Modified Lee Plate Equations for the Vibration Analysis of Piezoelectric Plates with Consideration of Stiffness and Mass of Electrodes

Lee plate equations for high frequency vibrations of piezoelectric plates have been established and improved over the decades with the sole objective to obtain the accurate prediction of frequency and mode shapes to aid crystal resonator design. The latest improvement includes extra terms related to derivatives of the flexural displacement to adjust the accuracy and for the consideration of the electrode for practical applications. As part of the efforts to make the equations more practical for resonator design with the improved of frequency accuracy and consideration of electrodes, the authors derived Lee plate equations for electroded plates by changing the integration limits in the dimension reduction procedure to signify the dominant role of the crystal plate. As a result, the equations are modified for the inclusion of the electrode effects. To improve the accuracy in the vicinity of thickness-shear vibration frequency of electroded plates, we modified the density terms in plate equations to reflect the contribution of both electrode stiffness and density, which makes the frequency more accurate for commonly used electrode materials