Influence of structural form on the acoustic black hole array coupled with damping layers Zhengcheng Yao 1 School of Transportation Science and Engineering, Beihang University South 3 rd Street, Gaojiaoyuan, Changping District, Beijing, 102200, China Xiandong Liu 2 School of Transportation Science and Engineering, Beihang University South 3 rd Street, Gaojiaoyuan, Changping District, Beijing, 102200, China Haoming Liang 3 School of Transportation Science and Engineering, Beihang University South 3 rd Street, Gaojiaoyuan, Changping District, Beijing, 102200, China Yue Bao 4 School of Transportation Science and Engineering, Beihang University South 3 rd Street, Gaojiaoyuan, Changping District, Beijing, 102200, China Yingchun Shan 5 School of Transportation Science and Engineering, Beihang University South 3 rd Street, Gaojiaoyuan, Changping District, Beijing, 102200, China Tian He 6 School of Transportation Science and Engineering, Beihang University South 3 rd Street, Gaojiaoyuan, Changping District, Beijing, 102200, China

ABSTRACT Because of the characteristics of manipulating bending wave, acoustic black hole (ABH) structures can efficiently concentrate vibration energy in the center. Other than single ABH structures, the ABH array or damping layers can be used to have a better performance in vibration reduction and noise control, which have great engineering application prospects of plate parts. In order to explore the influences of ABH array on vibration and noise attenuation, finite element models of two-dimensional ABH array coupled with damping layers are established in this paper to study the effects of arrangements and number of ABH cells on dynamic characteristics. Firstly, this paper analyses the influences of array embedded with three ABHs on the modal loss factor and vibration characteristics of the structures as well as the effects of different arrangements. Sec- ondly, the array with different number of ABH cells were established to study the effects of the number of elements on the vibration response of the plate. Finally, the structure-sound coupling

1 yaozc@buaa.edu.cn 2 liuxiandong@buaa.edu.cn 3 sy1913122@buaa.edu.cn 4 baoyue@buaa.edu.cn 5 shanych@buaa.edu.cn 6 hetian@buaa.edu.cn

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models were established to study the acoustic radiation characteristics and explain the noise re- sponse of the array plate. This study provides a reference for the design and application of the plates embedded with ABH array.

1. INTRODUCTION

In a variety of applications, there is an increasing demand for advanced structures with lightweight and high vibration and noise reduction capabilities [1]. Especially in automotive applications, it is often necessary to meet the requirements of interior noise and lightweight, and the exploration of relevant approaches needs new theories. Acoustic black hole (ABH), as a new wave manipulation method, provides a new method for vibration and noise control of structures [2]. Using the ABH structure on the plate can make the bending wave converge in a specific region, and better vibration and noise control can be achieved by using less damping material in the region with higher energy density, along with the improvement of energy dissipation efficiency of damping material. Therefore, ABH effect has the potential to solve the contradiction between vibration and noise control of body panel and structural lightweight.

The ABH effect was first proposed by Pekeris [3] in 1946. He found that no reflection of waves occurred in inhomogeneous stratified medium and the wave velocity decreased to zero as the depth of the profile increased. In 1988, Mironov [4] found that ABH effect also exists in wedge-shaped structures, that is, in a plate whose thickness changes in a specific power-law form, the bending wave velocity will decrease as the thickness decreases. Due to the limitation of machining, there will be some truncation at the end of the structure, which will obviously weaken the ABH effect and result in large amount of reflection of bending wave in the plate. Krylov [5,6] found that the presence of a small amount of damping material on wedge-shaped surfaces with a power-law profile can signifi- cantly reduce the reflection coefficient.

The ABH structure is mainly divided into one- and two- dimensional forms, in which the two- dimensional ABH is materialized by a cylindrically symmetrical indentation of power-law profile whose exponent is equal or larger than two. For practical mechanical structures such as automobiles and ships, it is not feasible to fabricate one-dimensional ABH structures at the outer edges. Mean- while, the two-dimensional ABH structures can be embedded in most positions of the plate, and its layout is more diverse and flexible, which has a wider application range and more research value. After Krylov [7] first proposed this form of ABH structure, Georgiev et al. [8] carried out the first experimental study, and verified that when fitted with the same damping material, the velocity con- ductivity and vibration attenuation of the elliptic plate structure with embedded two-dimensional ABH were significantly lower than those of the uniform plate structure. Conlon et al. [9] proposed the formula of the cut-on Frequency of two-dimensional ABH structure, indicating that the external radius of ABH should be enlarged to reduce the cut-on Frequency without changing the material. However, for automobile and other structures, due to the requirement of compactness, it is not prac- tical to simply expand the radius in order to achieve the effect of vibration and noise reduction in wide band, and the plate structure embedded with a single acoustic black hole is difficult to play a role in low band [10]. In this case, ABH arrays with flexible arrangement and the potential of broad- ening the effective control band of the bending wave has become a new research direction.

The study of two-dimensional ABH can be carried out by analytical methods. Krylov [7] and Yan et al. [11] theoretically solved the propagation trajectory of lamb wave in two-dimensional ABH structure by using geometric acoustic approximation theory, which proved the energy convergence effect of the ABH structure. Huang et al. [12] also used geometric acoustic approximation method to

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numerically integrate the eikon function equation, and then analyzed and studied the bending wave propagation trajectory and energy convergence phenomenon in the ABH structure, which can guide the design of ABH structure and provide a reference for the optimized attachment position of damping material. However, this method is difficult to analyze complex structures, especially for ABH array structures. The numerical simulation based on finite element method overcomes the shortcoming of analytical methods and can be applied to both frequency domain and time domain analysis of ABH structures. Conlon et al. [9,13,14] studied the vibration and noise performance of a plate embedded in the two-dimensional ABH array by using finite element and boundary element models, and pointed out that periodic ABH array can improve the low-frequency vibration control performance of the structure and the low order modes of the ABH are very important for the overall vibration reduction of the ABH plate. They also discussed the optimization method for vibration reduction through damp- ing arrangement. Bowyer and Krylov [15] machined two-dimensional ABH structures in composite plates and found that the composite has a large loss factor and can absorb most of the vibration energy without using damping materials. Prill et al. [16] applied the ABH to the main floor of a car, estab- lished a finite element model with 10 two-dimensional ABH arrays embedded in it, and studied the vibration response and sound radiation characteristics under wind excitation. The results showed that this model had a certain attenuation effect on noise in the frequency range above 200 Hz.

Currently, in the existing two-dimensional ABH research, the plate thickness is generally about 5mm, while the ABH research for the thin plate of autobody about 1-2mm is rarely studied. For the ABH array with additional damping layers, the influence of the arrangement of elements on the vi- bration and noise attenuation effect is still not clear. Based on the above questions, this paper uses the finite element method to study the influence of the arrangement, position and number of cells in the two-dimensional ABH array coupled with damping layers on vibration and noise attenuation. For the plate embedded with three ABHs, several layouts are designed to study the law of performance. Then the finite element model of the structure with different numbers of ABH embedded is estab- lished to study the effect of the number of cells on the vibration response of the plate. In addition, the acoustic radiation characteristics of ABH arrays are also studied. The results can provide reference for the design of ABH arrays and the application of ABH in the automobile.

2. EFFECT OF ELEMENT ARRANGEMENT OF ABH ON VIBRATION RESPONSE

2.1. Establishment of the ABH Array Model Coupling with Damping Material

In order to study the possibility of applying two-dimensional ABH structures in the automobile plate, this paper chooses to arrange the ABH structure in the steel plate with length and width of 800 mm and thickness of 2 mm. Due to the current machining accuracy, the thickness of ABH structures cannot be infinitesimal. Therefore, typical two-dimensional ABH structures usually has a hole or a circular platform of constant thickness at the center of the indentation [12], i.e., residual thickness, which could affect the performance of ABH structure. The damping materials can be added to reduce the reflection coefficient of the curved wave [5].

A schematic diagram of the embedded two-dimensional ABH plate with additional damping is shown in Figure 1. For ABH, the relation between section thickness and radius can be expressed as:

   =  −    ( ) ( ) 2 1 2 1

,0 ( ) + ( , . ) .

m h x x h x h x x x x x 

m h h x x = − − , (1)

1 1

1 1 1 2

1 h 2 h m where is the residual thickness, is the plate thickness of the uniform part, is the power index,

1 2 x x ， and are the inner radius and outer radius of the ABH respectively.

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Figure 1: Schematic diagram of ABH structure coupled with damping material

Pa.

1 2 1 =20 =100 0.2 x x h = ， ， In the thin plate structure studied in this paper, the design ABH size is , and the thickness profile is designed as

2.2 20 ( ) 0.2+1.8( ) ,20 100 80 x h x x − =   (2)

A cylindrical damping material LD-400 [17] with a radius of 40mm and a thickness of 0.6mm is fitted at the bottom of each ABH. The relevant material parameters are shown in Table 1.

Table 1: Material properties of plate and damping layer

Density (kg/m 3 ) Poisson’s Ratio Damping Loss

Property Modulus of Elasticity (GPa)

Factor

Steel 200 7800 0.3 0.01

Damping Layer 1.8 968 0.49 0.8

Cut-on frequency [9] is an indicative frequency at which the wavelength of the incident bending waves starts to be equal to the characteristic dimension (the diameter in this case) of the ABH pit. It can be defined as:



E h f

  − =

− (3)

0

( ) ( )

2 2

0 0 12 1 2 cut on

R

ABH

h ABH R In the formula, represents the thickness of the plate structure; represents the outer radius

0  0 E 0  of the ABH acoustic black hole; , , and represent the density, elastic modulus, and Pois-

son's ratio of steel plates, respectively. For the ABH structure studied in this paper, its cut-on fre- quency is calculated to be about 481 Hz.

Based on the dimensions above, six kinds of array plate structures embedded with three ABH cells are designed in this paper. When studying vibration characteristics, an excitation is applied at 50 mm away from the right boundary, and a row of nodes is set at 50 mm away from the left boundary as the output, as shown in Figure 2. In order to simulate the general boundary conditions of the body plate, the four edges of the plate are constrained by fixed supports. To accommodate the varying thickness of the ABH region in the structure, the mesh is divided in a non-uniform manner with a minimum element size of 0.3 mm and a maximum element size of 2 mm to ensure that there are at least 10 elements within each bending wave length. In addition, the second-order element C3D20R is used to discretize the structure to ensure the calculation accuracy.

（ a ） （ b ） （ c ） （ d ） （ e ） （ f ）

Figure2: ABH array structures in different arrangements

2.2. Damping Characteristics of ABH Array Structures

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In the actual situation, damping exists in all vibrating systems. Structural damping is used to sim- ulate the damping effect of steel plate structure and damping layer. Structural damping, also known as hysteresis damping or solid damping, is caused by internal friction of imperfectly elastic materials during vibration.

For the above ABH array plate structures, the complex modal analysis is carried out by applying the four sides fixed constraint. The modal loss factors distribution of ABH array and damping material coupling structures under different arrangements are obtained, and symmetric array structures 1 and 4 have the same modal loss factor distribution. The modal loss factors are compared with those of single ABH structure with the same size and condition, as shown in Figure 3. The larger the modal loss factor at each mode frequency in the 20-2000 Hz frequency band, the stronger the energy dissi- pation capacity of the plate structure after coupling with the damping material at this frequency.

(a) ABH array structure 1&4 (b) ABH array structure 2 (c) ABH array structure 3

(d) ABH array structure 5 (e) ABH array structure 6 Figure 3: Comparison of modal loss factors between different ABH arrays and single ABH It can be seen from Figure 3 that in the 20-2000 Hz band, the overall ABH array structures obvi- ously have a higher modal loss factor. The maximum modal loss factor of the single ABH coupled with the damping layer is 0.13, and most of the modal loss factors are distributed between 0.01 and 0.07. However, most of the modal loss factors of several array structures are distributed between 0.015 and 0.1, which are significantly higher than those of single ABH structure, especially in the frequency band above 500 Hz. In addition, within the 2000 Hz frequency band, the single ABH plate structure has the 201 order complex modes, while those ABH array structures all have the 240 order complex modes. This indicates that compared with the single ABH structure, the array structure en- riches the modal diversity of the original plate structure, which is more conducive to coupling the ABH structure with the damping material and increase the consumption of vibration energy by damp- ing layer. At the same time, there is no obvious difference between the modal loss factors of the array structures with different configurations in 2000 Hz frequency band, and the average values of their corresponding modal loss factors are very similar. This indicates that the arrangement of elements in the array has no great effect on the overall modal loss factor, but the number of ABH elements in the array may have a great effect.

In addition, in order to analyze the damping characteristics and vibration attenuation mechanism of ABH array plate in the lower frequency band below the cut-on frequency, the modal loss factors in the frequency band below 500 Hz are analyzed, as shown in Figure 4.

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Figure 4 Distribution of modal loss factors for different ABH structures below 500 Hz Figure 4 shows that modal loss factors of different ABH array structures are mostly distributed below 0.05 in the frequency band below 500 Hz. In addition, there are two maximum values of the loss factor in this frequency band, the first maximum value appears in the vicinity of 280-310 Hz, the second maximum value appears in the vicinity of 480-500 Hz. In order to further analyze the mech- anism of the occurrence of maxima, the modal shapes corresponding to the occurrence of maxima are respectively examined at the modal frequencies, as shown in Figure 5. Among them, the array 1 and 4 are symmetric, so they have the same complex mode frequency and shape.

—e

（ a ） 278.36 Hz of structures 1&4 （ b ） 483.86 Hz of structures 1&4

（ c ） 306.01 Hz of structures 2 （ d ） 498.86 Hz of structures 2

（ e ） 300.75 Hz of structures 3 （ f ） 483.97 Hz of structures 3

（ g ） 305.01 Hz of structures 5 （ h ） 492.9 Hz of structures 5

（ i ） 295.77 Hz of structures 6 （ j ） 513.17 Hz of structures 6

Figure5 Modal shapes of ABH array structures As can be seen from the modal shape diagrams in Figure 5, at the frequencies where the maximum modal loss factor appears, the amplitude of the modes in the ABH region is large, especially in the position of the damping layer, that is to say, local modes appear in all the structures. This indicates

Ti |

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that the local mode shapes in the ABH region of the plate structures can improve the damping capacity of the whole system below the cut-on frequency. Among them, the mode loss factor corresponding to array structure 6 is the smallest at the first-order local mode frequency. As can be seen from (i), this is because only two ABH cells have obvious local modes, while the ABH on the right has no local mode modes. At the second local mode frequency, the modal loss factor corresponding to array structure 1 and 4 is the largest, which is due to the obvious local mode of the three ABH cells as shown in (b). In the whole 500 Hz frequency band, the modal loss factors of the above different array structures are similar, indicating that the cell arrangement has little influence on the position of the first two important local mode shapes of the array structures.

2.3. Vibration Characteristics of ABH Array Structures

For several ABH structures shown in Figure 2, a simple harmonic excitation in the frequency band of 20-2000Hz is applied to calculate the mean square velocity of the left row of nodes according to the formula below, where N represents the number of nodes.

1 1 N

n n i v v v N = =   (4)

2 *

The vibration of the structures in the frequency band can be obtained, as shown in Figure 6. The vibration response curves of the uniform plate and the plate embedded with single ABH (fitted with the same damping layer) are also drawn in the figure for reference.

(a) ABH array structure 1 (b) ABH array structure 2

(c) ABH array structure 3 (d) ABH array structure 4

(e) ABH array structure 5 (f) ABH array structure 6 Figure 6: Comparison of mean square velocity between different ABH arrays and single ABH It can be seen from the Figure 6 that, compared with the single ABH plate, the array structures obviously have better vibration attenuation effect, especially in the frequency band above 500 Hz, no matter how it is arranged. The frequencies of the array structures with different arrangements are

different when they begin to produce significant vibration attenuation. Observe the frequency corre- sponding to the last resonant peak of the array structure which is the same as that of the uniform plate, that is, the cut-on frequency of the structure at the beginning of the obvious ABH effect. The average mean square velocity in the frequency band is also calculated, as shown in Table 2.

Table 2: Cut-on frequency and mean square velocity of different ABH structures

Different form ( ) Hz Cut-on frequency observed ( ) 2 2 / mm s Mean square velocity

Single ABH 740 3.4763

structure 1 460 1.2938

structure 2 380 0.9977

structure 3 620 1.4963

structure 4 400 1.1137

structure 5 390 0.9970

structure 6 390 0.9916

The cut-on frequencies of the array structures are significantly lower than that of the single ABH structure, and the cut-on frequency is roughly positively correlated with the mean square velocity. This shows that the array structure can reduce the cut-on frequency of the ABH effect and improve the vibration attenuation effect, which proves the superiority of the ABH array structure.

At the same time, it can be found that different arrangements have different influences on the vi- bration attenuation effect. In terms of the overall frequency band, array structure 6 is the best, and array structure 3 is the worst. In addition, comparing array structures 1,2 and 4, it can be seen that the more the arrangement of ABH tends to be in longitudinal arrangement, the better its vibration atten- uation effect is. The possible reason is that for the transverse arrangement as shown in array 1, after the vibration energy propagates to the left from the excitation position and passes through the first ABH cell, part of the energy bypasses the second ABH from the upper and lower directions, and the energy convergence effect is affected to some extent. By comparing array structures 2,3,5 and 6, it can be seen that the ABH cell nearest to the excitation position has the greatest influence on the overall vibration attenuation effect for the longitudinal array structure. When the ABH in the middle is far away from the excitation position, such as array structure 3, the initial frequency increases and the damping effect of the whole structure becomes significantly worse.

This indicates that the arrangement of ABH has a great influence on the vibration attenuation effect when designing the array structure composed of three ABHs. In order to enhance the overall damping effect of the plate structure, the ABH cells should form a triangular arrangement as shown in array structure 6, and the array should be arranged as close to the excitation position as possible.

3. EFFECT OF THE NUMBER OF ABH CELLS ON VIBRATION AND NOISE CHAR- ACTERISTICS

3.1. Vibration characteristics of ABH arrays with different numbers

In order to study the impact of the number of ABH cells on vibration reduction, finite element models of the plate structure with 3, 5 and 9 ABHs embedded were established respectively, in which

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the size of the plate structure, damping layer, excitation and boundary conditions remained the same as before, as shown in Figure 7. The mean square velocity distribution of the three structures is ob- tained through simulation, as shown in Figure8. In order to display the vibration velocity in lower frequency band more clearly, the distribution of mean square velocity in 20-600 Hz band is shown in Figure 8 (b). After that, the relevant vibration indicators of several structures are sorted out, as shown in Table 3.

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Figure7: ABH array structures

（ a ） 20-2000Hz （ b ） 20-600Hz Figure8: Vibration response of ABH array structures with different numbers

Table 3: Relevant vibration indicators of the structures

Different

Mean square veloc-

Mean square ve-

Cut-on fre-

The number of

( ) Hz quency

form

ity within 20-2000

locity within 20-

modal orders within 20-2000 Hz

( ) 2 2 / mm s Hz

( ) 2 2 / mm s 500 Hz

Single ABH 3.4763 1.3535 740 201

3 elements 1.4936 1.0396 620 240

6 elements 0.6335 0.7487 390 281

9 elements 0.3711 1.2361 390 358

It can be seen that the more ABH cells in the array structure, the better vibration attenuation effect and the lower mean square velocity will be in the frequency band in 20-2000Hz, especially in the frequency band above 500Hz. With the increase of ABH cells, the modal order of the whole plate structure will increase, which makes the local modes of the ABH region become more, and the energy dissipation effect of the structure becomes better. In addition, the cut-on frequency of the array struc- ture embedded with 5 and 9 ABHs is about 390 Hz, while that of 3 cells is about 620 Hz. This indi- cates that the array structure with more ABH cells is beneficial to reduce the cut-on frequency of the ABH effect and further improve the vibration response at low and medium frequencies.

However, it can be seen from Figure 8 (b) that, in the low frequency band below 400Hz, the array structures embedded with 3, 5 and 9 elements have no obvious advantage over the uniform plate structure in vibration response. In particular, the plate structure embedded with 9 ABHs has strong vibration in 200-300 Hz band and does not show a better effect. It can be seen from Table 3 that in

the lower frequency band of 20-500 Hz, when the number of ABH cells in the array structure is less than 5, the more cells there are, the lower the mean square velocity will be. However, the response of 9 ABHs plate structure is worse than that of 3 and 5 cells. This is because when the number of ABH cells increases, more material mass is removed from the original plate structure, and the stiffness of the plate structure decreases significantly, leading to stronger vibration of the structure at low fre- quencies. This indicates that the excessive number of ABH is not conducive to the vibration control of the plate structure at lower frequencies, and the number of ABH cells should be determined by considering the working state and working frequency of the plate when designing the array structure.

3.2. Acoustic radiation characteristics of ABH arrays with different numbers

In order to study the radiated noise of ABH array structures, the structure-acoustic coupling models are established in COMSOL software. The excitation position, matched damping layer and boundary conditions remain unchanged. The outer boundary of the air layer adopts perfectly matched layer (PML), whose main purpose is to construct a closed space without reflection. Scattering sound waves will be rapidly absorbed after reaching PML, so we have a reflectless boundary. The air block is divided by tetrahedral element, and there are more than 6 elements in each wavelength of air layer. In addition, the thickness of the air layer is 500 mm to ensure that the distance between the plate structure and the air boundary is large enough, generally exceeding 1/3 of the wavelength of the sound wave.

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The sound pressure level (SPL) distribution of the whole acoustic-vibration coupling model at each frequency in the 20-2000 Hz band can be obtained through simulation. For example, the SPL distri- bution of the model with three ABHs at the 1960 Hz frequency is shown in Figure 9(a). Then, the average SPL of the air layer surface directly above the plate structure is further calculated as the overall noise radiation index of the structure, and the frequency distribution curves of SPL of different array structures are obtained, as shown in Figure 9(b).

(a) (b) Figure9: Sound pressure level of the coupled model It can be seen from Figure 9 that in the frequency range below 600 Hz, the three structures all have relatively high level of noise radiation, while in the frequency range above 600 Hz, the noise radiation level of the three structures is controlled, and the average SPL is maintained near 63dB. In addition, the noise radiation of the 5 ABHs is weaker than that of the 3 ABHs in the frequency band below 600 Hz, but when the number of ABH increases further, the noise radiation of the 9 ABHs is stronger than that of the former two. This is because the ABH plate structure mainly relies on local modes for energy collection at lower frequencies. When the number of ABH cells increases, the number of local modes increases, which leads to the increase of energy convergence of the ABH structure and the enhancement of energy dissipation capacity of damping materials. However, when the number of ABH is too large, more materials are removed, and the stiffness of the original plate structure is greatly reduced, resulting in intensified radiation noise at low frequencies. Therefore, when designing the ABH array structures, the stiffness of the whole structure and the requirements of noise attenua- tion should be considered comprehensively.

4. CONCLUSIONS

In this paper, based on the model of plate structures embedded ABH arrays, the effect of the ar- rangement and number of ABH cells on the vibration and noise attenuation of the entire plate structure is studied by finite element method, and the regulation method of the ABH array structure is obtained. Firstly, the finite element model of plate structures with different arrangement are established. The loss factor, vibration response and local mode are compared by simulation to explain the influence of different arrangement on vibration response of plate structure. In addition, finite element model and acoustic-vibration coupling model of array structure with different number of ABH cells are estab- lished to study the effect of the number on vibration and noise response of plate structures. These studies lay a good foundation for the engineering application of ABH array structures.

5. ACKNOWLEDGEMENTS

The authors would like to thank the financial support provided by the Natural Science Foundation of China (Grant No. 11972003).

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