. The Bell System technical journal . BRATION 161 there will be many extensional modes which have resonant frequenciessomewhat above those given by Eq. Analysis of the motion showsthat for these modes the displacement along the thickness varies periodically(or breaks up) along the major dimensions of the plate. There again thedistortion pattern of the plate may become very complex. Shear Vibrations The second class of vibrations which will now be considered is the type of mode is of special importance because of the fact that piezo-electric plates vibrating in shear are w
. The Bell System technical journal . BRATION 161 there will be many extensional modes which have resonant frequenciessomewhat above those given by Eq. Analysis of the motion showsthat for these modes the displacement along the thickness varies periodically(or breaks up) along the major dimensions of the plate. There again thedistortion pattern of the plate may become very complex. Shear Vibrations The second class of vibrations which will now be considered is the type of mode is of special importance because of the fact that piezo-electric plates vibrating in shear are widely used for frequency control ofoscillators. For example, the AT quartz plate which is so much in demandutilizes a fundamental thickness shear mode in which particle motion isprincipally at right angles to the thickness. The distortion of the plate willbe similar to that shown in Fig. A simple, yet very useful formula for the resonant frequencies associatedwith the above type of displacement has been derived on the assumption. Fig. —Orientation of thin plate that the length and width of the plate are very large in comparison to thethickness. For the xy shear mode, the displacement u is assumed to beII = U cos ky, all other displacements being equal to zero. The only stressthat need be considered then, is the Xy shear which is proportional to sin conditions on this stress at the major surfaces of the plate areeasily satisfied by choosing k such that Xj, = 0 at y = 0 and y = t. (Refer to Fig. ) This will be the case if ^ = •—, where m is any integer, and t t is the thickness of the plate. By using the simplified equilibrium equation as reduced from equations , a formula for the resonant frequencies is obtained in much the same manner as for extensional thickness modes. 0, = 27r/ = — 4/? w = 1, 2, 3, etc. () t y p In this formula the shear modulus A appears instead of Youngs modulusas in the case of longitudinal modes. Harmonic modes are
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