Characteristics of powders that cause sound absorption in the low frequency range due to longitudinal vibration in lightweight and fine powders Shuich Sakamoto 1 Mechanical Engineering Program, Faculty of Engineering, Niigata University. 2-8050 Ikarashi, Nishi-ku, Niigata, 950-2181, Japan. Ren Saito 2 Graduate School of Science and Technology, Niigata University. Ryusuke Suzuki 3 Graduate School of Science and Technology, Niigata University. Keisuke Jindai 4 Graduate School of Science and Technology, Niigata University. Koki Ikeda 5 Graduate School of Science and Technology, Niigata University.

ABSTRACT In this paper, focus is placed on "lightweight and fine powders" which have small particle size and bulk density and exhibit unique sound absorption characteristics at low frequencies due to longitudinal vibration of powder particles. Theoretical analysis of the sound absorption coefficient of powder layers requires the peak frequency of sound absorption measured experimentally. In this study, we clarified the relationship between powder properties and sound absorption characteristics, and predicted the sound absorption coefficient of the powder layer from the experimental equation based on the accumulated data. This made it possible to classify powders in which sound absorption due to longitudinal vibration occurs based on the relationship between the "areal density per particle layer," which is indicated by "particle size multiplied by bulk density," and the peak sound absorption value. Moreover, from the plot of "areal density per particle layer" and "first-order sound absorption peak frequency from experiment," the experimental formula for the first-order sound absorption peak frequency was regressively obtained. By using the experimental equation obtained in this study, it was possible to show how to estimate the sound absorption coefficient from the properties of each powder alone.

1 sakamoto@eng.niigata-u.ac.jp 2 t18m904f@gmail.com 3 t15m044d@gmail.com 4 f21b090e@gmail.com 5 t18a011f@gmail.com

1. INTRODUCTION

The sound-absorption coefficient of a porous sound-absorbing material at medium and low frequencies can be increased by increasing the thickness of the material and adding an air space behind it. However, simply increasing the material thickness is not desirable because doing so increases the cost, weight, and dimensions. Therefore, the authors have focused on powders as a lightweight sound-absorbing material. In addition to sound absorption due to a boundary-layer viscosity similar to that of porous materials [1], powder layers attenuate acoustic energy via the interaction between particles whose vibrations are excited by sound waves [2]. In a previous report [3], a model was proposed to analyze the sound- absorption coefficient of a powder layer by combining the damping due to the boundary-layer viscosity between powder particles and that due to vibration, assuming that the powder layer is a continuum of multi-degree-of-freedom vibrations. The lightweight and fine powders [2][3] treated in this paper have a small particle size and bulk density, and exhibit significant sound-absorption characteristics at low frequencies due to the longitudinal vibration of the powder particles. However, it is unclear at which values of particle size and bulk density the material would exhibit significant sound absorption. Theoretical analysis of the sound-absorption coefficient of a powder layer [2] requires the peak frequency of sound absorption obtained experimentally, which is an obstacle in the search for new powders through model-based development. Therefore, the purpose of this study is to clarify the relationship between the powder’s specifications and sound-absorption characteristics. Furthermore, the sound-absorption coefficient of the powder layer is estimated from experimental equations based on accumulated data, rather than directly using the peak sound-absorption frequencies obtained experimentally.

2. THEORETICAL ANALYSIS

2.1. Analysis of Longitudinal Vibration of Powder Layer The powder layer is assumed to be a multi-degree-of-freedom forced-vibration problem of a continuum divided into five parts in the thickness direction, as shown in Figure 1 [2]. In Figure 1, m n , k n , and c n ( n = 1–5) are the mass, spring constant, and damping coefficient of each element after division, respectively. With mass matrix [ M ], stiffness matrix [ K ], damping matrix [ C ], and external force { F }, the equation of motion becomes Equation (1).

 [ ]{ } [ ]{ } [ ]{ } M x C x K x F   

(1)

The modal mass and stiffness are obtained for each mode by calculating a dynamic mass and stiffness matrix that considers the inertial forces and analyzing the eigenvalues of Equation (1) as an undamped free-vibration system. Because the damping mechanism of powders is complex and nearly impossible to obtain theoretically, the modal damping coefficients were obtained under the assumption of Rayleigh damping. The obtained modal mass, modal stiffness, and modal damping coefficient are used to replace the multi-degree-of-freedom vibration system with an equivalent one-degree-of-freedom vibration system. The specific acoustic impedance for each mode is given by Equation (2) from the equation of motion. By superimposing the obtained r th-order specific acoustic impedance up to the fifth order, the specific acoustic impedance of the entire powder layer can be obtained.

1 k p r Z c j m r r r x S                  

(2)

2.2. Analysis of Boundary-Layer Viscosity between Powder Particles If the packing structure of powder particles is assumed to be a simple cubic lattice, as shown in Figure 2, the boundary-layer viscosity of the gap between particles is analyzed using the transfer matrix method related to sound pressure and volume velocity based on the one-dimensional wave equation [4][5][6]. The particle packing structure is divided perpendicular to the direction of sound wave

incidence, the characteristic impedance and propagation constant of each divided element are calculated, and the transfer matrix method is applied. By connecting the obtained transfer matrixes of the divided elements with an equivalent circuit, the overall transfer matrix T boundary is obtained, as shown in Equation (3). Stinson’s method [7] is applied to derive the characteristic impedance Z c and propagation constant  , and the gap is approximated in two planes to account for attenuation in the gap. Because the end of the powder layer is a rigid wall, the specific acoustic impedance of the boundary- layer viscosity is calculated by Equation (4).

                   

Z l l A B S Tall S C D l l Z

c cosh( ) sinh( )

 

(3)

 

sinh( ) cosh( ) c

p A Z S boundary u C   (4)

0

0 0

x n F

S

m = M / n

x 3

S 0

x 2

z

x 1 c n

k n

y

x

Figure 1: Analysis model of powder layer. Figure 2: Analysis model of simple cubic lattice.

2.3. Calculation of Sound-Absorption Coefficient The specific acoustic impedance was calculated by taking into account the longitudinal vibration of the powder layer and the boundary-layer viscosity between the particles, respectively. By connecting them in parallel as in Equation (5), the overall specific acoustic impedance is obtained. Furthermore, the complex reflectance is calculated from the specific acoustic impedance to obtain the sound- absorption coefficient. Table 1: Materials used in the experiments.

Type of powder Particle size

[mm] Bulk density [g/cm 3 ] Areal density per layer

of particles [g/cm 2 ] Gran u lated silica 0.0428 0.057 2.43E-04 Hollow glass beads 0.0291 0.080 2.33E-04 Hollow gla ss beads d > 38 μm 0.0675 0.064 4.30E-04 Hollow plast i c beads d > 38 μm 0.0423 0.030 1.26E-04 PMMA b eads d ≤ 38 μm 0.0221 0.730 1.61E-03 PMMA beads 38 < d ≤ 53 μm 0.0470 0.735 3.45E-03 PMMA b eads d > 53 μm 0.0634 0.730 4.63E-03 PP beads 0.0767 0.480 3.68E-03 PTFE beads 0.1786 0.907 1.62E-02 Solid gl a ss beads φ0.05 0.0567 1.450 8.22E-03

   (5)

Z Z powder boundary Zall Z Z powder boundary

3. MEASURING EQUIPMENT AND SAMPLES

A Brüel & Kjær Type 4206 two-microphone impedance tube was used as the measurement device, and the normal incident sound-absorption coefficient was calculated according to ISO 10534-2. The 10 powder types listed in Table 1 were used. The sample tubes were small tubes with an inner diameter of 29 mm and sample lengths of 10, 20, 30, and 40 mm. 4. RELATIONSHIP BETWEEN POWDER SPECIFICATIONS AND SOUND- ABSORPTION COEFFICIENT

Figures 3(a)–(c) plot the relationship between the peak sound-absorption coefficient of the powder layer and the powder properties. The abscissa is the particle size, bulk density, and areal density per particle layer of the powder, respectively, and the ordinate is the peak sound-absorption coefficient. From Figure 3(a), it can be seen that in some cases, the sound-absorption coefficient is higher when the particle size is smaller, but in others, it is not. This is because even if the grain size is small, a vibration is not easily excited when the bulk density is large. Figure 3(b) shows that the sound- absorption peak tends to be higher when the bulk density is smaller. However, a small bulk density

1

frequency of the 1st peak ( l = 20 mm)

Sound absorption coefficient at

Granulated silica Hollow glass beads

0.8

Hollow glass beads d>38μm Hollow plastic beads d>38μm

0.6

PMMA beads d ≦ 38μm PMMA beads 38

0.4

PMMA beads d>53μm PP beads

0.2

PTFE beads Solid glass beads φ0.05

0

0.01 0.1 1

Particle size [mm]

(a) Relationship with particle size.

Light and fine powder

1

1

frequency of the 1st peak ( l = 20 mm)

frequency of the 1st peak ( l = 20 mm)

Sound absorption coefficient at

Sound absorption coefficient at

0.8

0.8

0.6

Conventional powder

0.6

Polystyrene foam  2

0.4

mm (0.016 g/cm 2 )

0.4

0.2

No vibration

0.2

0

0

0.0001 0.001 0.01 0.1

0.01 0.1 1 10

Areal density per layer of particles [g/cm 2 ]

Bulk density [g/cm 3 ]

(b) Relationship with bulk density. (c) Relationship with Areal density per layer

of particles.

Figure 3: Relationship between sound absorption coefficient and physical properties of powders

does not necessarily mean that vibration-induced sound absorption occurs. Here, we have plotted the results for expanded polystyrene foam beads, a  2-mm granular material, as a reference. The ease of sound absorption by longitudinal vibration must be determined from both the grain size and bulk density. Therefore, as shown in Figure 3(c), taking the areal density per particle layer, expressed as the particle diameter times the bulk density, as the horizontal axis, it is clear that a smaller areal density per particle layer yields a higher sound-absorption coefficient. 5. ESTIMATED SOUND-ABSORPTION COEFFICIENT, COMPARISON WITH EXPERIMENTAL VALUES

Estimating the sound-absorption coefficient of a powder layer requires the first-order peak sound- absorption frequency of the powder layer, which is obtained experimentally. Figure 4 plots the relationship between areal density per particle layer and normalized first-order peak sound-absorption frequency multiplied by the thickness of each layer. From these plots, the experimental equation for the first-order peak sound-absorption frequency was obtained through regression. Figure 5 compares the experimental values of sound-absorption coefficient with the estimated sound-absorption coefficient of the powder layer using the first-order peak sound-absorption frequency back-calculated by the regressive experimental equation. Although there are differences in the peak frequency, the peak sound absorption, dip value, and sound-absorption curve are close to the experimental values.

1

100000

normalized by multiplying by layer

Absorption coefficient

0.8

Frequency of the 1st peak

thickness [Hz×mm]

0.6

10000

0.4

0.2

1000

100 500 1000 5000 0

0 0.0001 0.0002 0.0003 0.0004 0.0005

6400

Areal density per layer of particles [g/cm 2 ]

Frequency [Hz]

Granulated silica Hollow glass beads Hollow glass beads d>38μm Hollow plastic beads d>38μm

Experimental Value

Theoretical Value

Calculation with f peak based on experimental formula 　　　 (this study)

Calculation with f peak based on actual measurements 　　　 (conventional)

6. CONCLUSIONS

Figure 4: Relationship between modulus of longitudinal elasticity and areal density per

Figure 5: Comparison between experimental and theoretical values for hollow plastic beads

layer of particles.

( d > 38μm, l = 20 mm).

To clarify the conditions necessary for lightweight and fine powders to exhibit unique powder-layer sound absorption, the relationship between the peak sound-absorption coefficient of the powder layer and the various specifications of the powder was investigated. As a result, lightweight and fine powders could be classified based on the relationship between the areal density per particle layer and the peak sound-absorption coefficient value. Furthermore, the experimental equation for the first- order peak sound-absorption frequency was obtained by regression from a plot of the areal density per particle layer versus the experimental first-order peak sound-absorption frequency normalized by multiplying by layer thickness. With the experimental equations obtained in this study, we were able to demonstrate a method for estimating the sound-absorption coefficient based solely on the

specifications of each powder. This method of classifying powders and estimating sound-absorption coefficients from the powder specifications alone enables the search for lightweight, fine powders that exhibit significant sound absorption. 7. ACKNOWLEDGEMENTS

We gratefully acknowledge this work was supported by JSPS KAKENHI Grant Number 20K04359. 8. REFERENCES

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experiment analysis on the sound absorption characteristics of a layer of fine lightweight powder, Journal of the Acoustical Society of America, 146(4) , 2253–2262 (2019). 3. Sakamoto, S., Takakura, R., Suzuki, R., Katayama, I., Suzuki, K., Theoretical and experimental

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