations occurring in the frequency-response curves of the primary nonlinear oscillator. 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.