MODES OF VIBRATION 177 



By dividing row 1 by cos A ;; , row 2 by cos A 7: ; row 3 by cos (3 - and 



by subtracting the elements of row 3 from those of row 2, considerable 

 simplification results. 



The full expansion gives the following equation, after further simplifying 

 operations are performed. 



\iaB{k' + /i + m) + A(l) tan A^Tifa - k' - 



m I 



+ 2fi/'3 A{k' ■\-m') tan/3^ = 



It should be noticed that (-:. has dropped out entirely. Actually 4 = 6 

 as previously explained. Also the expression 



[' 



■(■> / ,2 , ,?v 



7« J 



must not be zero, for in simplifying equation (7.51) it was used as a divisor. 

 Equation (7.51) may be rewritten to give 



h 

 '^^ ^2 ^ -2U.A{k' + m') 



^ . b [(7B{]^ + 't\ + m') + ACl\[Cl - k' -m^] 

 tan (3- 



In the above 



U + B) {k^- + /"I + m-) = fx,-] 



2 (7.53) 



2 



By letting 6- = —— and k- + m- = a- equations (7.52) and (7.53) above 

 become 



''"^^2* -2^.^3^a^ 



^_ .6 [<jB{(1 + a') + AllHd -a'] 

 tan ^3 ;^ 



(7.22) 



with l\ = 0^- 



A + B 



(I = (I = O" - a 



Equation (7.22) represents the general solution for normal thickness 

 vibrations in an isotropic plate of finite thickness extending to infinity in 

 both major directions. The analogy for plates of finite dimensions is 

 considered in the text. 



