MODES OF VIBRATION 157 



boundary condition on the Xx stress will be met if ^ = — so that the 



following solutions result. 



^ /p(l + <t)(1 - 2<t) ^ /' 



"V £(1-.) ^ ^ = ^o^-f , 



u = cos 0} A/ — — l...' ' V ' X = cos 0} A/ ■_ ■ a. 



. , WT /£ (1 - a) MT ^ A + 2n (7.11) 



m = 1, 2, 3, etc. 

 It is seen that this formula for resonant frequencies is the same as given by 

 equations 7.8, with E replaced by -q ^— r ^ , so that the frequency 



constant/-/ will be somewhat higher than f-^ for a long slender bar. 



It is recognized that the solutions derived above hold true only for the 

 limiting cases of a long slender bar, and a very thin plate respectively. It is 

 therefore of interest to trace the resonant frequencies corresponding to these 

 extensional modes of vibration as departure is made from the limiting cases 

 mentioned above. 



An experimental plot of the resonant frequencies of a thin plate of length / 

 and width w reveals that the frequency of the longitudinal mode first 

 discussed is gradually lowered as the width of the plate is increased. There 

 is also another frequency corresponding to an extensional vibration along 

 the width which for a very narrow plate corresponds to the second type of 

 extensional mode considered in the foregoing paragraphs, except that the 

 frequency constant will be slightly different because coplanar stresses are 

 involved. 



As seen from Fig. 7.6, the resonant frequency curves do not cross,' but 

 exhibit coupling effects. This is understandable from the fact that motion 

 in one direction is mechanically coupled to motion in the other as indicated 

 by Poisson's ratio a. 



In order to derive expressions for the u and v displacements associated 

 with the extensional mode along the length, taking into account the above 

 coupling effect, the following analysis proves interesting. 



Consider the infinite isotropic strip of width b as shown by figure 7.7. 

 As will be demonstrated presently, solutions can be found such that the 

 equilibrium equations and the boundary conditions are precisely satisfied. 

 Furthermore it will be found possible to cut a section out of this strip in 



' If the length and width of the plate are very large in comparison to the thickness, 

 the boundary conditions for the F„ and Z^ stresses may be neglected without causing 

 appreciable error. The quantity A + B has been evaluated in terms of E and a for 

 purposes of comparison. 



