164 BELL SYSTEM TECHNICAL JOURNAL 



The lowest order solution to this equation is found to correspond to 

 flexure vibrations in the infinite strip. A calculation of stresses, however, 

 reveals that boundary conditions cannot be satisfied properly even for the 

 case of a long narrow plate. It can be shown, however, that another 

 solution may be derived for the same value of frequency by letting k become 

 imaginary. This simply means that the u and v displacements become 

 hyperbolic functions of x instead of sinusoidal. The two complete solutions 

 for the infinite strip may then be superimposed and parameters adjusted so 

 that for definite values of length corresponding to fundamental and harmonic 

 modes the proper stresses reduce essentially to zero on the ends of the plate. 

 For plates having a ratio of width to length less than .5, this method gives 

 very accurate expressions for displacements and stresses. If only the 

 resonant frequency is required, ratios up to unity and beyond (for the 

 fundamental mode) may be considered. 



An example has been worked out to provide a complete picture of the 

 displacements for a bar of width = \,k = 1 and c = .2)2>. Use of equation 



7.20 yields the quantity 6- = — - = .166 from which the resonant frequency 



may be obtained. Using this value of 6'^, one finds that k"^ = —.800 also 

 satisfies equation 7.20. By making the total length of the bar equal to 

 4.50 the Xx extensional stress and the Xy shear stress may be made essen- 

 tially zero on the ends of the plate regardless of y. 

 The following expressions for u and v are obtained: 



u = (sinh .9132 y — 1.02 sinh .9718y) sin x 



-.160 (sin .98283' - .9568 sin .9250y) sinh .8944:*; 



V = (-1.094 cosh .9132y + .9915 cosh .9718) cos x 



-.160 (.9095 cos .9828y - .990 cos .9250^) cosh .8944x 



(7.21) 



Fig. 7.12 shows the distortion of the plate as calculated from the above 

 expressions. It is seen that there will be two points at which there is no 

 motion in either the x or y directions. These nodal points can be used in 

 holding the plate, since it may be clamped firmly there without altering 

 the displacements or resonant frequency. For the example shown, these 

 nodes are positioned a distance of .211^ from the ends of the plate as com- 

 pared to .224f for a long thin bar. 



' A graphical solution to determine t is most convenient in which parameters are 



/ h f 



adjusted so that X^ = at x = ±- and y = ±^; Xy = at a; = i^ and y = 0. These 



IV . 



stresses will remain essentially zero for all values of y if the ratio of j is not too great. 



