Flexoelectric effect on thickness-shear vibration of a rectangular piezoelectric crystal plate

Thickness-shear (TSh) vibration of a rectangular piezoelectric crystal plate is studied with the consideration of flexoelectric effect in this paper. The developed theoretical model is based on the assumed displacement function which includes the anti-symmetric mode through thickness and symmetric mode in length. The constitutive equation with flexoelectricity, governing equations and boundary conditions are derived from the Gibbs energy density function and variational principle. For the effect of flexoelectricity, we only consider the shear strain gradient in the thickness direction so as to simply the mathematical model. Thus, two flexoelectric coefficients are used in the present model. The electric potential functions are also obtained for different electric boundary conditions. The present results clearly show that the flexoelectric effect has significant effect on vibration frequencies of thickness-shear modes of thin piezoelectric crystal plate. It is also found that the flexoelectric coefficients and length to thickness ratio have influence on the thickness-shear modes. The results tell that flexoelectricity cannot be neglected for design of small size piezoelectric resonators.

Effects of Elliptical Ring Electrodes on Shear Vibrations of Quartz Crystal Plates

A theoretical analysis is performed on the thickness-shear vibrations of an AT-cut quartz piezoelectric crystal plate with elliptical ring electrodes. The scalar differential equation by Tiersten and Smythe is used. An analytical solution is obtained. Numerical results from the solution show that the thickness-shear mode of interest may be trapped by the ring electrodes and can have a convex, concave, or nearly flat vibration distribution near the plate center, which is fundamentally important when the plate is used as an acoustic wave mass sensor. The vibration distribution is found to be sensitive to both the geometric and physical parameters of the electrodes. Therefore, a careful design is needed to realize the desired trapped mode with suitable center convexity for sensor application

Response of Three Oscillators Coupled by a Weak Nonlinearity

As a simplified model of the exchange processes occurring among resonance modes in physical systems, such as a piezoelectric crystal plate or an acoustic interferometer, a study is made of the response of three oscillators that are coupled by a weak nonlinearity and whose frequencies satisfy the condition ω1 + ω2 ≅ ω3. The transient behavior is obtained by a perturbation expansion. There exist three integral constraints on the amplitude and phase variation of the oscillations for a conservative system, and the solution of the response can be reduced to quadrature. The phase diagram describing the motion indicates that the high frequency oscillation is unstable; the energy associated with it, under certain conditions, can be diverted to lower frequency oscillations. For nonconservative systems, the effects of dissipation and detuning are examined for their role in limiting the energy exchange among the oscillations and in determining the steady-state response to forcing. Predictions from this analysis are compared with results of a reported experiment in which a piezoelectric crystal plate is forced to oscillate at amplitudes sufficient to generate coupled subharmonics.