Vibration analysis of coupled flexural torsional mode tuning fork type quartz crystals resonator

The frequency-temperature characteristics of a tuning-fork-type quartz crystal resonator which couples the torsional mode vibration to the flexural mode vibration are analyzed, and the estimated values of the various constants for a resonator manufactured for trial experiments are presented. A secular equation coupling the torsional mode vibration to the flexural mode vibration is first derived and used for a theoretical analysis of the frequency-temperature characteristics of the resonator. The calculated values are shown to agree well with experimental values obtained for the manufactured resonator. The results indicate that resonators with excellent frequency-temperature characteristics, low crystal impedance, and a high quality factor are realized.<>