A method for damping flexural vibrations in thin polygonal plates using the acoustic black hole effect is proposed. Acoustic black holes in thin plates consist of a zone of decreasing thickness in which flexural waves slow down, acting as wave sinks. An acoustic black hole of axisymmetric thickness profile is here tested on different polygonal plates. A parabolic edge is added in order to focus waves into the black hole area. The experimental results show significant reduction of the vibration level. A model based on the image source method provides qualitative agreement with the experiments

CUENCA, Jacques

GAUTIER, François

PELAT, Adrien

I-Revues

20e`me Congre`s Franc¸ais de Me´canique Besanc¸on, 29 aouˆt au 2 septembre 2011
Vibration damping in polygonal plates using the
acoustic black hole effect: model based on the image
source method
Jacques Cuencaa,b, Adrien Pelata, Franc¸ois Gautiera
a. Laboratoire d’Acoustique de l’Universite´ du Maine, Av. O. Messiaen, 75085 Le Mans 9, France
b. Institute of Sound and Vibration Research, University of Southampton, Highfield, Southampton
S017 1BJ, United Kingdom
Abstract:
A method for damping flexural vibrations in thin polygonal plates using the acoustic black hole effect
is proposed. Acoustic black holes in thin plates consist of a zone of decreasing thickness in which
flexural waves slow down, acting as wave sinks. An acoustic black hole of axisymmetric thickness
profile is here tested on different polygonal plates. A parabolic edge is added in order to focus waves
into the black hole area. The experimental results show significant reduction of the vibration level. A
model based on the image source method provides qualitative agreement with the experiments.
Re´sume´ :
Cet article pre´sente une me´thode d’amortissement de vibrations de flexion de plaques polygonales en
utilisant l’effet de trou noir acoustique. Ce dernier a lieu lorsqu’une onde de flexion traverse une zone
d’e´paisseur de´croissante, ce qui a pour effet de la ralentir, agissant ainsi comme un pie`ge a` ondes. Dans
ce travail, diffe´rentes plaques polygonales munies d’un trou noir acoustique a` symme´trie de re´volution
sont e´tudie´es. Un bord parabolique est e´galement utilise´ afin de focaliser les ondes vers le trou noir.
Les re´sultats expe´rimentaux montrent une re´duction importante du niveau vibratoire de la structure.
Par ailleurs, un mode`le phe´nome´nologique base´ sur la me´thode des sources image est de´veloppe´ et
fournit des re´sultats en accord qualitatif avec l’expe´rience.
Keywords: acoustic black hole; polygonal plate; vibration damping
1 Introduction
Flexural vibrations are responsible for sound radiation and damage of a large number of thin structures,
such as vehicle cabins, industrial liquid containers and satellite panels during launch, for example. The
most common methods for passive vibration damping of thin structures rely on surface treatments us-
ing polymer-based viscoelastic layers. The main obstacle for improving the efficiency of such damping
solutions is the increase of the total weight of the structures due to the added layers.
Recently, attention has been paid to the acoustic black hole effect as a lightweight alternative for
vibration damping in thin structures [4–7]. An acoustic black hole in a thin vibrating structure
consists of a region of gradually decreasing thickness, typically in the form of a power-law profile. As
flexural waves enter such area of the structure, their phase velocity decreases and their wavelength
becomes smaller, such that in the ideal case of a thickness profile smoothly decreasing to zero the
waves stop propagating and never reflect back. This was first studied by Mironov [8] within the
approximation of geometrical acoustics. In practice, a perfectly smooth thickness profile is impossible
to achieve, yet it has been proven that the small truncations in the profile can be compensated by
using a small amount of damping film in the area of lower thickness [6].
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20e`me Congre`s Franc¸ais de Me´canique Besanc¸on, 29 aouˆt au 2 septembre 2011
The aim of this paper is to use the acoustic black hole effect for vibration damping in polygonal
plates. Since wave propagation takes place in the plane, a two-dimensional acoustic black hole is used,
consisting of an axi-symmetric pit of power-law thickness profile. In addition, in order to maximise the
effect, the waves are focused towards the region of varying thickness with the aid of a parabolic-shaped
edge in its vicinity.
Section 2 of the paper presents the measurements performed on two polygonal plates and in particular
a comparison of the driving-point mobilities with and without the acoustic black hole treatment. In
sec. 3, a phenomenological model using the image source method is developed as a potential tool for
rapid prototyping acoustic black hole damping solutions.
2 Experimental results on polygonal plates
The aim of this section is to show the vibration damping achieved using the acoustic black hole effect.
In the following, the driving-point mobility of a polygonal plate with all edges free is compared to
the mobility of a plate of same shape with one edge treated with an acoustic black hole extension.
The acoustic black hole extension consists of a parabolic-shaped edge intended to focus flexural waves
towards the acoustic black hole. A rectangular plate and an irregular polygonal plate, both with and
without acoustic black hole extension, are used for the experiments and are depicted in Fig. 1. The
dashed line represents the edges of the polygonal plates without black hole treatment. The dotted
circle represents the boundary of the acoustic black hole pit, which consists of an axi-symmetric
thickness profile of the form
h(r) = h0
(
r
rb
)2
, (1)
where h0 is the thickness of the plate and rb is the radius of the acoustic black hole.
-0.1
0
0.1
0.2
0.3
0.4
0.5
0 0.2 0.4 0.6 0.8 1
y
(
m
)
x (m)
r0
(a)
-0.1
0
0.1
0.2
0.3
0.4
0.5
0 0.2 0.4 0.6 0.8 1
y
(
m
)
x (m)
r0
(b)
Figure 1: Polygonal plates used for the experiments. The dashed line represents the boundary of the
reference plate, without the acoustic black hole treatment. The dotted circle represents the boundary of
the acoustic black hole. Distances are in metres.
The plates are made of 1.5 mm-thick aluminium and the black hole profile is carved using high-speed
machining, the resulting thickness at the thinest point being of approximately 10µm. A small amount
of damping film is placed in the centre of the pit, as described in Ref. [4]. Driving-point mobilities
are acquired using a shaker and an impedance head on point r0 (see Fig. 1) and shown in Fig. 2.
For both plates, the driving-point mobilities show up to 15 dB of reduction of the vibration level. In
particular, the resonant behaviour of the response of the plates is significantly reduced. This is due to
the fact that the black hole extension acts as an open boundary due to its low-reflecting properties,
which dims the interferences with the direct field from the source and the field reflected from the other
boundaries.
2
20e`me Congre`s Franc¸ais de Me´canique Besanc¸on, 29 aouˆt au 2 septembre 2011
-40
-30
-20
-10
0
10
20
30
0 1 2 3 4 5 6 7 8
|Y
0
|
(
d
B
)
f (kHz)
(a)
-40
-30
-20
-10
0
10
20
30
0 1 2 3 4 5 6 7 8
|Y
0
|
(
d
B
)
f (kHz)
(b)
Figure 2: Driving-point mobilities. (a) Rectangular plate; (b) irregular polygonal plate. · · · · · ·, polyg-
onal plate alone; - - - - - , plate with acoustic black hole treatment.
3 Model using the image source method
The image source method consists in simulating the vibrations of a polygonal domain excited by a
point source as the superposition of the vibrations emitted by virtual sources representing successive
reflections on the boundaries. Each boundary is described by its reflection properties, which allows
the method to compare the effect of different boundary conditions on the overall field of the structure.
In the following, the image source method is used as an underlying tool for simulating the vibrations
of the polygonal plates studied above, with and without the acoustic black hole treatment. The image
source method being restricted to homogeneous domains with polygonal boundaries, the acoustic black
hole extension is modelled as an equivalent boundary characterised by its reflection coefficient.
The computation of the flexural vibrations of polygonal plates has been recently studied by the authors
using the image source method [1–3]. The main ideas are summarised in the following. Considering a
polygonal plate Ω, as depicted in Fig. 3, the transverse displacement Green’s function for the flexural
vibrations can be written in the form
G˜Ω(r, r0; kf ) = G∞(r, r0; kf ) +
∞∑
s=1
Gs(r, rs; kf ), (2)
where r0 and r are respectively the source and observer positions and kf is the flexural wavenumber.
The Green’s function of the plate arises as a sum of different terms, where G∞ represents the direct
·
·
···
·
·
··
·
·
··
·
·
·
r0
Figure 3: Construction of the first image sources of a polygonal plate.
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20e`me Congre`s Franc¸ais de Me´canique Besanc¸on, 29 aouˆt au 2 septembre 2011
excitation from the source and the terms Gs represent the contributions of image sources to the field,
s being the image source index. Figure 3 shows the construction of the first image sources of the
irregular polygonal plate used for the experiments. The image sources are obtained by successive
reflections of the original source of the problem, located at r0.
A one-dimensional model developed in Ref. [4] allows to compute the reflection coefficient of a black
hole profile termination as seen from the virtual edge (at 0.08 m from the black hole pit centre).
Using the geometrical and material parameters of the manufactured plates, the black hole extension
is modelled by a reflection coefficient of 0.01. Fig. 4 shows the computed driving-point mobility of the
irregular polygonal plate with all edges free and with the acoustic black hole equivalent edge, using a
total of 2934 sources. The figure shows a reduction of 20 in the vibration level by adding the black
hole treatment. As in the experimental study, the resonant behaviour of the structure is reduced.
From the image source point of view, this can be understood from the fact that the image sources
involving one or more reflections on the equivalent black hole edge are weighted by a low reflection
coefficient, which diminishes the importance of their contribution.
-80
-70
-60
-50
-40
-30
-20
-10
0 500 1000 1500 2000 2500 3000
|Y
0
|
(
d
B
)
f (Hz)
Figure 4: Driving-point mobility of the irregular polygonal plate with and without virtual acoustic black
hole edge.
4 Conclusion
In this paper we presented an application of the acoustic black hole effect to the damping of flexural
vibrations in polygonal plates. In the studied case, the acoustic black hole consists of a pit of power-
law thickness profile acting as a wave sink. Additionally, a parabolic-shaped edge is employed as a
means to maximise the effect by focusing waves into the black hole region. By comparing polygonal
plates with and without an acoustic black hole treatment, it is seen that the measured driving-point
mobilities show a high vibration level reduction. Furthermore, a phenomenological model is developed
using the image source method and predicts a similar reduction of the vibration level. The equivalent
model by the image source method is a first step towards using it as a tool for rapid prototyping
structures with a black hole treatment.
Acknowledgements
The authors would like to thank Vasil B. Georgiev for previous work on the subject and the Institut
Technologique Universitaire of Le Mans for the manufacturing of the plates.
References
[1] J. Cuenca, F. Gautier, and L. Simon. The image source method for calculating the vibrations of
simply supported convex polygonal plates. Journal of Sound and Vibration, 322(4-5):1048–1069,
2009.
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20e`me Congre`s Franc¸ais de Me´canique Besanc¸on, 29 aouˆt au 2 septembre 2011
[2] J. Cuenca, F. Gautier, and L. Simon. Modelling the vibrations of convex polygonal plates by the
image source method. In Noise and Vibration: Emerging Methods (NOVEM), Oxford, 5-8 April
2009.
[3] J. Cuenca, F. Gautier, and L. Simon. Harmonic green’s functions of semi-infinite and convex
polygonal plates with arbitrary boundary conditions. Submitted for publication in the Journal of
Sound and Vibration, 2011.
[4] V.B. Georgiev, J. Cuenca, F. Gautier, L. Simon, and V.V. Krylov. Damping of structural vibrations
in beams and elliptical plates using the acoustic black hole effect. Journal of Sound and Vibration,
330(11):2497–2508, 2011.
[5] V.V. Krylov. New type of vibration dampers utilising the effect of acoustic ‘black holes’. Acta
Acustica United With Acustica, 90:830–837, 2004.
[6] V.V. Krylov and F.J.B.S. Tilman. Acoustic ‘black holes’ for flexural waves as effective vibration
dampers. Journal of Sound and Vibration, 274:605–619, 2004.
[7] V.V. Krylov and R.E.T.B. Winward. Experimental investigation of the acoustic black hole effect
for flexural waves in tapered plates. Journal of Sound and Vibration, 300:43–49, 2007.
[8] M.A. Mironov. Propagation of a flexural wave in a plate whose thickness decreases smoothly to
zero in a finite interval. Soviet Physics – Acoustics, 34:318–319, 1988.
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Vibration damping in polygonal plates using the acoustic black hole effect : model based on the image source method

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Vibration damping in polygonal plates using the acoustic black hole effect : model based on the image source method

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