Variational calculation of force-frequency constants of doubly rotated quartz resonators

The calculus of variations has been used to obtain solutions for the statics tress distributions in circular quartz resonator blanks caused by forces acting along the blank diameter. The resulting static stress distributionsh ave been used with nonlinear elastic calculations (third-order elastic constants) to predict the force sensitivity of the resonant frequency of thickness shear resonators. The present results are shown to agree more favorably with experiment than results obtained by previous theories where the quartz was assumed to be isotropic for the statics tress solution. General theoretical results and supporting experimental results argeiv en for the force-frequency constants of doubly rotated family (yxωl)ϕ,θ along the technologically important zero temperature coefficient branch which contains the AT-, FC-, IT-, SC-, and rotated X-cuts The results will be useful for choosing the location of bonding pads for two-point mountss o that the mounting stress effectsc an be minimized in frequency control applications or maximized in force transducer applications.