62 BELL SYSTEM TECHNICAL JOURNAL 



length, Mason has shown that it is necessary to consider the effects of rotary 

 and lateral inertia. His solution leads to the same frequency equation as 

 6.1 but with a different evaluation of the factor m which is obtained from the 

 transcendental equations 



tan m X = K tanh mX' for even modes 



tan m X = —-r^ tanh mX' for odd modes 

 A 



6.2 



where 





X 



Equation 6.2 holds only for the case of a rod free to vibrate on both ends. 

 The case of a clamp-free rod is somewhat more complicated since it cannot be 

 given by separate solutions for the even and odd modes. The interpretation 

 of m given in equations 6.2 will result in the same value as before [m = 



(n + h)ir] for values of — less than .05 but decrease considerabh' for larger 



values and ultimately as the bar becomes wider the effects of rotary inertia 

 result in the flexure frequency approaching the extensional mode as an 

 asymptote. As stated before measurements on quartz bars vibrating in 

 flexure departed from that predicted by the simple definition of m when the 



width of the bar was such that — > .1. By using the value of m defined by 



nw 

 equation 6.2 it is possible to predict the frequency for widths as great as — = 



.5. For widths greater than this, experiment shows a frequency lower than 

 that predicted by equation 6.2. This then leads one to believe that the effect 

 of shear plays an important part in the flexure of bars with appreciable width. 

 An investigation of the effect of shear on the flexure frequencies of beams 

 has been made by Jacobsen^ and his results lead to the same frequency 

 equation as 6.1 and to the same transcendental equations derived by Mason 

 (6.2) but with different values of A', X' and A' to account for the shearing 



- \V. P. ^lason, "Electromechanical Transducers and Wave Filters," Appendix A. 

 D. Van Nostrand Company, Inc. 



^Jour. Applied Mechanics, March 1938. 



