The tire/road noise is one of the major problems facing the tire industry due to the nuisance felt by the vehicle's driver and passengers. Moreover, it is expected in the coming years that this sound nuisance will be one of the main sources of vehicle noise due to the transition from combustion engine driven vehicles to electric vehicles. Tire manufacturers have therefore refined the design of their tire structures to find technological solutions to reduce traffic noise. Tire/road noise is also generated by various mechanisms which depend on different parameters such as the properties of the tire, road texture and driving conditions. This noise is partly caused by the acoustic radiation induced by the tire vibrations due to contacts with the road. The simulation and analysis of the tire's vibrations remains a challenge for the tire industry. Indeed, the simulation of the full dynamic response of a rolling patterned tire requires not only taking onto account various nonlinearities but also the multi-scale nature of the generated dynamic response. Contrary to a straightforward strategy that consists in using a time integrator to predict the multi-scale dynamic response, the strategy proposed in this study is based on a two-step approach to separate the dynamics occurring at different scales. The mathematical formulation of the proposed method is detailed as are the different modeling choices to simulate tire rolling under imposed load. The sensitivity of the tire's vibrations response is analyzed under different rolling conditions and with respect to certain key tire tread pattern parameters.