A matrix model of the axle vibration of a piezoelectric motor

In this work, a matrix model of the axle vibration of a piezoelectric motor is proposed. The stator of this motor is composed of a thin piezoelectric membrane and a steel axle fitted at the center of the membrane. The rotor consists of a cylinder-shaped permanent magnet, pressed in contact with the other end of the axle by means of the magnetic forces. A travelling wave is excited in the membrane by using four electrodes and four, properly delayed, driving signals. The rotating flexural displacement of the membrane produces a wide precessional motion of the axle. In this way, a continuous slipping takes place between the axle and the rotor, and therefore, a torque is transmitted to the rotor. In this paper, the precessional motion of the axle is modeled as the composition of two transverse vibrations belonging to two perpendicular planes passing through the axle. The axle, vibrating in its transverse mode, is modeled as a two-port system: the input is the bending moment supplied by the membrane, and the output is the transverse force at the terminal end of the axle. With this model, we have computed the trasmission transfer function as a function of frequency, and the transversal displacement along the axle at its resonance frequency. The computed results are in reasonable agreement with experimental interferometric measurements carried out on a prototype.