Nonlinear acceleration sensitivity of quartz resonators

The nonlinear effects of initial stress/strain of the quartz plate resonator on its acceleration sensitivity was studied. Finite element models were developed using a theory of small deformations superposed on finite initial deformations in a Lagrangian formulation. AT- and SC-cut quartz circular plate resonators were studied. The plates were respectively subjected to diametrical compression force and bending force. The initial strains due to the application of diametrical force represented initial strains due to in-plane acceleration, while the initial strains due to bending force represented initial strains due to out-of-plane acceleration. Our model results using nonlinear initial strains showed good agreement with measured data by Ballato, Mingins, and Fletcher and Douglas. The model results using linear initial strains compared well only with the measured data for plates subjected to diametrical force but not for plates subjected to bending forces. Hence our model results showed that for accurate prediction of out-of-plane acceleration sensitivity the nonlinear initial strains must be used. The linear initial stress/strain cannot fully capture rotation and bending effects. The acceleration sensitivity model using linear initial strains could only be employed for in-plane acceleration, or for very low g out-of-plane acceleration. The SC-cut crystals showed better linearity of frequency change with respect to applied bending forces than the AT-cut crystals. The principle of superposition for out-of-plane acceleration sensitivity in AT-cut crystals is in general not valid, especially in cases of high g accelerations.