Abstract: Based on the simplified parameter-lumped dynamical model which dominates the vertical vibration of wheel-rail system, the state equations of the wheel-rail coupling nonlinear dynamic model are derived, in which the Hertz nonlinear contact force is adopted to simulate the wheel-rail interactions. The self-adapted Runge-Kutta method is used to solve the state equations. The change rule of the overall displacements, velocities and accelerations of all parts changed with the vehicle speed is investigated and the critical speed which describes the dynamic stability of vehicle system is obtained. The numerical results show that the method can not only obtain the critical speed, but also may have the overview of the vibration and instability of all parts.