# A New Multi-Mechanism Synergistic Acoustic Structure for Underwater Low-Frequency and Broadband Sound Absorption

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Acoustic Model and Calculation Method

#### 2.1. Model of MMSC−AS

_{1}, and the radii of the multi-layered coating layers and spherical resonators are r

_{i}(i = 1–6). The height and radius of the cylindrical cavity are h

_{c1}and r

_{c1}, respectively. The thickness of gradient layer 2 is h

_{2}, and the radii of the single spherical resonator and coating layer are r

_{7}and r

_{8}, respectively. The height and radius of the cylindrical cavity are r

_{c2}and h

_{c2}, respectively. Additionally, the thickness of the steel plate is h

_{s}, and the distance between adjacent cavities in each layer is a

_{1}+ a

_{2}. The FGMs proposed in this paper were synthesized from epoxy and inorganic fillers. The gradient indices of gradient layers 1 and 2 are P

_{1}and P

_{2}, respectively. The materials of the coating layer and resonators are soft rubber and steel, respectively.

#### 2.2. Acoustic Calculation Model for MMSC−AS

**K**

_{f}and

**M**

_{f}are the stiffness matrix and mass matrix,

**p**is the sound pressure vector, R describes the fluid–structure coupling,

**u**is the node displacement of the structure at the fluid–structure coupling surface,

**f**is the fluid load vector, and ρ

_{0}is the density of water.

_{ij}are stiffness factors related to coordinates, which can be written as:

_{f}is the work performed by the fluid load and W is the work performed by mechanical loads.

**K**

_{s}and

**M**

_{s}are the stiffness matrix and mass matrix of the structure and

**F**

_{m}is the mechanical load vector.

**S**is the shape function vector of structural elements and

**B**

_{δ}represents the first-order partial derivative vector of the shape function.

## 3. Numerical Results and Analyses

#### 3.1. Acoustic Model Validation

_{1}is the volume fraction of each material h and can be described as:

_{1}= 30 mm, h

_{2}= 30 mm, and h

_{s}= 10 mm, respectively. Then, gradient layers 1 and 2 are evenly stratified into n

_{1}and n

_{2}thin layers, respectively. The distribution of the material properties of the matrix of the MMSC−AS along the thickness direction is shown in Figure 4. The material properties of each component of the MMSC−AS are listed in Table 1, which were the same as those in Ref. [49]. The sound speed and density of water were c = 1489 m/s and ρ

_{0}= 1000 kg/m

^{3}.

_{1}= n

_{2}= 1000 thin layers. The acoustic coefficients obtained through the transfer matrix method (TMM), FEM, and G-FEM are displayed in Figure 5 and Figure 6. It can be found that the acoustical coefficient curves obtained through the FEM and G-FEM were consistent with those obtained through the TMM. It proves that the G-FEM computation program is correct (The program runs in software Python2.7). However, the deviations between the TMM and FEM were obvious in the high frequency when P

_{1}= 5.0 and P

_{2}= 5.0, while the acoustic coefficients obtained through the G-FEM were consistent with TMM in the high frequency. Therefore, the accuracy of the G-FEM was better than that of the FEM when the number of mesh elements was the same.

#### 3.2. Sound Absorption Characteristics of the MMSC−AS

_{1}= 30 mm, h

_{2}= 30 mm, and h

_{s}= 5 mm, and the radii of multi-layered spherical resonators and coating layers were r

_{1}= 12 mm, r

_{2}= 11 mm, r

_{3}= 9.5 mm, r

_{4}= 8.5 mm, r

_{5}= 7 mm, r

_{6}= 6 mm, r

_{7}= 12 mm, and r

_{8}= 10 mm, respectively. The geometrical parameters of the cavity were r

_{c1}= 8 mm, h

_{c1}= 20 mm, r

_{c2}= 12 mm, and h

_{c1}= 20 mm, and the lattice constants were set as a

_{1}= 30 mm and a

_{2}= 30 mm. The SAC of the MMSC−AS could be obtained on the basis of the acoustic calculation model.

_{1}and P

_{2}. The SAC of the MMSC−AS at low frequencies was not affected by the changes in P

_{1}and P

_{2}, while those in the high-frequency range increased with the increase in the gradient index. Therefore, the sound absorption band of the MMSC−AS could be broadened by increasing the gradient index.

_{1}= 5.0 and P

_{2}= 5.0, the vibration mode of MMSC−AS at various frequencies is shown in Figure 8. It can be seen that, when f = 1010 Hz, corresponding to the first sound absorption peak (SAP), the vibration energy is mainly concentrated in the resonators in the low-frequency range, and the modulation of MMSC−AS on sound waves is mainly through the local resonance of resonators. As the frequency increases, when f = 1210 Hz and f = 1710 Hz (the second SAP), the vibration energy is mainly concentrated on gradient layers 1 and 2, the steel plate, and resonators, respectively, and the sound waves are modulated by the bending vibration of gradient layer and steel plate, coupling resonance between the multi-resonator and cavity resonance. In contrast, in the high-frequency range, when f = 4110 Hz (the third SAP) and f = 9110 Hz, the structural vibration energy was mainly concentrated in gradient layer 1, and the effect of MMSC−AS on sound waves was mainly through the bending vibration of gradient layer 1, which converted longitudinal waves into transverse waves. Therefore, the synergistic effect between the mechanisms enriched the sound wave modulation modes because of the multiple sound wave modulation mechanisms inside the MMSC−AS, and the sound absorption band could be broadened with effect.

#### 3.3. The Effects of Sound Absorption Characteristics

_{1}and P

_{2}on the SAC of MMSC−AS were studied. The effect of P

_{1}on the SAC of the MMSC−AS is displayed in Figure 11. The sound absorption characteristics of the MMSC−AS at low frequencies were not affected by the change in P

_{1}, the frequency of the first SAP was almost unchanged, and the second and the third SAPs moved to high frequencies.

_{2}on the SAC of the MMSC−AS. The acoustic characteristics of the MMSC−AS at low frequencies were not affected by P

_{2}, and the frequency corresponding to the first SAP was almost unchanged. However, the second and third SAPs moved to the low frequency, and the absorption coefficients of the MMSC−AS in the high-frequency band remained unchanged.

_{1}and P

_{2}of the matrix material had no effect on the sound absorption performance of the MMSC−AS at low frequencies. At the frequency corresponding to the second SAP, the effect of the MMSC−AS on sound waves mainly occurred through the bending vibration effect of functional gradient layers 1 and 2, multi-vibrator coupling resonance effect, and cavity resonance effect. Additionally, at the peak frequency corresponding to the third SAP frequency, the effect of MMSC−AS on sound waves mainly occurred through the bending vibration of gradient layer 1 and cavity resonance. It can be seen that, with the increase in P

_{1}, the equivalent stiffness of gradient layer 1 increased, causing the second and third SAPs to move to the high frequency. In addition, the increase in the equivalent Young’s modulus of gradient layer 1 increased the acoustical impedance of the material and enhanced the dissipation of the high-frequency sound wave.

_{2}decreased the equivalent stiffness of gradient layer 2. Therefore, increasing P

_{2}caused the second and third SAPs of the MMSC−AS to move to lower frequencies. The change in P

_{2}did not affect the material properties of gradient layer 1, so the SAC of the MMSC−AS at high frequencies remained unchanged. Additionally, the second SAP of the MMSC−AS shifted to lower frequencies, which enhanced the coupling between the first and second SAPs, increasing the absorption coefficient of the first SAP.

_{c1}is shown in Figure 13. It can be found that, as the cavity radius r

_{c1}of gradient layer 1 increased, the sound absorption performance of the MMSC−AS at low frequencies was almost unaffected, and the frequency of the first SAP was almost unchanged. However, the second and the third SAPs both shifted to lower frequencies, which increased the value of the first SAP, and the absorption coefficients of the MMSC−AS decreased at high frequencies.

_{c2}of gradient layer 2 on the SAC of the MMSC−AS under various gradient indexes. With the increase in the cavity radius r

_{c2}of gradient layer 2, the frequency corresponding to the first SAP was almost unchanged, and the second SAP moved to lower frequencies. Furthermore, the third SAP moved to lower frequencies with the increase in the cavity radius r

_{c2}of gradient layer 2, and the sound absorption performance of the MMSC−AS at high frequencies was rarely affected.

_{c1}and r

_{c2}of the matrix in the gradient layers 1 and 2 had no effect on the sound absorption performance of the MMSC−AS at low frequencies. The second and third SAPs of the MMSC−AS moved to lower frequencies with the increases in the cavity radius r

_{c1}, while the second SAP of the MMSC−AS moved toward the lower frequencies with increases in the cavity radius r

_{c2}. The second SAP of MMSC−AS shifted to lower frequencies, which enhanced the coupling between the first and second SAPs, increasing the value of the first SAP. In addition, the energy dissipation for high-frequency sound waves occurred through material damping, and the acoustical impedance of the matrix decreased with the increase in the cavity radius r

_{c1}of gradient layer 1. Therefore, the increase in the cavity radius r

_{c1}reduced the SAC of the MMSC−AS at high frequencies.

_{1}= 5.0 and P

_{2}= 5.0, the SAC of the MMSC−AS with different resonator distributions are displayed in Figure 16. It can be found that, when the multi-resonator was embedded in gradient layer 1, the frequencies corresponding to the first and second SAPs of the MMSC−AS were basically unchanged compared with S-S, and the absorption coefficient of the MMSC−AS increased at high frequencies. In Figure 16b, when the multi-resonator was embedded in gradient layer 2, the first and second SAPs of MMSC−AS moved to higher frequencies, while the frequency corresponding to the third SAP basically remained unchanged compared with S-S, and the absorption coefficient of the MMSC−AS at high frequencies basically remained unchanged. By comparing S-M with M-M, it can be observed from Figure 16c that, when the distribution of resonators inside gradient layer 1 was changed, the acoustic absorption behavior of the MMSC−AS at low frequencies was basically not affected. Additionally, the third SAP moved to higher frequencies, and the SAC of the MMSC−AS at high frequencies increased. In contrast, by comparing M-S with M-M, it can be found from Figure 16d that, when the distribution of resonators inside gradient layer 2 changed, the first and second SAPs of the MMSC−AS moved to higher frequencies.

#### 3.4. Optimization of Sound Absorption Characteristics

**x**represents the design variables and obj(

**x**) donates the objective function.

_{c1}and r

_{c2}of gradient layers 1 and 2 were important considerations affecting the sound absorption characteristics of the MMSC−AS. Taking the geometric parameters of the cavity of the MMSC−AS as variables, the range of geometric parameters for the cavities of gradient layers 1 and 2 and the selection of initial values are shown in Table 2. Taking P

_{1}= 2.0, P

_{2}= 2.0 and P

_{1}= 5.0, P

_{2}= 5.0 as examples, the optimization of the sound absorption characteristics is carried out in software COMSOL5.6, and the optimal solution of the geometric parameters and the SAC after optimization can be achieved.

_{1}= 2.0 and P

_{2}= 2.0, the geometric parameters of MMSC−AS after optimization were r

_{c11}= 4.4 mm, r

_{c12}= 4.4 mm, h

_{c1}= 25.0 mm, r

_{c21}= 10.2 mm, r

_{c22}= 9.7 mm, and h

_{c2}= 22.1 mm, and the comparison of SAC before and after optimization of the MMSC−AS is shown in Figure 17a. When gradient indexes P

_{1}= 5.0 and P

_{2}= 5.0, the geometric parameters of the MMSC−AS after optimization were r

_{c11}= 6.2 mm, r

_{c12}= 6.2 mm, h

_{c1}= 21.1 mm, r

_{c21}= 10.5 mm, r

_{c22}= 8.8 mm, and h

_{c2}= 25.0 mm. A comparison of SAC of MMSC−AS before and after optimization is displayed in Figure 17b.

_{1}= 2.0, P

_{2}= 2.0) and 0.8269 (P

_{1}= 5.0, P

_{2}= 5.0) by optimizing the geometric parameters of the cavities of gradient layers 1 and 2, respectively. In addition, by optimizing the geometric parameters of the cavities of gradient layers 1 and 2, the sound absorption characteristics of the MMSC−AS were obviously improved at high frequencies, but the sound absorption characteristics at low frequencies were almost unchanged. This was mainly due to the fact that the effect of the MMSC−AS on sound waves was mainly through the local resonance of resonators. Therefore, optimizing the geometric parameters of the cavities of gradient layers 1 and 2 had almost no influence on the sound absorption behavior of the MMSC−AS at low frequencies.

^{8}Pa, density ρ = 2863 kg/m

^{3}, Poisson ratio ν = 0.49, gradient indexes P

_{1}= 5.0, and P

_{2}= 4.13, respectively. The SACs of the MMSC−AS before and after optimization are shown in Figure 18. The SAPs at low frequencies were almost unchanged, and the third SAP shifted to higher frequencies. The sound absorption performance of the MMSC−AS at high frequencies was significantly improved after the optimization of the material parameters of the MMSC−AS. According to the sound absorption mechanism of the MMSC−AS, the effect of the MMSC−AS on sound waves mainly occurred through local resonance. Therefore, the optimization of material parameters of the MMSC−AS had almost no influence on the acoustic absorption coefficients at low frequencies.

^{8}Pa, density ρ = 3100 kg/m

^{3}, Poisson ratio ν = 0.45, gradient index P

_{1}= 5.0, P

_{2}= 2.0, r

_{c11}= 3.0 mm, r

_{c12}= 3.0 mm, h

_{c1}= 10 mm, r

_{c21}= 3.0 mm, r

_{c22}= 3.0 mm, and h

_{c2}= 10 mm, respectively. The sound absorption curves before and after the comprehensive optimization of the MMSC−AS are shown in Figure 19, the frequency corresponding to the first SAP was almost immutable, the second and third sound SAPs shifted to higher frequencies after the comprehensive optimization of the SAC of MMSC−AS, and the SAC increased notably at high frequencies.

_{2}decreased after optimization, which is the reason for the SAC of MMSC−AS increasing at high frequencies. Moreover, the coupling effect between the first and second acoustic peaks was weakened as a result of the deviation of the second SAP of the MMSC−AS moving to higher frequencies, which reduced the first SAP value.

## 4. Conclusions

- (1)
- The accuracy of the G-FEM was better than that of the FEM when calculating the acoustic coefficients of the FGAS under the condition of the same number of finite elements meshes;
- (2)
- Owing to the presence of multiple sound wave modulation mechanisms inside the MMSC−AS, the synergy between various mechanisms enriched the energy dissipation modes of sound waves, which widened the sound absorption frequency band with effect, thus achieving the underwater low-frequency and broadband sound absorption performance;
- (3)
- The MMSC−AS could be regulated by adjusting material or structural parameters that affect sound absorption characteristics based on the sound wave modulation mechanism of the MMSC−AS in different frequency bands;
- (4)
- The sound absorption characteristics of the MMSC−AS could be effectively ameliorated by optimizing the material or the structural parameters, while the sound absorption characteristics at low frequencies could be ameliorated by optimizing the distribution or filling ratio of the resonators.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 4.**The gradient distribution of the material properties of the matrix of the MMSC−AS along the z direction.

**Figure 9.**Composite functionally graded acoustic structure containing resonator and cavity structure.

**Figure 15.**Cross-sectional view of a unit cell of the MMSC−AS with different resonator distribution forms on the xoz plane.

**Figure 16.**SAC of the MMSC−AS with different resonator distributions when P

_{1}= 5.0 and P

_{2}= 5.0.

Materials | Density (kg/m ^{3}) | Young’s Modulus (Pa) | Poisson’s Ratio | Loss Factor | |
---|---|---|---|---|---|

Impedance match layer | Epoxy | 1100 | 1.4 × 10^{8} | 0.49 | 0.6 |

Functionally graded layer | Epoxy | 1100 | 1.4 × 10^{8} | 0.49 | 0.6 |

Inorganic fillers | 1700 | 3.4 × 10^{8} | 0.49 | 0.6 | |

Steel plate backing | Steel | 7800 | 2.07 × 10^{11} | 0.3 | 0 |

Variables | r_{c11}/mm | r_{c12}/mm | h_{c1}/mm | r_{c21}/mm | r_{c22}/mm | h_{c1}/mm |
---|---|---|---|---|---|---|

Range | [3, 10] | [3, 10] | [10, 25] | [3, 12] | [3, 12] | [10, 25] |

Initial value | 8 | 8 | 20 | 12 | 12 | 20 |

Variables | Young’s Modulus (Pa) | Density (kg/m^{3}) | Poisson’s Ratio | P_{1} | P_{2} |
---|---|---|---|---|---|

Range | [2.4, 5.4] ×10^{8} | [2.1, 4.1] ×10^{3} | [0.45, 0.49] | [0.5, 5.0] | [0.5, 5.0] |

Initial value | 2.4 × 10^{8} | 3100 | 0.45 | 1.0 | 1.0 |

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## Share and Cite

**MDPI and ACS Style**

Shi, K.; Li, D.; Hu, D.; Shen, Q.; Jin, G.
A New Multi-Mechanism Synergistic Acoustic Structure for Underwater Low-Frequency and Broadband Sound Absorption. *J. Mar. Sci. Eng.* **2023**, *11*, 2373.
https://doi.org/10.3390/jmse11122373

**AMA Style**

Shi K, Li D, Hu D, Shen Q, Jin G.
A New Multi-Mechanism Synergistic Acoustic Structure for Underwater Low-Frequency and Broadband Sound Absorption. *Journal of Marine Science and Engineering*. 2023; 11(12):2373.
https://doi.org/10.3390/jmse11122373

**Chicago/Turabian Style**

Shi, Kangkang, Dongsheng Li, Dongsen Hu, Qi Shen, and Guoyong Jin.
2023. "A New Multi-Mechanism Synergistic Acoustic Structure for Underwater Low-Frequency and Broadband Sound Absorption" *Journal of Marine Science and Engineering* 11, no. 12: 2373.
https://doi.org/10.3390/jmse11122373