Torsional vibrations of quartz crystal beams

An exact solution of a partial differential equation including elastic compliance constant s'/sub 56/, with respect to stress function /spl psi/ has been found for torsional modes of vibration of an arbitrary (singly, doubly, triply) rotated beam with a pair of parallel free edges. The solution is obtained by relaxing the condition that the edge planes are perpendicular to the main faces of the beam. That is, the edges are off perpendicular by the angle /spl Theta/=arctan(-s'/sub 56//8'/sub 55/). The exact solution can reduce the difference of the calculated and measured values for a thickness-to-width ratio which gives the first-order temperature coefficient /spl alpha/=0. Also, a comparatively large inclination of the edge cuts is required to reduce the unwanted complicated mode shapes to simple ones.