Vibrations of a Z-cut quartz resonator-structure in the vicinity of the third overtone of extensional mode

Abstract

Vibrations of a miniature Z-cut, third overtone, extensional quartz resonator were studied. The resonator-structure consisted of a long vibrating beam, a short supporting beam, and a tuning fork. The vibrating beam and supporting beam intersected each other perpendicularly at their middle points. The ends of the supporting beam were in turn connected to the ends of tines of the tuning fork. Two corners near the base of the tuning fork were mounted on supports which had a fixed width but variable lengths. The vibrating beam is mainly excited at the third overtone of extensional mode. The resonator was produced by chemical etching of a Z-cut quartz plate. An anisotropic, four-node quadrilateral, Mindlin plate element model was employed for the study. Incompatible modes were included in the stiffness matrix to prevent shear locking. The frequency spectrum of the resonator as a function of the mounting length and its mode shapes were discussed. The resonator motional resistances for different mounting lengths were measured. Normalized strain energy ratios in the base area and tine area were computed and compared with the normalized motional resistance at various mounting lengths. Good correlations were found: the peaks in strain energy ratio corresponded with the peaks in motional resistance. These peaks occurred at regions in the frequency spectrum which showed strong coupling with other modes of vibration. Large scatterings of the measured motional resistances were also observed at these regions. The strain energy ratio in the base and tines was proposed as a qualitative criterion for evaluating the relative changes in motional resistance of resonators with different mounting lengths and geometries.