Model for a quartz-crystal tuning fork using plate spring approximated to torsion spring adopted at the joint of the arm and the base

We developed a method for analyzing the frequency of a quartz-crystal tuning fork using a plate spring approximation. In order to calculate the actual frequency of the tuning fork, we approximated the right half of the tuning fork to an L-shaped bar, which two bars acting as the base and the arm undergo bending vibration, and a torsion spring at the joint of beams A corresponding to the base and B corresponding to the arm combined to form. Furthermore, we approximated the torsion spring with torsion spring constant R to a plate spring composed of the base. R was expressed in terms of both the sizes and the bending stiffness of the tuning fork. We could obtain a series of equations for calculating the frequency of the tuning fork by plate spring approximation and the convenient new frequency equation for a cantilever with the length l₂+w₁/10 by considering the bend of a plate spring. A comparison was made of frequencies calculated by these methods and experimental results.