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BELL SYSTEM TECHNICAL JOURNAL 



the 14th flexure which breaks up once along the z dimension and the shear 

 which breaks up once along z — etc. A few of these flexures which break up 

 along s are shown for the 16th ordinal flexure. 



2200 



2100 



2000 



1900 



2 1600 



If) 



uj 1700 



y 1600 



1500 



1400 



1300 



1200 



10 



13 14 



LENGTH 

 THICKNESS 



16 



Fig. 7.16 — -XY thickness flexure modes for square plate 



7.6. Summary 



Three main classes or families of vibrational modes are found to exist in 

 rectangular elastic plates free on all surfaces; namely, the extensional, the 

 shear, and the flexural. In general, the associated displacements are 

 functions of all three dimensions and may vary in such a manner as to make 

 the distortion of such plates quite complex. 



For certain limiting cases, approximate solutions for the resonant fre- 

 quencies and displacements (from which strains and stresses may be cal- 

 culated) can be derived. Though there are a number of methods that can 

 be used for specific problems, it has been found very convenient to utilize 

 the classical formulation. For this reason the basis of this method has been 

 discussed briefly. In essence it requires that displacements and stresses 

 occurring within the elastic solid satisfy conditions of equilibrium as de- 

 rived from Newton's Law. At the boundaries, certain other relations must 

 be satisfied in order that conditions of clamping might be fulfilled. For 

 plates entirely unrestrained the latter requires that all forces (tractions) 

 acting through the free surfaces must vanish. 



For thin rectangular plates (such as quartz crystal oscillator plates) the 

 modes of greatest practical consequence are plate modes, for which all 



