Analysis of nonlinear resonance in thickness-shear and trapped-energy resonators

Abstract

Electroelastic equations containing terms up to cubic in the small mechanical displacement field, but no higher than linear in the electric variables, are applied in the analysis of nonlinear resonance in rotated Y-cut quartz oscillators. Both pure thickness-shear vibrators and essentially thickness-shear trapped-energy resonators are treated. This is a natural continuation of earlier work on intermodulation in the same resonators. Since in each equation each nonlinear term is negligible compared to an associated linear term, the solutions are obtained by employing an asymptotic iterative procedure and expanding in the eigensolutions of the associated linear problem and, in the vicinity of a resonance, retaining only that nonlinear term correcting the dominant eigensolution. Lumped parameter representations of the solutions, which are valid in the vicinity of a resonance and relate the amplitude of the dominant mode nonlinearly to the voltage across the crystal, are presented for both the pure thickness-shear and trapped energy thickness-shear problems. In each instance the expression for the current through the crystal is determined, the influence of the external circuitry is included in the analysis and, ultimately, an expression cubic in the mode amplitude and linear in the driving voltage is obtained. The analyses hold for the fundamental and odd overtone thickness-shear modes. Nonlinear resonance curves are presented for AT-cut quartz using the nonlinear coefficient γ determined in earlier work on intermodulation. Subject Classification: [43]40.24, [43]40.30; [43]85.52.