Abstract
The frequencies and mode shapes of rolling rings with radial and circumferential displacement constraints are investigated. The displacement constraints practically come from the point contact, e.g., rolling tire on the road, or other applications. The proposed approach to analysis is calculating the natural frequencies and modes of a non-contacted spinning ring, then employing the receptance method for displacement constraints. The frequency equation for the constrained system is hence obtained, and it can be solved numerically or graphically. The receptance matrix developed for the spinning ring is surprisingly found not symmetric as usual. Moreover, the cross receptances are discovered to form complex conjugate pairs. That is a feature that has never been described in literature. The results show that the natural frequencies for the spinning ring in contact, as expected, higher than those for the non-contacted ring. The variance of frequencies to rotational speeds are then illustrated. The analytic forms of mode shapes are also derived and sketched. The traveling modes are then shown for cases.
