Analysis of Q-Value of Quartz-Crystal Tuning Fork by Use of Thermoelastic Coupling Equations Including an Effect of Thermal Radiation

Abstract

We investigated a method of predicting the actual Q-value of a quartz-crystal tuning fork on the basis of thermoelastic coupling equations by considering an effect of thermal radiation from the flexural vibrating arms of the tuning fork to the external environment. We developed the method of calculating the quartz-crystal tuning fork's Q-value by use of a new temperature distribution solution, indicating an intermediate situation between adiabatic and isothermal solutions by use of a parameter α1 (0 ≦α1 ≦1; isothermal solution is given at α1 = 0 and adiabatic solution is given at α1 = 1) for the coupled heat conduction equation without a thermal radiation term, which was adjusted so as to have almost the same curve as the approximated temperature distribution including the zero-order and first-order temperature distribution solutions of the coupled heat conduction equation with the thermal radiation term, given by the perturbation method. Our calculated Q-value was closer to the measured value for an etching-processed quartz-crystal tuning fork than that obtained by the coupled heat conduction equation without the thermal radiation term under an adiabatic boundary condition.