# Acoustic Insulation Characteristics and Optimal Design of Membrane-Type Metamaterials Loaded with Asymmetric Mass Blocks

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## Abstract

**:**

## 1. Introduction

## 2. Model and Method

_{M}, and the coordinate origin is located at one vertex of the rectangular membrane while the x-axis and y-axis coincide with both sides of the rectangle, whose lengths are L

_{x}and L

_{y}, respectively. The periphery of the membrane is fixed, and its surface density is m

_{M}. A uniform tension force per unit length, T, acts on the inside of the membrane. As shown in Figure 1, an asymmetric mass is loaded on the membrane. The geometric shape of this added mass is a cylinder, whose cross-sectional shape is an asymmetric ring, which is composed of a large circle minus a non-concentric small circle. The large circle is located at the center of the membrane with radius R, and the small circle, with radius r, is offset in the x-direction from the large one by l

_{0}. The height of the mass block is h

_{k}. A plane wave with amplitude of P

_{0}is incident on the surface of the MAM. We used the analytical method presented by Langfeldt et al. [20] to analyze this MAM. The sound transmission loss (STL) of this MAM can be calculated according to the mass law:

_{0}and c

_{0}are the density and speed of the medium through which the sound wave propagates, and m’ is the effective surface density of the membrane and can be calculated by:

**b**is the membrane excitation vector.

**B**is a matrix in the standard eigenvalue equation:

**Ax**= k

^{2}

**Bx**, obtained from the equation system of this mass-loaded membrane [20]. The standard eigenvalue problem can be solved by the Arnoldi method [28] to obtain the eigenvalues and the eigenvectors of the coupled system.

**Λ**is a diagonal matrix composed of the eigenvalues and

**X**is a matrix in which the eigenvectors are arranged.

**I**is the identity matrix.

## 3. Verification and Comparison

_{_in}is the input sound pressure set as 1 Pa, and p

_{_out}is the transmitted sound pressure, which can be obtained by calculating the average sound pressure of the sound exit boundary.

_{M}= 1270 kg/m

^{3}, Young’s modulus E

_{M}= 2.9 GPa, and Poisson’s ratio ν

_{M}= 0.44, was selected as the membrane material. The added mass material was steel, with density ρ

_{k}= 7860 kg/m

^{3}, Young’s modulus E

_{k}= 210 GPa, and Poisson’s ratio ν

_{k}= 0.3. The configuration was set with membrane length L

_{x}= L

_{y}= 20 mm, membrane thickness h

_{M}= 0.025 mm, large-circle radius R = 2.5 mm, small-circle radius r = 1.5 mm, small-circle offset l

_{0}= 0.5 mm, and mass height h

_{k}= 2 mm. The internal tension of the membrane was T = 120 N/m. In the calculation, standard atmospheric conditions were assumed, with ρ

_{0}= 1.225 kg/m

^{3}and c

_{0}= 340 m/s. The calculation frequencies ranged from f = 50 to 1000 Hz.

_{0}= 5 mm, whose eccentric directions were the same. The distribution position was symmetrical, but due to the asymmetry of the mass, the whole structure was asymmetric to the y-axis. Figure 6a shows the STL curve calculated by the finite element method and the analytical method. As can be seen from the figure, both the analytical result and the finite element result can reflect three large peaks. In the finite element results, there was a small sound insulation peak at about 300 Hz, which could not be found in the analytical result. It is speculated that there are two modes with similar frequencies at this position, but the analytical model cannot accurately identify them, so a peak is ignored. This mode is presumed to be caused by the eccentric arrangement of the mass blocks, because there is no such special peak in the FEM model of any MAM with centrally arranged mass blocks. However, this small peak did not significantly affect the sound insulation performance of the MAM, so we will ignore it in subsequent discussions and consider the analytical model to be practical in most cases. Figure 7 shows three eigenmodes of this model at 278.87 Hz, 567.69 Hz, and 585.96 Hz, which were in good agreement with the frequencies of the STL valleys. It can be seen from Figure 7 that the first mode exhibited symmetry of vibration while the other two exhibited asymmetry, and all three modes caused the MAM to generate STL valleys.

_{0}= 5 mm and y

_{0}= 5 mm. Figure 9a shows the finite element and analytical calculation results, and they were in excellent agreement in most frequency bands. The four-mass model has similarity to the two-mass model in that there are two masses distributed in the y-direction in the same way as those distributed in the x-direction. Due to the orthogonal symmetry of the square membrane, the effects of the two groups of masses on the membrane were also similar, resulting in a set of orthogonally symmetric modes, in the form of strengthening the anti-resonance modes. Therefore, the number of obvious sound insulation peaks did not change (only a new small peak appeared, but it is not discussed in this paper and we consider that it only has two obvious sub-peaks), but the two sub-peaks have been significantly improved and widened. There was also a small peak at about 300 Hz in the FEM model, which did not exist in the analytical model. This is for the same reason as the two-mass model, and we also chose to ignore it. Figure 10 shows the five eigenmodes of this MAM. It can obviously be found that the second and third eigenmodes were close in frequency and had orthogonal symmetry in shape. This phenomenon is also reflected in the fourth and fifth eigenmodes. In addition, compared with the two-mass model, the three obvious peaks were shifted to high frequencies when the four-mass model was added. The reason for this phenomenon is that the masses and the elastic force of the membrane formed a spring oscillator system, whose resonant frequency increased when the added mass increased.

_{0}≠ y

_{0}. We left x

_{0}= 5 mm unchanged and changed y

_{0}= 4 mm to draw the STL plot, as shown in Figure 9c. As can be seen from the figure, the number of distinct peaks did not change. This phenomenon is normal because in this form of MAM, one mass block in the x-direction and one mass block in the y-direction can also be regarded as a group, thus forming an oblique line symmetric structure. Therefore, the shape of the STL curve was similar to that when x

_{0}= y

_{0}. However, the frequency of each peak has changed. Compared with the case of x

_{0}= y

_{0}, the main peak has moved forward, while the two sub-peaks have moved backward.

## 4. Discussion

_{0}= 0 mm, 0.2 mm, 0.4 mm, 0.6 mm, and 0.8 mm were calculated, respectively, as shown in Figure 11. It can be found from the figure that when l

_{0}= 0 mm, which means the masses are completely centrosymmetric rings, the STL curve showed two large peaks at around 500 Hz and 640 Hz, and two valleys at around 200 Hz and 550 Hz, which is in line with the characteristics of the symmetrical MAMs. As l

_{0}gradually increased, the frequency of the first valley did not significantly change, and the positions of the first peak and the second valley only underwent a small backward shift. The main changes were concentrated at the second peak. With the emergence of the asymmetry of the mass block, the original second peak was split into two, and with the increase of l

_{0}, the two new peaks showed a tendency to offset away from each other. The main reason for the frequency shift comes from the change of the position of the mass centroid, which caused the distribution of the loading force of the mass to change, so the whole coupling structure produced different modes. This shows that by increasing the eccentricity, l

_{0}, the distribution distance of the two secondary peaks can be made farther, which is an effective way to regulate the sub-peaks.

_{0}= y

_{0}= 4 mm, 5 mm, 6 mm, and 7 mm are calculated, respectively,, as shown in Figure 12. It can be seen that with the change of the distribution position, all the peaks and valleys of the MAMs have significantly changed. According to Figure 12, the following conclusions can be drawn: With the increase of x

_{0}and y

_{0}, the first and second valleys and the first peaks all tended to shift backwards, and the distance between the two sub-peaks relative to the main peak generally tended to decrease. The shift of the peaks was also caused by the change of the position of the loading force, but to a larger extent, so the shift was more obvious. It is concluded that the distribution position can be used as a means to tune the main peak of metamaterials. Combined with the conclusion in the above section, by matching the eccentricity with the distribution position of the asymmetric masses, it is feasible to regulate the overall sound insulation performance of the MAM and design it for specific usage needs.

## 5. Optimization

_{0}and x

_{0}on the STL performance of the MAM. However, the influence of other structural parameters on sound insulation characteristics has not been explored. Furthermore, we hoped to find a set of optimized structural parameters that can maximize the performance of this MAM. Therefore, we used Isight software for optimization analysis. The Isight platform can integrate a variety of software for engineering optimization problems [29,30,31]. For the structure of this MAM, we selected the membrane thickness, h

_{M}, the membrane side length, L

_{x}(L

_{x}= L

_{y}), the mass external radius, R (r = 0.6 R), the mass height, h

_{k}, the mass eccentricity, l

_{0}, the mass position, x

_{0}(x

_{0}= y

_{0}), and internal tension, T, as design variables. Their value ranges were 0.02 mm ≤ h

_{M}≤ 0.03 mm, 15 mm ≤ L

_{x}≤ 30 mm, 2 mm ≤ R≤3 mm, 1.5 mm ≤ h

_{k}≤ 3 mm, 0 ≤ l

_{0}≤ 1 mm, 4 mm ≤ x

_{0}≤ 8 mm, and 90 N/m ≤ T ≤ 150 N/m. We attached importance to the sound insulation performance of low-frequency broadband and lightweight conditions of the MAM, so we took the average STL in the calculated frequency band, the frequency bandwidth higher than 30 dB, and the average surface mass density of MAM as the optimization objectives. They are defined as TLs, frq

_{30}, and m, respectively. Therefore, the multi-objective optimization problem can be described as:

_{Kmin}, x

_{Kmax}] of the K(K∈[1, N])-dimension, the K-dimensional space is divided into M intervals, each cell is recorded as [x

_{Ki-1}, x

_{Ki}], and then a sample point is randomly selected from each intervals, so a total of M sample points is selected. The optimal Latin hypercube design is an improvement on the basis of the Latin hypercube design, which enhances the space-filling ability and the uniformity of samples. Therefore, the optimal Latin hypercube was used as the DOE method in this paper. In the design variable space, 150 groups of sample variables were selected for calculation. The pareto graphs and main effect graphs obtained by statistics according to the sample calculation results are shown in Figure 13 and Figure 14. These two kinds of graphs represent the ways in which design variables affect the objective function. In the Pareto graphs, the size of the bar indicates the percentage contribution of the design variable to the influence of the objective function, where blue indicates a positive effect and red indicates a negative effect. It can be found from the figure that L

_{x}, T, and R had obvious effects on both TLs and frq

_{30}, while the others had weak effects. The meaning of the main effect graph representation is similar to that of the Pareto graph, which can show how the objective functions changed with the design variables. Obviously, the larger the absolute value of the slope of the curve, the greater the influence of the design variables on the objective functions. Therefore, the same conclusion can be drawn from the main effects graphs.

^{2}), and the closer R

^{2}is to 1, the higher the accuracy and credibility of the approximate model. The response surface model (RSM), radial basis function (RBF) neural network model, orthogonal polynomial model, and the Kriging model are commonly used approximate models. Combined with the DOE data above, the four models were all used to construct an approximate model of the objective functions. Here, 40 groups of data from the DOE samples were randomly selected for error analysis. The values of the model accuracy parameter R

^{2}are shown in Table 1. An approximation model can be considered as credible if the R

^{2}value is greater than 0.9, so these approximation models all met the requirements. Among them, the RSM model had the best approximate effect on TLs, and the Kriging model had the best approximate effect on freq

_{30}. Therefore, these two methods were selected to establish approximate models for the two optimization objectives for the subsequent optimization calculation. Figure 15 shows the scatter plot of the actual value and the predicted value of the approximate models. The black line is the contour line, and the blue line is the actual average. Most of the points in the figure are close to the contour line, which also indicates that the approximate models can meet the requirements.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 3.**(

**a**) Comparison of the analytical and numerical results for the STL curve of the MAM with an asymmetric ring. (

**b**) Comparison of the MAMs with an asymmetric ring and a cylinder for the STL curve.

**Figure 5.**Definitions for the configuration of the MAM loaded with two masses: (

**a**) asymmetric arrangement and (

**b**) symmetric arrangement.

**Figure 6.**(

**a**) Comparison of the analytical and numerical results for the STL curve of the MAM with two asymmetric rings. (

**b**) Comparison of the MAMs with two asymmetric rings and two cylinders for the STL curve. (

**c**) Comparison of the MAMs with two asymmetric rings, asymmetrically and symmetrically arranged.

**Figure 7.**Eigenmodes of the double-mass-loaded membrane: (

**a**) 278.87 Hz, (

**b**) 567.69 Hz, and (

**c**) 585.96 Hz.

**Figure 9.**(

**a**) Comparison of the analytical and numerical results for the STL curve of the MAM with four asymmetric rings. (

**b**) Comparison of the MAMs with four asymmetric rings and four cylinders for the STL curve. (

**c**) Comparison of the MAMs with mass positions x

_{0}= y

_{0}and x

_{0}≠ y

_{0}for the STL curve.

**Figure 10.**Eigenmodes of the four-mass-loaded membrane: (

**a**) 192.72 Hz, (

**b**) 571.13 Hz, (

**c**) 575.69 Hz, (

**d**) 632.59 Hz, and (

**e**) 644.25 Hz.

RSM | RBF | Orthogonal | Kriging | |
---|---|---|---|---|

TLs | 0.98627 | 0.98303 | 0.984588 | 0.91226 |

frq_{30} | 0.90442 | 0.90444 | 0.90559 | 0.93409 |

Variables | Original Values | Optimized Values |
---|---|---|

L_{x} | 20 mm | 17.8635 mm |

h_{M} | 0.025 mm | 0.02386 mm |

R | 2.5 mm | 2.1138 mm |

h_{k} | 2 mm | 1.5534 mm |

T | 120 N | 148.6532 N |

x_{0} | 5 mm | 4.5826 mm |

l_{0} | 0.5 mm | 0.6565 mm |

Optimization Objectives | Original Values | Optimized Values |
---|---|---|

TLs | 23.334 dB | 26.997 dB |

frq_{30} | 338 Hz | 377 Hz |

m | 2.006 kg/m^{2} | 1.404 kg/m^{2} |

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## Share and Cite

**MDPI and ACS Style**

Jiang, R.; Shi, G.; Huang, C.; Zheng, W.; Li, S.
Acoustic Insulation Characteristics and Optimal Design of Membrane-Type Metamaterials Loaded with Asymmetric Mass Blocks. *Materials* **2023**, *16*, 1308.
https://doi.org/10.3390/ma16031308

**AMA Style**

Jiang R, Shi G, Huang C, Zheng W, Li S.
Acoustic Insulation Characteristics and Optimal Design of Membrane-Type Metamaterials Loaded with Asymmetric Mass Blocks. *Materials*. 2023; 16(3):1308.
https://doi.org/10.3390/ma16031308

**Chicago/Turabian Style**

Jiang, Renjie, Geman Shi, Chengmao Huang, Weiguang Zheng, and Shande Li.
2023. "Acoustic Insulation Characteristics and Optimal Design of Membrane-Type Metamaterials Loaded with Asymmetric Mass Blocks" *Materials* 16, no. 3: 1308.
https://doi.org/10.3390/ma16031308