Model for a quartz-crystal tuning fork using torsion spring at the joint of the arm and the base and analysis of its frequency

We developed a method for analyzing the frequency of a quartz-crystal tuning fork using a torsion-spring model. In order to calculate the frequency near the actual frequency of the turning fork, we approximated the right half of the tuning fork as an L-shaped bar in which two bars acting as the base and the arm undergo bending vibration. Furthermore, we applied the torsion spring model with torsion spring constant R to the joint of beam A corresponding to the base and beam B corresponding to the arm. A comparison was made between frequencies calculated by this method and experimental results, and the relationship between R and the width of the base was discussed.