Nonlinear extensional vibrations of quartz rods

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Abstract
The one-dimensional scalar differential equation describing the extensional motion of thin piezoelectric rods is obtained from the general nonlinear three-dimensional description. Only the elastic nonlinearities are considered. The relations between the quadratic and cubic coefficients of the rod and the fundamental anisotropic elastic constants of various orders are derived. The quadratic rod coefficients are calculated for various orientations of quartz rods, but not the cubic rod coefficients because the fundamental elastic constants of fourth order, which are required for the calculation, are not presently known. The nonlinear equation and boundary conditions are applied in the analyses of both intermodulation and nonlinear resonance of quartz rods. In each instance a lumped parameter representation of the solution, which is valid in the vicinity of a resonance, is obtained and the influence of the external circuitry is included in the treatment.

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