Elsevier

Journal of Sound and Vibration

Volume 375, 4 August 2016, Pages 38-52
Journal of Sound and Vibration

Scattering of flexural waves by a pit of quadratic profile inserted in an infinite thin plate

https://doi.org/10.1016/j.jsv.2016.04.034Get rights and content

Abstract

An acoustic black hole (ABH) is a pit of power law profile inserted in a plate with internal damping. Such a pit with varying thickness leads to local variations of wave propagation properties and has been found to be an efficient vibration damper. In this paper, the ABH is seen as a penetrable obstacle and is studied through its scattering properties. A wave based model is developed within the framework of the Kirchhoff theory for a thin plate of locally varying thickness. It is shown that analytical solutions can be found in the case of a pit of quadratic profile and for uniform damping properties (without added layer). A major outcome of the model is the derivation of the dispersion relations giving a detailed analysis of the behavior of the waves within the ABH. Particularly, cut-on frequencies below which no wave propagates within the ABH are derived and thus give interpretation of the well known typical ABH efficiency threshold frequency for damping vibrations. The analysis is led from numerical computations of the dispersion curves, the scattering diagrams and the scattering cross-section, which are compared to the case of a simple hole. The results show that the ABH behaves as a resonant scatterer, which is a key outcome of this study. The so-called trapped modes, which describe free damped oscillations of the ABH, are responsible for variations of the scattering cross-section with frequency. These variations are investigated thanks to a parametric study on the geometrical properties of the ABH.

Introduction

The lightening of mechanical structures constitutes a major practical issue in transport industry and in mechanical engineering since it generally leads to an increase in vibrational and acoustical levels. The usual passive methods, based on the use of viscoelastic coatings, cannot always be used due to the additional mass they involve. In this context, a proposition consists in modifying the geometry of the structure to provide the so-called acoustic black hole (ABH) effect able to lead to large reduction of vibrational levels without mass increase of the structure. The ABH effect is related to bending wave properties in a beam of decreasing thickness and has been first theoretically shown by Mironov [1] in the context of geometrical acoustics. At the neighborhood of the edge of a beam, if the thickness smoothly decreases, the wave slows down. If the thickness profile follows a power law and the thickness strictly vanishes at the edge, it can be shown that the needed travel time for a wave to reach the edge becomes infinite. Thus, the wave is not reflected at the edge and the reflection coefficient vanishes. In practice, ABH manufacture always involves non-zero residual thickness at the edge (called truncation thickness) never small enough and so the reflection coefficient of the ABH area becomes large. Krylov showed [2] that the addition of a thin layer of viscoelastic material at the extremity can significantly reduce the effect of the truncation. This idea has been extended to two-dimensional structures at the edge of rectangular plates [3], [4] or inserted on plates of various geometries [5], which provide particularly attractive reductions of vibrational levels [6], [7].

When an incident wave interacts with an obstacle, a scattering phenomenon occurs: the obstacle behaves as a secondary source that radiates waves going out of itself. Over the last century, wave scattering by different kinds of obstacles has been extensively studied: Morse and Feshbach [8] gave the general theoretical framework for analyzing the scattering from cylinders and spheres of both electromagnetic and acoustic waves. An application to the case of flexural waves in thin plates has been done by Mow and Pao [9] within the framework of the Kirchhoff theory to analyze scattering from a circular rigid inclusion. Norris and Vemula [10] studied analytically the scattering properties of soft and rigid obstacles inserted in an infinite thin plate. They gave results for the limiting cases of a circular rigid inclusion and a simple hole. The same authors [11] also analyzed the same scatterers within the framework of the Mindlin plate theory and pointed out the difference between these two plate theories. Squire and Dixon [12] applied a similar method to analyze the flexural wave scattering from a coated cylindrical obstacle in an infinite thin plate. In this case, since the scatterer is penetrable, the coupling between internal and external displacement fields is described by the continuity relations at the interface. The ABH belongs to this class of penetrable scatterers, for which there exists a field internal to the obstacle.

Although O׳boy and Krylov [13] developed a numerical approach to the calculation of mobilities for a circular plate with a central ABH, a theoretical description of the more general problem of the scattering of an incident plane wave from an ABH is not available in the literature. This paper aims to apply the theoretical formulation from [10] to the particular case of a penetrable scatterer that consists of a circular pit of quadratic profile inserted in an infinite plate of uniform thickness. In order to reach a detailed understanding of the wave field inside and around the ABH, an analytical solution of the problem is sought. Such an analytical solution requires the power-law profile to be quadratic and of uniform material properties, that is without added damping layer. Hence, the physical understanding provided by this work would then be useful to develop other numerical approaches dedicated to the design of ABH inserted on industrial mechanical structures.

In Section 2, the governing equations are presented and the resolution method is developed to model the scattering from an ABH. The known case of a simple hole is also recalled to be compared with the ABH case. In Section 3, results from the model are presented to analyze the main ABH scattering characteristics. Trapped modes are obtained and are studied in more detail in Section 4. Parametric studies of the scattering properties are reported in Section 5. The conclusion finally gives the main outcomes and further work.

Section snippets

Statement of the problem

An infinite plate of constant thickness h0, made of a material of mass density ρ, complex Young׳s modulus E and Poisson׳s ratio ν, is considered. A circular scatterer of external radius b is placed at the origin of a polar coordinates system (r,θ). This scatterer can be a simple hole (Fig. 1(a)) or an ABH of external radius b and internal radius a (Fig. 1(b)).

The ABH scatterer consists of a pit of power-law radial thickness profile given byh(r)={h0(raba)mwithm2,a<r<bh0,r=b.

Since the ABH

Numerical results

All the numerical results presented in Section 3 are obtained for an aluminum plate for which geometrical and material properties are given in Table 1. These geometrical settings involve a thickness ratio ht/h0=1/100 with ht being the truncation thickness of the ABH.

Trapped modes of flexural waves

The presence of a penetrable scatterer in an infinite medium can give rise to local resonance phenomena, related to its so-called trapped modes. These modes correspond to the free oscillations of finite energy localized around and eventually inside the scatterer, if penetrable. Trapped modes have been observed and studied in different fields of physics. In the case of a two-dimensional acoustic waveguide, Callan et al. [23] studied the local free oscillations in the vicinity of a circular rigid

Parametric studies

The material and geometrical parameters of the ABH are numerous and their effect on the scattering properties would be evaluated in order to set some designing rules able to tailor the ABH scattering properties to specific mechanical engineering applications. Since the response of the ABH is the result of a complex combination of circumferential orders, to find explicit links between the ABH design and its scattering properties is difficult in the framework of this model. Nevertheless,

Conclusion

In this paper, the scattering of flexural waves from an acoustic black hole, seen as a penetrable scatterer, is modeled within the framework of the Kirchhoff theory. In the case of a quadratic profile (m=2) with uniform internal damping properties, the analytical derivation of the wave dispersion relations inside the ABH is a major outcome of the model. Cut-on frequencies are analytically obtained, giving a theoretical interpretation of the threshold frequency for damping vibration observed

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