# Modelling of Microperforated Panel Absorbers with Circular and Slit Hole Geometries

## Abstract

**:**

## 1. Introduction

_{1}, which is complex. A good absorber must have impedance matched to that of the air, Z

_{0}, which is real. Therefore, in order that such an MPP provides enough absorption, other complex impedance is needed to compensate for the reactive part of Z

_{1}. This can be carried out by adding an air cavity of thickness D in front of the perforated panel. Thus, an MPP consists of a parametric, tunable absorber which depends on (d,t,φ,D) [4].

## 2. Single-Layer Microperforated or Microslit Panels

_{1}. When a plane wave, propagating in air, with characteristic impedance Z

_{0}, reaches the MPP or MSP, the impedance contrast (Z

_{1}− Z

_{0}) causes a reflection, and hence, an absorption. At normal incidence, the reflection, R, and absorption, α

_{0}, coefficients are

- viscous–thermal dissipation within the perforations, characterized by impedance Z
_{hole}for an MPP, and Z_{slit}for an MSP; - flow distortion in the perforation edges, characterized by impedance Z
_{edge}; - resonances in the air cavity, with impedance Z
_{D}; - structural vibrations of the panel impinged by the incident acoustic field, with impedance Z
_{vib}.

_{hole}and Z

_{edge}is named Z

_{MPP}for an MPP (or Z

_{MSP}for an MSP). The air cavity impedance is:

_{vib}, can be obtained as a function of the elastic parameters of the panel. The input impedance, Z

_{1}, can be obtained as [29]:

_{vib}→∞ and

_{MPP}in Equation (4b) must be changed for Z

_{MSP}. In the following, equations for Z

_{MPP}and Z

_{MSP}will be analyzed.

#### 2.1. Maa Model

_{hole}. However, the exact Equation (8) will be used in the following.

_{s}, is also named the surface resistance or resistance end correction, and the reactive term, X

_{m}, is called the mass reactance or reactance end correction.

_{m}, through:

_{1}is the overperforation correction factor of Equation (13).

#### 2.2. Randeberg–Vigran (RV) Model

_{3}is the overperforation correction factor for slits in Equation (23).

#### 2.3. Equivalent Fluid (F) Model

_{0}is the ambient pressure, B

^{2}is the Prandtl number, and σ is the flow resistivity. For air at 18 °C, γ = 1.4, K

_{0}= 1.0132 10

^{5}Pa, and B

^{2}= 0.71. The hole geometry is introduced in this formulation through two constants, C

_{1}and C

_{2}[20]:

_{1}and C

_{2}for different cross-sectional hole geometries [20].

_{c}, and propagation constant, k

_{c}, can be obtained from the equations

_{c}and k

_{c}are the characteristic impedance and the propagation constant, respectively, of the equivalent fluid of the perforated panel given by Equations (25) and (26), respectively.

#### 2.4. Comparison of Models

- When the perforation ratio increases (overperforation) and the perforation diameter decreases, the absorption peak moves towards higher frequencies and the absorption bandwidth becomes wider [6].
- It seems that the Maa model of MPP always provides more absorption than the RV model for the equivalent MSP. The absorption peaks may be displaced towards lower frequencies for low perforation ratios, although the bandwidth of the Maa curve is wider than that of the RV model for the equivalent MSP, for the considered combinations of parameters.
- There are discrepancies between the absorption curves provided by the Maa model and the EF model for circular holes at low perforation ratios (Figure 2A,B). In this case, the EF curve is slightly displaced towards higher frequencies with respect to the Maa curve. However, when the perforation ratio increases (Figure 2C,D), both curves tend to match each other.
- The discrepancies between the absorption curves provided by the RV and the EF models for slits are large for all combinations of parameters. For low perforation ratios (Figure 2A,B) the EF curve is displaced towards higher frequencies. For high perforation ratios (Figure 2C,D) the EF curve has a higher peak and wider bandwidth.

## 3. Results

_{1}= 15 mm). Slotted panels were manufactured by successive pouring of hot PolyLactic Acid (PLA) bioplastic. Table 2 summarizes the combination of the nominal parameters of these MSPs. The three MSPs were overperforated. The first one, MSP8, had 6 turns of a slit 0.44 mm wide, resulting in a perforation ratio of 16%. The second MSP, MSP9, had 7 turns of a slit of hydraulic diameter 0.44 mm. Its perforation ratio was 19%. The slit of the third MSP, MSP11, was 0.35 mm wide and was machined in a spiral of 8 turns, resulting in a porosity of 18%. The normal incidence absorption coefficient of these MSPs was measured in an impedance tube by means of the Transfer Function method [36]. The transfer function between two ¼ inch condenser microphones separated by 50 mm was used to calculate the normal incidence sound absorption coefficient of the three MSPs with spiral-shaped slits, mounted at one end of an impedance tube with an inner diameter of 30 mm, in the frequency range of 200 to 4000 Hz. Special care was taken to ensure the linearity of microphone measurements in the tube. These absorption curves are shown in Figure 4. They have peaks at 1200 Hz (0.88), 1440 Hz (0.65), and 1350 Hz (0.81), respectively.

## 4. Discussion

_{1},C

_{2}) affecting the parameter s and the flow resistivity, σ (see Table 1). Looking at Figure 5, Figure 6 and Figure 7, it seems that discrepancies between the EF model’s predicted and measured absorption curves are not too large. Therefore, the predictions of the above-described EF model could be improved by the empirical fitting of (C

_{1},C

_{2}).

## 5. Conclusions

_{1},C

_{2}) with spirally machined MSPs of different (d,t,φ) values.

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Absorption curves of MPPs and MSPs with varoius combinations of parameters: (

**A**) (d,t,φ,D) = (0.5 mm, 0.5 mm, 0.5%, 3 cm); (

**B**) (d,t,φ,D) = (0.35 mm, 0.65 mm, 1%, 3 cm); (

**C**) (d,t,φ,D) = (0.25 mm, 1 mm, 5%, 3 cm); (

**D**) (d,t,φ,D) = (0.2 mm, 5 mm, 20%, 3 cm).

**Figure 5.**Comparison of the measured sound absorption curve of MSP8 with those provided by the three models.

**Figure 6.**Comparison of the measured sound absorption curve of MSP9 with those provided by the three models.

**Figure 7.**Comparison of the measured sound absorption curve of MSP11 with those provided by the three models.

Circle | Square | Equilateral Triangle | Slit | |
---|---|---|---|---|

C_{1} | 1 | 1.07 | 1.11 | 0.81 |

C_{2} | 8 | 7 | 6.5 | 12 |

MSP | d (mm) | N | φ (%) |
---|---|---|---|

0.44 | 6 | 16 | |

0.44 | 7 | 19 | |

0.35 | 8 | 18 |

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**MDPI and ACS Style**

Cobo, P. Modelling of Microperforated Panel Absorbers with Circular and Slit Hole Geometries. *Acoustics* **2021**, *3*, 665-678.
https://doi.org/10.3390/acoustics3040042

**AMA Style**

Cobo P. Modelling of Microperforated Panel Absorbers with Circular and Slit Hole Geometries. *Acoustics*. 2021; 3(4):665-678.
https://doi.org/10.3390/acoustics3040042

**Chicago/Turabian Style**

Cobo, Pedro. 2021. "Modelling of Microperforated Panel Absorbers with Circular and Slit Hole Geometries" *Acoustics* 3, no. 4: 665-678.
https://doi.org/10.3390/acoustics3040042