Application of Catastrophe Theory to Nonlinear Forced Vibration

In mechanical system, there are many discontinuous phenomena such as the bistable vibration of rotor caused by unbalance, the nonlinear torsional vibration of power transmission system, the jump phenomenon of nonlinear dynamic vibration absorber and the jump phenomenon of conical spring. Catastrophe theory provides a natural synthesis between the practical and theoretical aspects of such a phenomenon. Metal spring such as the nonlinear coil spring and progressive spring has nonlinear restoring force. Diaphragm air spring has nonlinear spring characteristics. These springs are effective for vibration isolation. In this study, nonlinear forced vibration with one-degree-of-freedom is taken up, and the characteristics of Duffing's equation are explained by means of the bifurcation set and catastrophe manifold of cusp catastrophe. And, it is clarified that the jump phenomenon of this system is described by the bifurcation set and catastrophe manifold of Cusp Catastrophe.