MASS-FREQUENCY INFLUENCE SURFACE, MODE SHAPES, AND FREQUENCY-SPECTRUM OF A RECTANGULAR AT-CUT QUARTZ PLATE

The mass-frequency influence surface and frequency spectrum of a rectangular AT-cut quartz plate is studied. The mass-frequency influence surface is defined as a surface giving the frequency change due to a small localized mass applied on the plate surface. Finite-element solutions of Mindlin's two-dimensional (2-D) plate equations for thickness-shear, thickness-twist, and flexural vibrations are given. Spectrum splicing, and an efficient eigenvalue solver using the Lanczos algorithm were incorporated into the finite-element program. A convergence study of the fundamental thickness-shear mode and its first symmetric, anharmonic overtone was performed for finite-element meshes of increasing fineness. As a general rule, more than two elements must span any half-wave in the plate or spurious mode shapes will be obtained. Two-dimensional (2-D) mode shapes and frequency spectrum of a rectangular AT-cut plate in the region of the fundamental thickness-shear frequency are presented. The mass-frequency influence surface for a 5-MHz rectangular, AT-cut plate with patch electrodes is obtained by calculating the frequency change due to a small mass layer moving over the plate surface. The frequency change is proportional to the ratio of mass loading to mass of plate per unit area, and is confined mostly within the electrode area, where the magnitude is of the order 10(8) Hz/g.