# A Basic Study on the Design of Dotted-Art Heterogeneous MPP Sound Absorbers

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Background: Microperforated Panels (MPPs)

#### 1.2. Heterogeneous MPPs and Design-Oriented MPPs

#### 1.3. Outline of the Present Work

## 2. Prediction Method for the Absorption Characteristics of Heterogeneous MPPs

_{1}, d

_{2}, …, d

_{n}, and that its surface impedance is defined as Z

_{MPP}. Then, it is assumed that for each hypothetical MPP with holes of specific diameters only in surface area S, the impedance is defined as Z

_{MPP,1}, Z

_{MPP,2}, …, Z

_{MPP,n}; that is, the impedance of a hypothetical MPP with a hole of diameter d

_{1}is Z

_{MPP,1}, and the impedance of that with a hole of diameter d

_{2}is Z

_{MPP,2}, etc. (see Figure 1). In this study, Guo’s theory [21,22,23,24] (see Appendix A) is employed to calculate Z

_{MPP,1}, Z

_{MPP,2}, ….

_{MPP}is considered to be the impedance that is obtained when synthesizing the impedances of the hypothetical MPPs, Z

_{MPP,1}, Z

_{MPP,2}, …, Z

_{MPP,n}. Therefore, Z

_{MPP}is derived by the following equation:

_{MPP,i}is the impedance of a hypothetical MPP with holes of diameter d

_{i}. The impedances of the hypothetical MPPs are calculated as follows: a hypothetical impedance Z

_{MPP,i}is considered to be the synthesized impedance of the part with the holes and that of the part without holes, which is acoustically rigid. The surface area of the part with holes is S

_{i}, and its impedance is Z

_{i}. The surface area of the part without holes is (S-S

_{i}), and its impedance is Z

_{rigid}, which is assumed as infinity. Therefore, Z

_{MPP,i}is expressed by the following equation, and this calculation procedure is shown in the diagrams in Figure 2:

_{MPP,1}, Z

_{MPP,2}, …, Z

_{MPP,n}are calculated using Equation (2) and substituted into Equation (1).

_{i}

^{−1}as Y

_{i}, the acoustic admittance, and factor r

_{i}, the equation can be written in the form of the admittance sum method (ASM). In the following calculation, S

_{i}is set to the area occupied by a row of holes, which is determined by the spacing of the holes.

## 3. Preliminary Study: Applicability of the Synthetic Impedance Method to Various Heterogeneous MPPs

#### 3.1. Experiment

#### 3.1.1. Specimens

- Specimen 1 consists of four types of holes with diameters of 0.3 mm, 0.5 mm, 0.7 mm, and 0.9 mm, which are arranged in a gradient pattern so that the holes become larger every four rows (due to the limitation of the specimen size, only the 0.9 mm holes are arranged into three rows).
- Specimen 2 consists of two types of holes with diameters of 0.3 mm and 0.9 mm, which are arranged in a solidified manner.
- Specimen 3 consists of two types of holes with diameters of 0.3 mm and 0.9 mm, which are arranged into alternating rows.
- Specimen 4 consists of holes with diameters of 0.3 mm, 0.5 mm, 0.7 mm, 0.9 mm, and 1.1 mm, which are arranged in increasing order from the inner side.
- Specimen 5 consists of two types of holes with diameters of 0.3 mm and 0.9 mm, which are arranged in a checkerboard pattern so that the different holes are adjacent to each other.
- Specimen 6 consists of four types of holes with diameters of 0.3 mm, 0.5 mm, 0.7 mm, and 0.9 mm, arranged into rows of one.
- Specimen 7 consists of five types of holes with diameters of 0.3 mm, 0.5 mm, 0.7 mm, 0.9 mm, and 1.1 mm, which are arranged on circumferences of 10 mm, 30 mm, 50 mm, 70 mm, and 90 mm, respectively, in order from the inside.
- Specimen 8 consists of four types of holes with diameters of 0.5 mm, 0.7 mm, 0.9 mm, and 1.1 mm, which are arranged on circumferences of 16 mm, 29 mm, 42.5 mm, and 57 mm, respectively, in order from the inner side.
- Specimen 9 consists of 1.0 mm diameter holes arranged on a square of 60 mm per side.
- Specimen 10 consists of 1.0 mm diameter holes arranged on a circumference of 60 mm.
- Specimen 11 consists of holes with a diameter of 1.0 mm arranged on a circumference of 80 mm.
- Specimen 12 consists of 0.5 mm diameter holes arranged in the central area of a 50 mm square.

#### 3.1.2. Experimental Setup

#### 3.2. Prediction Error Indices and Classification According to the Error

_{error}; the relative error of the maximum value of sound absorption, i.e., the value of the resonance peak, α

_{error}; the RMS error of the sound absorption coefficient between 125 and 1700 Hz.

_{error,}of the frequency of the maximum sound absorption coefficient is calculated as follows:

_{pre}is the frequency at which the predicted sound absorption coefficient is maximum, and f

_{mea}is that at which the measured sound absorption coefficient is maximum.

_{error}, of the maximum sound absorption coefficient is calculated as follows:

_{max,pre}is the maximum value of the predicted sound absorption coefficient, and α

_{max,mea}is the maximum value of the measured sound absorption coefficient.

_{pre,1}.

_{25i}is the predicted sound absorption coefficient at 1.25i Hz, and α

_{mea,1}.

_{25i}is the measured sound absorption coefficient at 1.25i Hz.

- Group (1) has a relative error f
_{error}of the frequency with the maximum sound absorption coefficient and a relative error α_{error}of the maximum sound absorption coefficient, both below 5%, and an RMS error of 0.05 or less. - Group (3) has a relative error f
_{error}of the frequency of the maximum sound absorption or a relative error α_{error}of the maximum value of sound absorption greater than 10% or an RMS error greater than 0.1. - Specimens that do not fall into either group (1) or (3) were classified into group (2).

_{error}and α

_{error}were both less than 10%, and the RMS error was less than 0.1, and these were the reference values used to determine the predictability of the SIM.

#### 3.3. Results and Discussion

- Group (1).

- 2.
- Group (2).

- 3.
- Group (3).

- The number of hole types with different diameters is between 2 and 4;
- The holes are distributed over the entire surface of the specimen surface;
- The hole spacing is constant.

- The number of different hole types with different diameters is between 2 and 5;
- The holes are distributed over the entire surface of the specimen;
- The hole spacing is not always constant.

- The number of hole types with different diameters ranges from 1 to 5;
- The holes are distributed only in some parts of the specimen surface;
- The hole spacing is not constant.

## 4. Investigation by Prototyping Dotted-Art Heterogeneous MPPs

#### 4.1. Design Concept of Dotted-Art Heterogeneous MPPs

- The holes are distributed over the entire surface of the absorber;
- The hole separation is constant over the entire surface of the absorber.

#### 4.2. Preparation of the Specimens

- Specimen A is made using 0.2 mm and 0.8 mm diameter holes in a star pattern and with a hole spacing of 4.0 mm.
- Specimen B is made using 0.2 mm and 1.0 mm diameter holes in a star pattern and with a hole spacing of 4.0 mm. The arrangement of the holes is identical to that of specimen A.
- Specimen C is a star pattern with 0.2 mm and 0.8 mm diameter holes, with a hole spacing of 6.0 mm. Although the hole spacing is different, the size of the star is almost the same as that in specimens A and B.
- Specimen D is made with 0.2 mm and 1.0 mm diameter holes and a hole spacing of 6.0 mm to express a star pattern. The arrangement of the holes is identical to that in specimen C.
- Specimen E is made using 0.2 mm and 0.8 mm diameter holes, with a hole spacing of 4.0 mm to produce a star pattern. The hole spacing and the type of holes used are the same as those in specimen A, but the size of the star pattern is smaller than that in specimen E.
- Specimen F is made using 0.2 mm and 0.8 mm diameter holes, with a hole spacing of 4.0 mm and expressing the pattern of a bear face.
- Specimen G is made of 0.2 mm and 0.8 mm holes and a hole spacing of 4.0 mm, which illustrate the bold letters ‘KOBE’.
- Specimen H is the outer frame of the bold letters ‘KOBE’, with holes of 0.2 mm and 0.8 mm in diameter and a hole spacing of 4.0 mm.
- Specimen I is a zigzag pattern, with holes of 0.2 mm and 0.8 mm in diameter and a hole spacing of 4.0 mm.

#### 4.3. Method of Investigating Predictability

#### 4.3.1. Experiment

#### 4.3.2. Results and Discussion

#### 4.4. Effect of Hole Distribution on Prediction Accuracy

_{av}(%). Then, the proportion of the larger holes x

_{i}(%) to the total number of holes in each part can be calculated to obtain the standard deviation, SD.

#### 4.5. Effect of the Absence of Holes in the Background on Prediction Accuracy

## 5. Conclusions

_{error}, α

_{error}, and RMS error. From the investigations, the following were found:

- The SIM can predict the sound absorption characteristics of heterogeneous MPPs that satisfy the following two conditions: (1) the holes are distributed over the entire surface of the specimen, and (2) the hole spacing is constant.
- The condition of constant hole spacing is not essential for the application of the SIM, but it is considered to be a condition that enables a more reliable prediction.
- Heterogeneous MPPs with holes that are distributed in only a limited part of the specimen surface are likely to be outside the scope of the application of the SIM.
- When using the SIM to predict the sound absorption properties of heterogeneous MPPs, a number of different hole sizes is unlikely to influence prediction accuracy.

- The prediction accuracy for the specimens of dotted-art heterogeneous MPPs, designed according to the concept described above, tends to be good. This is because holes of the same diameter are unbiasedly distributed over the surface; i.e., the distribution of holes of the same diameter is less biased. Therefore, for the design of a dotted-art heterogeneous MPP, the distribution of holes of the same diameter should be as unbiased as possible.
- For dotted-art heterogeneous MPPs with a combination of two types of holes with different diameters: If the smaller holes in the background area are removed and only one type of hole is used in the patterned area of the heterogeneous MPP, the SIM is still applicable to the specimens (without smaller holes); however, prediction accuracy decreases. Therefore, a dotted-art heterogeneous MPP with two types of holes is better in terms of prediction accuracy.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Outline of Guo’s Theory

_{t}, which is expressed as

_{e}is the effective density, which is expressed by the following equation in the case of a circular hole:

_{0}and J

_{1}are the zeroth- and first-order Bessel functions of the first kind, respectively. The parameter s in Equation (A2) in the case of a circular hole is defined as follows:

_{e}is given as follows:

^{2}); this becomes σϕ = 8η/r

_{p}

^{2}in the case of a square hole, with r

_{p}as the parameter defined by the perimeter of the cell l (m) and the cross-sectional area S (m

^{2}), which is the equivalent radius r

_{p}= 2S/l (m). G

_{c}(s) is given as follows:

_{s}is given by the following equation:

_{c}, as follows:

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**Figure 1.**Diagrams of the hypothetical MPPs. The upper MPP is the original MPP, and the lower MPPs are the hypothetical MPPs, which have holes of different diameters only in surface area S.

**Figure 2.**A diagram of the calculation procedure for the impedance of a hypothetical MPP. The area surrounded by the dashed line is the part with holes, and the hatched area is the part without holes, which is acoustically rigid.

**Figure 5.**Comparison of the measured and predicted values of the normal incidence sound absorption coefficient of specimen 1. The orange solid line represents the measured values, and the blue dotted line represents the predicted values.

**Figure 6.**Comparison of the measured and predicted values of the normal incidence sound absorption coefficient of specimen 3. The orange solid line represents the measured values, and the blue dotted line represents the predicted values.

**Figure 7.**Comparison of the measured and predicted values of the normal incidence sound absorption coefficient of test specimen 8. The orange solid line represents the measured values, and the blue dotted line represents the predicted values.

**Figure 8.**Schematic diagram of the design concept for a dotted-art heterogeneous MPP. In the diagram, the orange dots represent the larger holes of the pattern, and the grey dots represent the smaller holes that comprise the background part.

**Figure 10.**Comparison of the measured and predicted values of the normal incidence sound absorption coefficient of test specimen A. The orange solid line represents the measured values, and the blue dotted line represents the predicted values.

**Figure 11.**Schematic diagram of the division into parts to calculate the standard deviation of the number of the larger holes: (

**a**) 25 horizontal and 25 vertical rows; (

**b**) 17 horizontal and 17 vertical rows.

Specimen | Diameter (mm) and (Number of Holes) | Hole Separation (mm) | Average Perforation Ratio (%) | Thickness of the Plate (mm) |
---|---|---|---|---|

1 | 0.3 (60), 0.5 (60), 0.7 (60), 0.9 (45) | 6.0 | 0.6774 | 0.5 |

2 | 0.3 (120), 0.9 (105) | 6.0 | 0.7528 | 0.5 |

3 | 0.3 (120), 0.9 (105) | 6.0 | 0.7528 | 0.5 |

4 | 0.3 (4), 0.5 (12), 0.7 (20), 0.9 (28), 1.1 (36) | 10.0 | 0.6236 | 0.5 |

5 | 0.3 (113), 0.9 (112) | 6.0 | 0.7924 | 0.5 |

6 | 0.3 (60), 0.5 (60), 0.7 (60), 0.9 (45) | 6.0 | 0.6774 | 0.5 |

7 | 0.3 (10), 0.5 (30), 0.7 (50), 0.9 (70), 1.1 (90) | 3.142 on all circumferences | 1.559 | 0.5 |

8 | 0.5 (45), 0.7 (45), 0.9 (45), 1.1 (60) | On all the circumferences from inside to outside: 1.117, 1.518, 2.225, 2.985 | 1.118 | 1.0 |

9 | 1.0 (48) | 5.0 | 0.3770 | 0.5 |

10 | 1.0 (60) | 3.142 | 0.4712 | 0.5 |

11 | 1.0 (100) | 2.513 | 0.7854 | 0.5 |

12 | 0.5 (121) | 5.0 | 0.2376 | 0.5 |

**Table 2.**Mean values of the error indices for the four conditions of air-back cavity depth of specimens 1 and 2.

Specimen | f_{error} (%) | α_{error} (%) | RMS Error |
---|---|---|---|

1 | 1.540 | 3.223 | 0.04191 |

2 | 3.675 | 2.632 | 0.03966 |

**Table 3.**Mean values of the error indices for the four conditions of air-back cavity depth of specimens 3, 4, 5, 6, and 7.

Specimen | f_{error} (%) | α_{error} (%) | RMS Error |
---|---|---|---|

3 | 3.591 | 5.730 | 0.04853 |

4 | 2.704 | 4.656 | 0.05104 |

5 | 1.038 | 5.588 | 0.04313 |

6 | 2.242 | 5.036 | 0.03980 |

7 | 7.938 | 3.657 | 0.04662 |

**Table 4.**Mean values of the error indices (Equations (4)–(6)) for the four conditions of air-back cavity depth of specimens 8–12.

Specimen | f_{error} (%) | α_{error} (%) | RMS Error |
---|---|---|---|

8 | 13.69 | 11.95 | 0.1424 |

9 | 4.467 | 13.39 | 0.08091 |

10 | 2.310 | 13.24 | 0.08296 |

11 | 9.604 | 14.29 | 0.09839 |

12 | 11.18 | 8.632 | 0.09974 |

**Table 5.**Parameters of specimens A–I. Note that the larger hole is used for the pattern, and the smaller hole is used for the background *.

Specimen | Diameter (mm) and (Number of Holes) | Hole Separation (mm) | Average Perforation Ratio (%) | Thickness of the Plate (mm) |
---|---|---|---|---|

A | 0.2 (469), 0.8 (156) | 4.0 | 0.9315 | 0.5 |

B | 0.2 (469), 1.0 (156) | 4.0 | 1.373 | 0.5 |

C | 0.2 (216), 0.8 (73) | 6.0 | 0.4348 | 0.5 |

D | 0.2 (216), 1.0 (73) | 6.0 | 0.6412 | 0.5 |

E | 0.2 (546), 0.8 (79) | 4.0 | 0.5686 | 0.5 |

F | 0.2 (470), 0.8 (150) | 4.0 | 0.6774 | 0.5 |

G | 0.2 (318), 0.8 (307) | 4.0 | 1.643 | 0.5 |

H | 0.2 (417), 0.8 (208) | 4.0 | 1.177 | 0.5 |

I | 0.2 (425), 0.8 (200) | 4.0 | 1.139 | 0.5 |

**Table 6.**Mean absolute values of the error indices (Equations (4)–(6)) for the four air-back cavity depth conditions for specimens A–I.

Specimen | f_{error} (%) | α_{error} (%) | RMS Error |
---|---|---|---|

A | 8.598 | 3.640 | 0.05230 |

B | 7.356 | 3.001 | 0.04935 |

C | 6.846 | 3.147 | 0.08007 |

D | 1.100 | 8.184 | 0.06381 |

E | 3.205 | 2.295 | 0.04886 |

F | 0.8354 | 5.409 | 0.03031 |

G | 2.718 | 4.792 | 0.03080 |

H | 1.842 | 4.409 | 0.02988 |

I | 1.631 | 2.206 | 0.02379 |

**Table 7.**Comparison of the standard deviation (SD) and error indices (Equations (4)–(6)) of specimens A–I.

Specimen A | Specimen B | Specimen C | |||
---|---|---|---|---|---|

SD | 33.45 | SD | 33.45 | SD | 30.72 |

f_{error} (%) | 8.598 | f_{error} (%) | 7.356 | f_{error} (%) | 6.846 |

α_{error} (%) | 3.640 | α_{error} (%) | 3.001 | α_{error} (%) | 3.147 |

RMS error | 0.05230 | RMS error | 0.04935 | RMS error | 0.08007 |

Specimen D | Specimen E | Specimen D | |||

SD | 30.72 | SD | 23.96 | SD | 11.81 |

f_{error} (%) | 1.100 | f_{error} (%) | 3.205 | f_{error} (%) | 0.8354 |

α_{error} (%) | 8.184 | α_{error} (%) | 2.295 | α_{error} (%) | 5.409 |

RMS error | 0.06381 | RMS error | 0.04886 | RMS error | 0.03031 |

Specimen G | Specimen H | Specimen I | |||

SD | 19.02 | SD | 10.16 | SD | 5.426 |

f_{error} (%) | 2.718 | f_{error} (%) | 1.842 | f_{error} (%) | 1.631 |

α_{error} (%) | 4.792 | α_{error} (%) | 4.409 | α_{error} (%) | 2.206 |

RMS error | 0.03080 | RMS error | 0.02988 | RMS error | 0.02379 |

**Table 8.**Mean absolute values of the error indices (Equations (4)–(6)) for the four air-back cavity depth conditions for specimens A–I and A’–I’. ‘With’ means ‘with holes in the background parts’, and ‘Without’ means ‘without holes in the background parts’. To compare the ‘With’ and ‘Without’ conditions, the smaller values are hatched.

f_{error} (%) | α_{error} (%) | RMS Error | ||||
---|---|---|---|---|---|---|

With | Without | With | Without | With | Without | |

Specimens A, A’ | 8.598 | 9.361 | 3.640 | 8.239 | 0.05230 | 0.07576 |

Specimens B, B’ | 7.356 | 8.016 | 3.001 | 9.256 | 0.04935 | 0.08741 |

Specimens C, C’ | 6.846 | 6.413 | 3.147 | 8.076 | 0.08007 | 0.08845 |

Specimens D, D’ | 1.100 | 1.478 | 8.184 | 12.66 | 0.06381 | 0.07217 |

Specimens E, E’ | 3.205 | 3.675 | 2.295 | 16.06 | 0.04886 | 0.1076 |

Specimens F, F’ | 0.8354 | 1.112 | 5.409 | 7.030 | 0.03031 | 0.05351 |

Specimens G, G’ | 2.718 | 2.236 | 4.792 | 5.564 | 0.03080 | 0.04364 |

Specimens H, H’ | 1.611 | 4.373 | 4.640 | 8.494 | 0.02988 | 0.04858 |

Specimens I, I’ | 1.631 | 2.506 | 2.206 | 7.442 | 0.02379 | 0.03993 |

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## Share and Cite

**MDPI and ACS Style**

Sakagami, K.; Kusaka, M.; Okuzono, T.
A Basic Study on the Design of Dotted-Art Heterogeneous MPP Sound Absorbers. *Acoustics* **2022**, *4*, 588-608.
https://doi.org/10.3390/acoustics4030037

**AMA Style**

Sakagami K, Kusaka M, Okuzono T.
A Basic Study on the Design of Dotted-Art Heterogeneous MPP Sound Absorbers. *Acoustics*. 2022; 4(3):588-608.
https://doi.org/10.3390/acoustics4030037

**Chicago/Turabian Style**

Sakagami, Kimihiro, Midori Kusaka, and Takeshi Okuzono.
2022. "A Basic Study on the Design of Dotted-Art Heterogeneous MPP Sound Absorbers" *Acoustics* 4, no. 3: 588-608.
https://doi.org/10.3390/acoustics4030037