Abstract
The problem of vibrations of rotating rings has been of interest for its wide applications in engineering, such as the vibratory ring gyroscopes. For the vibratory ring gyroscopes, the vibration of a micro ring is usually actuated and sensed by means of electrostatics. The analytical models of electrostatic microstructures are complicated due to their non-linear electromechanical coupling behavior. Therefore, this paper presents for the first time the free vibration of a rotating ring under uniform electrical field and the results will be helpful for extending our knowledge on the problem of vibrations of rotating rings, helping the design of vibratory ring gyroscopes, and inspiring the feasibilities of other engineering applications. An analytical model, based on thin-ring theory, is derived by means of energy method for a rotating ring under uniformly distributed electrical field. After that, the closed form solutions of the natural frequencies and modes are obtained by means of modal expansion method. Some valuable conclusions are made according to the results of the present analytical model. The electrical field causes not only an electrostatic force but also an equivalently negative electrical-stiffness. The equivalent negative electrical-stiffness will reduce either the natural frequencies or critical speeds of the rotating ring. It is known that the ring will buckle when its rotational speed equals its natural frequencies. The introduction of electrical field will further reduce the buckling speeds to a value less than the natural frequencies. The rotation effect will induce the so-called traveling modes, each one travels either in the same direction as the rotating ring or in the opposite direction with respect to stationary coordinate system. The electrical field will reduce the traveling velocities of the traveling modes.