# Addition of Two Substantial Side-Branch Silencers to the Interference Silencer by Incorporating a Zero-Mass Metamaterial

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Measuring Equipment and Samples

#### 2.1. Transmission Loss Measurement

#### 2.2. Zero-Mass Metamaterial and Delay Tube of Interferometric Silencer

^{2}orifice plate fabricated by a photo fabrication type 3D printer, Formlabs Form2 (Somerville, MA, USA), was used, and photocuring resins were used. The membrane material was attached and fixed to the orifice plate coated with Vaseline without tension.

_{d}) was divided into the upstream (L

_{da}) and downstream (L

_{db}) lengths, respectively, at the position where the zero-mass metamaterial was installed. The main tube length (L

_{m}) was 60.4 mm.

## 3. Theoretical Analysis

#### 3.1. Analysis of Zero-Mass Metamaterial

_{mem}denotes the mass of the membrane attached to the orifice hole; M

_{air}, the mass of the air in the orifice hole; k

_{mem}, the spring constant of the membrane; and b

_{eff}, the damping coefficient of the zero-mass metamaterial. Here, the spring constant of the thin film k

_{mem}was measured using the method of Lee et al. [13].

_{i}is the incident sound pressure; p

_{r}, the reflected sound pressure; p

_{t}, the transmitted sound pressure; u

_{i}, the incident particle velocity; u

_{r}, the reflected particle velocity; and u

_{t}, the transmitted particle velocity. Furthermore, S is the cross-sectional area of the acoustic tube. Note that p

_{1}and u

_{2}are given by Equations (1)–(4) expressing p

_{2}and u

_{2}, respectively.

_{0}) of air, the sound pressure can be expressed as Equations (6)–(8), respectively.

_{eff}denotes the effective mass defined by Equation (11), and b

_{eff}is the effective damping coefficient of the zero-mass metamaterial. To simplify the calculations, b

_{eff}is set to zero because its effect on the theoretical value of transmission loss is negligible.

_{m}of the zero-mass metamaterial is given by Equation (15).

#### 3.2. Theoretical Analysis of Delay Tube with Built-In Zero-Mass Metamaterials

_{m}in Equation (15).

_{0}and speed of sound c

_{0}in the air.

#### 3.3. Transfer Matrix for Entire Interference Silencer

_{f}, the length of the upstream delay tube; l

_{b}, the length of the downstream delay tube; and S, the cross-sectional area of the main and delay tubes. If the transfer matrices T

_{delay}and T

_{2}are expressed by Equation (19), T after a parallel connection is expressed by Equation (20).

_{1}-, T-, and T

_{1}-connected cascades are T

_{all}, and are expressed by Equation (21).

_{all}–D

_{all}in Equation (21), the transmission loss (TL) of the interference silencer can be expressed as in Equation (22).

^{2}aperture.

#### 3.4. Peak Frequency of Sound Attenuation for Ordinary Interference-Type Silencers and Side-Branch Silencers

_{side}denotes the length of the branch tube of the side-branch silencer.

_{peak}satisfying Equations (28)–(31) and its odd-numbered multiples [16]. Therefore, it is sufficient to determine the delay tube length of the interference-type silencer so that the “dip frequency of the TL of the zero-mass metamaterial in the interference-type silencer” and the “peak frequency of the sound reduction of the interference-type silencer” coincide.

## 4. Experimental and Theoretical Values for Transmission Loss

#### 4.1. Transmission Loss of Single Zero-Mass Metamaterial

_{m}of 11 μm.

#### 4.2. Comparison of Experimental and Theoretical Values

_{da}= 50 and 60 mm, respectively.

_{d}= 180.9 mm (Figure 9 and Figure 10). Therefore, it can be assumed that the acoustic mass of the zero-mass metamaterial approaches zero as expected.

_{da}= 50 and 60 mm, respectively (redrawn from Figure 9 and Figure 10).

_{d}= 50 and 125.9 mm, respectively, and the experimental values for a zero-mass metamaterial installed at L

_{da}= 50 mm (redrawn from Figure 13).

_{d}= 60 and 115.9 mm, respectively, and the experimental values for a zero-mass metamaterial installed at L

_{da}= 60 mm (redrawn from Figure 14). Exactly as in Figure 13, the experimental values of the proposed silencer exhibit sound reduction peaks at 735, 1420, and 2200 Hz, in addition to the sound reduction peak near 1425 Hz of the interference silencer. Similarly, the 735 and 2200 Hz sound reduction peaks correspond to the respective peak frequencies of the side-branch silencer with a 115.9 mm branch tube length (blue line in Figure 14). The 1420 Hz peak of the proposed silencer corresponds to the 1420 Hz peak of the side-branch silencer with a 60 mm branch tube length (red line in Figure 14).

## 5. Conclusions

- The incorporation of zero-mass metamaterial into an interference-type silencer can introduce the silencing effect of a side-branch silencer with two different branch tube lengths without increasing the volume of the interference-type silencer. Consequently, the total volume of the interference and two side-branch silencers could be reduced by half.
- The incorporation of zero-mass metamaterials in the interference silencer increased the sound reduction peaks.
- Theoretical analysis of the interference-type silencer with built-in zero-mass metamaterial was conducted using the Equations of motion, and theoretical values were obtained using the transfer matrix. Consequently, the theoretical and experimental values were close, enabling us to predict the TL of the proposed silencer.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Conflicts of Interest

## Nomenclature

ASTM | American Society for Testing and Materials |

A-D, A_{all}–D_{all} | Four-terminal constants |

b_{eff} | Damping coefficient |

c_{0} | Speed of sound in the air (m/s) |

d | Orifice plate diameter (mm) |

f, f_{peak}, f_{side} | Peak frequency (Hz) |

FFT | Fast Fourier transform |

I | Imaginary unit |

k | Wavenumber |

k_{mem} | Spring constant of the thin film (N/m) |

L_{d}, L_{da}, L_{db}, L_{m}, L_{side} | Length of the tube (mm) |

M_{air} | Mass of the air in the orifice hole |

M_{eff} | Effective mass |

M_{mem} | Mass of the membrane attached to the orifice hole |

p_{1}, p_{2}, p_{i}, p_{r}, p_{t} | Sound pressure (Pa) |

r | Orifice hole radius (mm) |

S | Cross-sectional area of the acoustic tube (mm^{2}) |

t | Orifice plate thickness (mm) |

t_{m} | Film thickness (mm) |

T, T_{1}, T_{2}, T_{11}, T_{12}, T_{21}, T_{22}, T_{all}, T_{b}, T_{delay}, T_{f}, T_{m} | Transfer matrix |

TL | Transmission loss (dB) |

u_{1}, u_{2}, u_{i}, u_{r}, u_{t} | Particle velocity (m/s) |

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**Figure 2.**(

**a**) Delay tube section of the interference silencer with incorporated zero-mass metamaterial; (

**b**) dimensions of the delay tube.

**Figure 6.**Acoustic elements of the delay tube: (

**a**) acoustic elements of the delay tube and (

**b**) equivalent circuit of the delay tube.

**Figure 7.**Acoustic elements and equivalent circuit of an interference silencer: (

**a**) division into acoustic elements and (

**b**) equivalent circuit.

**Figure 9.**Experimental and theoretical values of the interference silencer with incorporated zero-mass metamaterial (zero-mass metamaterials installed at L

_{da}= 50 mm).

**Figure 10.**Experimental and theoretical values of the interference silencer with incorporated zero-mass metamaterial (zero-mass metamaterials installed at L

_{da}= 60 mm).

**Figure 11.**Comparison of the experimental values with and without zero-mass metamaterial (zero-mass metamaterials at L

_{da}= 50 mm).

**Figure 12.**Comparison of the experimental values with and without zero-mass metamaterial (zero-mass metamaterials at L

_{da}= 60 mm).

**Figure 13.**Comparison with simple side-branch silencer (experimental: interference silencer with zero-mass metamaterial; theoretical: side-branch silencer with 50 mm and 125.9 mm branch tube length).

**Figure 14.**Comparison with simple side-branch silencer (experimental: interference silencer with zero-mass metamaterial; theoretical: side-branch silencer with 60 mm and 115.9 mm branch tube length).

Material of Membrane | Nominal Density (kg/m ^{3}) | Measured Membrane Thickness (μm) |
---|---|---|

Low-density polyethylene | 920 | 11 |

Material of Orifice | Thickness of Plate (mm) | Diameter of Hole (mm) |
---|---|---|

Photocurable resin | 5 | 10 |

Element | Ratio (%) | Element | Ratio (%) |
---|---|---|---|

Aluminum | Balance | Copper | 0.10 max |

Magnesium | 2.2–2.8 | Manganese | 0.10 max |

Chromium | 0.15–0.35 | Zinc | 0.10 max |

Silicon | 0.25 max | Others, each | 0.05 max |

Iron | 0.40 max | Others, total | 0.15 max |

Length of Side-Branch TubeL_{side}(mm) | |||||

50.0 | 60.0 | 125.9 | 115.9 | ||

Peak Frequency f _{side} (Hz) | n = 1 | 1701.5 | 1418.0 | 675.7 | 734.0 |

n = 2 | 5104.5 | 4253.8 | 2027.2 | 2202.1 |

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**MDPI and ACS Style**

Sakamoto, S.; Shin, J.; Abe, S.; Toda, K.
Addition of Two Substantial Side-Branch Silencers to the Interference Silencer by Incorporating a Zero-Mass Metamaterial. *Materials* **2022**, *15*, 5140.
https://doi.org/10.3390/ma15155140

**AMA Style**

Sakamoto S, Shin J, Abe S, Toda K.
Addition of Two Substantial Side-Branch Silencers to the Interference Silencer by Incorporating a Zero-Mass Metamaterial. *Materials*. 2022; 15(15):5140.
https://doi.org/10.3390/ma15155140

**Chicago/Turabian Style**

Sakamoto, Shuichi, Juung Shin, Shota Abe, and Kentaro Toda.
2022. "Addition of Two Substantial Side-Branch Silencers to the Interference Silencer by Incorporating a Zero-Mass Metamaterial" *Materials* 15, no. 15: 5140.
https://doi.org/10.3390/ma15155140