Thickness-shear and flexural vibrations of linearly contoured crystal strips with multiprecision computation

Wang, Ji; Lee, P.C.Y.; Bailey, David H.

doi:10.1016/s0045-7949(98)00189-8

Cited by 35 publications

(19 citation statements)

References 12 publications

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“…The procedure has been used to derive approximate plate theories for both elastic and piezoelectric crystal plates with uniform [15][16][17] and nonuniform thickness [18][19][20][21]. Following the same procedure, we derive a set of approximate equations for axisymmetrically contoured elastic plates in this section, and develop analytical solutions for torsional vibrations in stepped and linearly contoured circular plates in Sections III and IV, respectively.…”

Section: Two-dimensional Plate Equationsmentioning

confidence: 99%

“…In thickness-shear mode quartz crystal resonators, partial electrodes are used to achieve energy trapping, for which the size of the electrodes has an upper limit given by Bechmann's number in order to eliminate anharmonic overtones [4][5][6][7][8][9]. The effectiveness of energy trapping can be further improved by using contoured crystal plates with decreasing thickness from the center to the edges, such as beveled and convex plates [18][19][20][21]. It was demonstrated that torsional waves can be trapped in stepped elastic cylinders [13], in which case the torsional waves are propagating in the axial direction in the region with a greater diameter but nonpropagating (evanescent) in the regions with a smaller diameter.…”

Section: B First-order Torsional Modesmentioning

confidence: 99%

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Trapped Torsional Vibrations in Elastic Plates

Kang

Huang

Knowles³

2006

2006 IEEE International Frequency Control Symposium and Exposition

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Abstract-We have observed that torsional vibrations can be trapped in elastic plates with circular regions of slightly thicker steps or with smooth convex contoured surfaces. An electromagnetic acoustic transducer (EMAT) was used to generate oscillatory surface traction. The resonant frequencies and Q-values were measured. It was found that these trapped torsional modes have Q-values exceeding 100,000 with pure inplane motion, which is of practical importance for acoustic sensor applications.In this paper, a set of approximate two-dimensional equations is developed to study vibrations in axisymmetrically contoured or stepped elastic plates. By assuming circumferentially independent motion, the first-order equations are decoupled into four groups, with torsional modes uncoupled from flexural and extensional modes. Analytical solutions for torsional modes are obtained for stepped and linearly contoured circular plates. It is found that the firstorder torsional modes can be trapped in an infinite plate with a stepped or contoured region if critical conditions for the geometrical parameters are met. The analytical results are compared to experiments and finite element analyses with good agreements. I. INTRODUCTIONMechanical resonators with energy trapping typically have low losses and hence high quality factors, and are therefore sensitive to surface loading. An example of this is the thickness-shear mode quartz crystal microbalance (QCM), which has found broad applications as detectors for mass deposition, for chemical and biochemical absorption, and for liquid phase sensing [1][2][3]. Energy trapping in QCM is usually achieved by confining thickness-shear mode vibrations under a thin-film electrode deposited on part of crystal surface, which eliminates crystal edges and mounting structures as sources of energy loss. The quartz plate can be regarded as an acoustic waveguide, with the electrode acting as a mass load. The mass reduces the cut-off frequency and results in a frequency band, within which at least one trapped resonance exists [4,5]. To eliminate undesirable overtones, the upper limit of the ratio between the electrode size and the quartz thickness is set by Bechmann's number, as given by Mindlin and Lee [6] and others [7][8][9].

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Section: Two-dimensional Plate Equationsmentioning

confidence: 99%

Section: B First-order Torsional Modesmentioning

confidence: 99%

Trapped Torsional Vibrations in Elastic Plates

Kang

Huang

Knowles³

2006

2006 IEEE International Frequency Control Symposium and Exposition

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“…For example, the behaviors of cylindrical TSh motions or the so-called straight-crested waves depending on one of the two in-plane spatial variables of the crystal plates only are now reasonably well understood. [5][6][7] Finite element procedures based on the 2-D plate equations were also developed. 8,9 More references can be found in a review article by Wang and Yang.…”

Section: Introductionmentioning

confidence: 99%

Electrically Forced Vibrations of Partially Electroded Rectangular Quartz Plate Piezoelectric Resonators

Chen¹,

Wang²,

Du³

et al. 2017

IJAV

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We performed a theoretical analysis on electrically forced, coupled thickness-shear, and flexural vibrations of a rectangular AT-cut quartz resonator, which is electroded in its central part only. Mindlin's first-order theory for piezoelectric plates is used and simplified by the flexure-twist approximation due to Mindlin and Spencer. An analytical solution is then obtained. The admittance of the resonator under a time-harmonic driving voltage, which is an important property in resonator design, is calculated and examined. The mode shapes at resonances are also presented and discussed. It is found that for a perfectly symmetric resonator only cylindrical modes depending on one of the two in-plane spatial coordinates can be excited by the applied voltage. This supports the conventional approximate analyses based on the assumption of cylindrical deformations.

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“…By mass loading of quartz crystal plates with electrodes, thickness-shear vibrations can be trapped under the electrodes with amplitude decreasing exponentially away from the electrodes in the unplated region [12]- [16]. It also has been observed that, by decreasing the plate thickness from center to edge or contouring, the thickness-shear vibrations can be confined in the center portion of the plate, which improves the resonator performance by reducing edge leakage due to boundary mismatch and mode conversion [17]- [19]. Much less attention has been paid to torsional vibrations of plates.…”

Section: Introductionmentioning

confidence: 99%

Torsional vibrations of circular elastic plates with thickness steps

Kang

Huang

Knowles³

2006

IEEE Trans. Ultrason., Ferroelect., Freq. Contr.

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Abstract-This paper presents a theoretical study of torsional vibrations in isotropic elastic plates. The exact solutions for torsional vibrations in circular and annular plates are first reviewed. Then, an approximate method is developed to analyze torsional vibrations of circular plates with thickness steps. The method is based on an approximate plate theory for torsional vibrations derived from the variational principle following Mindlin's series expansion method. Approximate solutions for the zeroth-and first-order torsional modes in the circular plate with one thickness step are presented. It is found that, within a narrow frequency range, the first-order torsional modes can be trapped in the inner region where the thickness exceeds that of the outer region. The mode shapes clearly show that both the displacement and the stress amplitudes decay exponentially away from the thickness step. The existence and the number of the trapped first-order torsional modes in a circular mesa on an infinite plate are determined as functions of the normalized geometric parameters, which may serve as a guide for designing distributed torsionalmode resonators for sensing applications. Comparisons between the theoretical predictions and experimental measurements show close agreements in the resonance frequencies of trapped torsional modes.

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Thickness-shear and flexural vibrations of linearly contoured crystal strips with multiprecision computation

Cited by 35 publications

References 12 publications

Trapped Torsional Vibrations in Elastic Plates

Trapped Torsional Vibrations in Elastic Plates

Electrically Forced Vibrations of Partially Electroded Rectangular Quartz Plate Piezoelectric Resonators

Torsional vibrations of circular elastic plates with thickness steps

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