It is well understood that the strong coupling of thickness-shear and flexural vibrations in piezoelectric crystal plates only occurs at specific length at which the vibration mode conversion, like the flexural mode gradually converting to thickness-shear mode while the thickness-shear mode converting to higher-order flexural mode, happens. It is important to avoid the strong coupling of modes in a crystal resonator that uses thickness-shear vibrations to enhance the energy trapping. To achieve such a design goal, the length of a crystal blank should be carefully chosen such that the coupling is at its weakest, which usually is in the middle of two strong coupling points. Through a closer examination of the frequency spectra, or the frequency-length relationship in this study, we can see that the strong coupling points appear periodically. This implies that we can find exact locations with the plate theory that predicts the resonance frequency. Based on this observation, we first use the first-order Mindlin plate theory with the precise thickness-shear frequency, which is normalized to one, to find corresponding wavenumbers. Then the length as a variable is solved from the coupled frequency equation for exact coupling points in a crystal plate of AT-cut quartz. The optimal length of a crystal blank in the simplest resonator model is calculated for the coupled thickness-shear, flexural, and extensional vibrations. The solutions and the method will be important in the determination of optimal length of a crystal blank in the resonator design process.

We study electromechanical fields in a piezoelectric semiconductor rod nonuniformly doped with impurities producing holes and electrons. The phenomenological theory of piezoelectric semiconductors consisting of the equations of piezoelectricity and the conservation of charge for holes and electrons is used, which was reduced to a one-dimensional model for thin rods in a previous paper. In this paper the one-dimensional theory is linearized for low electric potential or voltage. Solutions from the linearized one-dimensional equations are obtained for three specific doping profiles: linear doping, piecewise linear doping with fundamentally important applications in PN junctions, and sinusoidal doping. Various electromechanical fields produced by the doping are calculated and examined. The results are fundamental to piezoelectric semiconductor devices or piezotronics.

An extension has been made with the popular Rayleigh-Ritz method by integrating the energy functional over the time of one period of vibration to eliminate the harmonics from the deformation function. An eigenvalue problem is obtained for the frequency-amplitude dependence of nonlinear vibrations of small deformation. This is the extension of Rayleigh-Ritz method of the energy formulation and Galerkin method with the inclusion of time, which is equivalent to the harmonics matching in nonlinear vibrations. The extended Rayleigh-Ritz method can be utilized for the analysis of free and forced nonlinear vibrations of structures as a new technique with significant advantages.