MODES OF VIBRATION 169 



stresses are essentially coplanar and independent of the thickness, and 

 thickness modes, for which all dimensions must be considered except in 

 limiting cases. 



Because of their great utility, simplified formulae have been derived for 

 the resonant frequencies associated with long, narrow bars vibrating longi- 

 tudinally, thin plates with extensional motion along the thickness dimension, 

 and thin plates vibrating with shearing motion at right angles to the 

 thickness. 



Exact solutions for the infinite strip have been derived, and used in 



obtaining the displacements and resonant frequencies for flexural and 



longitudinal modes. Such solutions take account of the fact that the width 



of the plate may become appreciable. WTiile limiting cases of plate shear 



w 

 may be analyzed, solutions for ratios of — approaching unity have not 



proved very satisfactory. This is attributable to the fact that coupling to 

 flexural modes is severe. 



Thickness flexural modes which exhibit displacement variations along 

 both length and width dimensions of the plate have been analyzed by 

 extending the "infinite strip" theory to three dimensions. Solutions 

 obtained are fairly accurate if the harmonic order of the flexure is sufficiently 

 great. 



7.7. Nomenclature 



p = density 



E = Young's modulus 



cr = Poisson's ratio 



A = Shear modulus = 



E 



= M 



2(1 + cr) 



B = -7- — ; — r-7- --7 = X + M for 3 dimensions 



2(1 + (r)(l — Iff) 



for plane stress 



2(1 - ff) 

 o) = angular velocity = 27r/ 



6^ = 4 



A 



u, V, w = displacements in r, y and z directions 



_ du . dv , dw 

 dx dy dz 



