Vibration characteristics optimization of a rectangular plate embedded with two-dimensional acoustic black holes Xiaofei Du 1 School of Mechanical Engineering, Nanjing Institute of Technology No. 1 Hongjing Avenue, Jiangning District, Nanjing 211167, China Qing Gu 2 Nanjing Institute of Technology No. 1 Hongjing Avenue, Jiangning District, Nanjing 211167, China

ABSTRACT The acoustic black hole (ABH) structures have the potential to achieve structural vibration suppres- sion and noise reduction through the effect of the ABH on concentration and manipulation of flexural waves. In this paper, FEM models of the damping ABH plate with different geometric parameters of acoustic black holes and damping layers are established to investigate the comprehensive effect of the 2-D ABHs and the damping layers on the vibration characteristics of the plate through vibration response analysis and the Kriging surrogate model was used to obtain predictive values of the vibra- tion response of the ABH plate. Based on the constructed surrogate model, the vibration velocities of the four monitoring points of the ABH plate were optimized by using multi-objective genetic algo- rithms. This paper can provide guidance for the optimal design of acoustic black hole structure in engineering applications.

1. INTRODUCTION

In recent years, due to the convergence and manipulation of flexural waves in the structure, the acoustic black hole (ABH) has become a new type of vibration and noise reduction technology. The characteristics of the acoustic black hole are that the local thickness of the solid structure gradually decreases in the form of a power index and finally approaches zero, then this region can gather the flexural waves spreading in the structure here, thereby reducing the vibration of other regions of the structure [1]. Different researchers have carried out many researches on the spread of flexural waves, vibration characteristics, and acoustic characteristics of acoustic black hole structures [2~4]. The acoustic black hole is becoming an important research direction in the field of sound and vibration, and it is expected to play an important role in the vibration and noise reduction of various mechanical structures in engineering.

The research on vibration reduction applications of the acoustic black hole in different engineering fields mainly includes: Bowyer et al. [5] studied the vibration suppression effect of one-dimensional ABH on engine turbine blades under excitation of mechanical force and airflow through experiments. He et al. designed a new type of box-type acoustic black hole structure for the beam structure of the

1 duxiaofei@njit.edu.cn

2 1442808104@qq.com

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aircraft wing and studied the vibration reduction effect within the broadband range through the FEM simulation and experiments [6].

In our previous research, the vibration suppression effect of the acoustic black hole on the refrig- erator rectangular support plate has been analyzed [7]. On this basis, the effects of different acoustic black hole parameters and damping layer parameters on the vibration reduction effect of the rectan- gular support plate of the refrigerator were further analyzed in this paper and the Kriging surrogate model was used to obtain predictive values of the vibration response of the ABH plate. Based on the constructed surrogate model, the vibration velocities of the four monitoring points of the ABH plate were optimized by using multi-objective genetic algorithms.

2. FEM MODELING OF THE ABH PLATE

The rectangular plate embedded with 8 two-dimensional acoustic black holes is shown in Figure 1, and the size of the rectangular plate is 901×300 mm. The upper and lower rows of ABHs are symmetrical about the middle line of the rectangular plate, and the distance between the centres of the two row ABHs is 182.2 mm. For the single ABH in Figure 1 (a), Figure 1 (b) shows its profile

1.8 mm p h = view. The thickness of the uniform region of the rectangular plate is , the relationship between the thickness and radius of the ABH region is:

( ) m h x x b ε = − (1)

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Where ε is a constant, and m is the power exponent of the thickness change of the ABH region

2 m ≥ ( ), b is the distance between the center of the ABH region and the coordinate origin in y -axis direction. However, due to the manufacturing technology, there will always be a truncation existing in the actual ABH structure. The thickness of the truncated region can be expressed as:

p t h h b = − (2)

2

When the flexural wave propagates from the uniform region of the ABH plate to the ABH region, as the thickness of the plate decreases sharply with a power exponent, the velocity of the flexural wave will also gradually decrease, so as to achieve the concentration effect of the flexural wave. Utilizing the aggregation effect of the ABHs on the flexural wave, the vibration suppression of the rectangular plate can be achieved [8].

(a)

(b) Figure 1: Schematic diagram of the rectangular plate embedded with ABHs. The second-order tetrahedral unit C3D10 is used to mesh the damping ABH support plate. The grid size of the ABH regions and damping layers is 1 mm, and the mesh in the small curved area of the ABH regions and damping layers is encrypted to ensure a smooth transition from small cells to large cells and the minimum cell size is 0.2 mm during encryption. The contour curve of the damping layers is the same as the curve of the acoustic black hole region. Therefore, the damping layer can be fully attached on the acoustic black holes. The thickness of the damping layer is 0.89 mm, and the maximum diameter of the damping layer is 33.14 mm. The material parameters of the rectangular plate and the damping layer are shown in Table 1.

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Figure 2: The FEM model of the damping ABH plate. Table 1: The material parameters of the rectangular plate and the damping layer.

oO

Material Young’s Modulus

Density (kg/m 3 )

Poisson’s

Ratio Loss Factor

(Pa)

Viscoelastic material 9 × 10 6 1812 0.45 0.4 Steel 2.1× 10 11 7800 0.3 0.005

The boundary condition of the rectangular plate is that: the left and right faces are clamped, and the rest faces and edges are free. The force of 6 N is applied at the exciting point 1~4, respectively, and the frequency of the excitation force is 1 Hz~3 kHz (for 1 Hz interval). The vibration velocity responses of the four monitoring points on the damping ABH plate were obtained through calcula- tions in the commercial FEM software LMS Virtual. Lab. 3. OPTIMAL DESIGN BASED ON KRIGING SURROGATE MODEL

Figure 3 shows a schematic diagram of the maximum diameter of the ABH D ABH , the truncation thickness of the ABH h t , the maximum diameter of the damping layer D damping and the thickness of

EY INNA ON

the damping layer h damping . Considering the power index m of the acoustic black hole region, the above five parameters are set as variables. The upper and lower limits of the five variables are shown in Table 2, and the five variables are sampled using the Latin Hypercube Sampling method to obtain the design space. The design space contains fifty combinations of different parameters of the maximum ABH diameter, the truncated thickness of the ABH, the power index of the ABH, and the damping layer thickness, the maximum diameter of the damping layer. The values of the above five parameters were changed respectively, and then different CAD models of the damping ABH plates were estab- lished. The CAD models were meshed with the same sizes of grids in Seciton 2, and then the meshes were imported into the LMS Virtual. Lab to set the material properties and boundary conditions. The material properties and boundary conditions are the same as the FEM model in Section 2. In addition, the position, amplitude value of the excitation force and position of the monitoring points remain unchanged. After the FEM calculations were performed, the vibration velocity responses of the four monitoring points around 600 Hz in the whole design space were obtained. Fifteen groups of the vibration velocities of the four monitoring points under different parameters of the maximum ABH diameter, the truncated thickness of the ABH, the power index of the ABH, and the damping layer thickness, the maximum diameter of the damping layer were set as test values. The R 2 values of the established surrogate model is calculated and shown in Table 3. It can been seen in Table 3 that the surrogate model constructed is highly accurate and can accurately fit the vibration velocities of the four monitoring points of the ABH plate.

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Figure 3: Schematic diagram of the geometric parameters of acoustic black holes and damping lay- ers.

Table 2: The geometric parameters of acoustic black holes and damping layers in the design space.

Variables Description Lower Bound

Upper Bound Continuity Unit

X1 The truncation thickness of

ABH region, h t 0.1 1 Consequent mm

The power exponent of the thickness change of the ABH

2 6 Value interval 0.5 /

X2

region, m

X3 The diameter of the ABH re-

gion, D ABH 40 100 Consequent mm

X4 The thickness of the damping

layer, h damping 0.8 1.6 Consequent mm

X5 The diameter of the damping

layer, D damping 20 60 Consequent mm

Deasspine Irsping

Table 3: R 2 of the established surrogate model.

No. Monitoring

Monitoring

Monitoring

Monitoring

point 1

point 2

point 3

point 4 R 2 0.8357 0.8530 0.8029 0.8391

Based on the constructed surrogate model, the vibration velocities of the four monitoring points of the ABH plate were optimized by using multi-objective genetic algorithms. The Pareto front of the optimization is shown in Figure 4. Five groups of the optimized ABH and damping layer geometric parameters and vibration responses of the four monitoring points were selected in the Pareto front, as shown in Table 4.

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Figure 4. Pareto front of the optimization.

Table 4: Optimized ABH and damping layer geometric parameters and vibration responses of the four monitoring points.

Value of

Value of

Value of

Value of

X1 X2 X3 X4 X5

M1 (m/s)

M2 (m/s)

M3 (m/s)

M4 (m/s) 0.238 4.007 58.387 1.247 32.265 0.094 0.059 0.070 0.015 0.189 4.683 61.225 1.308 40.372 0.051 0.066 0.053 0.027 0.238 4.216 58.122 1.245 32.370 0.093 0.060 0.073 0.016 0.239 4.014 62.237 1.236 33.147 0.089 0.074 0.070 0.016 0.177 3.550 58.589 1.236 37.947 0.066 0.047 0.038 0.018

Pareto front Objective 3 ° 2 g 8 014 0.06 on 0.08 0.08 oT one 0.06 Objective 1 0.14 0.04 Objective 2

4. CONCLUSIONS

In this paper, FEM models of the damping ABH plate with different geometric parameters of acoustic black holes and damping layers are established to investigate the comprehensive effect of the 2-D ABHs and the damping layers on the vibration characteristics of the plate through vibration response analysis and the Kriging surrogate model was used to obtain predictive values of the vibra- tion response of the ABH plate. Based on the constructed surrogate model, the vibration velocities of the four monitoring points of the ABH plate were optimized by using multi-objective genetic algo- rithms. Five groups of the optimized ABH and damping layer geometric parameters and vibration responses of the four monitoring points were selected in the Pareto front after optimization. 5. ACKNOWLEDGEMENTS

This work was supported by the National Key Research and Development Program of China (grant number 2019YFB2006402) and Scientific Research Foundation for High-level Talents of Nanjing Institute of Technology (grant number YKJ202102). 6. REFERENCES

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