# Effect of Sheet Vibration on the Theoretical Analysis and Experimentation of Nonwoven Fabric Sheet with Back Air Space

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## Abstract

**:**

## 1. Introduction

## 2. Measurement Equipment and Samples

#### 2.1. Measurement Equipment

_{n}of nonwoven fabrics was measured using a Kato Tech KES-F8-AP1 permeability tester. This is an air permeability tester that injects a constant flow of air into a sample via a plunger and cylinder. A precise pressure gage was used to quantify the pressure loss caused by the sample at a constant flow rate of 4 cc/cm

^{2}/s (4 × 10

^{−2}m/s). The measurement result allowed the ventilation resistance R

_{n}(kPa s/m) to be calculated directly. Flow resistivity σ

_{n}was calculated by dividing the ventilation resistance R

_{n}by the thickness of the nonwoven fabric.

#### 2.2. Measurement Samples

_{n}is the ventilation resistance R

_{n}divided by the thickness t

_{n}of the nonwoven sheet.

_{f}= 109.90 mm, thickness t

_{f}= 1.00 mm, the width of the frame was 1.0 mm, and aperture was a regular hexagonal honeycomb shape, where each face was 3.44 mm long. The aperture ratio was as large as 0.73 to ensure that the mesh was acoustically negligible.

_{i}= 100 mm, outer diameter 120 mm, and length of the back air space L = 100 mm.

## 3. Theoretical Analyses

#### 3.1. Analysis Model Corresponding to the Measured Sample

#### 3.2. Transfer Matrix Based on a One-Dimensional Wave Equation

_{1}and u

_{1}, respectively, the sound pressure and particle velocity at the end of the measurement tube are p

_{2}and u

_{2}, respectively, and the cross-sectional area of the measurement tube is S. The transfer matrix T between the incident and end surfaces can be expressed using the one-dimensional wave equation as follows [13]:

_{c}is the characteristic impedance, and l is the length of the measurement tube. When Equation (2) is substituted for A, B, C, and D in Equation (1) and expressed as a series of equations for p

_{1}and u

_{1}, they are one-dimensional wave equations for sound pressure and particle velocity.

#### 3.3. Transfer Matrix for Nonwoven Fabrics as Porous Materials

_{n}and characteristic impedance Z

_{n}for a nonwoven fabric can be expressed as in Equations (3) and (4) using the Miki model [2]. Note that j is an imaginary unit.

_{n}of the nonwoven fabric in Equations (3) and (4) can be expressed as Equation (5).

_{n}is the ventilation resistance of the nonwoven fabric and t

_{n}is the thickness of the nonwoven fabric.

_{n}of the nonwoven fabric is obtained by substituting Equations (3) and (4) into the transfer matrix in Equation (2) as follows to yield:

#### 3.4. Transfer Matrix for the Vibration of Nonwoven Sheet

_{p}for a vibrating sheet is expressed using Equation (7) [12].

_{n}is the area of the nonwoven fabric, ρ

_{A}is the area density of the nonwoven fabric, ξ is the damping ratio, and ω is the angular frequency of the sound wave (ω = 2πf, f: frequency). The attenuation constant b of the nonwoven fabric, mass m of the nonwoven fabric, natural angular frequency ω

_{0}, and spring constant k

_{a}of the air layer can be expressed using Equations (8)–(11), respectively, as follows

_{0}is the atmospheric pressure, S

_{t}is the cross-sectional area of the measurement tube (S

_{t}= S

_{n}), and L is the length of the back air layer. The damping ratio ξ is set to 0.1.

_{p}in the following equation, the transfer matrix T

_{p}of the vibrating sheet is expressed using the right side of the following equation:

#### 3.5. Transfer Matrix for Back Air Space

_{B}for the back air space can be expressed as follows, assuming that damping in the transfer matrix based on the one-dimensional wave equation in Equation (2) is negligible:

#### 3.6. Equivalent Circuit and Transfer Matrix Corresponding to the Analytical Model

_{1}can be obtained via the cascade connecting the transfer matrix T

_{n}using the Miki model in Equation (6) and the transfer matrix T

_{B}for the back air space in Equation (13), as follows:

_{n}using the Miki model in Equation (6) and the transfer matrix T

_{p}for the vibration of the nonwoven sheet in Equation (12) are connected in parallel [12,14,15]. The parallel connection occurs because there are two sound-absorbing principles for one incident surface [12,14]. Owing to the parallel connection, sound pressure acts equally on both sound-absorbing principles and particle velocity is diverted more toward the sound-absorbing principle with lower impedance. Next, the transfer matrix T

_{2}is obtained via a cascade connecting the transfer matrix T

_{B}of the back air layer in Equation (13) to them, as shown in the following Equation [13].

#### 3.7. Derivation of the Sound Absorption Coefficient

_{1}and T

_{2}obtained in Section 3.6 was calculated. The four-terminal constants of the transfer matrices T

_{1}and T

_{2}correspond to the four-terminal constants A, B, C, and D of the transfer matrix T in Equation (1).

_{1}and T

_{2}are rigid walls, u

_{2}= 0; therefore, Equation (1) can be expressed as:

_{1}and Su

_{1}= S

_{t}u, the specific acoustic impedance Z is expressed as in Equation (18).

## 4. Comparisons of Experimental and Theoretical Values

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Schematic diagram of a two-microphone impedance tube for absorption coefficient measurements.

**Figure 5.**Equivalent circuits corresponding to test samples: (

**a**) framed nonwoven fabric; (

**b**) nonwoven fabric sheet.

**Figure 6.**Comparison of experimental and theoretical values (length of back air space L = 100 mm): (

**a**) RW2100; (

**b**) 3501BD; (

**c**) 3A01A; (

**d**) 3701B; (

**e**) 3A51AD; (

**f**) RW2250.

Name | Ventilation Resistance ${\mathit{R}}_{\mathit{n}}\text{}(\mathbf{kPa}\text{}\times \text{}\mathbf{s}/\mathbf{m})$ | Thickness ${\mathit{t}}_{\mathit{n}}\text{}\left(\mathbf{mm}\right)$ | Flow Resistivity ${\mathit{\sigma}}_{\mathit{n}}{=\mathit{R}}_{\mathit{n}}/{\mathit{t}}_{\mathit{n}}\text{}(\mathbf{kPa}\text{}\times \text{}\mathbf{s}/{\mathbf{m}}^{2})$ | Area Density ${\mathit{\rho}}_{\mathbf{A}}\text{}(\mathbf{g}/{\mathbf{m}}^{2})$ | Material |
---|---|---|---|---|---|

RW2100 | 0.1767 | 0.58 | 303.6 | 100 | Polypropylene |

3501BD | 0.2021 | 0.19 | 1064 | 50 | Polyester |

3A01A | 0.3363 | 0.39 | 862.2 | 100 | Polyester |

3701B | 0.3763 | 0.24 | 1568 | 70 | Polyester |

3A51AD | 0.5283 | 0.45 | 1174 | 152 | Polyester |

RW2250 | 1.350 | 0.82 | 1646 | 250 | Polypropylene |

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**MDPI and ACS Style**

Sakamoto, S.; Iizuka, R.; Nozawa, T.
Effect of Sheet Vibration on the Theoretical Analysis and Experimentation of Nonwoven Fabric Sheet with Back Air Space. *Materials* **2022**, *15*, 3840.
https://doi.org/10.3390/ma15113840

**AMA Style**

Sakamoto S, Iizuka R, Nozawa T.
Effect of Sheet Vibration on the Theoretical Analysis and Experimentation of Nonwoven Fabric Sheet with Back Air Space. *Materials*. 2022; 15(11):3840.
https://doi.org/10.3390/ma15113840

**Chicago/Turabian Style**

Sakamoto, Shuichi, Ryo Iizuka, and Takumi Nozawa.
2022. "Effect of Sheet Vibration on the Theoretical Analysis and Experimentation of Nonwoven Fabric Sheet with Back Air Space" *Materials* 15, no. 11: 3840.
https://doi.org/10.3390/ma15113840