Calculating the full leaky Lamb wave spectrum with exact fluid interaction.

Daniel A. Kiefer,Stefan J. Rupitsch,Michael Ponschab,Michael Mayle

doi:10.1121/1.5109399

Abstract

Lamb waves are elastodynamic guided waves in plates and are used for non-destructive evaluation, sensors, and material characterization. These applications rely on the knowledge of the dispersion characteristics, i.e., the frequency-dependent wavenumbers. The interaction of a plate with an adjacent fluid leads to a nonlinear differential eigenvalue problem with a square root term describing exchange of energy with the surrounding medium, e.g., via acoustic radiation. In this contribution, a spectral collocation scheme is applied to discretize the differential eigenvalue problem. A change of variable is performed to obtain an equivalent polynomial eigenvalue problem of fourth order, which is linear in state-space and can reliably be solved using modern numerical methods. Traditionally, the leaky Lamb wave problem has been solved by finding the roots of the characteristic equations, a numerically ill-conditioned problem. In contrast to root-finding, the approach described in this paper is inherently able to find all modes and naturally handles complex wavenumbers. The full phase velocity dispersion diagram and attenuation curves are presented and are shown to be in excellent agreement with solutions of the characteristic equation as well as computations made with a perturbation method. The procedure is applicable to anisotropic, viscoelastic, inhomogeneous, and layered plates coupled to an inviscid fluid.

Highlights

Lamb waves are guided modes confined inside a plate
Hayashi and Inoue22 solved the fluid loaded plate problem by discretizing the waveguide with finite elements and assuming plane harmonic waves in the surrounding medium. They further use a priori knowledge on the propagation symmetry of leaky Lamb waves to reduce the nonlinear eigenvalue problem to a generalized eigenvalue problem in terms of the transversal wavenumber
Mazzotti et al.23,37 calculated the dispersion characteristics of leaky waveguides of arbitrary crosssection. They combine a finite element discretization of the waveguide with a boundary element discretization of the fluid domain. They solve the resulting nonlinear eigenvalue problem by a contour integral method, which requires a priori knowledge about the regions in the complex plane where solutions may lie

Summary

INTRODUCTION

Lamb waves are guided modes confined inside a plate. Understanding their propagation characteristics is important for diverse applications. Hayashi and Inoue solved the fluid loaded plate problem by discretizing the waveguide with finite elements and assuming plane harmonic waves in the surrounding medium. They further use a priori knowledge on the propagation symmetry of leaky Lamb waves to reduce the nonlinear eigenvalue problem to a generalized eigenvalue problem in terms of the transversal wavenumber. They combine a finite element discretization of the waveguide with a boundary element discretization of the fluid domain They solve the resulting nonlinear eigenvalue problem by a contour integral method, which requires a priori knowledge about the regions in the complex plane where solutions may lie. The resulting computational method is implemented and able to reliably calculate the full dispersion characteristics of the leaky Lamb wave problem with analytically exact fluid interaction

PROBLEM FORMULATION

DISCRETIZATION

CHANGE OF VARIABLE

CALCULATION RESULTS

CONCLUSIONS