3-DIMENSIONAL ANALYSIS OF SPURIOUS VIBRATIONS OF RECTANGULAR AT-CUT QUARTZ PLATES

We have developed a technique for the three-dimensional (3-D) analysis of all the spurious vibrations of rectangular AT-cut quartz plates. T he 3-D displacements are described with a linear combination of modes guided along the X crystal axis of an AT-cut quartz strip. We accurate ly calculate each guided mode Ly applying the one-dimensional finite e lement method to the two-dimensional strip waveguide problem. To uniqu ely determine the amplitudes of guided modes in the 3-D solutions we e mploy tile variational expression for the stress-free conditions on th e X-length edges, The present technique makes it possible to perform 3 -D analysis of spurious vibrations within a reasonable computational t ime. Frequency spectra for rectangular AT-cut quartz plates are comput ed as a numerical example and are compared with the numerical results obtained using an extended version of the previous method and with Kog a's experimental data. Vibrational pal iel rls of spurious modes are s hown, and a convergence study is carried out.