Thickness-twist Vibrations Research Articles

Thickness-shear and thickness-twist vibrations of rectangular quartz crystal plates with nonuniform thickness

ABSTRACTAn analysis was performed on the vibration of a rectangular contoured quartz plate resonator. The surface of the resonator has different curvatures in the two in-plane directions. The analysis was based on the equation by Tiersten and Smythe, which has variable coefficients. The power series method was used. Thickness-shear and thickness-twist vibration frequencies and modes were obtained. The effects of the curvatures of the resonator surface were examined. Two types of boundary conditions were considered from which the proper size of the resonator was determined. The results are useful in the understanding of these resonators and their design.

Analysis of thickness-shear and thickness-twist modes of AT-cut quartz acoustic wave resonator and filter

The thickness-shear (TS) and thickness-twist (TT) vibrations of partially electroded AT-cut quartz plates for acoustic wave resonator and filter applications are theoretically studied. The plates have structural variations in one of the two in-plane directions of the plates only. The scalar differential equations derived by Tiersten and Smythe for electroded and unelectroded AT-cut quartz plates are used, resulting in free vibration resonant frequencies and mode shapes for both fundamental and overtone families of modes. The trapped modes with vibrations, mainly confined in the electroded areas, are found to exist in both the resonator and the filter structures. The numerical results for the trapped modes are presented for different aspect ratios of electrodes and material properties, providing a reference to the design and optimization of quartz acoustic wave resonators and filters.

Thickness-shear and thickness-twist modes in an AT-cut quartz acoustic wave filter

We studied thickness-shear and thickness-twist vibrations of a monolithic, two-pole crystal filter made from a plate of AT-cut quartz. The scalar differential equations derived by Tiersten and Smythe for electroded and unelectroded quartz plates were employed which are valid for both the fundamental and the overtone modes. Exact solutions for the free vibration resonant frequencies and modes were obtained from the equations. For a structurally symmetric filter, the modes can be separated into symmetric and antisymmetric ones. Trapped modes with vibrations mainly under the electrodes were found. The effect of the distance between the two pairs of electrodes was examined.

Coupled extension and thickness-twist vibrations of lateral field excited AT-cut quartz plates

In this paper, the coupled extension and thickness-twist vibrations are studied for AT-cut quartz plates under Lateral Field Excitation (LFE) with variations along the x1-direction. Mindlin’s two-dimensional equations are used for anisotropic crystal plates. Both free and electrically forced vibrations are considered. Important vibration characteristics are obtained, including dispersion relations, frequency spectra, and motional capacitances. It is shown that, to avoid the effects of the couplings between extension and thickness-twist vibrations, a series of discrete values of the length/thickness ratio of the crystal plate need to be excluded. The results are of fundamental significance for the design of LFE resonators and sensors. Open image in new window

Analysis of the electrically forced vibrations of piezoelectric mesa resonators

We study the electrically forced thickness-shear and thickness-twist vibrations of stepped thickness piezoelectric plate mesa resonators made of polarized ceramics or 6-mm class crystals. A theoretical analysis based on the theory of piezoelectricity is performed, and an analytical solution is obtained using the trigonometric series. The electrical admittance, resonant frequencies, and mode shapes are calculated, and strong energy trapping of the modes is observed. Their dependence on the geometric parameters of the resonator is also examined.

Analysis of a Monolithic, Nonperiodic Array of Quartz Crystal Microbalances

We study thickness-shear and thickness-twist vibrations of a crystal plate with a nonperiodic array of surface mass layers for mass sensor applications. A theoretical analysis is performed. The equations of anisotropic elasticity are used. An analytical solution is obtained for the free vibration eigenvalue problem using trigonometric series from which the resonant frequencies and mode shapes are calculated. Results show that, depending on the geometric and physical parameters of the mass layers, the vibration may be mainly trapped under some of the mass layers but not under others.

Thickness-shear and thickness-twist vibrations of circular AT-cut quartz resonators

Exact solutions for free vibration frequencies and modes are obtained for thickness-shear and thickness-twist vibrations of unelectroded circular AT-cut quartz plates governed by the two-dimensional scalar differential equation derived by Tiersten and Smythe. Comparisons are made with experimental results and the widely-used perturbation solution by Tiersten and Smythe under the assumption of weak in-plane anisotropy. Our solution is found to be much closer to the experimental results than the perturbation solution. For the frequency of the fundamental thickness-shear mode, the error of the perturbation method is 0.4549%, significant in resonator applications.

Overtone frequency spectra for ξ3-dependent modes in AT-cut quartz resonators [Correspondence

We study straight-crested waves and vibration modes with spatial variations along the x <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sub> direction only in an AT-cut quartz plate resonator. The equations of anisotropic elasticity are used. Dispersion relations for face-shear and thickness-twist waves in unbounded plates are plotted. Frequency spectra are obtained for face-shear and thickness-twist vibrations of finite plates in which these modes are coupled by boundary conditions. Most importantly, our analysis produces the frequency spectra for overtone modes which do not seem to have been obtained before for x <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sub> -dependent modes. Numerical results for third- and fifth-overtone AT-cut quartz resonators are presented, showing that higher-order overtone modes are associated with more mode couplings.

Energy trapping of thickness-shear and thickness-twist modes in a partially electroded AT-cut quartz resonator

The thickness-shear and thickness-twist vibrations of a finite and partially electroded AT-cut quartz resonator are investigated. The equations of anisotropic elasticity are used with the omission of the small elastic constant c56. An analytical solution is obtained using Fourier series from which the free vibration resonant frequencies, mode shapes, and energy trapping are calculated and examined.

Simultaneous determination of thin-film inertia and shear stiffness using thickness-twist and face-shear modes of an AT-cut quartz resonator

We propose the use of thickness-twist (TT) and face-shear (FS) vibration modes of an AT-cut quartz crystal plate resonator for simultaneous determination of the inertia and stiffness of a thin film deposited on a crystal surface. A theoretical analysis using Mindlin's first-order theory for crystal plates is performed to demonstrate the idea. Expressions for the stiffness ratio and mass ratio between the thin film and the resonator are presented in terms of frequency shifts of FS and TT modes, which are experimentally measurable. A numerical example is given.

Thickness-shear and thickness-twist vibrations of an AT-Cut quartz mesa resonator

We study thickness-shear and thickness-twist vibrations of an AT-cut quartz plate mesa resonator with stepped thickness. The equations of anisotropic elasticity are used with the omission of the small elastic constant c(56). An analytical solution is obtained using Fourier series from which the resonant frequencies, mode shapes, and energy trapping are calculated and examined. The solution shows that a mesa resonator exhibits strong energy trapping of thickness-shear and thickness-twist modes, and that the trapping is sensitive to some of the structural parameters of the resonator.

Effects of mass layer dimension on a finite quartz crystal microbalance

In this paper, we investigate free thickness-shear and thickness-twist vibrations of a finite crystal resonator of AT-cut quartz carrying a thin mass layer at the central portion of its top surface operating as a quartz crystal microbalance. The equations of anisotropic elasticity are used with the omission of the small elastic constant c56. An analytical solution is obtained using Fourier series from which the resonant frequencies, mode shapes and vibration confinement by the mass layer are calculated and examined.

An exact analysis of forced thickness-twist vibrations of multi-layered piezoelectric plates

This paper deals with the thickness-twist vibration of a multi-layered rectangular piezoelectric plate of crystals of 6 mm symmetry or polarized ceramics. An exact solution is obtained from the three-dimensional equations of linear piezoelectricity. The solution is useful to the understanding and design of composite piezoelectric devices. A piezoelectric resonator, a piezoelectric transformer, and a piezoelectric generator are analyzed as examples.

Electrically forced vibration of a thickness-twist mode piezoelectric resonator with non-uniform electrodes

We study the effect of electrodes with varying thickness on thickness-twist vibration of a piezoelectric plate resonator of crystals of 6 mm symmetry. An exact theoretical analysis is performed. Results show that non-uniform electrodes have a strong effect on mode shapes, and suggest the possibility of using nonuniform electrodes for strong energy trapping.

Analysis of a rectangular ceramic plate in electrically forced thickness-twist vibration as a piezoelectric transformer

A rectangular ceramic plate with appropriate electrical load and operating mode is analyzed for piezoelectric transformer application. An exact solution from the three-dimensional equations of linear piezoelectricity is obtained. The solution simulates the real operating situation of a transformer as a vibrating piezoelectric body connected to a circuit. Transforming ratio, input admittance, and efficiency of the transformer are obtained.

Mass sensitivity of thickness-twist modes in a rectangular piezoelectric plate of hexagonal crystals

Mass sensitivity of thickness-twist vibration modes in a rectangular plate piezoelectric resonator of 6 mm crystals are obtained from the three-dimensional equations of linear piezoelectricity. The boundary conditions at the finite-plate boundaries all have been taken into consideration. The solutions obtained are exact, which is relatively few for piezoelectric problems over finite domains. The results are fundamental and useful to the understanding and design of piezoelectric resonators and acoustic wave sensors made of polarized ceramics, ZnO and AlN.

Propagation of thickness-twist waves in a multi-sectioned piezoelectric plate of 6 mm crystals

We study thickness-twist vibrations and waves in an unbounded, multi-sectioned piezoelectric plate of crystals with 6 mm symmetry or polarized ceramics. An exact solution from the three-dimensional equations of piezoelectricity is obtained. Basic vibration and wave propagation characteristics are calculated based on the solution. The results are useful in the understanding and design of plate resonators, filters and acoustic wave sensors of ZnO, AlN and polarized ceramics.

An exact analysis of a rectangular plate piezoelectric generator

We study thickness-twist vibration of a finite, piezoelectric plate of polarized ceramics or 6-mm crystals driven by surface mechanical loads. An exact solution from the three-dimensional equations of piezoelectricity is obtained. The plate is properly electroded and connected to a circuit such that an electric output is generated. The structure analyzed represents a piezoelectric generator for converting mechanical energy to electrical energy. Analytical expressions for the output voltage, current, power, efficiency, and power density are given. The basic behaviors of the generator are shown by numerical results.

Thickness-twist modes in a rectangular piezoelectric resonator of hexagonal crystals

Exact, simple expressions are obtained for thickness-twist vibration modes in a rectangular plate piezoelectric resonator of hexagonal crystals. The results are fundamental and useful to the understanding and design of piezoelectric resonators and acoustic wave sensors made of polarized ceramics, ZnO, and AlN.

In-plane rotational and thickness-twist vibrations of polygonal plates and spherical shallow shells

In-plane rotational and thickness-twist vibrations of polygonal plates and spherical shallow shells