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.

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].

Abstract-Thickness-shear mode quartz crystal microbalance (QCM) has been widely used as liquid-phase sensors, such as viscometers and bio-detectors. However, due to coupling between the in-plane shear motion and the out-of-plane flexure, when used in contact with or immersed in a liquid, the out-of-plane motion generates compressional waves in the liquid that reflect off the liquid surface and return to the crystal. This interference effect causes depth-sensitive perturbations in the sensor response, often undesirable. In this study, we show that torsional-mode resonators may be used for liquid sensing without the depth effect. Samples in form of stepped plates, circular decals, and convex contoured faces are machined in elastic plates (e.g., cast aluminum, stainless steel, and brass). A non-contact electromagnetic acoustic transducer (EMAT) was employed to drive torsional-mode vibrations. Efficient energy trapping was observed for first-order torsional modes, leading to high quality factors. When placed in contact with water, the resonance frequency of the torsional mode was found to be independent of the water depth, in contrast to depth-dependent frequency oscillation for the thickness-shear mode. Finite element analyses are conducted to understand the torsional-mode vibrations as well as the effect of material anisotropy.

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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].