0885–3010/$25.00 © 2010 IEEE 1673 IEEE TRANSACTIONS ON ULTRASONICS, FERROELECTRICS, AND FREQUENCY CONTROL, . 57, . 7, JULY 2010 Abstract—A partial differential equation (PDE) model for the dynamics of a thin piezoelectric plate in an electric field is presented. This PDE model is discretized via the finite volume method (FVM), resulting in a system of coupled ordinary dif- ferential equations. A static analysis and an eigenfrequency analysis are done with results compared with those provided by a commercial finite element (FEM) package. We find that fewer degrees of freedom are needed with the FVM model to reach a specified degree of accuracy. This suggests that the FVM model, which also has the advantage of an intuitive inter- pretation in terms of electrical circuits, may be a better choice in control situations. I. I M  the size of electrical servo-drives used in small-scale actuation has become an important issue. Micro-satellites, micro-robot actuation, or servo-drives used in the automotive industry can exploit the properties of linear or rotary ultrasonic motors. These motors are actuators which are composed of piezoelectric materials. The piezoelectric materials can also be employed as sen- sors. When used as actuators, piezoelectric materials de- liver force and/or displacement proportional to an applied voltage, and when used as sensors, they deliver a voltage proportional to the applied force. To use these devices as an actuator or as a sensor, one has to know how to model and control them properly. The dynamics of a piezoelectric plate in an electric field can be modeled with a pair of coupled partial differential equations (PDE): a hyperbolic PDE for the displacements and Gauss’ law for the electric displacement field. The finite element method (FEM) has been a popular choice for numerical studies of piezoelectric structures; some of the earlier works are [1]–[7]. One approach is to apply Hamilton’s variational principle to the Lagrangian for the fully 3-D problem, which results in a variational structure within which one can use the FEM [1]. For piezoelectric materials that are thin in one direction, one can make ap- proximations via a Taylor expansion in the thin direction, resulting in 2-D problems which can be studied with the FEM [3]. Also, for thin domains one can simply assume uniformity of the electric field and strain in the thin direc- tion, resulting in a 2-D problem which fits into the varia- tional formulation and can then be studied with the FEM [7]. The finite volume method (FVM), popular in various other areas of engineering, does not appear to have been applied to piezoelectric structures before. In this paper, the finite volume method is used to dis- cretize the PDE model, producing a system of ordinary differential equations (ODEs) whose solution approxi- mates the solution of the PDE model. One can choose the discretization to meet a specified accuracy. Some reasons to consider the FVM discretization are: The FVM ODEs can be interpreted intuitively in • terms of of equivalent electrical circuits of the piezo- electric system [8]. These circuits can then be imple- mented using schematic capture packages. This makes it easier to interface the FVM model of the piezoelec- tric system with the control circuits. In discretizing the PDE, one may have to make choic- • es in how to apply the boundary conditions. As will be shown in Section V, the choices made in applying the boundary conditions can have significant impact on the rate of convergence of the discretized model to the true solution. The FVM approach allows one to imple- ment these choices in a straightforward manner. The FVM works easily with surface integrals, making • it easier to deal with phenomena that occur at the boundary between two different materials. Therefore, this method may be more suitable to model an ul- trasonic motor because the operating principle of the motor is based on the friction mechanism that takes place at the common contact boundary between the stator and the rotor. In this article, the FVM is applied to a thin piezoelectric plate, allowing the electric field to be assumed constant in space. Approximating the electric field in this way is common for thin piezoelectric materials and has the ad- vantage of not only capturing the physics well but of also providing a simpler case study for numerical methods. Specifically, the system of coupled PDEs is reduced to a single PDE, resulting in a smaller system of ODEs after the discretization. However, the FVM could have equally well been applied to the full problem without this simpli- fying assumption. II. T PDE M The constitutive equations for a linear piezoelectric ma- terial are Modeling of Piezoelectric Devices With the Finite Volume Method Valentin Bolborici, Member, IEEE, Francis P. Dawson, Fellow, IEEE, and Mary C. Pugh Manuscript received July 23, 2009; accepted March 20, 2010. V. Bolborici and F. P. Dawson are with the Department of Electrical and Computer Engineering, University of Toronto, Toronto, ON, Canada (e-mail: valentin.bolborici@utoronto.ca). M. C. Pugh is with the Department of Mathematics, University of Toronto, Toronto, ON, Canada. Digital Object Identifier 10.1109/TUFFC.2010.1598 Authorized licensed use limited to: The University of Toronto. Downloaded on July 19,2010 at 16:36:36 UTC from IEEE Xplore. Restrictions apply.