Analysis of thickness modes of contoured doubly rotated, quartz resonators

It is the intent of this work to provide a working resource for calculating all three mode families of a doubly rotated, contoured quartz resonator. It is shown that the theoretical development of Stevens and Tiersten [1986] can be used for this purpose. Their approach uses a transformation of the mechanical displacement vector to the eigenvector triad of the pure thickness solution. The solution methodology here reorganizes the transformation matrix Q in their formulation to calculate the other two mode families. Calculations compare well with experimental results for the three mode families of an SC-cut crystal and an FC-cut crystal and with published calculations for the SBTC-cut mode family with major displacement along the as blank axis. The key constants for the SC-cut are presented for workers to use in the future. In addition, the equations of motion and boundary conditions are derived for the two additional mode families using assumptions parallel to those used by Stevens and Tiersten. Calculations with these equations are presented for completeness to support the present conclusions by showing the equivalence of either simply reorganizing the Q matrix or using separate equations for each of the three mode families.