A layerwise plate theory for the vibrations of electroded crystal plates

Electrodes on a crystal resonator has been traditionally considered as mass addition to the crystal plate, thus resulting the neglect of their stiffness. This assumption is considered reasonable if electrodes are thin in comparison with the crystal in terms of the mass ratio, the relative mass of electrodes. For thicker electrodes of high frequency resonators, this assumption has to be reexamined for better prediction on their effects on the resonance frequency and the frequency-temperature characteristics. In this study, the electrodes are considered as layers of plates with their own stiffness and mass properties. As a result, the deformation of electrodes also involve independent ones in addition to the crystal deformation. The theory employed also enables us to consider the deformation to a higher order of degree. The layerwise plate theory is derived and implemented for the finite element method. Numerical results are compared with the other computing schemes with and without electrodes considerations.