Effects of Electrode Off Centre on Trapped Thickness-Shear Modes in Contoured AT-Cut Quartz Resonators

We investigated thickness-shear vibrations of a contoured, AT-cut quartz resonator with a pair of electrodes displaced from the resonator centre. The scalar differential equations by Stevens and Tiersten for thickness-shear vibrations of electroded and unelectroded quartz plates were employed. Based on the variational formulation of the scalar differential equations established in a previous paper and the variation-based Ritz method with trigonometric functions as basis functions, free vibration resonance frequencies and trapped thickness-shear modes were obtained. The effects of the electrode off centre on resonance frequencies and mode shapes were examined. When the electrode off centre is about one hundredth of the electrode length, the relative frequency shift is of the order of one part per million, significant in certain resonator design and applications. The electrode off centre also causes the loss of symmetry of modes, which has an adverse effect on resonator frequency stability under a normal acceleration