MODES OF VIBRATION 163 



Beam theory has been used quite extensively to derive the equations 

 which yield the resonant frequencies and displacements for bars vibrating 

 in flexure. To obtain reasonably accurate results for ratios of width to 

 length approaching unity, however, the effects of lateral contraction, rotary 

 inertia, and shearing forces must be considered. This leads to a rather 

 complicated solution which is much more accurate than that derived by the 

 use of simple beam theory only, though it is still approximate in nature. 



For two dimensional plates free on all edges a method of analysis may be 

 used which is similar to that described under extensional modes. While 

 it is somewhat involved it yields direct expressions for the two displacements 

 u and V, so that all stresses may be calculated, and the extent to which 

 boundary conditions are satisfied determined. 



Fig. 7.11 — Distortion of plate in first shear mode 



Solutions for u and v are assumed to be of the form 



u = U sin Cv ccs kx 



(7.19) 

 V = V cos {y sin kx 



For the infinite strip previously considered a transcendental equation is 

 obtained which is the same as equation 7.13 with the exception that the 

 left-hand expression is inverted. 



(Refer to Eq. 7.14 also.) 



^ This is an extension of DoerfTler's analysis used to obtain harmonic flexure frequenceis 

 for plates — "Bent and Transverse Oscillations of Piezo-Electrically Excited Quartz 

 Plates"— Zeitschrift Fiir Physik, v. 63, July 7, 1930, p. 30. Also refer to "The Distribu- 

 tion of Stress and Strain for Rectangular Isotropic Plates Vibrating in Normal Modes of 

 Flexures" — New York Univ. Thesis by Author, June 1940. 



