High−frequency vibrations of crystal plates under initial stresses

Abstract

A system of six two−dimensional plate equations is derived for motions of small−amplitude waves or vibrations superimposed on finite, elastic deformations due to static, initial stresses. In the stress−strain relations, the nonliner terms associated to the third−order elastic stiffness coefficients are included. These equations accomodate the coupling of the six lowest modes of vibration, i.e., the flexure, extension, face−shear, thickness−shear, thickness−twist, and thickness−stretch modes, and all their anharmonic overtones. The new equations are applied to the rotated Y cuts of quartz in studying the thickness−shear and flexural vibrations. The changes in the resonance frequencies of the fundamental thickness−shear vibrations are computed as functions of the direction of initially applied force and of the angle of rotated Y cuts. The predicted results are compared with experimental data and with existing computed results. An explicit formula is obtained for the change of fundamental thickness−shear frequencies in terms of initial deformations and the second− and third−order elastic stiffness coefficients. Subject Classification: 40.24.

Special features of quartz oscillator dynamics numerical analysis

“…В работе [5] отмечено, что влияние внешнего кинематического воздействия на частоту кварцевого резонатора может быть определе-но численно только при использовании нелинейных уравнений меха-ники пьезоэлектрической сплошной среды. При этом тензор дефор-маций имеет вид ( ) Расчеты проведены для амплитуд ускорений a в интервалах от 0 до ±5g (g = 9,81 м/с 2 -ускорение свободного падения), действую-щих в направлениях x и y (см.…”

High frequency vibrations of quartz crystals subject to initial thermo-mechanical bias

“…He demonstrated that the temperature dependence of the force-frequency coe cient depends on the azimuth angle, , between the X crystallographic axis and the direction of the applied force. The theoretical aspect of the problem belongs to the general theory of incremental elastic deformations superposed on initial nite deformations [11]. The initial deformation is produced by applying diametric forces and also depends on the change in temperature of the medium.…”

Eect of anisotropy and piezoelectricity on the force-frequency coecient of AT-cut quartz crystals

“…This error arises from the approximation included in power series expansion of thickness shear deformation along the thickness of plate. More details on the calculation of can be found in [15].…”

“…Azimuth angle, , de nes the location where the force is applied on the perimeter of disk and variation of changes the force-frequency coe cient. One of the main methods to analyze the forcefrequency e ect has been proposed by Lee [15]. Lee derived the basic equations of the theory in Lagrangian formulation to analyze the force-frequency e ect at room temperature.…”

“…Lee developed the rst successful theoretical analysis of the force-frequency e ect [15]. Lee derived an explicit formula to predict the frequency changes by employing the theory of incremental elastic deformations superimposed on initial nite deformations.…”