MODES OP VIBRATION 



155 



X = XJ 4- Xyin + X,n 

 Y = Yyin + Y,n + XJ 

 Z = ZiU + X^C + FiW^ 



* = for free surfaces 



(7.5) 



{(, m, and w are direction cosines of the normal to the surface at the point in 

 question). 



The general problem is now seen to be one of finding solutions for the 

 displacements «, v, and w such that both the equilibrium and boundary- 

 conditions are satisfied. In the following section several interesting solu- 

 tions will be considered for rectangular plates having all surfaces free, this 

 being the case of greatest interest in so far as this paper will be concerned. 



7.3. EXTENSIOXAL ViBEATIOKS 



One of the most useful modes of vibration of practical interest is the 

 extensional, in which particle motion takes place in essentially one direction 

 so as to alternately stretch and compress the elastic medium. Piezoelectric 



X = 1 



Fig. 7.4 — Longitudinal bar 



plates vibrating in this manner, and of the shapes shown in figures 7.4 and 

 7.5 have been used extensively in wave filter and oscillator circuits. The 

 approximate resonant frequencies corresponding to this type of motion are 

 easily obtained by a consideration of equations 7.1 and 7.2. For the 

 longitudinal bar of Fig. 7.4 the only stress that need be considered is the Xx 

 extensional, all other stresses being so small that they can be neglected. 

 The equilibrium equation then becomes 



or, smce 



(7.6) 



(7.7) 



