Effect of electrodes on the acceleration sensitivity of crystal resonators

Based on Mindlin´s first-order equations of plated crystal plates, a set of six coupled equations is derived in which the effect of the mass of the electrodes, the extensional elastic stiffness of the electrodes, and the thermal expansions of the plate and electrodes are taken into account. The principle of virtual displacements corresponding to these equations is obtained. Strains and displacement gradients in an electroded crystal disk subject to a steady acceleration are obtained from the finite-element solutions of the derived equations. By inserting these strains and displacement gradients in the frequency equation of the fundamental thickness vibrations of the disk and solving the frequency equation by a perturbation method, acceleration sensitivities of crystal resonators with various electrode arrangements are obtained. It is found that for small plate backs, say R⩽0.5%, the percent increase of acceleration sensitivity of an electroded plate as compared with that of an unelectroded plate is essentially proportional to the ratio of the total mass of the attached electrodes to the total mass of the plate