Electrically forced thickness-shear vibrations of quartz plate with nonlinear coupling to extension

We study electrically forced nonlinear thickness-shear vibrations of a quartz plate resonator with relatively large amplitude. It is shown that thickness-shear is nonlinearly coupled to extension due to the well-known Poynting effect in nonlinear elasticity. This coupling is relatively strong when the resonant frequency of the extensional mode is about twice the resonant frequency of the thickness-shear mode. This happens when the plate length/thickness ratio assumes certain values. With this nonlinear coupling, the thickness-shear motion is no longer sinusoidal. Coupling to extension also affects energy trapping which is related to device mounting. When damping is 0.01, nonlinear coupling causes a frequency shift of the order of 10−6 which is not insignificant, and an amplitude change of the order of 10−8. The effects are expected to be stronger under real damping of 10−5 or larger. To avoid nonlinear coupling to extension, certain values of the aspect ratio of the plate should be avoided.