Simulating the full dynamic response of a rolling sculpted tire requires not only taking into account various non-linearities but also considering the multi-scale nature of the dynamic response itself. On one hand, there is the macroscopic rolling dynamic behavior that operates around the rotating frequency with relatively high amplitudes. On the other hand, the vibratory response operates in a larger frequency window with relatively low amplitudes. In contrast to a straightforward strategy that consists of using an energy-conserving stable time integrator to predict the multi-scale dynamic response, the proposed strategy is based on a two-steps approach to separate the dynamics operating at different scales. This methodology is applied to simulate the nonlinear vibrations of a hyperelastic solid undergoing large deformations in contact with a rigid plane. In order to illustrate the potential of the proposed numerical method, the nonlinear vibrations response of a grooved cylinder rolling on a rigid plane is investigated.