Sound absorption model of thin woven fabric with tortuosity effects by equivalent circuit method

The fabric has been widely used as a noise-reduction material. This paper presents a revised equivalent circuit model to simulate the acoustical behavior of woven fabric with backing an air gap, with a special focus on including a tortuosity parameter. The simulated sound absorption of three fabrics with and without the tortuosity parameter was experimentally validated. The equivalent circuit model including the tortuosity parameter predicts the absorption curve better, particularly at the local minima in comparison to the existing models without the tortuosity effects. It is beneficial to the structural design of woven fabric with enhanced sound absorption.


Introduction
Resonant sound absorbers with membrane-like surfaces backed by an air gap were considered as good noise-reduction materials.The commonly used membrane-like materials mainly include plastic films, perforated plates and textiles [1][2][3].As sheet materials, the thickness of thin textiles is lower than the wavelength of acoustic waves which has the advantages of being lightweight, good flexibility and cost-effective [4].Recently, several models were developed to simulate the attenuation of acoustic waves.For instance, Pieren's model is related to airflow resistivity, areal density and the distance of the backing air gap [5,6].
Thin woven fabric is made of fibrous materials which has similar acoustical behavior with porous absorbers to some extent.According to previous studies, fabrics can be modeled as equivalent fluid media by tortuosity, which is a measure of the porous frame to indicate the effects of sound waves traveling through the non-straight pores [7,8].This tortuosity parameter is related to corrected length which can be calculated by the radiation of perforated pores in free air [9], which has been neglected in predicting sound absorption coefficients.According to reported studies [10,11], tortuosity is closely related to the position of the quarter-wavelength peaks and also determines the sound absorption behavior at the high frequency of porous materials.Therefore, these models could not well describe the minimum absorption valley value of sound absorption curves at a high-frequency range [5,6,12].
In this work, the tortuosity parameter is used to describe the micro-structures of thin woven fabric.Additionally, Miki's model is taken to evaluate the acoustical impedance and wavenumber of the fabric layer [13].The present model needs four parameters: airflow resistivity, areal density, the tortuosity of the fabric layer and air gap depth.The validation is accomplished through sound absorption measurements on three different woven fabrics.

Theoretical analysis
As shown in Figure . 1 (a), the resonant sound absorber system includes a membrane-like layer and a backed air gap with the distance of D. The weaving structure of the woven fabric is shown in Figure . 1 (b), where a is length and b is the width of perforated rectangularly shaped patulous pores.In the previous work of Pieren [5], a model is established to predict the noise reduction of textile gap through an equivalent circuit method.As for the fabric, the surface impedance ZT can be calculated by: where Z0 = ρc is air characteristic impedance, ρ is air density (kg/m 3 ), c corresponds to the velocity of acoustic waves (m/s) and k0 represents air wave number, which is k0 = ω/c and ω = 2πf, D is the distance of backing air gap (m), j denotes the imaginary unit.Rs is airflow resistivity (Pa s/m) and m is fabric areal density (kg/m 2 ).Z f = jωmR s jωm+R s is the parallel connection impedance of fabric, and Z D = −jZ 0 cot(k 0 D) is the impedance of rigid backing air gap.Normal incidence absorption coefficients α is: where z = Z T /Z 0 and it denotes the normalized total surface acoustic impedance.For a better comparison of different models, this model includes the airflow resistivity Rs and the areal density m.The air gap D is labeled as Model-1.In this study, an equivalent circuit method with the Π-type network was taken to simulate the sound absorption of fabrics.In Figure . 1 (c), the airflow through the textile layer and the air gap are both modeled by an equivalent fluid approach, which are represented as lumped elements.The Π-type network is particularly for describing the parallel connection of the mechanical impedance, thus the vibration effects of the textile layer can be easily included in the analytical formulas [6].The used generalized impedance for the textile layer is shown as [6]: Z c is acoustical characteristic impedance and k c is wavenumber of the textile layer which are estimated from Miki's model [13].d mea is the measured thickness, and it is effective to characterize the acoustic characteristics.As for the backing air gap, the formulas of impedances are written as follows [6,14]: Using Z D and the above-mentioned equivalent circuit, the surface acoustic impedance can be derived.Auxiliary variable Z p can be defined as the impedance of the parallel connection of Z α A and the mechanical impedance jωm of the fabric: where ∥ represents the parallel connection of impedance components.By inserting Z p into the Πtype network equivalent circuit model, the analytical expression for the total surface impedance Z T of the structure is found as: In this work, the sound absorption can be obtained by Equation ( 2).This model is denoted as Model-2, which includes airflow resistivity Rs, areal density m, air gap D and the measured thickness d mea .
Tortuosity is a measure to account for how far the pores deviate from the normal cylinder and the pores lengths other than straight, which is also known as structure form factor.It is effective in characterizing the inertial effects in the pores and side branches of fibrous materials [6,9].According to Atalla and Sgard's work, screen and perforated sheet are effective to be simulated as equivalent fluid through the Johnson-Allard microstructure model considering the tortuosity parameter.And expression of tortuosity is shown as [7]: (9) where ε = 0.48√πr 2 (1 − 1.14√ϕ) as the corrected length, ϕ is perforation ratio and r = ab (a + b) ⁄ is hydraulic radius of the fabric pores.Tortuosity α ∞ is a dimensionless quantity and d eff = d mea ⋅ α ∞ , which is used to account for the effective thickness of the fabric.This model is denoted as Model-3, which includes all the parameters of Model-2 and also the tortuosity α ∞ .

Measurements and calculations
In this work, the airflow resistivity of specimens was tested based on ISO 9053.The thickness of textiles was determined by the YG141D digital fabric thickness test.Both pore size (a and b) and the distance between pores were tested through a fabric density mirror.For calculating hydraulic radius r and perforation ratioϕ, tortuosity parameter was calculated from Equation (5).

Validations and comparisons
In this section, the experimental validation is carried out through three thin woven fabrics and their physical parameters are listed in Table 1.In this work, the calculated surface acoustic impedance Z T of the absorber system consisting of fabric and air gap is shown in Figure 2. The values of the three fabrics are gradually increased with increasing frequency.As for the imaginary part, the value from Pieren's model at the resonant frequency region is higher than that of the reported work.In this work, both the real part and the imaginary part of impedance are increased due to the tortuosity effect except for the resonant frequency region.Therefore, it has been suggested that the current model is better at describing acoustical behavior when considering the fabric thickness and the micro-structures by tortuosity parameter.To verify the proposed method of this work, the tested and calculated normal incidence coefficients of different fabrics were compared.As for fabrics with 3 cm and 10 cm backing air gap, the measured curves are exhibited in Figure.3 and Figure. 4 respectively.The tested absorption curve is generally higher than the predicted result of the present model without considering tortuosity.In addition, both the previous Pieren model and the present model agree with the measured absorption coefficients.Because of the test uncertainties of sound absorption and the influence of mechanical resistance at this region, a small margin of errors may occur at lower frequencies [5,14].Furthermore, the present model outperforms the previous models at the local valley of the measured curves when resonance frequency at the approximation of nc 2D ⁄ where D is 0.03 m and n denotes air gap mode number which is equal to one in this experiment.This improved prediction at the local minima can be attributed to the well description of woven fabric fibrous characteristics by considering tortuosity.Thus, it is suggested to forecast the sound absorption of the fabric absorber by an equivalent circuit method by considering the tortuosity and the estimated impedance and wavenumber by Miki's model of the textile layer.

Conclusion
In summary, a novel developed equivalent circuit approach of predicting the acoustical behavior of thin woven fabric under different air gap conditions was proposed by including tortuosity.The sound absorption of fabrics is illustrated by an equivalent circuit of the Π-type network, where airflow characteristics of the textile layer and air gap are modeled by an equivalent fluid approach as lumped elements.Compared with the previous model, the proposed equivalent circuit model agrees better with impedance tube measurement at the local valley of the curve with an air gap of 3 cm.The results show that using the estimated impedance and wavenumber by Miki's model, it is feasible to predict the sound absorption of the woven fabric layer by considering tortuosity through an equivalent circuit method.

Figure 1 .
Figure 1.Diagrammatic sketch of fabric resonant absorber: (a) Membrane materials and air gap structure; (b) Yarn interleaving structure of woven structure; (c) Equivalent circuit diagram.

Figure 2 .
Figure 2. Real and imaginary part of acoustic impedance of front textile with air gap of 3 cm.

Figure 3 .
Figure 3. Measured and predicted curves of three validated specimens with 3 cm air gap.

Figure 4 .
Figure 4. Measured and predicted curves of three validated specimens with 10 cm air gap.

Table 1 .
Physical parameters of three validated thin woven fabrics.