Hybrid Modelling of a Traveling Wave Piezoelectric Motor

Abstract

This thesis considers the modeling of the traveling wave piezoelectric motor (PEM). The rotary traveling wave ultrasonic motor "Shinsei type USR60" is the case study considered in this work. The traveling wave PEM has excellent performance and many useful features such as high holding torque, high torque at low speed, quiet operation (ultrasonic), simple structure, compactness in size and no electromagnetic interferences. However, the mathematical model of the PEM is complex and difficult to derive due to its driving principle based on high-frequency mechanical vibrations and frictional force. Despite many attempts a lumped motor model of the PEM is unavailable so far. The dynamical characteristics of the PEM are complicated, highly nonlinear, and the motor parameters are time varying due to temperature rise and changes in motor drive operating conditions. Therefore it is difficult to predict the performance characteristics of the PEM under various working conditions. The main objective of this PhD project is to derive a suitable model for investigating some nonlinear control strategies in a simulated environment. Most of the existing modeling approaches are inappropriate for the control community due to their many drawbacks. The established analytical methods have high computational demands, whereas the equivalent circuit method lacks the ability to capture some of the important dynamics such as the frictional behavior of the contact mechanism driving this type of motors. This thesis reports on the attempt to solve the highly demanding problem of performance prediction of the PEM. The emphasis is on the combination of the electrical network method, the physics underlying piezoelectric phenomena, the variational work and elasticity theory (Hamilton's principle), besides contact mechanics (friction) and finally the basic laws of dynamics. In order to overcome some of the drawbacks of the existing methods, and thereby meet the needs of the control community, three main approaches are considered in this modeling task. First, the equivalent circuit method is investigated in order to derive a lumped model of an ultrasonic traveling wave rotary piezoelectric motor. This approach is carried out on the basis of the experimental investigation combined with the electrical network method. Consequently, an insight in the analysis of the electromechanical coupling force factor, which is responsible for the electrical to mechanical energy conversion, is obtained. Thereby, the difference between the effective force factor and the modal force factor is highlighted, and how these parameters should be integrated in the equivalent circuit model is emphasized. Furthermore, the effect of temperature on the mechanical resonance frequency is considered and thereby integrated in the final equivalent circuit model for long term operations. Second, the laws of physics based on the energy balance method are used for the purpose of predicting, a priori, the performance of the motor as a function of the design parameters and thereby a theoretical model is derived. Since the dynamic characteristics of the real motor are difficult to capture in an analytical model, and the parameters of the motor are time varying and highly nonlinear, then some assumptions are required in order to simplify the modeling task and thus provide a suitable model for control purposes. Consequently, a general state space model is derived on the basis of the special design of the motor of interest, which is a two phase symmetrical system. Furthermore, a simplified model is derived within the framework of various assumptions on the behavior of the stator, which makes it possible to decouple the assumed excited modes and thereby predict the performance of the stator by a single second order system. The frictional behavior at the interface contact between the stator and the rotor constitutes the main problem in the modeling approach and therefore its modeling is treated with care in order to avoid large discrepancies with the physical plant. Consequently, the stick-slip behavior of the driving mechanism is integrated in the time varying parameters of the model. This makes it possible to predict most of the behavior and performance characteristics of the motor as a function of the external loading parameters. The outcome of this modeling approach is a less computationally demanding model suitable for control investigation purposes. Finally, a hybrid model which combines the strength of the equivalent circuit method and the analytical method is derived. The analogy between the equivalent circuit model and the analytically reduced model of the unforced stator, which models the behavior of one phase of the stator, is highlighted. This makes it possible to substitute the parameters of the equivalent circuit model in the framework of the simplified analytical model. Consequently, the large uncertainties on the values of the dielectrical and electromechanical constants are avoided. The simplified hybrid model of the complete motor is thereby derived in terms of the forced stator model, the spinning motion model and the vertical motion model. Finally the effect of temperature on the mechanical resonance frequency is considered and thereby integrated in the final hybrid model for long term operations. The validity of the model has been established by simulations and comparison to the performance characteristics of the real system. The results achieved by the different approaches are compared and a final conclusion is drawn. The main achievement of this work is the combination of the strength of the equivalent circuit method and the analytical method in a hybrid simplified model providing the ability to predict most of the performance characteristics of the motor under various working conditions