An analysis of the resonance frequencies of stepped, clamped-free torsional mode resonators and their applications to quartz crystal tuning fork

This paper describes torsional mode quartz crystal resonators of the tuning fork type with stepped vibration bars. A frequency equation and a transcendental equation were derived from the ''clamped-free'' boundary condition and the stepped portion of the continuity condition, respectively. Because these equations were given as a function of torsional rigidity, the torsional rigidity was derived from the equation of equilibrium. From related equations, it was shown that high or low frequency could be obtained without a change in length. The calculated results were then compared with the measured values so that both results would be in good agreement.