162 BELL SYSTEM TECHNICAL JOURNAL 



In addition to the resonant frequencies predicted by the foregoing analysis, 

 there will be others corresponding to shear vibrations in which the principal 

 shear stress varies periodically along the length and width of the plate. 

 A formula which yields the approximate frequencies for these modes is 

 developed in Section 7.9. It is shown that if the length and width are 

 large in comparison to the thickness, the following expression may be used: 



co = 27r/=:ry-y,,,^+ -^ + ^ (7.1/) 



In this formula which has been derived for xy shears the c constants are 

 the standard elastic constants for aeolotropic media. For isotropic plates 

 such as have been considered up to this point 



Cl\ = TT-o - A + 2/i 



1 — 2a^ — (J 

 and 



^55 = c^6 = A, the shear modulus (7.18) 



Various combinations of the integers m, n, and p may be chosen, with the 

 restriction that neither m nor n can equal zero. It is seen that if € and w 

 are very large the formula reduces to that of Eq. 7.16 which was derived on 

 precisely that basis. Also, it is seen that the more complex modes all lie 

 somewhat above the fundamental shear obtained by setting m = n = I 

 and p = 0. 



Plate shear modes are also of considerable interest, particularly the one 

 of lowest order. For a plate having a large ratio of length to width a formula 

 similar to that given by equation 7.17 (but for two dimensions only) may 

 be developed. If the plate is nearly square, however, this formula no longer 

 yields sufficiently accurate values for the resonant frequencies. Coupling 

 to other modes of motion* complicates the problem so much that only 

 experimental results have been of much practical consequence. Fig. 7.11 

 shows in an exaggerated way the distortion of a nearly square plate vibrating 

 in the first shear mode. 



7.5. Flexural Vibrations 

 7.51. Plale Flexures 



One of the most studied types of vibrations has been the flexural. Perhaps 

 this is true because it is the most apparent and comes within the realm of 

 ex-perience of nearly everyone. The phenomena of vibrating reeds, xylo- 

 phone bars, door bell chimes, tuning forks, etc. are quite well known. 



* It is found ejcperimentally that odd order shears are strongly coupled to even order 

 flejcures; similarly, even order shears and odd order flexures are coupled. 



