Thickness-shear modes of an elliptical, contoured at-cut quartz resonator

Abstract

We study free vibrations of an elliptical crystal resonator of AT-cut quartz with an optimal ratio between the semi-major and semi-minor axes as defined by Mindlin. The resonator is contoured with a quadratic thickness variation. The scalar equation for thickness-shear modes in an AT-cut quartz plate by Tiersten and Smythe is used. Analytical solutions for the frequencies and modes to the scalar equation are obtained using a power series expansion that converges rapidly. The frequencies and modes are exact in the sense that they can satisfy the scalar differential equation and the free edge condition to any desired accuracy. They are simple and can be used conveniently for further studies on other effects on frequencies and modes of contoured resonators.

The investigation of trapped thickness shear modes in a contoured AT-cut quartz plate using the power series expansion technique

“…It seems convincing that the energy trapping also can be induced by partial bulge thickness in a contoured quartz plate. From view of physical mechanism, the partial bulge thickness is equivalent to the additional mass inertia [21][22][23][24][25]. However, from view of mathematics, the thickness variation provides much complexity for the theoretical solution of thickness shear waves, which is due to the fact that the contoured resonator is characterized by differential equations with spatially varying coefficients [23][24].…”

“…From view of physical mechanism, the partial bulge thickness is equivalent to the additional mass inertia [21][22][23][24][25]. However, from view of mathematics, the thickness variation provides much complexity for the theoretical solution of thickness shear waves, which is due to the fact that the contoured resonator is characterized by differential equations with spatially varying coefficients [23][24]. A general solution is hard to be achieved, and some analytical solutions can be obtained only for a few rare cases with specific thickness variations, such as hyperbolic [23], quadratic [24], cubic [25], or stepped function [3,26].…”

“…However, from view of mathematics, the thickness variation provides much complexity for the theoretical solution of thickness shear waves, which is due to the fact that the contoured resonator is characterized by differential equations with spatially varying coefficients [23][24]. A general solution is hard to be achieved, and some analytical solutions can be obtained only for a few rare cases with specific thickness variations, such as hyperbolic [23], quadratic [24], cubic [25], or stepped function [3,26]. For example, the energy trapping has been proved to exist in an oblate elliptical cylinder with the aid of the Fourier series technique [21] and the Mathieu function [22].…”

Overtone Modes in an At-Cut Quartz Trapped-Energy Resonator with Elliptical Electrodes

“…Due to the in-plane anisotropy of quartz plates, the Mindlin optimal electrodes are nearly elliptical, with the major axis exceeding the minor axis by 25%. Subsequent studies on elliptical electrodes can be found in [2][3][4][5][6][7]. For at-cut quartz plates, the optimal electrodes [1,4], the frequencies and mode shapes in plates with elliptical electrodes [5] were all determined using the Mindlin firstorder plate theory and thus were limited to the fundamental TSh modes with one nodal point along the plate thickness only.…”

Effects of free edge on the vibration characteristics of a contoured, beveled quartz resonator

“…This scalar equation is simple and accurate, and has been widely used in theoretical analysis of quartz resonators, e.g. [6][7][8][9][10][11][12][13][14][15][16][17][18][19]. In this paper the scalar equation [5] for transversely varying thickness modes in doubly-rotated quartz resonators referred to the coordinate system in which no mixed derivatives occur is applied in the analysis of the steady-state vibrations of contoured resonators with beveled cylindrical edges.…”