A Theoretical Model for Analyzing the Thickness-Shear Vibration of a Circular Quartz Crystal Plate With Multiple Concentric Ring Electrodes.

A dynamic model to analyze the thickness-shear vibration of a circular quartz crystal plate with multiple concentric ring electrodes on its upper and bottom surfaces is established with the aid of coordinate transformation. The theoretical solution is obtained, which can be written in a superposition form of Mathieu functions and modified Mathieu functions. The convergence of the solution is demonstrated, and the correctness is numerically validated via results from the finite element method (FEM). Subsequently, a systematic investigation is carried out to quantify the effect of the electrode size on the energy trapping phenomenon, i.e., the resonant frequency and mode shape, which reveals that the ring electrode has a great influence on the work performance of resonators. With the increase of the electrode inertia, i.e., the radius and mass ratio, new trapped modes emergence with the vibration mainly focused on the plate with partial electrodes. Besides, owing to the anisotropy, degenerated trapped modes have different resonant frequencies and the frequency discrepancy between them will become smaller for higher modes. Finally, the influence of multiple ring electrodes is investigated, and the qualitative analysis and quantitative results demonstrate that multiple ring electrodes will lead to a more uniform mass sensitivity compared with a single ring electrode. The outcome is widely applicable, which can provide theoretical guidance for the structural design and manufacturing of quartz resonators, as well as a thorough interpretation about the underlying physical mechanism.