MODES OF VIBRATION 159 



As shown in Section 7.9, two solutions of this type will satisfy the equi- 

 librium equations precisely. One corresponds to e = in the wave equation 

 7.3, while for the other e ^ 0. Superposition of these two solutions and 

 proper evaluation of parameters make it possible to satisfy the boundary 



conditions at the edge of the strip; namely, that at y = ± -, F„ = and 



Xy = 0. (Refer to equations 7.5). The following transcendental equation 

 is obtained 



(7.13) 



in which 



(7.14) 



This equation may be solved graphically to yield values of frequency 

 corresponding to given values of k. For our discussion of the length ex- 

 tensional mode of vibration, the first root only will be considered. 



Fig. 7.8 shows a plot oi d-b against b-k assuming that Poisson's ratio is 



.33.^ If ^ = 1, and b = I, for example, 6 = a/ -00 = 1.62. 



The equations for the displacements when determined as explained in 

 Section 7.9 become: 



u = Vi [cosh -344 y -f- .402 cos 1.278 y] cos x 



(7.15) 

 V = Ui [.344 sinh .344 y + .315 sin 1.278 y] sin x 



All three stresses Xx, Yy, and Xy may be calculated from the above 

 equations. If the length of our plate is made equal to niT, where m is an 

 integer, the extensional stress Xx will equal zero regardless of y at the 

 boundaries x = and x = t since X^a sin x = when x = mr. Also it 

 can be shown by calculation that Xy is so small in comparison to the exten- 

 sional stresses as to be entirely negligible; hence our solution is complete. 



If ^ = TT, the plate will be vibrating in its fundamental longitudinal mode. 

 The distortion which results is shown by Fig. 7.9. It is seen that most of 



* Plotted in this way, the same curve results regardless of the value of b chosen for the 

 purpose of solving Eq. 7.13. 



