MODES OF VIBRATION 



167 



For reasonably high order flexures it may be reasoned that the true dis- 

 ])lacements will be very nearly the same as those for the doubly infinite 

 I)late as derived by the above method since the correction necessary to 

 fulfill the boundary conditions will only apply very close to the edges of the 

 plate. It will then be sufficient to choose values for k and m such that 



k = p ^ and m — — where p and q are integers. 

 I w 



The values of a~ obtained 

 in this way determine the corresponding resonant frequencies. 



5 

 4 



3 



ab=bVkV^ 

 Fig. 7.15 — d-b versus «•& for thickness flexures 



If it is desired to solve for the ordinal xy flexures, for example, m should 

 be set equal to zero. The displacements in this case will be independent 

 of the z dimension. When q is assigned values other than zero however, 

 the resulting modes may be considered as xy flexures which vary or break 

 up along the third major dimension. If q is small the resonant frequencies 

 will lie only slightly higher than that of the corresponding ordinal flexure 

 for which q — 0. 



Fig. 7.16 shows a few of the resonant frequencies as calculated for values 

 of shear modulus and density corresponding to AT quartz. The effects of 

 couphng to the fundamental thickness shear are shown by dotted lines for 

 the 14th xy flexure. As might be expected there is similar coupling between 



