Scalar Differential Equations for Transversely Varying Thickness Modes in Doubly Rotated Quartz Crystal Sensors

For time-harmonic motions, we generalize the 2-D scalar differential equation by Stevens and Tiersten for transversely varying thickness-shear vibrations of doubly rotated quartz plate sensors to include the effects of surface acoustic loadings. Starting from the 3-D equations of piezoelectricity, the governing equations are first transferred into a new coordinate system formed by the three eigenvibration directions; then, by neglecting the terms pertaining to the third and higher order in-plane wavenumbers, which are considered to be small compared to the thickness wavenumber, a scalar differential equation for the quasi-thickness shear modes is derived. The surface impedance tensor was introduced to represent the mechanical effects of surface mass layers and viscous fluid. Both unelectroded and electroded plates are treated. For electroded plates, both free and electrically forced vibrations under a time-harmonic driving voltage are analyzed. In addition to the fast thickness variation, which was considered in conventional 1-D models such as Sauerbrey's equation and Kanazawa's equation, the new equations properly accounted for the slow transverse variation of the finite sensors. Numerical examples for partially electroded fluid viscosity sensors are presented to show practical applications of the new equations for design and analysis of quartz crystal sensors.