Torsional vibrations of quartz crystal beams

摘要

An exact solution of a partial differential equation includingnelastic compliance constant s'56, with respect to stressnfunction Ψ has been found for torsional modes of vibration of annarbitrary (singly, doubly, triply) rotated beam with a pair of parallelnfree edges. The solution is obtained by relaxing the condition that thenedge planes are perpendicular to the main faces of the beam. That is,nthe edges are off perpendicular by the angle Θ=arctan(-s'56n/8'55). The exact solution can reduce the difference ofnthe calculated and measured values for a thickness-to-width ratio whichngives the first-order temperature coefficient Α=0. Also, ancomparatively large inclination of the edge cuts is required to reducenthe unwanted complicated mode shapes to simple ones