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].

The theory of a trapped degenerate mode resonator

“…7 There are several important examples of resonant sensors where the resonating element is formed from isotropic material and actuation is performed electromagnetically. 3,4 The highly successful Hemispherical Resonator Gyroscope (HRG) manufactured over several decades by Northrop Grumman and SAGEM uses isotropic fused quartz and electrostatic actuation. Piezoelectric actuation, although common in planar form resonators, are anisotropic and their use would compromise the high degree of symmetry required for exploiting modal degeneracy.…”

Effective phononic crystals for non-Cartesian elastic wave propagation

“…We demonstrate how to generally realize PC properties in non-Cartesian systems for elastic waves using radial axisymmetric torsional 2-D waves, which are relevant for e.g. rotating machinery [23] and liquid sensing [24]. Still, our approach can be applied to other polarizations, wave propagation directions, and dimensions.…”