On the normal acceleration sensitivity of contoured quartz resonators with the mode shape displaced with respect to rectangular supports

It is shown that the normal acceleration sensitivity of contoured quartz resonators with rectangular supports vanishes when the centers of the mode shape and support rectangle coincide. This result is a consequence of symmetry and applies to many other shapes. Since it is essentially impossible to realize this situation in practice, an analysis of the influence of an offset of the centers on the normal acceleration sensitivity is performed. The biasing deformation is determined by means of a variational approximation procedure using the variational principle with all natural conditions for anisotropic static flexure. The very important accompanying strains varying quadratically across the thickness are determined recursively, as in earlier work. The resulting flexural biasing states are employed in an existing perturbation equation along with the equivalent trapped-energy-mode shapes of the contoured resonators to calculate the normal acceleration sensitivities. It is shown that for small offsets the acceleration sensitivity increases linearly with offset, and orientations for which this effect is minimized are found.