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Waveguides of annular acoustic black holes for the
conduction and focusing of flexural waves in plates
Jie Deng, Oriol Guasch, Ling Zheng
To cite this version:
Jie Deng, Oriol Guasch, Ling Zheng. Waveguides of annular acoustic black holes for the conduction
and focusing of flexural waves in plates. Forum Acusticum, Dec 2020, Lyon, France. pp.661-662,
�10.48465/fa.2020.0127�. �hal-03240222�
WAVEGUIDES OF ANNULAR ACOUSTIC BLACK HOLES FOR THE
CONDUCTION AND FOCUSING OF FLEXURAL WAVES IN PLATES
Jie Deng
1,2Oriol Guasch
2Ling Zheng
11
State Key Laboratory of Mechanical Transmission,
College of Automotive Engineering Chongqing University, Chongqing 400044, PR China
2
GTM - Grup de Recerca en Tecnologies M`edia, La Salle,
Universitat Ramon Llull C/ Quatre Camins 30, 08022 Barcelona, Catalonia (Spain)
dengjie@cqu.edu.cn, oriol.guasch@salle.url.edu
ABSTRACT
Arrays of acoustic black hole (ABH) indentations on plates exhibit some amazing properties in terms of flexural wave manipulation, constituting an alternative to some single-phase metamaterials. In this work, we focus on collimation effects and present some waveguide designs consisting of annular ABH arrangements. The waveguides are able to conduct flexural waves on simple curved paths and to fo-cus waves for energy concentration. The cases of infinite and finite plates are addressed. First, the vibration of in-finite phononic ABH plates is accomplished by means of the semi-analytical Gaussian expansion method, which is herein extended to deal with periodic boundary conditions. Equi-frequency contours are inspected to find frequency bands prone to collimation and the influence of annular ABH geometrical parameters on the bands is reported. The information is then used to design ABH waveguides on fi-nite plates that can collimate waves at the identified fre-quencies, to achieve conduction and energy convergence.
1. INTRODUCTION
Acoustic black holes (ABHs) achieved by tailoring struc-tural thickness following power law have drawn increas-ing attention in recent years. The enterincreas-ing wave expe-riences a slowdown of the group and phase velocities, together with an amplification of wave peak along the thickness reduction direction. This has been proved in many one-dimensional investigations [1]. Usually in two-dimensional cases, the wave propagation direction will gradually change in the ABH area, resulting in the wave energy concentrating in the ABH center for perfect ABHs or gathering at a point behind the center for imperfect ABHs [2]. Despite this, multiple or periodic embedded ABHs, instead of only one ABH, could be further adopted for manipulate flexural waves in plates.
Several pioneering endeavour have been performed to focusing and collimating waves using periodic ABHs [3], which is of great interest to reveal the potential of ABHs in terms of wave manipulations. However, the existent works are always based on finite element method (FEM) which is hard for parametric analysis due to the large amount of
computation. Thus it is very urgent to develop an analytical or semi-analytical model towards this goal.
In this paper, we resort to the Gaussian expansion method (GEM) as a semi-analytical approach to recover the displacement field of plates with multiple ABHs [4]. Next, four Bloch-Floquet boundary conditions are applied for a single ABH cell to obtain the Equi-frequency con-tours (EFCs) that guide us to select frequencies of colli-mation effects. As the final task, some waveguides are de-signed for conducting or focusing waves.
2. MODELLING METHODOLOGY 2.1 Equations of motion
In this paper, we adopt the Kirchhoff-Love plate theory to describe the displacements of the plate. Only the bending wave is considered, which can be decomposed as
w(x, y, t) =X
i
ai(t)ϕi(x, y) =: a>ϕ = ϕ>a, (1)
where ϕ = α⊗β, the vectors α and β respectively contain Gaussian basis functions in the x and y direction. They are
αjk i = 2 j/2e−(2jx−k i)2/2, βp ri= 2 p/2e−(2py−r i)2/2. (2)
Then we can incorporate the kinetic K, potential U ener-gies and external work W , and the Euler-Lagrange equa-tions to reach the equaequa-tions of motion of the system
−ω2M + Kˆ
A = ˆF . (3)
2.2 Bloch-Floquet boundary conditions
To reveal the equi-frequency contours of the ABH lattice, four Bloch-Floquet boundary conditions are applied at the edges of a single ABH. We first consider the periodici-ties of displacement, rotation, bending moment, and force, along x direction [5]
w(−a, y) = w(a, y)e−2ikxa, (4)
∂xw(−a, y) = ∂xw(a, y)e−2ikxa, (5)
Ω 0.50 0.52 0.54 0.56 0.58 0.60 0.62 -0.5 -0.3 -0.1 kx 0.1 0.3 0.5 -0.5 -0.3 -0.1 0.1 0.3 0.5 ky huni 2b a rout hc rin rc (a) (b)
Figure 1. (a) The geometry of a single annular ABH cell. (b) The EFC for the third band.
∂xx2 w(−a, y) = ∂2xxw(a, y)e−2ikxa, (6)
∂xxx3 w(−a, y) = ∂
3
xxxw(a, y)e−2ikx a
, (7)
where 2a is the lattice constant and kxthe wavenumber is
x direction. These boundary conditions require the shape function in Eq. (2) to be reconstructed. The similar pro-cedure can be performed in y direction. Thus we could reach the eigenvalue problem in the wavenumber domain to calculate the EFCs
−ω2M (k
x, ky) + K(kx, ky)
ˆ
A = 0. (8)
3. RESULTS
3.1 Equi frequency contours for collimation
In this paper, annular ABH [6] is adopted to achieve high collimation level, its geometry is sketched in
Fig. 1a. The ABH thickness variation complies the
power law h(r) = ε|r − rc|m + hc, where rc =
(rout + rin)/2. Parameters [a, b, rout, rin, huni, hc] =
[0.05, 0.05, 0.05, 0, 0.001, 0.067] m and m = 2.5 are cho-sen for the ABH. The plate is made of structural steel. The third band EFC of the annular ABH is computed and plotted in Fig. 1b (note the non-dimensional frequencies Ω are normalized to the bending wavelength). It is seen that the EFC is mainly square-shaped, which shows very strong collimation effect. This is very useful for conduct-ing waves usconduct-ing periodic ABH arrays. From the color scale of Fig. 1b, it is observed that the collimation effect ranges from [0.5,0.54], thus the central frequency Ω = 0.52 is se-lected in the forthcoming analysis for typically exhibiting the wave manipulation.
3.2 Waveguides for manipulations
In this section, a square plate of dimension 1.6 m × 0.8 m × 0.01 m is considered for the host plate. The lattice constant 2a = 0.04 m is chosen for the annular ABHs. As plotted in Fig. 2a, the absolute displacement is cal-culated under a line force of strength 1 N/m, and length
lf = 0.12 m. A perfectly matched layer is set surrounding
the plate to cancel the boundary reflections. It is seen that wave strength is lowered and divergent as propagation dis-tance increases. Three rows of ABHs are then embedded
(a)
(c)
(b)
(d)
Figure 2. Absolute displacement of (a) uniform plate, (b) straight waveguide, (c) curved waveguide, and (d) conver-gent waveguide.
in the plate. As seen in Fig. 2b, the wave is perfectly con-ducted by the straight waveguide. Besides, the curved and
convergent (lf = 0.4 m) waveguides are further designed
for change the wave propagation direction and focusing, respectively (see Figs. 2c and 2d).
4. CONCLUSIONS
In this paper, a semi-analytical model is developed to col-limate the waves in waveguides. To that goal, the EFC is calculated by inserting the periodic boundary conditions,
which helps finding the available frequency band.
Fi-nally, several waveguides have been tested, showing very promising application of the present method and designs.
5. REFERENCES
[1] J. Deng, L. Zheng, P. Zeng, Y. Zuo, and O. Guasch, “Passive constrained viscoelastic layers to improve the efficiency of truncated acoustic black holes in beams,” Mech. Syst. Signal Pr., vol. 118, pp. 461–476, 2019. [2] W. Huang, H. Ji, J. Qiu, and L. Cheng, “Analysis of ray
trajectories of flexural waves propagating over gener-alized acoustic black hole indentations,” J. Sound Vib., vol. 417, pp. 216–226, 2018.
[3] H. Zhu and F. Semperlotti, “Two-dimensional
structure-embedded acoustic lenses based on periodic acoustic black holes,” J. Appl. Phys., vol. 122, no. 6, p. 065104, 2017.
[4] J. Deng, L. Zheng, O. Guasch, H. Wu, P. Zeng, and Y. Zuo, “Gaussian expansion for the vibration analysis of plates with multiple acoustic black holes indenta-tions,” Mech. Syst. Signal Pr., vol. 131, pp. 317–334, 2019.
[5] J. Deng, O. Guasch, and L. Zheng, “A semi-analytical method for characterizing vibrations in circular beams with embedded acoustic black holes,” J. Sound Vib., p. 115307, 2020.
[6] J. Deng, O. Guasch, and L. Zheng, “Ring-shaped acoustic black holes for broadband vibration isolation in plates,” J. Sound Vib., vol. 458, pp. 109–122, 2019.
