Bifurcation analysis of two-DOF nonlinear vibration isolation system with internal resonance

Abstract

The equations of motion of a two-degree-of-freedom nonlinear vibration isolation system were formulated where the nonlinear restoring force was approximated as a polynomial. The averaging method was applied to obtain the bifurcation equations for the two cases: 1) quadratic nonlinear stiffness with primary resonance and 1:2 internal resonance; 2). quadratic, cubic nonlinear stiffness with primary resonance and 2:1 internal resonance. By means of singularity theory, the bifurcation behaviours of the amplitude with respect to a parameter (which is related to the amplitude of the external force) were studied. The high-codimensional universal unfoldings were given and the transition sets in the parameter planes and the bifurcation diagrams were plotted.