Thickness Shear Mode Research Articles

Properties of single crystal piezoelectric Ca3TaGa3Si2O14 and YCa4O(BO3)3 resonators at high-temperature and vacuum conditions

Piezoelectric single crystals Ca3TaGa3Si2O14 (CTGS) and YCa4O(BO3)3 (YCOB)were grown by the Czochralski pulling method and specially cut into three types of resonator samples with different vibration modes (CTGS lateral mode, YCOB thickness shear mode and planar mode). The dielectric, elastic and piezoelectric properties of the resonators have been investigated as functions of temperature (up to 800°C) and vacuum pressure (down to 10−10Bar). It was found that all the crystal resonators have excellent stability in the test range of vacuum pressure level; and their fundamental resonant frequencies exhibit temperature dependent characteristics, some of which were linear. It is also found that for CTGS lateral mode, YCOB thickness shear mode and YCOB planar mode resonators, the relative dielectric permittivity ε11T/ε0, ε22T/ε0, ɛT/ɛ0 are 17.0–18.2 (17.6±0.6), 13.5–24.4 (19.0±5.5), 14.0–16.9 (15.4±1.5); the elastic coefficients S11E, S66E, SE are 8.8–9.3 (9.1±0.3), 16.3–17.3 (16.8±0.5), 8.3–9.2 (8.8±0.5) (pm2/N); and the piezoelectric coefficients d11, d26, d are 3.7–3.9 (3.8±0.1), 8.8–10.9 (9.9±1.0), 3.3–3.9 (3.6±0.3)(pC/N), respectively. Mechanical quality factor Q was found to be 4500–7000 and 6000–11,000 for CTGS and YCOB, respectively. The excellent property stability of these CTGS and YCOB piezoelectric resonators, especially the thickness shear mode, make them very promising candidates for sensing applications at high-temperature and high-vacuum conditions.

Effects of Electrode Configuration on Vibration Characteristics of Quartz Thickness-Shear Mode Trapped-Energy Resonators

We study the effects of electrode configuration of quartz thickness-shear mode trapped-energy resonators on their basic vibration characteristics. Free and electrically forced vibration analyses are performed on a rectangular trapped-energy resonator by COMSOL, focusing on the effects of the extended parts of the electrodes leading from the main rectangular parts of the crystal plate. Results show that the effect of the extended parts on resonant frequencies is of the order of 100 ppm. They also displace the mode center which has further implications in resonator frequency stability. These effects are nonnegligible, and should be taken into consideration in resonator design.

FEM Simulation of Quartz Thickness Shear Mode Resonator for Gas Sensing Applications

The objective of this paper is to present simulation results of the Thickness Shear Mode (TSM) resonator based on quartz using finite element simulation method. 3D model of quartz resonator and simulations were completed using finite element method in CoventorWare software suite for design and simulation of MEMS devices. Different techniques for simulation of adsorption effect on selective layer were studied: influence of change in mass of the sensitive layer and influence of change in density of the sensitive layer. Analyses of resonant modes were performed for both cases and displacement profiles in selected modes were determined for the resonator under study. Impedance and phase characteristics were calculated and measured for clean sample and sample with selective layer coated. The adsorption model calculates the frequency shift in basic resonant frequency with adsorbed amount of sensed gas. The simulation results were used in design of gas sensors for dangerous substances detection.

An exact frequency equation in closed form for Timoshenko beam clampled at both ends

Abstract The author has discovered several errors which are not typographical in the frequency equations for a Timoshenko beam clamped at both ends by Huang who presented the frequency equations and normal mode equations for all six common types of simple, finite beams in closed form for the first time. The exact frequency equations in closed form for Timoshenko beams clamped at both ends are derived based on his analysis. And then in order to justify the amended solutions of Huang, two versions of the closed form exact method and the Ritz method are applied. The frequency equations by the previous researcher present frequencies for only the flexural modes, while the closed form exact method and the Ritz method give ones for the thickness–shear modes as well as the bending modes. The purpose of the present study is to reveal the errors, correct them, and give some numerical results.

QCM-Based Measurement of Bond Rupture Forces in DNA Double Helices for Complementarity Sensing

After fixing the DNA molecule in the form of a double helix on the surface of a thickness shear mode resonator (QCM), mechanical oscillations at increasing amplitude cause detorsion of the helix. The force necessary for detorsion can be determined from the voltage applied to the QCM at the rupture moment. The high sensitivity of this method is due to the fact that measurements are carried out in the frequency region around the QCM resonance, where any (even very weak) distortions of the consistent oscillating system cause noticeable distortions of the amplitude-frequency dependence, and these distortions are used to fix the rupture moment. The measured rupture forces were within 30-40 pN, and the sensitivity was 10(8) molecules. It was demonstrated that the proposed procedure allows one to determine the factors that affect the stability of the DNA double helix. This procedure can be the basis for the development of a new method of rapid DNA analysis. Experiments performed with model DNA showed that it is possible to reveal complementarity between two DNA samples.

A Piezoelectric Gyroscope with Self-equilibrated Coriolis Force Based on Overtone Thickness-shear Modes of a Lithium Niobate Plate with an Inversion Layer

We propose a new structure for piezoelectric gyroscope application. It is made from a Z-cut lithium niobate plate with a ferroelectric inversion layer. It is shown theoretically that when the plate is electrically driven into thickness-shear vibration in one of the two in-plane directions of the plate and is rotating about the plate normal, the Coriolis force due to rotation causes a thickness-shear vibration in a perpendicular direction with an electrical output, which can be used to measure the angular rate of the rotation. Different from existing thickness-shear mode piezoelectric gyroscopes, which are based on the fundamental thickness-shear modes, the proposed gyroscope operates with the second overtone thickness-shear modes because of the inversion layer for which the Coriolis force is a self-equilibrated system and does not transmit to the mounting structure. This implies higher quality factor and better performance.

A Composite Steel Plate Shear Walls for Offshore Constructions

A new-type of weldable composite steel plate shear wall, which consists of a steel plate sandwiched by either of two or one composite panels at each side or at one side, has been proposed. An analytical model for such shear wall - via shell model is derived and the vibrational modes are discussed. Truss reinforcement is used to increase the integration between the steel and composite layers and the cross sectional properties were graded by magnetic nanoparticles fillers. The thickness shear modes at the composite wall appear higher than those of thickness stretch modes, but they are varied in a very orderly manner with respect to the vibrational mode. Also, some of characteristics are examined.

Variational formulation of the Stevens-Tiersten equation and application in the analysis of rectangular trapped-energy quartz resonators

The two-dimensional scalar differential equation for transversely varying thickness modes in quartz crystal resonators operating with thickness-shear modes is formulated into variational form for trapped-energy resonators with both electroded and unelectroded regions. A theoretical analysis of rectangular trapped-energy resonators of singly rotated quartz is performed using the Ritz method based on the variational formulation. Free vibration resonant frequencies and modes are obtained. The results show the existence of trapped modes under the electrodes. The effects of various geometric and physical parameters on the trapped modes are examined. It is also found that the classical frequency prediction given by Tiersten and Smythe from an approximate analysis using the scalar differential equation has an inaccuracy on the order of 100 ppm for the fundamental mode, significant in resonator design.

Electro‐elastic properties of YCa4O(BO3)3 piezoelectric crystals

YCa4O(BO3)3 (YCOB) crystal was reported to be a potential material for high temperature piezoelectric sensing applications. In this paper, the full matrix of electro‐elastic constants, including dielectric permittivities, elastic constants, and piezoelectric coefficients were determined. The relative dielectric permittivities were measured to be , , , and , respectively. The piezoelectric coefficients were determined to be d11 = 1.7, d12 = 3.9, d13 = −4.2, d15 = −1.1, d24 = 4.4, d26 = 7.9, d31 = −0.77, d32 = −2.5, d33 = 1.4, and d35 = −5.0 pC N−1. In addition, piezoelectric cross‐talks of YCOB crystals were investigated, where (XZlw)45°/35° and (YXt)−30° crystal cuts were found to possess pure thickness shear mode, with relatively large effective piezoelectric coefficients, being on the order of d15 = 8.8 and d26 = 9.0 pC N−1, respectively.

Scalar Differential Equation for Slowly-Varying Thickness-Shear Modes in AT-Cut Quartz Resonators With Surface Impedance for Acoustic Wave Sensor Application

For time-harmonic motions, we generalize a 2-D scalar differential equation derived previously by Tiersten for slowly-varying thickness-shear vibrations of AT-cut quartz resonators. The purpose of the generalization is to include the effects of surface acoustic impedance from, e.g., mass layers or fluids for sensor applications. In addition to the variation of fields along the plate thickness, which is considered in the usual 1-D acoustic wave sensor models, the equation obtained also describes in-plane variations of the fields, and therefore can be used to study the vibrations of finite plate sensors with edge effects. The equation is compared with the theory of piezoelectricity in the special cases of acoustic waves and pure thickness vibrations in unbounded plates. An example of a finite rectangular plate is also given.

Amplitude evolution equation and transient effects in piezoelectric crystal resonators

We study transient effects in AT-cut quartz plate thickness-shear mode acoustic wave resonators. Mindlin's equations for thin crystal plates are employed. Based on the relatively slow evolution of the vibration amplitude compared to the high-frequency thickness-shear vibration, a first-order differential equation governing the amplitude evolution is obtained. The equation allows simple and accurate solutions for transient processes in quartz resonators. A few numerical examples of resonator startup and turning off are given. The equation can be applied to other transient processes in crystal resonators such as driving voltage fluctuations and generalized to resonator-based acoustic wave sensors.

Magnetoelectric effect at thickness shear mode in ferrite-piezoelectric bilayer

This manuscript discusses first observation of magnetoelectric (ME) coupling at thickness shear modes in a magnetostrictive-piezoelectric bilayer. The measured value of magnetoelectric voltage coefficient is optimized by adjusting the geometry of layers. Theoretical estimates for the bilayer of Y-cut langatate and yttrium iron garnet are in satisfactory agreement with data. The obtained results are expected to facilitate observation of enhanced ME coupling at overlapping of electromechanical and magnetic resonance due to energy transfer between spin waves and phonons.

Formation of Bending-Wave Band Structures in Bicoupled Beam-Type Phononic Crystals

Beam-type phononic crystals as one kind of periodic material bear frequency bands for bending waves. For the first time, this paper presents formation mechanisms of the phase constant spectra in pass-bands of bending waves (coupled flexural and thickness-shear waves) in bicoupled beam-type phononic crystals based on the model of periodic binary beam with rigidly connected joints. Closed-form dispersion relation of bending waves in the bicoupled periodic binary beam is obtained by our proposed method of reverberation-ray matrix (MRRM), based on which the bending-wave band structures in the bicoupled binary beam phononic crystal are found to be generated from the dispersion curves of the equivalent bending waves in the unit cell due to the zone folding effect, the cut-off characteristic of thickness-shear wave mode, and the wave interference phenomenon. The ratios of band-coefficient products, the characteristic times of the unit cell and the characteristic times of the constituent beams are revealed as the three kinds of essential parameters deciding the formation of bending-wave band structures. The MRRM, the closed-form dispersion relation, the formation mechanisms, and the essential parameters for the bending-wave band structures in bicoupled binary beam phononic crystals are validated by numerical examples, all of which will promote the applications of beam-type phononic crystals for wave filtering/guiding and vibration isolation/control.

Frequency shift of a crystal quartz resonator in thickness-shear modes induced by an array of hemispherical material units.

A two-dimensional model was established to study the dynamic characteristics of a quartz crystal resonator with the upper surface covered by an array of hemispherical material units. A frequency-dependent equivalent mass ratio was proposed to simulate the effect of the covered units on frequency shift of the resonator system. It was found that the equivalent mass ratio alternately becomes positive or negative with change of shear modulus and radius of each material unit, which indicates that the equivalent mass ratio is strongly related to the vibration mode of the covered loadings. The further numerical results show the cyclical feature in the relationship of frequency shift and shear modulus/radius as expected. The solutions are useful in the analysis of frequency stability of quartz resonators and acoustic wave sensors.

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

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.

Characterization of functionally graded elastic materials using a thickness-shear mode quartz resonator

We propose the use of a quartz crystal plate thickness-shear (TSh) mode resonator to measure material property variations in a functionally graded material (FGM). A theoretical analysis is performed on TSh vibrations of an AT-cut quartz plate carrying a layer of an FGM whose density and stiffness vary along its thickness. The effect of the material property gradient on the resonant frequencies of the two-layer plate as a compound resonator is examined. It is shown that this effect may be used to measure the material property gradient of the FGM.

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.

Vapor Discrimination With Single- and Multitransducer Arrays of Nanoparticle-Coated Chemiresistors and Resonators

This study explores whether arrays of vapor sensors assembled from two different types of transducers provide greater response diversity than arrays of a single transducer (ST) type. Calibrated sensitivities to five vapors of four chemiresistors (CRs) and four thickness-shear mode resonators (TSMR) coated with matching interface films of four different thiolate-monolayer-protected gold nanoparticles (MPNs) were considered. Test vapors consisted of toluene, nitromethane, 2-butanone, <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$n$</tex></formula> -propanol, and <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$n$</tex></formula> -octane. The pooled set of 40 vapor-sensor sensitivities was analyzed using principal components regression models in conjunction with Monte Carlo simulations to evaluate the classification performance with different levels of error superimposed on the sensor responses. Recognition rates (RR) were estimated for the individual vapors and their binary mixtures with virtual arrays consisting of all possible combinations of MPNs and transducer types. The best overall performance was obtained with a multitransducer (MT) array of <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$n=4$</tex></formula> sensors, which provided average RRs of 99.7% for individual vapor discrimination and 74.7% for discrimination of the 10 binary mixtures from their components, both with 5% superimposed error. MT-array RR values did not improve for <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$n&gt;4$</tex></formula> . The corresponding average RRs for the all-CR and all-TSMR 4-sensor ST arrays were both <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${\sim}97\%$</tex></formula> for the individual vapors and both <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${\sim}69\%$</tex></formula> for the binary mixtures, respectively. Results demonstrate that MT arrays can provide modestly greater diversity than ST arrays of similar dimension.

Effects of the viscoelastic interface bonding on a thickness-shear mode circular cylindrical piezoelectric transformer

A cylindrical piezoelectric transformer with an imperfect interface operating with thickness-shear modes is used within the three-dimensional equations of piezoelectricity. The shear–slip interface model is used to simulate the effect of viscoelastic imperfect interface. The influence of interface parameters on the transforming ratio, input admittance, power density, efficiency, and the displacement and stress distributions along the radius direction is discussed. Numerical results show that the weak interface lowers the performance of the transformer overall, which may provide theoretical guidance for the design of piezoelectric transformers.

Thickness-shear vibration of a rectangular quartz plate with partial electrodes

We study free vibration of a thickness-shear mode crystal resonator of AT-cut quartz. The resonator is a rectangular plate partially and symmetrically electroded at the center with rectangular electrodes. A single-mode, three-dimensional equation governing the thickness-shear displacement is used. A Fourier series solution is obtained. Numerical results calculated from the series show that there exist trapped thickness-shear modes whose vibration is mainly under the electrodes and decays rapidly outside the electrodes. The effects of the electrode size and thickness on the trapped modes are examined.