Trapped torsional vibrations in elastic plates

Abstract

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].

Trapped Torsional Vibrations in Elastic Plates

“…The uncoupling of torsional modes from flexural modes allows pure in-plane motion of the resonator, which is advantageous for applications requiring immersion in or in contact with liquids [12]. It is evident from Eq.…”

“…The theoretical results from this study were compared to experimental measurements [12,23]. Circular steps and contoured regions were machined in a 6000 series cast aluminum plate.…”

“…This interference effect causes depth-sensitive perturbations in the sensor responses [10,11]. Recently, we have discovered that torsional vibrations can be trapped in elastic plates with circular regions of slightly thicker steps or with smooth convex contoured surfaces [12]. In an elastically isotropic plate, torsional modes do not couple to flexural modes, have no out-of-plane motions, and hence no interference effect in contact with liquids, allowing an improved sensor response compared to thickness-shear-mode resonators.…”

Torsional vibrations of circular elastic plates with thickness steps

“…Such plates are of interest because torsional vibrations may be trapped near the steps, similar to that in stepped solid cylinders [11] and trapping of thickness-shear vibrations in contoured plates [17]- [19]. Recent experiments by Knowles et al [26] have observed trapped torsional modes in both stepped and contoured aluminum plates, which may be used to design sensors with improved performance in the presence of liquids. This paper is organized as follows.…”

Energy-Trapping Torsional-Mode Resonators for Liquid Sensing

“…Consequently, no compressional waves are generated in the adjacent liquid, and the sensor response can be related to the liquid properties without concerning the liquid thickness. Previously we have demonstrated that energy-trapping torsional-mode vibrations can be excited in elastic plates by using an electromagnetic acoustic transducer (EMAT) [10,11]. In this paper, experiments are conducted to study effect of liquid thickness on the torsional-mode resonators, in comparison with thickness-shear mode.…”