A general model of the axle vibration in piezoelectric motors

In the present work a general model of the vibrational behavior of the axle of a piezoelectric motor is proposed. In this motor, a cylinder-shaped permanent magnet, which act as a rotor, is pressed in contact with an end of a steel axle by means of the magnetic forces. The other end of the axle is fitted at the center of a rotating traveling wave generator. A piezoelectric membrane, vibrating in a flexural anti-symmetrical mode, or a thick disk, vibrating in a radial anti-symmetrical mode, can be exploited as traveling wave generators. In the first case a bending moment, in the second case a transverse force is applied to the axle. In both cases, if the driving frequency coincides with a resonance frequency of the axle, the axle acts as a resonant displacement amplifier; a continuous slipping takes place between the axle and the rotor, and a torque is transmitted to the rotor. The proposed model is able to describe the axle vibrational behavior when it is excited by a bending moment, by a transverse force, and also when these two excitations are simultaneously applied. The axle is modeled as a four-port system and all its transfer functions, as well as the transversal displacement along the axle at each frequency can be easily computed. Computed results have been compared with experimental measurements carried out on two motor prototypes that exploit as traveling wave generators a membrane and a disk, respectively. A good agreement was obtained by properly taking into account the loading effect of the generator on the axle.