171 



Xy AXy 



7.9. Mathematical Derivations 

 7.91. Longitudinal Vibrations in Two-Dimensional Plates 



As explained in the text, solutions for the infinite strip of Fig. 7.7 are 

 first derived. Let 



u = U cos kx cos 4) 



ir • J, ■ f ( (7.12) 



V = K sin kx sin lyj 



where U and V are constant. From these expressions e = — + — can 



ox ay 



be obtained and substituted into the equilibrium equations 7.2. Two 

 expressions as follows result after dividing through by the common term 

 cos kx cos /y . 



A(k' -\-f)-t^ (-kU + ^V) = pJ 



A{k' + n +^ i-kU + m = pco' 



(7.25) 



Subtracting the second from the first of these equations, it is seen that 



(f + f ) ^-^^ + m =0 (7.26) 



Either or both of these factors equal to zero will satisfy 7.26, so that two 



V . 



values of - are obtained. By substituting back into equations (7.25), 



conditions on co- are found. The two solutions will be 



^= -^-lwith(A+B)(k' + tl) = pJ 

 Ui k 



(7.27) 



jj- = -J with A{k^ + (I) = p(a 



By superimposing the two solutions the u and v displacements now become 

 II = [Ui cos Cij -\- Uz cos ^ y] cos kx 



k .. . . 1 . , (7.28) 



V = \ — — f/i sin 4 y + — £/2 sin 4 y si 

 \_ k h J 



