Another Special Case of Vibrations of a Rolling Tire

We investigate a special case of vibrations of a loaded tire rolling at constant speed. 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 tread is a circular cylinder. The tread is reinforced with inextensible cords. The tread is the part of the tire that makes actual contact with the ground plane. In the undeformed state, the sidewalls are represented by parts of two tori and consist of incompressible rubber described by the Mooney –Rivlin model. The previously obtained partial differential equation which describes the tire radial in-plane vibrations about 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 the sixth degree, the rare case of a root of multiplicity three is discovered. 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.