A simplified model of a rotating tire using cylindrical shells with free ends supported by an elastic foundation

In this paper a simplified model of a rotating tire is developed and free vibrations of the tire are analysed on an example case. The tire belt is modelled as a rotating cylindrical shell. The shell ends are assumed to be with free boundary conditions. A flexible foundation is considered under the belt surface in order to take into account the stiffness of the air inside the tire as well as the tire sidewall. Moreover, the tire inflation pressure is accounted for. This is necessary since it generates an important tangential pre-stress in the tire belt. Finally, the effects of rotation onto the resonance frequencies and the mode shapes of the tire are considered. This is done by introducing Coriolis and centrifugal terms into the equations of motion plus the tangential centrifugal pre-stress. The boundary conditions of the elastically supported shell are satisfied exactly and the corresponding mode shapes and the resonance frequencies are calculated for an example case. Unlike the shells with simply supported ends, the shells with free ends have the functional form of the mode shapes changing with frequency. In particular, the dependence of a mode shape on the axial coordinate exhibits strong generic variation with frequency. Consequently, different combinations of transcendental functions (hyperbolic and circular) occur at different frequency ranges. The Coriolis accelerations and the centrifugal tension are shown to play an important role in how the resonance frequencies of the tire veer with rotation speed. In fact, the modes rotating backward and forward along the tire circumference tend to slow down or speed up with the increase in rotation speed. Thus they no longer superimpose into standing waves. In contrast, they keep propagating as travelling modes.