Resonant mode calculations of contoured doubly-rotated quartz crystals using a variational approach

A technique has been developed using variational calculus for calculating resonant modes in doubly-rotated contoured quartz crystal. Basis functions used are similar to past analytical solutions for the slow-shear, fast-shear, and extensional thickness modes. Flexural motion is included in the analysis to ensure rotational balance. Calculated frequencies and mode shapes are consistent with past analytical results for trapped modes. However the present approach includes coupling between the different mode families. In addition, mode shapes are available for weakly trapped modes that are not predicted by past analytical methods. The present approach offers the future opportunity to analyze the impact of loss of symmetry of the crystal from electrode patterns and imperfect contouring processes.