Resonance bifurcation of a three-degree-of-freedom vibratory system with symmetrical rigid stops

Abstract

Abstract A three-degree-of-freedom vibratory system impacting symmetrical rigid stops is considered. Dynamics of the vibroimpact system is studied by use of a mapping derived from the equations of motion, supplemented by transition conditions at the instants of impacts. Two-parameter bifurcations of fixed points in the vibroimpact system, associated with 1 : 3 strong resonance, are analyzed. Neimark–Sacker bifurcations of period-1 double-impact symmetrical orbits, near 1 : 3 resonance point, are found, and the transition of quasi-periodic impact motions to unstable period-3 six-impact motions, are obtained numerically. The results conform to 1 : 3 resonance bifurcation theory of maps available. More complicated “tire-like” quasi-periodic attractor and torus bifurcation associated with the transition of the closed invariant curve to torus, near 1 : 3 resonance point, are found to exist in the nonlinear system. The results imply that there exist possibly more complicated bifurcation sequences near 1 : 3 resonance points of nonlinear dynamical systems.