Mass-frequency influence surface and frequency spectrum of an AT-cut quartz plate

The mass-frequency influence surface is defined as a surface yielding the frequency change due to a small localized mass applied on the plate surface. Finite-element solutions of R.D. Mindlin's (1963) two-dimensional 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 mode shapes and the 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 frequency change is on the order of 10⁸ Hz/(g/mm²)/mm².