JP4182111B2 - Method for predicting acoustic properties of fiber assemblies - Google Patents

Description

  The present invention relates to a method for predicting acoustic characteristics such as sound absorption characteristics in a fiber assembly such as felt and glass wool by calculation.

  The acoustic characteristics such as the sound absorption characteristics of the fiber assembly or the porous synthetic resin layer (hereinafter referred to as porous material) are measured by measuring the flow resistance of air flowing through the porous material and the porosity called porosity. The ratio of air in the porous material, energy conversion value called viscosity and thermal characteristic length, air passage length in porous material called tortoise (maze degree), and elastic modulus that is the hardness index of porous material Prediction is possible by actually measuring and adding the apparent bulk density and material thickness of the porous material to the calculation.

Reference 1 below discloses the theory of acoustic characteristics of a porous material (Johnson model).
Allard, J .; F, “Propagation of Sound in Porous Media”, Elsevier Applied Science, 1993.

By the way, the above-mentioned flow resistance is obtained by flowing a slight wind at a wind speed of 0.05 cm / s through a porous material and measuring the differential pressure before and after the material. There is a problem that measurement errors are likely to occur, for example, it is necessary to pay sufficient attention to the environment of the apparatus.
The porosity is obtained by placing a porous material in a fixed volume container and sealing it, changing the volume slightly, and measuring the volume change and pressure change at that time. The characteristic characteristic length requires measurement equipment and man-hours, such as using an ultrasonic pulse transmitter / receiver and placing a porous material in a test room where the atmosphere is replaced with helium gas. Otherwise, measurement errors are likely to occur, and it is not easy to predict acoustic characteristics with high accuracy.
Therefore, the present invention intends to provide a method that makes it possible to predict acoustic characteristics such as normal incidence sound absorption coefficient, acoustic impedance, propagation constant, and characteristic impedance with high accuracy for fiber assemblies such as felt and glass wool. It is.

上式中、ρ₀は空気の密度で１８℃の場合ρ₀=1.2[kg/m³]、μは空気の粘性係数で常温の空気ではμ≒1.84×10^-5[N・ｓ/m²]、ωはω＝2πｆで表される角速度（ｆは周波数）、Λは繊維材料の粘性特性長、Λ’は繊維材料の熱的特性長である。

上式中、ρ₀は空気の密度、c₀は空気中の音速、γは空気の比熱比である。また、ρ₀c₀ ²は空気の体積弾性率で、大気圧P₀[P_a]を用いてρ₀c₀ ²=γP₀の関係式があり、18℃の空気の場合ρ₀=1.2[kg/m³]、c₀=342[m/s]、γ=1.4である。標準気圧P₀は1013×10²[Pa]である。また、ｉは虚数単位（ｉ^２＝−１）である。

上式中、μは空気の粘性係数で常温の空気ではμ≒1.84×10^-5[N・ｓ/m²]、Ｄは実効繊維径である。

In order to solve the above problems, the invention of the acoustic property prediction method for a fiber assembly described in claim 1 calculates the effective density ρ _eff by the following equation (Equation 26), and the bulk modulus K _f by the following equation (Equation 27). The constant C determined by the fiber diameter is calculated by the following equation (Equation 30), and the characteristic impedance of the fiber aggregate such as felt and glass wool is calculated using the effective density ρ _eff , the bulk modulus K _f and the constant C. A method for predicting acoustic characteristics of a fiber assembly, characterized in that Zc is calculated by the following equation (Equation 31).

In the above equation, ρ ₀ is the density of air at 18 ° C, ρ ₀ = 1.2 [kg / m ³ ], μ is the viscosity coefficient of air, and μ≈1.84 × 10 ^-5 [N · s / m for normal temperature air ² ], ω is an angular velocity (f is a frequency) represented by ω = 2πf, Λ is a viscous characteristic length of the fiber material, and Λ ′ is a thermal characteristic length of the fiber material.

In the above equation, ρ ₀ is the density of air, c ₀ is the speed of sound in the air, and γ is the specific heat ratio of air. Ρ ₀ c ₀ ² is the bulk modulus of air, and there is a relational expression of ρ ₀ c ₀ ² = γP ₀ using the atmospheric pressure P ₀ [P _a ]. In the case of air at 18 ° C., ρ ₀ = 1.2 [kg / m ³ ], c ₀ = 342 [m / s], and γ = 1.4. The standard atmospheric pressure P ₀ is 1013 × 10 ² [Pa]. Further, i is an imaginary unit (i ² = −1).

In the above equation, μ is the viscosity coefficient of air, μ≈1.84 × 10 ⁻⁵ [N · s / m ² ] in normal temperature air, and D is the effective fiber diameter.

上式中、ｉは虚数単位（ｉ^２＝−１）、ωはω＝2πｆで表される角速度（ｆは周波数）である。 In the invention described in claim 2, the propagation constant Γ of a fiber aggregate such as felt or glass wool is calculated by the following equation (Equation 32) using the effective density ρ _eff and the bulk modulus K _f described in claim 1. A method for predicting acoustic characteristics of a fiber assembly, characterized in that:

In the above formula, i is an imaginary unit (i ² = −1), ω is an angular velocity (f is a frequency) represented by ω = 2πf.

上式中、ｄは繊維集合体の厚みである。 The invention described in claim 3 is characterized in that the acoustic impedance Z ₀ on the condition of the back rigid wall of the fiber assembly such as felt or glass wool is the characteristic impedance Zc described in claim 1 and the propagation constant described in claim 2. A method for predicting acoustic characteristics of a fiber assembly, wherein Γ is calculated by the following equation (Expression 33).

In the above formula, d is the thickness of the fiber assembly.

上式中、ρ₀c₀は空気の固有音響抵抗で気温18℃の場合はρ₀c₀＝415Ns/m³である。 In the invention described in claim 4, the normal incident sound absorption coefficient α ₀ of the fiber assembly such as felt or glass wool is calculated using the acoustic impedance Z ₀ described in claim 3 by the following equation (Equation 34). A method for predicting acoustic characteristics of a featured fiber assembly.

In the above equation, ρ ₀ c ₀ is the specific acoustic resistance of air, and ρ ₀ c ₀ = 415 Ns / m ³ when the temperature is 18 ° C.

  According to the present invention, the acoustic characteristics of fiber aggregates such as felt and glass wool can be predicted relatively easily and accurately, so that the design and development of a new acoustic component such as a sound absorber can be facilitated and its cost can be reduced. It brings convenience such as mitigation.

上式中、ρ₀は空気の密度、c₀は空気中の音速、γは空気の比熱比である。また、ρ₀c₀ ²は空気の体積弾性率で、大気圧P₀[P_a]を用いてρ₀c₀ ²=γP₀の関係式から置き換えることも可能である。18℃の空気の場合ρ₀=1.2[kg/m3]、c₀=342[m/s]、γ=1.4である。標準気圧はP₀=1013×10²[Pa]である。また、ｉは虚数単位（ｉ^２＝−１）である。また、ｓおよびｓ'はせん断力波数で次式で定義される。

上式中、Rは円筒管の内半径、B²はプラントル数で常温空気の場合B²=0.71で扱われる。 Next, an embodiment of the present invention will be described. First, the outline of the acoustic characteristic theory (hereinafter referred to as the Johnson-Allard model) of the porous material currently published by Non-Patent Document 1 and the like will be described.
The Johnson-Allard model is based on a cylindrical tube model in which the inner radius of the cylindrical tube is replaced by the physical properties of the porous material. Flow resistance, porosity, tortoise, viscosity characteristic length, and thermal characteristic length are used as the physical characteristic values. The effective density ρ _eff and bulk modulus K _f of air inside the cylindrical tube are defined by the following equations using the Bessel function based on Kirchhoff's theory.

In the above equation, ρ ₀ is the density of air, c ₀ is the speed of sound in the air, and γ is the specific heat ratio of air. Ρ ₀ c ₀ ² is the bulk modulus of air, and can be replaced from the relational expression of ρ ₀ c ₀ ² = γP ₀ using the atmospheric pressure P ₀ [P _a ]. In the case of air at 18 ° C., ρ ₀ = 1.2 [kg / m 3], c ₀ = 342 [m / s], and γ = 1.4. The standard atmospheric pressure is P ₀ = 1013 × 10 ² [Pa]. Further, i is an imaginary unit (i ² = −1). Further, s and s ′ are shear force wave numbers and are defined by the following equations.

In the above equation, R is the inner radius of the cylindrical tube, B ² is the Prandtl number, and B ² = 0.71 for room temperature air.

また、円筒管の内半径Rと流れ抵抗σの関係はHagen−Poiseuilleの法則を基に次式が成り立つ。

上式中、μは空気の粘性係数で、常温の空気ではμ≒1.84×10^-5[N・ｓ/m²]となる。α_∞はトーチュオジティ、φはポロシティである。
Johnson-Allardモデルでは上記式（数５）のR₁を上記式（数７）のRに置き換えた式となる。 In the Johnson-Allard model, the Bessel function part is replaced by the following equation.

The relationship between the inner radius R of the cylindrical tube and the flow resistance σ is based on the Hagen-Poiseuille law.

In the above equation, μ is the viscosity coefficient of air, and μ≈1.84 × 10 ⁻⁵ [N · s / m ² ] for air at normal temperature. α _∞ is tortoise, and φ is porosity.
In the Johnson-Allard model, R ₁ in the above equation (Equation 5) is replaced with R in the above equation (Equation 7).

式（数７）に式（数８）を代入し、Λ'について書き換えると、

Johnson-Allardモデルでは式（数５）のR₂を式（数９）のΛ'に置き換えた式となる。
このため、実効密度式（数１）にトーチュオジティα_∞を考慮（α_∞を積算）し、式（数６）のR₃、R₄を熱的特性長Λ'に置き換えることで、以下のようにJohnson-Allardモデルの実効密度ρ_effと体積弾性率K_fの式が完成する。

Further, it is assumed that the following equation holds between the viscous characteristic length Λ, the thermal characteristic length Λ ′, and the inner radius R.

Substituting Equation (Equation 8) into Equation (Equation 7) and rewriting for Λ ′,

In the Johnson-Allard model, R ₂ in the equation (Equation 5) is replaced with Λ ′ in the equation (Equation 9).
For this reason, by considering the tortility α _∞ in the effective density equation (Equation 1) (accumulating α _∞ ) and replacing R ₃ and R ₄ in the equation (Equation 6) with the thermal characteristic length Λ ′, the following is obtained. The formulas for the effective density ρ _eff and bulk modulus K _f of the Johnson-Allard model are completed.

多孔質材料がフェルト，グラスウールなどの繊維集合体である場合、その実効材料密度ρ_sと実効繊維径Ｄを用い、熱的特性長Λ'は次式に書き変えられる。

上式中、ρは多孔質材料の嵩密度を示す。
なお、粘性特性長Λは、繊維材料が円柱状で、その繊維の長手方向に対して垂直に音波が入る場合、Allardは次式が成り立つとしている。

粘性特性長Λは、繊維の向きにより変化し、その繊維の長手方向に対して平行に音波が入る場合、２つの特性長は理論上一致する。インピーダンス管（定在波管）を用いる場合、一般的な繊維材料はその繊維の長手方向に対して垂直に音波が入ると仮定でき、上式（数１４）が成り立つと考えられる。ただし、上式（数１３）、上式（数１４）は繊維間の接触面積の減少分が考慮されていないことから嵩密度が高くなるにつれて、この計算値は実際より低めの値になることに注意する必要がある。 Next, the problem of this Johnson-Allard model is explained.
The thermal characteristic length Λ ′ is defined by the following equation by the volume V of the internal air of the porous material and the surface area A where the solid and the air contact.

When the porous material is a fiber aggregate such as felt or glass wool, the thermal characteristic length Λ ′ can be rewritten as follows using the effective material density ρ _s and the effective fiber diameter D.

In the above formula, ρ represents the bulk density of the porous material.
Note that the viscosity characteristic length Λ is such that when the fiber material is cylindrical and a sound wave enters perpendicularly to the longitudinal direction of the fiber, Allard satisfies the following equation.

The viscosity characteristic length Λ varies depending on the direction of the fiber, and when a sound wave enters parallel to the longitudinal direction of the fiber, the two characteristic lengths theoretically match. When an impedance tube (standing wave tube) is used, it can be assumed that a general fiber material enters a sound wave perpendicular to the longitudinal direction of the fiber, and the above equation (Equation 14) is considered to hold. However, since the above equation (Equation 13) and the above equation (Equation 14) do not take into account the decrease in the contact area between the fibers, the calculated value becomes a lower value as the bulk density increases. It is necessary to pay attention to.

FIG. 1 and FIG. 2 are graphs showing the theoretical formula (Formula 13) of the thermal characteristic length Λ ′, the theoretical formula (Formula 14) of the viscous characteristic length Λ, and the fitting value according to the Johnson model. Here, a polyester nonwoven fabric having an effective fiber diameter D of 31.48 [μm] and an effective material density ρ _s of 1380 [kg / m ³ ] was used. In addition, the fit value (Johonson-Allard model) of two characteristic lengths was calculated | required by regression analysis from 25 data by crushing a polyester nonwoven fabric with a heating board and changing bulk density.
Thus, there is a large difference between the theoretical value and the fit value (Johnson-Allard model) at low bulk density. And the fit value (Johnson-Allard model) shows that the two characteristic length values approach each other as the bulk density decreases, and when the bulk density is lower than 20 kg / m ³ , the viscous characteristic length is larger than the thermal characteristic length. Value. This state is physically contradictory. It can be inferred that this phenomenon is caused by applying the relational expression (Equation 7) between the flow resistance and the inner radius of the pipe, which is formed only in the cylindrical pipe, to the porous material. As the correction terms, the two characteristic lengths are treated as coefficients that ignore the physical definition.
Although there are technologies for measuring two characteristic lengths, the measurement values cannot be used because of the explanations given here, and at present, a method of obtaining two characteristic lengths by fitting them to the result of the sound absorption rate has become common. ing.
Since it is possible to match the sound absorption rate with the two characteristic lengths obtained in this way, this problem has been overlooked, or there is no best model to replace the Johnson-Allard model, so this Johnson-Allard has traditionally been A model has been used.

円筒管の場合、上記sとs'に用いられる特性長は内半径Rそのものであり理論的にもΛとΛ'はRと同一値となることが知られている。
また、Johnson-AllardモデルではBessel関数部を式（数５）、（数６）に置き換えている。しかし、この近似式は管内半径Rが1mm以上で一致するが、一般的な多孔質材料では管内半径に換算すると0.4mm以下［式（数７）から換算すると流れ抵抗が1000Ns/m⁴以上］に相当し、この近似式は妥当とは言えない。
そこで、管内半径Rが0.4mm以下（これを毛細管と定義する）で良く一致する近似式を上式（数１５）、（数１６）を使って次に示す。

Johnson-Allardモデルにおける近似式（数５）、（数６）との違いは、同式中にある数値16を12に変更しただけである。この変更により、管内半径Rが0に近づくと近似式（数１７）、（数１８）とBessel関数部は限りなく近づくことが確認された。 Therefore, in the model according to the present invention (hereinafter referred to as the modified model), the effective density ρ _eff and the bulk modulus K _f are calculated based on the cylindrical tube model formulas (Equation 1) and (Equation 2) as in the Johnson-Allard model. Ask for. Here, the in-pipe radius R used for the shear force wave number s in Equation (3) is replaced with a viscous characteristic length Λ as a characteristic related to the viscous resistance. The radius R in the tube used for the shear force wave number s ′ in the equation (Equation 4) is replaced with a thermal characteristic length Λ ′ as a characteristic related to heat conduction. This means that viscous friction is considered in effective density and heat conduction is considered in bulk modulus.

In the case of a cylindrical tube, the characteristic length used for the above s and s ′ is the inner radius R itself, and it is theoretically known that Λ and Λ ′ have the same value as R.
In the Johnson-Allard model, the Bessel function part is replaced by equations (5) and (6). However, this approximate expression agrees when the pipe inner radius R is 1 mm or more, but in the case of general porous materials, 0.4 mm or less when converted to the pipe inner radius [flow resistance is 1000 Ns / m ⁴ or more when converted from the formula (Equation 7)] This approximation is not valid.
Therefore, an approximate expression that agrees well when the tube inner radius R is 0.4 mm or less (this is defined as a capillary tube) is shown below using the above expressions (Expression 15) and (Expression 16).

The only difference between the approximate equations (Equation 5) and (Equation 6) in the Johnson-Allard model is that the numerical value 16 in the equation is changed to 12. As a result of this change, it was confirmed that the approximate expressions (Equation 17) and (Equation 18) and the Bessel function part approach as much as possible when the pipe radius R approaches 0.

この式（数１９）、（数２０）が毛細管における内部流体の実効密度と体積弾性率を求めるモデル式となる。即ち、毛細管の場合、粘性特性長Λと熱的特性長Λ'が同一値（管内半径R）となるのに対し、繊維材料では上式（数１４）に示すように異なる値となる。これが毛細管と繊維材料との違いである。そこで繊維材料では特性長の理論式（数１３）、（数１４）が成り立つと仮定し、毛細管と繊維材料ではどの様な形で違いが現れるか、毛細管のモデル式（数１９）、（数２０）を用い試験をした。この試験には繊維径5種類（１５〜３７μm）、嵩密度（15〜200kg/m³）のポリエステル不織布を用い、120個のデータを元に特性インピーダンスと伝搬定数を計測し、実効密度、体積弾性率、垂直入射吸音率の計測値と解析値を比較した。なお、計測における実効密度は特性インピーダンスZcと伝搬定数Γから次式で定義される。

Further, substituting the above equations (Equation 17) and (Equation 18) into the equations of (Equation 1) and (Equation 2) and rewriting the effective density ρ _eff and the bulk modulus K _f yields the following.

These equations (Equation 19) and (Equation 20) are model equations for obtaining the effective density and bulk modulus of the internal fluid in the capillary tube. That is, in the case of a capillary tube, the viscosity characteristic length Λ and the thermal characteristic length Λ ′ have the same value (inner tube radius R), but the fiber material has different values as shown in the above equation (Equation 14). This is the difference between capillaries and fiber materials. Therefore, it is assumed that the theoretical formulas (Equation 13) and (Equation 14) of the characteristic length hold in the fiber material, and how the difference appears between the capillary and the fiber material. The model equation (Equation 19), (Equation 20). For this test, polyester nonwoven fabrics with five fiber diameters (15 to 37 μm) and bulk density (15 to 200 kg / m ³ ) were used. The characteristic impedance and propagation constant were measured based on 120 pieces of data, and the effective density and volume were measured. The measured values and analytical values of elastic modulus and normal incidence sound absorption coefficient were compared. The effective density in measurement is defined by the following equation from the characteristic impedance Zc and the propagation constant Γ.

繊維径が太い場合は1/ωΛ'に近づき、細い場合は1/ωΛに近づくことを確認した。この関係式を一般化するためs''を次式のようにモデル化する。

2つの特性長が同一値（毛細管の状態）の時、このs''の項が消えることを意図してモデル化を行なっている。ここで、Ｃは繊維径により定まる定数で、検証データ（繊維径15〜37μm）では１〜２の値となる。太い繊維は１、細い繊維は２に近づく。上式（数２３）は体積弾性率の式（数２２）にも適用する。 FIG. 3 is a graph showing a comparison between measured values of effective density and analysis values obtained by a capillary model. In this analysis example, the effective material density ρ _s = 1380 [kg / m ³ ], the effective fiber diameter D = 37.1 [μm], and the bulk density ρ = 50.0 [kg / m ³ ]. As shown in the figure, a difference appears in the negative number (imaginary part) of the effective density. This phenomenon was also confirmed for other fiber diameters and bulk densities. As a result of investigating this difference in detail, we found that the following difference appears in the imaginary part of the Bessel function. That is, although the conditions are limited within a fiber diameter of 15 to 37 μm, when the difference is s ″, the following relationship is obtained.

It was confirmed that when the fiber diameter was thick, it approached 1 / ωΛ ', and when it was thin, it approached 1 / ωΛ. In order to generalize this relational expression, s '' is modeled as follows.

When the two characteristic lengths have the same value (capillary state), modeling is performed with the intention of eliminating this s ″ term. Here, C is a constant determined by the fiber diameter, and is 1 to 2 in the verification data (fiber diameter 15 to 37 μm). Thick fibers approach 1 and thin fibers approach 2. The above equation (Equation 23) is also applied to the equation (Equation 22) of the bulk modulus.

Bessel関数の近似式（数１９）、（数２０）を用い、上記式（数２３）を使って書き換えると、

この式（数２６）、（数２７）が本発明に係る繊維集合体における実効密度と体積弾性率の修正モデルである。 Next, the measured value of the effective density is compared with the analysis value by the modified model.
The effective density ρ _eff and the bulk modulus K _f of the capillary tube and the porous material can be expressed by the following equations by rewriting the above equations (Equation 1) and (Equation 2) using the relationship of Equation (Equation 23).

Using approximate equations (Equation 19) and (Equation 20) of the Bessel function and rewriting using the above equation (Equation 23),

These formulas (Equation 26) and (Equation 27) are modified models of the effective density and bulk modulus in the fiber assembly according to the present invention.

ここでφはポロシティで、実効材料密度ρ_sと嵩密度ρから次式で定義される。

この式（数２８）、（数２９）から体積弾性率の計測値が得られる。 FIG. 4 and FIG. 5 show a comparison between an analysis value and a measurement value obtained by using a polyester nonwoven fabric having the same specifications as described above and calculating a constant C by 1 using a correction model. The bulk modulus K _f in the measurement is defined by the following equation from the characteristic impedance Z _c and the propagation constant Γ.

Here, φ is porosity and is defined by the following equation from the effective material density ρ _s and the bulk density ρ.

A measured value of the bulk modulus can be obtained from the equations (Equation 28) and (Equation 29).

In addition, about the constant C decided by a fiber diameter, the empirical formula was calculated | required from the polyester nonwoven fabric used for verification. And as a result of confirming with the material used for verification, the following formula was derived.

また、伝搬定数Γをこの実効密度ρ_effおよび体積弾性率K_fを用い次式（数３２）により算出する。

上式中、ｉは虚数単位（ｉ^２＝−１）、ωはω＝2πｆで表される角速度（ｆは周波数）である。
また、背面剛壁を条件とする音響インピーダンスZ₀をこの特性インピーダンスZcおよび伝搬定数Γを用い次式（数３３）により算出する。

上式中、ｄは繊維集合体の厚みである。
さらに垂直入射吸音率α₀をこの音響インピーダンスZ₀を用い次式（数３４）により算出する。

上式中、ρ₀c₀は空気の固有音響抵抗で気温18℃の場合、ρ₀c₀＝415Ns/m³である。 Next, the modified model is verified. Here, the effective density ρ _eff is calculated using the modified model formulas (Equation 26) and (Equation 27) from the theoretical formulas (Equation 13) and (Equation 14) of the characteristic length and the constant equation (Equation 30) determined by the fiber diameter. The bulk modulus _Kf is calculated, and the characteristic impedance Zc is calculated from the obtained effective density and bulk modulus according to the following equation (Equation 31).

Further, the propagation constant Γ is calculated by the following equation (Equation 32) using the effective density ρ _eff and the bulk modulus K _f .

In the above formula, i is an imaginary unit (i ² = −1), ω is an angular velocity (f is a frequency) represented by ω = 2πf.
Further, the acoustic impedance Z ₀ subject to the back rigid wall is calculated by the following equation (Equation 33) using the characteristic impedance Zc and the propagation constant Γ.

In the above formula, d is the thickness of the fiber assembly.
Further, the normal incident sound absorption coefficient α ₀ is calculated by the following equation (Equation 34) using the acoustic impedance Z ₀ .

In the above equation, ρ ₀ c ₀ is the specific acoustic resistance of air, and when the temperature is 18 ° C., ρ ₀ c ₀ = 415 Ns / m ³ .

6 to 10 are graphs in which the acoustic characteristics of the fiber assembly are converted to the normal incident sound absorption coefficient α ₀ and the results are listed for five different fiber diameters. That is, FIG. 6 shows a normal incidence sound absorption coefficient at an effective fiber diameter D = 37.1 μm, FIG. 7 shows a normal incidence sound absorption coefficient at an effective fiber diameter D = 31.5 μm, and FIG. 8 shows a normal incidence at an effective fiber diameter D = 28.6 μm. FIG. 9 shows the normal incident sound absorption coefficient at an effective fiber diameter D = 23.3 μm, and FIG. 10 shows the normal incident sound absorption coefficient at an effective fiber diameter D = 15.7 μm. The materials used for the verification were all 100% polyester nonwoven fabric, and the effective material density ρ _s was 1380 kg / m ³ . In the graph, the bulk density ρ of the fiber assembly, the constant C determined by the fiber diameter, and the material thickness d are shown.

  In order to measure the acoustic impedance and the normal incidence sound absorption coefficient, as shown in FIG. 11, the fiber assembly 2 as a sample is formed in the inner low part of the bottomed acoustic impedance measuring tube 1 whose back is a rigid wall. The sound of the speaker 3 provided at the opening of the measurement tube 1 is taken in and analyzed by two microphones 4 and 5 provided at a predetermined interval at the front of the fiber assembly 2. A measurement system is used.

  The effectiveness of the modified model according to the present invention has been clarified from the verification results shown in FIGS. The graph shown in the lower right of FIG. 10 considers the contact area between the fibers, and sets the two characteristic length values 20% larger than the theoretical values for the graph on the left side (using the theoretical value of the characteristic length). Shows the results. Thus, by considering the contact area between the fibers, it becomes a model that can match even a material with a high bulk density of fine fibers.

個々の材料密度をρ_sxとすると実効材料密度ρ_sは次式で定義される。

また、実効繊維径Ｄについては、個々の繊維径をＤ_xとすると次式で定義される。

この式（数３７）中のρ_sは、上式（数３６）における実効材料密度である。
この式（数３６）および式（数３７）を用いることにより、音響的に等価な（特性長を変化させない）１本の繊維材料に置き換えることができる。この検証に用いたポリエステル不織布は、主原料70％にバインダー材30％を配合して製造されたもので、式（数３６）と式（数３７）に示した実効材料密度ρ_sと実効繊維径Ｄを用いている。 Next, supplementary explanation will be given on the effective material density and the effective fiber diameter of the fiber material.
When manufacturing non-woven fabrics using fiber materials, they are generally not created with a single material density and fiber diameter, but are thick fibers for forming a skeleton and thin fibers for ensuring acoustic performance, and these It is made by combining binder fibers for tying. On the other hand, the acoustic characteristic is determined by the characteristic length as shown in the modified model equations (Equation 26) and (Equation 27). Therefore, a method of replacing a plurality of materials with one material without changing the porosity, surface area, and bulk density from the definition of the characteristic length will be described.
Regarding the effective material density ρ _s , generally, when a fiber material is blended, the blending is controlled by the weight ratio of each material. Therefore, the weight ratio of the materials to Y _x.

If each material density is ρ _sx , the effective material density ρ _s is defined by the following equation.

Also, the effective fiber diameter D, is defined an individual fiber diameter by the following equation when the D _x.

Ρ _{s in} this equation (Equation 37) is the effective material density in the above equation (Equation 36).
By using this formula (formula 36) and formula (formula 37), it can be replaced with one acoustically equivalent (not changing the characteristic length) fiber material. The polyester nonwoven fabric used for the verification was manufactured by blending 70% of the main raw material with 30% of the binder material. The effective material density ρ _s and the effective fiber shown in the equations (36) and (37) were used. The diameter D is used.

この式（数３８）と面積減少を考慮した特性長を用いることにより、高嵩密度の材料まで精度の高い解析が可能なことを確認している。 Thus, the above formulas (Equation 26) and (Equation 27) are modified models according to the present invention that can determine the effective density and bulk modulus without breaking the physical definition of the characteristic length. This modified model does not require the flow resistance required by the Johnson-Allard model, and the physical definition of characteristic length can be used, so in the case of fiber assemblies, the fiber diameter and fiber density can be analyzed as a database. it can. As a result, when the acoustic material is produced by crushing the fiber material to a certain thickness, the thermal characteristic length Λ ′ and the viscosity characteristic can be obtained only by changing the bulk density from the characteristic length definition formulas (Formula 13) and (Formula 14). Since the length Λ can be defined, the acoustic characteristics can be easily obtained. In addition, when the bulk density is increased, it is suggested that the decrease in the contact area between the fibers needs to be reflected in the characteristic length. However, if the bulk density is less than 100 kg / m ³ , the formula (Equation 13), ( It was confirmed that the equation (14) can be used within an error range without any problem. Incidentally, although the tortueity α _∞ is not considered in this modified model, the effective density ρ _eff is expressed by the following equation when the tortuousity α _∞ is considered using the same concept as the Johnson-Allard model.

By using this equation (Equation 38) and the characteristic length in consideration of area reduction, it has been confirmed that high-accuracy analysis can be performed even for materials with a high bulk density.

Further effects that can be expected from this modified model include the following.
(1) Although the flow resistance is a characteristic that varies depending on the temperature due to the influence of the viscosity of the medium, the corrected model does not use the flow resistance, and therefore can accurately analyze even the temperature at which the flow resistance cannot be measured.
(2) In the case of a fiber material, the value of the viscosity characteristic length varies depending on the direction of the fiber. By quantifying this value, oblique incidence and random incidence can be analyzed with high accuracy as an anisotropic material.

The graph which shows the relationship between thermal characteristic length and bulk density. The graph which shows the relationship between viscosity characteristic length and bulk density. The graph which shows the comparison of the measured value of an effective density, and the analysis value by a capillary model. The graph which shows the comparison of the measured value of an effective density, and the analysis value by a correction model. The graph which shows the comparison of the measured value of a bulk modulus, and the analysis value by a correction model. The graph which shows normal incidence sound absorption coefficient in the effective fiber diameter of 37.1 micrometers. The graph which shows normal incidence sound absorption coefficient in the effective fiber diameter of 31.5 micrometers. The graph which shows the normal incidence sound absorption coefficient in the effective fiber diameter of 28.6 micrometers. The graph which shows the normal incidence sound absorption coefficient in the effective fiber diameter of 23.3 micrometers. The graph which shows the normal incidence sound absorption coefficient in effective fiber diameter 15.7 micrometers. The longitudinal cross-sectional view of an acoustic impedance measuring tube.

Explanation of symbols

ρ _eff Effective density inside fiber material
Bulk modulus of elasticity inside _Kf fiber material C constant
Zc Characteristic impedance Γ Propagation constant
Z ₀ Acoustic impedance α ₀ Normal incident sound absorption coefficient D Effective fiber diameter ρ ₀ Air density μ Air viscosity coefficient ω Angular velocity (ω = 2πf)
c ₀ Sound velocity in air γ Specific heat ratio of air ρ ₀ c ₀ ² Volume modulus of air
P ₀ Atmospheric pressure Λ Viscosity characteristic length of fiber material Λ ′ Thermal characteristic length of fiber material ρ Bulk density of fiber assembly ρ _s Effective material density D Effective fiber diameter i Imaginary unit (i ² = −1)
d Thickness of fiber assembly ρ ₀ c ₀ Specific acoustic resistance of air

Claims (4)

The effective density ρ _eff is calculated by the following equation (Equation 26), the bulk modulus K _f is calculated by the following equation (Equation 27), and the constant C determined by the fiber diameter is calculated by the following equation (Equation 30). A method for predicting acoustic characteristics of a fiber assembly, wherein the characteristic impedance Zc of the fiber assembly such as felt or glass wool is calculated by the following equation (Equation 31) using the density ρ _eff , the bulk modulus K _f and the constant C: .

In the above equation, ρ ₀ is the density of air, μ is the viscosity coefficient of air, ω is the angular velocity (f is the frequency) expressed by ω = 2πf, Λ is the viscosity characteristic length of the fiber material, and Λ ′ is the heat of the fiber material Characteristic length.

In the above equation, ρ ₀ is the density of air, c ₀ is the speed of sound in the air, and γ is the specific heat ratio of air. Ρ ₀ c ₀ ² is the bulk modulus of air, and there is a relational expression of ρ ₀ c ₀ ² = γP ₀ using the atmospheric pressure P ₀ [P _a ]. Further, i is an imaginary unit (i ² = −1).

In the above equation, μ is the viscosity coefficient of air, and D is the effective fiber diameter.

The propagation constant Γ of a fiber assembly such as felt or glass wool is calculated by the following equation (Equation 32) using the effective density ρ _eff and the bulk modulus K _f described in claim 1, and the acoustic property of the fiber assembly: Characteristic prediction method.

In the above formula, i is an imaginary unit (i ² = −1), ω is an angular velocity (f is a frequency) represented by ω = 2πf. The acoustic impedance Z ₀ subject to the back rigid wall of a fiber aggregate such as felt or glass wool is calculated by the following equation (Equation 33) using the characteristic impedance Zc described in claim 1 and the propagation constant Γ described in claim 2. A method for predicting acoustic characteristics of a fiber assembly, characterized in that:

In the above formula, d is the thickness of the fiber assembly. A method for predicting acoustic characteristics of a fiber assembly, wherein the normal incident sound absorption coefficient α ₀ of a fiber assembly such as felt or glass wool is calculated by the following equation (Equation 34) using the acoustic impedance Z _{0 according} to claim 3. .

In the above equation, ρ ₀ c ₀ is the specific acoustic resistance of air.

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