Hopf–Hopf bifurcation analysis based on resonance and non-resonance in a simplified railway wheelset model

This paper mainly investigates the dynamics of the non-resonant and near-resonant Hopf–Hopf bifurcations caused by the interaction of the lateral and yaw motion in a simplified railway wheelset model, which involves local and global dynamical scenarios, respectively. This study aims to clarify the resonances due to the wheelset instability. Firstly, the ratio of longitudinal suspension stiffness and the square of natural frequency in yawing direction denoted as the parameter k22\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k_{22}$$\end{document} has an important impact on the transitions of distinct Hopf–Hopf bifurcations, and the ratio of the oscillation frequencies ω1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega _1$$\end{document}/ω2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega _2$$\end{document} at the Hopf–Hopf singularity point will reduce with the decrease in k22\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k_{22}$$\end{document} within a certain range. Secondly, the absence of strong resonance under the non-resonant condition indicates that the operation wheelset will not produce the maximum oscillation amplitude triggered by the resonance point, and several torus solutions arisen from the wheelset are obtained by numerical simulation. Thirdly, five near-resonant Hopf–Hopf bifurcations reveal that the global dynamical scenario becomes much more complex than other cases as k22\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k_{22}$$\end{document} decreases. In particular, near the 1:4 resonant Hopf–Hopf interaction occurs when ω1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega _1$$\end{document}/ω2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega _2$$\end{document} is close to 1:4, which has the most marked effect on wheelset hunting motions and resonances. Finally, the cyclic bifurcation behaviors under the near-resonant conditions indicate the coexistence of multiple limit cycles, and the loop of equilibria and limit cycles detected between two Hopf bifurcation points reveals that the wheelset will perform a cyclical motion in lateral and yaw direction. These results show that the change in frequency ratio induced by the intersection of the lateral and yaw motion of the unbalanced wheelset will greatly affect the hunting motions and resonances of railway vehicles. Therefore, appropriately increasing the value of k22\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k_{22}$$\end{document} is helpful to maintain the vehicle stability.