Scattering of flexural waves by a pit of quadratic profile inserted in an infinite thin plate
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 . 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 by
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
References (26)
- et al.
Acoustic ‘black holes’ for flexural waves as effective vibration dampers
Journal of Sound and Vibration
(2004) - et al.
Experimental investigation of the acoustic black hole effect for flexural waves in tapered plates
Journal of Sound and Vibration
(2007) - et al.
Damping of flexural vibrations in rectangular plates using the acoustic black hole effect
Journal of Sound and Vibration
(2010) - et al.
Damping of structural vibrations in beams and elliptical plates using the acoustic black hole effect
Journal of Sound and Vibration
(2011) - et al.
Scattering of flexural waves on thin plates
Journal of Sound and Vibration
(1995) - et al.
Flexural wave propagation and scattering on thin plates using Mindlin theory
Wave Motion
(1997) - et al.
Scattering of flexural waves from a coated cylindrical anomaly in a thin plate
Journal of Sound and Vibration
(2000) - et al.
Damping of flexural vibrations in circular plates with tapered central holes
Journal of Sound and Vibration
(2011) - et al.
Modal overlap factor of a beam with an acoustic black hole termination
Journal of Sound and Vibration
(2014) Trapped waves in thin elastic plates
Wave Motion
(2007)
Propagation of a flexural wave in a plate whose thickness decreases smoothly to zero in a finite interval
Soviet Physics
Cited by (80)
Energy transfer for enhanced acoustic black hole effect through a cable-induced mechanical nonlinearity
2024, International Journal of Non-Linear MechanicsWave propagation and vibration attenuation in spiral ABH metamaterial beams
2024, International Journal of Mechanical SciencesBroadband shock vibration absorber based on vibro-impacts and acoustic black hole effect
2024, International Journal of Non-Linear MechanicsExperimental verification of additively manufactured stacked multi-wedge acoustic black holes in beams for low frequency
2024, Mechanical Systems and Signal ProcessingNumerical solution of the cavity scattering problem for flexural waves on thin plates: Linear finite element methods
2024, Journal of Computational PhysicsDamping vibration in three-dimensional helically tapered rod with power-law thickness
2023, International Journal of Mechanical Sciences