MODES OF VIBRATION 161 



there will be many extensional modes which have resonant frequencies 

 somewhat above those given by Eq. 7.11. Analysis of the motion shows 

 that for these modes the displacement along the thickness varies periodically 

 (or "breaks up") along the major dimensions of the plate. There again the 

 distortion pattern of the plate may become very complex. 



7.4. Shear Vibrations 



The second class of vibrations which will now be considered is the shear. 

 This type of mode is of special importance because of the fact that piezo- 

 electric plates vibrating in shear are widely used for frequency control of 

 oscillators. For example, the AT quartz plate which is so much in demand 

 utilizes a fundamental thickness shear mode in which particle motion is 

 principally at right angles to the thickness. The distortion of the plate will 

 be similar to that shown in Fig. 7.2. 



A simple, yet very useful formula for the resonant frequencies associated 

 with the above type of displacement has been derived on the assumption 



Fig. 7.10 — Orientation of thin plate 



that the length and width of the plate are very large in comparison to the 

 thickness. For the xy shear mode, the displacement u is assumed to be 

 II = U cos ky, all other displacements being equal to zero. The only stress 

 that need be considered then, is the Xy shear which is proportional to sin ky. 

 Boundary conditions on this stress at the major surfaces of the plate are 

 easily satisfied by choosing k such that Xj, = at y = and y = t. (Refer 



to Fig. 7.10.) This will be the case if ^ = • — , where m is any integer, and 



t 



t is the thickness of the plate. By using the simplified equilibrium equation 



as reduced from equations 7.1, a formula for the resonant frequencies is 



obtained in much the same manner as for extensional thickness modes. 



0, = 27r/ = — 4/? w = 1, 2, 3, etc. (7.16) 



t y p 



In this formula the shear modulus A appears instead of Young's modulus 

 as in the case of longitudinal modes. Harmonic modes are given by values 

 of m greater than unity. 



