Edge waves and resonance on elastic structures: An overview (original) (raw)
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On the edge-wave of a thin elastic plate supported by an elastic half-space
2017
In this contribution, we consider edge-wave propagating in a thin elastic semiinfinite plate which is bilaterally supported by a homogenenous isotropic elastic half-space. The problem is formulated in terms of a eigenproblem constituted by a system of five linear PDEs in the plate transverse displacement and in the scalar and vector elastic potentials subject to mixed boundary conditions accounting for plate-fundation displacement continuity under the plate and zero normal stress outside. Zero tangential stress is envisaged throughout. The problem could be reduced to an inhomogenenous Wiener-Hopf functional equation in terms of the half-space surface displacement and of the plate-to-fundation contact pressure only. The kernel function is analyzed and the Rayleigh wave speed is obtained together with a novel dispersion equation. Finally, kernel factorization is performed.
Edge bending wave on a thin elastic plate resting on a Winkler foundation
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2016
This paper is concerned with elucidation of the general properties of the bending edge wave in a thin linearly elastic plate that is supported by a Winkler foundation. A homogeneous wave of arbitrary profile is considered, and represented in terms of a single harmonic function. This serves as the basis for derivation of an explicit asymptotic model, containing an elliptic equation governing the decay away from the edge, together with a parabolic equation at the edge, corresponding to beam-like behaviour. The model extracts the contribution of the edge wave from the overall dynamic response of the plate, providing significant simplification for analysis of the localized near-edge wave field.
The edge wave on an elastically supported Kirchhoff plate
The Journal of the Acoustical Society of America, 2014
This Letter deals with an analysis of bending edge waves propagating along the free edge of a Kirchhoff plate supported by a Winkler foundation. The presence of a foundation leads to a nonzero cutoff frequency for this wave, along with a local minimum of the associated phase velocity. This minimum phase velocity corresponds to a critical speed of an edge moving load and is analogous to that in the classical 1D moving load problem for an elastically supported beam.
Three-dimensional edge waves in plates
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2008
The paper describes the propagation of three-dimensional symmetric waves localised near the traction free edge of a semi-infinite elastic plate with either traction free or fixed faces. For both types of boundary conditions, we present a variational proof of the existence of the low order edge waves. In addition, for a plate with traction free faces and zero Poisson ratio, the fundamental edge wave is described by a simple explicit formula, and the first order edge wave is proved to exist. Qualitative variational predictions are compared with numerical results, which are obtained using expansions in three-dimensional Rayleigh-Lamb and shear modes. It is also demonstrated numerically that whatever non-zero Poisson ratio in a plate with traction free faces, the eigenfrequencies related to the first order wave are complex valued.
Zeitschrift für angewandte Mathematik und Physik, 2004
Edge-vibration, and associated resonance phenomena, is investigated in respect of a semi-infinite strip composed of pre-stressed incompressible elastic material. The strip is assumed to have a traction free outer edge, with the upper and lower edges subject to some simple mixed boundary conditions. The frequency of the modes of free edge-vibration are shown to depend on the surface wave speed. Moreover, when the normal pre-stress approaches one of two critical values, associated with the vanishing of the surface wave speed, the edge spectrum density of the boundary value problem increases significantly. This problem then provides an example for which the famous Weyl's hypothesis, stating the edge spectrum is secondary in comparison with the whole body's spectrum, is not true. However, the corresponding theorem's statement is valid only with imposition of the Shapiro-Lopatinsky condition, which is not satisfied in this case. Variation of the pre-stress is also shown to greatly influence the resonance frequency arising in the forced vibration problem, to the extent that the phenomenon of resonance may be totally removed.
The edge bending wave on a plate reinforced by a beam (L)
The Journal of the Acoustical Society of America
The edge bending wave on a thin isotropic semi-infinite plate reinforced by a beam is considered within the framework of the classical plate and beam theories. The boundary conditions at the plate edge incorporate both dynamic bending and twisting of the beam. A dispersion relation is derived along with its long-wave approximation. The effect of the problem parameters on the cutoffs of the wave in question is studied asymptotically. The obtained results are compared with calculations for the reinforcement in the form of a strip plate.
The edge waves on a Kirchhoff plate bilaterally supported by a two-parameter elastic foundation
Journal of Vibration and Control, 2015
In this paper, the bending waves propagating along the edge of a semi-infinite Kirchhoff plate resting on a two-parameter Pasternak elastic foundation are studied. Two geometries of the foundation are considered: either it is infinite or it is semi-infinite with the edges of the plate and of the foundation coinciding. Dispersion relations along with phase and group velocity expressions are obtained. It is shown that the semi-infinite foundation setup exhibits a cut-off frequency which is the same as for a Winkler foundation. The phase velocity possesses a minimum which corresponds to the critical velocity of a moving load. The infinite foundation exhibits a cut-off frequency which depends on its relative stiffness and occurs at a nonzero wavenumber, which is in fact hardly observed in elastodynamics. As a result, the associated phase velocity minimum is admissible only up to a limiting value of the stiffness. In the case of a foundation with small stiffness, asymptotic expansions ar...
Journal of Mechanics of Materials and Structures, 2021
The paper is concerned with a bending edge wave on a thin orthotropic elastic plate resting on the Winkler-Fuss foundation. The main focus of the contribution is on derivation of a specialised reduced model accounting for the contribution of the bending edge wave to the overall dynamic response, allowing simplified analysis for a number of dynamic problems. The developed formulation includes an elliptic equation associated with decay over the interior, and a beam-like equation on the edge governing wave propagation accounting for both bending moment and modified shear force excitation, thus highlighting a dual parabolic-elliptic nature of the bending edge wave. A model example illustrates the benefits of the approach.