Theoretical Analysis of Modes of Vibration for Isotropic Rectangular Plates Having All Surfaces Free

The comparatively recent advent of crystal controlled oscillators and of wave filters employing piezoelectric elements has resulted in an extensive study of the ways in which plates made of elastic materials such as quartz or rochelle salt can vibrate. Of special interest have been the resonant frequencies associated with these modes of motion. As will be indicated in subsequent paragraphs, the general solution to the problem of greatest interest is quite complex, and has not been forthcoming, (i.e., as applied to rectangular plates completely unrestrained at all boundary surfaces). For this reason numerous approximate solutions have been developed which yield useful information in spite of their limitations. Several of these solutions will be discussed in the following sections. The three general types of modes (i.e., the extensional, shear, and flexural) will be analyzed in some detail. Also, as a preliminary step the formulation of the general problem along classical lines will be developed. For the most part, the solutions obtained here are limited to those for an isotropic body. However, such solutions provide considerable guidance for the modes of motion existing in an aeolotropic body such as quartz. 7.2. METHOD OF ANALYSIS