MODES OF VIBRATION 173 



by the crossover points. Values of 6^ corresponding to different values of 

 k^ may also be found in this way and a curve plotted for 6" versus k-, (or for 

 9-b versus k-b). 



7.92. Thickness Shear Vibrations 



To obtain a formula for the approximate resonant frequencies of thickness 

 lear for a plate having 

 following displacements: 



W L 



shear for a plate having large ratios of — and — , one may consider the 



u = U sin kx cos (y cos rz 



V = (7.34) 



w = 



If there are no cross couplings between shear stresses or between shear 

 and extensional stresses one may write :^ 



Xx = Cn — + Ci2 — + Ci3 — = Cii kU cos kx cos ly cos rz 

 dx oy oz 



Xy — C66 ( — + — ) = —ct&tU sin kx sin ly cos rz (7.35) 



\dy dx/ 



Xz = c^ii — + — ) = —Chh rU sin kx cos ly sin rz 

 \dz dx I 



Substituting into the first of the equiUbrium equations (7.1) and dividing 

 through by common factors 



c\\ k- + Ceo I- + ^65 r"^ = pco^ (7.36) 



The other two of equations (7.1) may be neglected if k and r are quite 

 small so that it will only be necessary to consider equation (7.36) which 

 can be solved for w-. It will be noticed that the .Y„ shear stress will pre- 



dominate under these restrictions on k and r. Letting ^ = — , ^ = — - , 



and r — — {ii, m and p are integers) in order to satisfy the boundary condi- 

 W 



tions for this stress and also for Xz , one obtains the following formula. 



(This choice of k, I, and r is also required if the shear stress is to vary in 



essentially the same manner as is experimentally observed.) 



0^ -lirf = 7r y-y jj + -^ + ^ (7.17) 



in which L and IF must be much larger than T. 



* Refer to equation (7.18) for values of c constants for isotropic case. 



