56 BELL SYSTEM TECHNICAL JOURNAL 



motion must be dynamically balanced, a node will appear at the center of 

 the bar, and the bar will grow longer and shorter as shown by the solid and 

 dotted lines. In the case of the second order of motion, as shown in Fig. 

 6.3, it consists essentially of two 1st order modes joined together at their 

 ends and of opposite phase. That is to say, when one half of the bar is 

 expanding, the other half is contracting. In the case of the 3rd mode, as 

 can be seen from Fig. 6.3, the central element is contracting while the exter- 

 nal elements are expanding. From this we may state generally, that for 

 odd order types of motion, the extreme ends of the bar will be expanding 

 or contracting in phase and for even order modes, the extreme ends will be 

 expanding or contracting in opposite phase. Fig. 6.3 illustrates extensional 

 motion in its simplest form. In a practical case an extension in one direction 

 is accompanied by a contraction in one or both of the other two dimensions. 

 This of course is due to elastic coupling and will be considered more in detail 

 later. If we consider a rectangular plate it is not difficult to imagine that it 

 would have three series of extensional modes of vibration due to the three 

 principal dimensions. 



6.23 Shear 



The low frequency of face shear type of motion of a plate is somewhat 

 more complicated than either the flexure or longitudinal and, as shown in 

 Fig. 6.4, consists simply of an expansion and compression in opposite phase 

 along the two diagonals of the plate. This motion is shown in Fig. 6.4 

 labeled m = I, n = 1. The two phases are shown, one a solid curve and 

 the other a dotted curve to illustrate the distortion with respect to the 

 original plate. One peculiarity of shear motion in plates is that it may 

 break up into motions similar to its fundamental along either the length or 

 the width. For example, if we take the motion associated with w = 1, 

 11= 1, and superimpose two of these in opposite phase on the same plate, 

 we would get the tynpe of motion illustrated by m = 2,n = I. In a similar 

 manner, the motion may reverse its phase any number of times along either 

 the length or the width. One particular case is shown for m = 6, n = 3. 

 As can be seen from the case of m = 1, n = 1, the distortion is not that of a 

 parallelogram as it is in the static case because here we are concerned only 

 with the dynamic case. While the equation of motion of a free plate vibrat- 

 ing in shear has not been completely solved, a microscopic analysis indicates 

 that the actual motion of the plate edges appear to be somewhat as shown 

 for the case m = 1, « = 1 when driven in this mode. 



The shear mode of motion in the case of a thin plate is somewhat diflferent 

 for the high frequency case than for the low frequency case. In the case of 

 high frequency shear modes of motion in thin plates, the motion of a particle 

 is at right angles to the direction of propagation which in this case would be 



