MODES OF VIBRATION 175 



W 

 Solving for — , using (7.41) and first of equations (7.38) 



The first ratio is the same as (7.39). For the second solution to the equilib- 

 rium equations, then the following relationships exist. 



V -e , w -k - 1 .^ ,,, 



When the above are substituted back into (7.38), it is found that 



A{k'~-{- ^2 + ^2) = p^2 (7,44) 



W fit 

 In a similar way, using 77 = -r the following are obtained: 

 U k 



W m V k -\- m 



(7.45) 



U k' U kC 



with A{k- -\- (" -\- m^) = pitT as before. This is the second solution for 

 e = 0. 



The three different solutions may now be combined or superimposed to 

 give 



11 = [Ui sin lij + U2 sin (-ly + Uz sin 4y] sin kx cos mz 



= — Z7i J cos Ay — U2J cos Ay 



[' 



+ — ^^"1 cos Ay cos kx cos mz .^ ... 



hk -^J (7.46) 



w = \ Ui — sm Ay — U2 7 — sm Ay 



m 



1 



-\- Us 1 sin Ay cos kx sin mz 



In the above equations A = A because of the double requirement 

 of 7.44.'" 



It is now possible to calculate the stresses existing at any point. It is 

 desired to choose Ui , U2 , and Us in such a manner that the boundary 

 conditions at the two major surfaces of the plate are satisfied. By using 

 the relations given in Section 7.8, the extensional stress Yy , and the two 

 shear stresses -V^ and Yz are calculated with the use of 7.46. They are then 



^ It should also be noticed that e = for this solution. 

 10 ki = ka = h = k 

 mi = 7«2 = mi = m 



