Mechanical Couplings Involving Discontinuities of the Frequency-Temperature Curves of Contoured Quartz Resonators

Abstract

In this paper, frequency jumps occuring in the frequency temperature curves of contoured resonators are studied. Besides vibration trapping effect, contouring the plate involves some mechanical couplings which may be strongly dependent on temperature. This will occur whenever two or several coupled modes are very close in frequency, while exhibiting different therrral sensitivities. Because contouring generates anharmonic modes in large frequency range, such an event frequently occurs. We have investigated some examples of such couplings in an AT-cut resonator vibrating in slow shear C-mode (seventh overtone) coupled with fast shear B-mode (sixth overtone). Each coupling is very strong in a range of a few degrees. Our theoretical analysis is built on Green's 5ormalism : the mode pattern of the actual vibration is assumed to be a rnlxing of simple mode patterns. Jumps of the frequency temperature curves are induced by the thermal dependency of this mixing. Use of separated X-Rays topographs for the components of the mechanical displacement enables us to verify the predicted mode patterns at various temperatures.