Alternate Analysis of a Strip Resonator with Tilted Edges in Crystal Class 32

In a previous paper, it was shown that resonator characteristics (desirable in practical applications) of an infinite plate of rotated Y-cut of a crystal in class 32 vibrating in thickness-twist mode can be realized in a miniature strip resonator by tilting side surfaces parallel to the X axis. In the case of quartz AT cut, billions of miniature strip resonators have been made. Similar benefits are expected to be obtained in resonators of such crystals as gallium orthophosphate (GaPO₄) and Langasite family (LGS, LGN and LGT). The original design was inspired by Mindlin´s exact solution of an infinite quartz strip. In practice, however, many refinements were required, because tilting angle given by the exact solution, which was called Mindlin´s angle, was valid only at a certain temperature. Sekimoto et al. developed an analytical method based on finite element method (FEM) and successfully applied to both two and three-dimensional cases of rotated cuts as well as doubly rotated cuts. In the present paper, an alternate analysis based on Mindlin´s solution is presented. It is similar to an analysis by Shimizu et al. of quartz AT/DT-cut. A few modifications are made and results are applied to resonators of other crystals. The analysis based on FEM is general enough to handle couplings between many modes of vibration, whereas the present analysis is intuitive and suitable to see details of coupling between only a few modes