A Special Case of Rolling Tire Vibrations

We investigate a special case of vibrations of a loaded tire rolling at constant speed without slipping in the contact area. A previously proposed analytical model of a radial tire is considered. The surface of the tire is a flexible tread combined with elastic sidewalls. In the undeformed state, the sidewalls are represented by parts of two tori and consist of incompressible rubber described by the Mooney – Rivlin model. In the undeformed state, the tread is a circular cylinder. The tread is reinforced with inextensible cords. The tread deformations are considered taking into account the exact nonlinear conditions of inextensibility of reinforcing cords. Due to nonlinear geometric constraints in the deformed state, the tread retains its cylindrical shape, which is not circular for a typical configuration. The contact between the wheel and the ground plane occurs by a part of the tread. The previously obtained partial differential equation which describes the tire radial in-plane vibrations about the steady-state regime of rolling is investigated. Analyzing the discriminant of the quartic polynomial, which is the function of the frequency of the tenth degree and the function of the angular velocity of sixth degree, the rare case of two pairs of multiple roots is discovered. If the geometry of the tire and the internal tire pressure are known, then the angular velocity of rotation, the tire speed and the natural frequency, corresponding to this case, are determined analytically. The mode shape of vibration in the neighborhood of the singular point is determined analytically.