Normal form method for large/small amplitude instability criterion with application to wheelset lateral stability

Subcritical and supercritical bifurcations are two typical behaviors that exist in high speed railway vehicles. In the presence of instability, the former and the latter behaviors may lead to large amplitude oscillation and small amplitude swaying, respectively. The normal form (NF) method of Hopf bifurcation provides a way to study the supercritical and subcritical bifurcation. The wheelset is a key component in the vehicle system and it plays an important role in vehicle lateral stability. To study the lateral stability problems, three wheelset models are considered, which involve the NF theory. This method is an algebraic approach as opposed to the integration approach. Like the sign of Re (λ) that determines the stability of linear system, the sign of Re c₁(0) determines the two bifurcation modes, meaning that Re c₁(0) >0 for supercritical bifurcation and Re c₁(0)< 0 for subcritical bifurcation. Furthermore, if the ordinary differential equation (ODE) is local linear near the equilibrium position, it leads to the condition of Re c₁(0) = 0, resulting in the jumping phenomenon. Besides, the expression of the 1/2-order approximation of limit cycle can be further obtained.