CHAPTER VII 



Theoretical Analysis of Modes of Vibration for Isotropic 

 Rectangular Plates Having All Surfaces Free 



By H. J. McSKIMIN 



7.1. Introduction 



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 Hnes 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 



In order to set up the desired mathematical statement of our problem it 

 will be necessary to consider first of all two very fundamental relationships. 

 The first of these is the well known law of Newton which states that a 

 force /acting on a mass m produces an acceleration a in accordance with the 

 formula 



/ = m-a 



The second relationship which we shall need is Hooke's law relating the 

 strains in a body to the stresses. If forces are applied to the ends of a long 

 slender rod made of an elastic material such as steel (Fig. 7.1) a certain 

 amount of stretching takes place. If the forces are not too great, a linear 



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