Analysis of thickness modes of contoured, doubly-rotated quartz resonators

It is shown that the theoretical development of D.S. Stevens and H.F. Tiersten (1986) can be used for calculating all three mode families of a doubly-rotated, contoured quartz resonator. 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 x₃ blank axis. The key constants for the SC-cut are presented for workers to use in the future