Subcritical Hopf and saddle-node bifurcations in hunting motion caused by cubic and quintic nonlinearities: experimental identification of nonlinearities in a roller rig

Abstract

Railway vehicles suffer from hunting motion, even when traveling below the critical speed obtained by linear analysis, due to the nonlinear characteristics of the wheel system. Nonlinear characteristics in Hopf bifurcations can be characterized as subcritical or supercritical, depending on whether the cubic nonlinearity is softening or hardening, respectively. In a system with softening cubic nonlinearity, third-order nonlinear analysis cannot detect nontrivial stable steady-state oscillations because they are affected by quintic nonlinearity. Therefore, in such a system, it is necessary to apply fifth-order nonlinear analysis to a system model in which quintic nonlinearity is taken into account. In this study, we investigated the cubic and quintic nonlinear phenomena in hunting motion with a roller rig that is widely used for hunting motion research. Previous experimental studies using a roller rig were restricted to the linear stability and the cubic nonlinear stability. We clarified that roller rig experiments can observe the hysteresis phenomenon and the existence of subcritical Hopf and saddle-node bifurcations, indicating that not only the cubic but also the quintic nonlinearity of the wheel system plays an important role. In addition, we obtained the normal form governing the nonlinear dynamics. We developed an experimental identification method to obtain the coefficients of the normal form. The validity of our method was confirmed by comparing the bifurcation diagrams obtained from the experimental time history and the normal form whose coefficients were experimentally identified using the proposed method.