Bifurcations in a simplified and smoothed model of the dynamics of a rolling wheelset

A simplified wheelset model with a nonlinear smooth equivalent conicity function is taken into account. The goal is to investigate the influence of the different nonlinear equivalent conicity functions on the bifurcation characteristics. The equivalent conicity functions of different types are fitted smoothly by the measured data, especially when the lateral displacement of the wheelset is less than 3 mm. In addition to qualitative analysis of the stability and Hopf bifurcation of the equilibrium, the mechanism that the type of the nonlinear equivalent conicity affects the Hopf bifurcation characteristics is explained using the normal form theory. Analytical studies reveal that if the second-order derivative of the equivalent conicity function with respect to the lateral displacement of the wheelset at the equilibrium is positive (negative), the Hopf bifurcation of the wheelset system is subcritical (supercritical). The limit cycle motion caused by the Hopf bifurcation is analyzed, such as the fold bifurcation, the period-doubling bifurcation, the cusp bifurcation, and the fold-flip bifurcation. It is noted that the concept of equivalent conicity applied in this paper is a purely heuristic engineering concept, which has been applied by many railway engineers, while it is still short of a strict mathematical basis.