Vibration of a metaplate made of an acoustic black hole and local resonators

Jie Deng 1

School of Marine Science and Technology, Northwestern Polytechnical University 127# West Youyi Road, Xi’an, China

Oriol Guasch 2

Department of Engineering, La Salle, Universitat Ramon Llull C / Quatre Camins 30, 08022 Barcelona, Catalonia (Spain)

Laurent Maxit 3

INSA–Lyon, Laboratoire Vibrations-Acoustique (LVA) 25 bis, av. Jean Capelle, F-69621 Villeurbanne Cedex, France

Nansha Gao 4

School of Marine Science and Technology, Northwestern Polytechnical University 127# West Youyi Road, Xi’an, China

ABSTRACT In recent years, acoustic black holes (ABHs) have revealed as a very e ffi cient means of reducing vibrations in the mid-high frequency range. However, below the cut-on frequency an ABH fails to work. Besides, locally resonant acoustic metamaterials can generate stopbands and reduce vibrations at a subwavelength scale. In this paper we combine the advantages of metamaterials and ABHs to diminish the vibrations of a plate covering the whole frequency range, from low to high frequencies. The proposed solution, the MMABH plate, consists of a uniform plate with an embedded ABH such that the removed material from the indentation is used to build the resonators and the total weight remains constant. To characterize the MMABH performance, a Gaussian expansion component mode synthesis (GECMS) method is used, based on the modal coupling between the resonators and the ABH plate. The bandgaps of an infinite periodic MMABH plate as well as the modes of a finite one can be accurately predicted with the GECMS. Numerical results show that the low-frequency peaks of an ABH plate can be substantially suppressed when resonators with proper loss factors are attached to it, and tuned at their first resonant frequency. The proposed MMABH shows great potential as a light-weight option to achieve broadband vibration reduction in structures.

1 dengjie@nwpu.edu.cn

2 oriol.guasch@salle.url.edu

3 laurent.maxit @insa-lyon.fr

4 gaonansha@nwpu.edu.cn

a slaty. inter.noise 21-24 AUGUST SCOTTISH EVENT CAMPUS O ¥, ? GLASGOW

(a) (b)

Δ M r k r

r v

r abh h uni

h c h v

Figure 1: The geometrical model of the proposed metaplate (MMABH): (a) 36 resonators attached to the plate surface, (b) the thickness profile of the ABH plate.

1. INTRODUCTION

Light-weight and highly e ffi cient damping solutions are greatly needed for vibration reduction in thin-walled structures. In the past 15 years, acoustic black holes (ABHs) have revealed as a good candidate to that goal. An ABH is achieved by tailoring an indentation on plates such that its thickness diminishes following a power-law profile, h ( r ) = ε r m + h c , in which ε, m , h c and r respectively represent the ABH slope, order, central thickness and distance to the ABH center. Recently, several new types of ABHs have been proposed and examined for vibration reduction [1,2], noise control [3, 4], and wave manipulation [5–8], which are e ff ective in the mid-high frequency range. However, ABHs do not perform well for lower frequencies, when the wavelength is larger than the ABH diameter, r abh . This results in a cut-on frequency, f cut − on , below which the ABH is inoperative [9]. To date, some attempts have been conducted to alleviate this problem. For instance, some authors have resorted to non-linearity to transfer energy from lower to higher frequencies [10–14], while others have considered exploiting periodicity of tangled ABG cells [7], or designing non-local ABH structures [15]. Metamaterials o ff er another possibility to control low-frequency vibrations. A metamaterial is generally achieved by periodically or randomly arranging sub-wavelength structures. In acoustics, they basically rely on local resonances or Bragg scattering for functioning. Let us focus on the former. Locally resonant [16] acoustic metamaterials with periodic resonators have been extensively studied for vibration reduction [17, 18], sound suppression [19–21], and wave manipulation [22, 23]. In this paper, we combine the advantages of this type of metamaterials and ABHs to achieve broadband vibration reduction on a plate, by setting the stopbands of the metamaterial below the cut-on frequency of the ABH structure. This idea was recently proposed by the authors in [24] and hereafter we will present it together with some additional examples. The metamaterial-ABH design (MMABH plate) is plotted in Figure 1. To characterize the performance of the MMABH, we make use of the semi-analytical Gaussian expansion method (GEM) developed by some of the authors in [25], which was proven very e ffi cient to capture the local behavior of ABHs at a very reasonable computational cost. Likewise, to include the influence of the resonators on the ABH plate, we consider the coupling between the ABH modes and the resonators. This is done by taking advantage of component mode synthesis (CMS) in combination with an artificial spring method, which will be referred to as the GECMS (Gaussian Expansion Component Mode Synthesis) approach in this work. The ASCMS facilitates mode synthesis in the framework of the Rayleigh-Ritz method and easily allows for mode truncation to speed up the numerical simulations.

2. THE GAUSSIAN EXPANSION COMPONENT MODE SYNTHESIS (GECMS) METHOD

The GECMS method essentially deals with the ABH plate and the resonators separately, and then assembles them in the Lagrangian of the total system. Let us first consider the bending motion of the ABH plate in harmonic regime and expand in series as,

N X

i = 1 a i ( t ) χ i ( x , y ) = a ⊤ ( t ) χ ( x , y ) = ˆ A ⊤ e i ω t χ ( x , y ) , (1)

w p ( x , y , t ) =

where χ ( x , y ) (with N entries) is a two-dimensional Gaussian function vector and a ( t ) = ˆ A e i ω t its corresponding coe ffi cient. From the kinetic, T p , and potential, V p , energies we get the Lagrangian of the plate,

L = T p − V p = 1

2 ˙ a ⊤ M p ˙ a − 1

2 a ⊤ K p a , (2)

where M p and K p respectively represent the mass and sti ff ness matrices determined by the Gaussian functions. Applying the Euler-Lagrange equations ∂ t ( ∂ ˙ a L ) − ∂ a L = 0 to Equation 2 provides the equations of motion,  K p − ω 2 M p  ˆ A = 0 . (3)

The solution to the eigenvalue pro ble m in Eq uat ion 3 gives the modal matrix ˆ P = [ ˆ A 1 , ˆ A 2 , ˆ A 3 , ..., ˆ A N ], which we can truncate to the first N modes ( N ≪ N ) to work with a reduced order model and speed

up computations. We are therefore left with P = [ A 1 , A 2 , A 3 , ..., A N ], where A i = ˆ A i / q

ˆ A ⊤ i M p ˆ A i ( ∀ i = 1 . . . N ) denotes a mass-normalized vector. The generalized mass and sti ff ness matrices of the problem are then given by,

M p = P ⊤ M p P , K p = P ⊤ K p P . (4)

Projecting the coe ffi cient vector of Equation 1 into the modal space yields a = P ε p , where ε p stands for the modal participation factor. We can now express the kinetic energy of the ABH plate as,

T p = 1

2 ˙ a ⊤ M p ˙ a ≡ 1

2 ˙ ε ⊤ p M p ˙ ε p . (5)

On the other hand, the kinetic energy of a single resonator is

T r = 1

2 ˙ w r ∆ M r ˙ w r = 1

2 ˙ w r ˙ w r , (6)

with w r = w r / √ ∆ M r . If we attach N r resonators to the ABH plate, the total kinetic energy of the MMABH reads,

N r X

N r X

1 2 ˙ w ri ˙ w ri ≡ 1

i = 1 T ri = 1

2 ˙ ε ⊤ p M p ˙ ε p +

2 ˙ ε ⊤ f M ˙ ε , (7)

T = T p +

i = 1

where in the last equality we have introduced the assembled modal participation factors of the ABH plate and resonators, ε , and the total mass matrix, f M , of the built-up system.

Table 1: Geometrical and material parameters of the ABH plate.

Geometry parameters Material parameters

m = 2 . 5 ρ p = 7800 kg / m 3

L x = 0 . 6 m E p = 210 GPa

L y = 0 . 6 m η p = 0 . 01

h uni = 0 . 005 m ν p = 0 . 3

r abh = 0 . 24 m

ε = 0 . 1595 m − 1 . 5 ρ v = 950 kg / m 3

h c = 0 . 0005 m E v = 5 GPa

r v = 0 . 24 m η v = 0 . 5

h v = 0 . 0015 m ν v = 0 . 3

As regards the potential energy stored in the i -th spring, it is given by

V ri = 1

2 k r h w p ( x ci , y ci ) − w ri i 2

= 1

2 k r h w 2 p ( x ci , y ci ) − w p ( x ci , y ci ) w ri − w ri w p ( x ci , y ci ) + w 2 ri i

⊤  

⊤  k r χ i χ ⊤ i − k r χ i − k r χ ⊤ i ω 2 r



 ε p w ri















 ε p w ri

P 0

P 0

= 1

2

0 1

0 1

≡ 1

2 ε i ⊤ K ri ε i , ∀ i = 1 . . . N r , (8)

with χ i = χ ( x ci , y ci ) being the shape function vector at the position of the i -th resonator. The total potential energy of the MMABH plate becomes

N r X

N r X

1 2 ε i ⊤ K ri ε i ≡ 1

i = 1 V ri = 1

2 ε ⊤ p K p ε p +

2 ε ⊤ f Kε , (9)

V = V p +

i = 1

where in the last equality we have defined the assembled sti ff ness matrix. If we finally apply the Euler-Lagrange equations to the Lagrangian built from Equation 5 and Equation 9, and assume ε = ˆ ε exp(i ω t ), we arrive at f K − ω 2 f M  ˆ ε = 0 , (10)

from which we can synthesize the modes of the MMABH plate. From modal superposition it is also possible to get the vibration of the MMABH under external forced excitation. The reader is referred to [24] for more details on the above exposed mathematical developments.

3. BROADBAND VIBRATION REDUCTION OF THE MMABH

Let us consider the damped ABH plate whose geometrical and material parameters are listed in Table 1. Applying a unit force at (0 . 5 , 0 . 5) m we can calculate the mean square velocity (MSV) of the total surface, which corresponds to the red curve in Figure 2. For comparison, we have also computed the MSV of a reference uniform (UNI) plate (gray curve in the figure) having the same

-40

214

-50

MSV (dB)

-60

-70

-80

UNI ABH MMABH

100 1000 20

Frequency (Hz)

Figure 2: Mean square velocity (MSV) of the MMABH, ABH and reference uniform (UNI) plates.

damping layer treatment. The cut-on frequency of the current ABH is f cut − on = 214 Hz, so the MSV peaks beyond that frequency get substantially suppressed by the ABH. However, the ABH clearly fails to work below f cut − on , especially at the first resonant peak ( f 1 = 61 . 7 Hz), which is even higher than that of the uniform plate. As explained before, to tackle this problem we resort to periodic tuned mass dampers. Taking into account that when building the ABH plate from a uniform one we need to remove mass from it (in the current example ∆ M = 3.5286 kg), we can recycle that material for the resonators. In particular, we distribute ∆ M between N r = 36 equal resonators so that each one has mass ∆ M r = 0 . 098 kg. Having done that, we can next determine their sti ff ness. As we target the ABH plate’s first peak at f 1 = 61 . 7 Hz, we tune the spring sti ff ness to k r = (2 π f 1 ) 2 ∆ M r = 14731 N / m. In addition, some structural damping is also added to the resonators by setting k ∗ r = k r (1 + 1i η r ), where η r = 0 . 2 is the spring loss factor. The ABH plate plus resonators constitutes the MMABH plate. The green curve in Figure 2 shows the MSV of the MMABH. As observed in the figure, the vibration near f 1 = 61 . 7 Hz has been greatly reduced. An improvement is also appreciatted for the other two peaks of the ABH plate contained between f 1 = 61 . 7 Hz and f cut − on = 214 Hz, thanks to the damping e ff ect of the resonators. To better understand the performance of the MMABH plate, in Figure 3 we have plotted the vibration shape of the uniform, ABH and MMABH plates at the first resonance of two former ones. If one compares Figure 3a with Figure 3b it is found that the vibration of the ABH plate is stronger than that of the uniform plate because we are below the cut-on frequency. However, the vibration is substantially reduced for the MMABH plate (see Figure 3c) because the energy is transferred to the resonators (see Figure 3d). It is to be noticed that resonators have little impact on the performance of the MMABH above f cut − on = 214 Hz (see Figure 2). This is because the eigenfrequency of the resonators is far from that range. To better show this, in Figure 4 we have plotted the analogous vibration shapes of Figure 3, but now for the higher frequency of 854 Hz. As seen, the vibration of the ABH plate mainly concentrates at its center, as one could expect, and it is obviously much smaller than that of the uniform plate thanks to the dissipation by the viscoelastic layer at the ABH central region. If we next have a look at the vibration of the MMABH plate in Figure 4c, we observe that it is almost identical to that of Figure 4b. This is because the resonators remain still (see Figure 4d) at high frequencies. The above analysis indicates that the resonators can e ffi ciently help reducing the vibrations at low frequencies, where the ABH fails to work, without a ff ecting the ABH performance at high frequencies.

2.4 × 10 -6 (m)

7.3 × 10 -7 (m)

(a)

0.6 (b)

0.6

1.2

3.7

0.4

0.4

y (m)

y (m)

0.0

0.0

0.2

0.2

-1.2

-3.7

0

0

-2.4

-7.3

0 0.2 0.4 0.6 x (m)

0 0.2 0.4 0.6 x (m)

7.3 × 10 -7 (m)

7.3 × 10 -7 (m)

(c)

0.6 (d)

0.6

3.7

3.7

0.4

0.4

y (m)

y (m)

0.0

0.0

0.2

0.2

-3.7

-3.7

0

0

-7.3

-7.3

0 0.2 0.4 0.6 x (m)

0 0.2 0.4 0.6 x (m)

Figure 3: Vibration shape of the (a) uniform, (b) ABH and (c) MMABH plates and (d) displacement of the resonators. The vibration in subfigure (a) corresponds to a frequency of 68.2 Hz, while that in subfigures (b)-(d) corresponds to 61.7 Hz. The asterisks represent the locations of the resonators.

4. CONCLUSIONS

In this paper we have shown that broadband vibration reduction on a plate can be achieved by combining two e ff ects. The first one is to embed an ABH indentation on the plate, which is known to e ffi ciently suppress high-frequency vibration components. The drawbacks of ABHs are less rigidity and stronger vibrations at lower frequencies. The second e ff ect aims at fixing that problem by attaching a set of periodically distributed resonators to the ABH plate. This yields to bandgap formation and suppression of the ABH plate first eigenfrequency (the resonators are tuned for that purpose). Other low frequency peaks are also lowered thanks to the structural damping of the resonators. Given that we use the material that should be removed from a uniform plate for the masses of the resonators, the result is a metaplate (MMABH) having the same mass as the uniform one but with substantial vibration reduction over the whole frequency range. The analysis in this work is numerical and has relied on the GECMS method. In future works it would be worthwhile to experimentally tests some of the conclusions reported herein.

3.8 × 10 -7 (m)

3.8 × 10 -7 (m)

(a)

0.6 (b)

0.6

1.9

1.9

0.4

0.4

y (m)

y (m)

0.0

0.0

0.2

0.2

-1.9

-1.9

0

0

-3.8

-3.8

0 0.2 0.4 0.6 x (m)

0 0.2 0.4 0.6 x (m)

3.8 × 10 -7 (m)

3.8 × 10 -7 (m)

(c)

0.6 (d)

se .*"*

0.6

1.9

1.9

0.4

0.4

y (m)

y (m)

0.0

0.0

0.2

0.2

-1.9

-1.9

0

0

-3.8

-3.8

0 0.2 0.4 0.6 x (m)

0 0.2 0.4 0.6 x (m)

Figure 4: Vibration shape at 854 Hz of the (a) uniform, (b) ABH and (c) MMABH plates and (d) displacement of the resonators.

ACKNOWLEDGEMENTS

This work was supported by the National Natural Science Foundation of China (Grant No. 52171323) and the China Postdoctoral Science Foundation (Grant Nos. 2018M631194 and 2020T130533).

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