Finite-Difference Method for the Investigation of the Vibrations of Solids and the Evaluation of the Equivalent-Circuit Characteristics of Piezoelectric Resonators. I

The differential equations of motion and boundary conditions of a piezoelectric resonator are first described in general terms, and then developed in a form suitable for the analysis of the free or forced vibrations of contour modes of a thin rectangular plate. Simplifications made possible by elastic and piezoelectric constants of certain symmetries are discussed. A finite-difference representation of these equations is then developed, particular attention being given to the difficulties of dealing with the boundary conditions at the corners. The paper then deals with methods of handling the resulting set of simultaneous equations so as to determine the resonant frequencies, particle displacements, vibrational nodes, and equivalent-circuit characteristics of various modes of vibration.