Analysis of the transmission loss of double-leaf panels with an equivalent spring–mass model for studs

The purpose of this study was to investigate the sound insulation of double-leaf panels. In practice, double-leaf panels require a stud between two surface panels. To simplify the analysis, a stud was modeled as a spring and mass. Studies have indicated that the stiffness of the equivalent spring is not a constant and varies with the frequency of sound. Therefore, a frequency-dependent stiffness curve was used to model the effect of the stud to analyze the sound insulation of a double-leaf panel. First, the sound transmission loss of a panel reported by Halliwell was used to fit the results of this study to determine the stiffness of the distribution curve. With this stiffness distribution of steel stud, some previous proposed panels are also analyzed and are compared to the experimental results in the literature. The agreement is good. Finally, the effects of parameters, such as the thickness and density of the panel, thickness of the stud and spacing of the stud, on the sound insulation of double-leaf panels were analyzed.


INTRODUCTION
In noise control engineering, panels are widely used as sound insulation materials. Sewell [1] proposed that mass is the most critical factor of sound insulation of a panel. Thus, for a singleleaf homogeneous panel, sound insulation can be estimated using the empirical mass law for frequencies less than the coincidence frequency. Double-leaf panels are widely used in sound insulation engineering because of their light weight and excellent sound insulation performance. To understand the sound transmission loss of double-leaf panels, Mulholland et al. [2] analyzed double panels without a stud by calculating surface impedance. Dym and Lang [3] proposed a method to investigate the sound insulation property of sandwich panels and demonstrated that the sound insulation of double-leaf panels was superior to that of single-leaf panels. Tseng [4] used the transfer matrix and Biot theory [5] to calculate the transmission loss of panels with a sound-absorbing material inside. Tang [6] analyzed the sound insulation of panels with honeycomb cores. Different sound waves in air and honeycomb structures were modeled using Biot theory. The effect of honeycomb cores on sound transmission loss was analyzed and compared with experimental results. However, studs, which are required to connect two surface panels, were not considered in all these experiments.
In practice, studs provide support to surface panels. They affect sound propagation in double-leaf panels and should be considered in sound insulation analysis. Wang et al. [7] modeled the stud as a combination of spring and mass to investigate the sound insulation of double-leaf panels and revealed that studs provide a path for sound propagation and alter the sound insulation property considerably. However, they only presented theoretical results. Davy [8][9][10] proposed that stud connection with double panels is categorized into point and line connections. Davy's results revealed that double-leaf panels with point connection have better sound insulation than line connection. Davy compared results with the experimental measurements of Halliwell et al. [11] and DuPree [12] to verify his analysis. Legault and Atalla [13] proposed boundary element analysis to investigate sound transmission through a double-panel structure. They changed the stiffness of the stud between double panels to simulate point and line connections and estimate sound transmission curves. Vigran adopted the transfer matrix method [14] to analyze sound transmission through multilayered structures. Bradley et al. [15] experimentally analyzed the effect of the stud material, length and depth on the sound insulation of wood stud exterior walls. Wyngaert et al. [16] investigated the effects of inserting sound-absorbing materials in the space between studs on the sound insulation of a double-leaf panel.
This study analyzed the sound insulation of a double-leaf panel. The stud connecting two surface panels was modeled by a combination of mass and a spring. The stiffness of the spring was considered a frequency-dependent variable. The wave propagation method instead of the vibrational energy transformation derived by Davy [8][9][10] is used in this study. This approach does not need to separate the above and below critical frequency ranges as done in Davy's investigation. All analyzed results were compared with experimental measurements to verify the accuracy of the proposed approach. Then, parameters, namely the thickness and density of panels, thickness of the stud and spacing of the stud, that influence the sound transmission of a doubleleaf panel are discussed.

THEORETICAL FOR MUL ATION
The schematic of the infinite double-leaf panel used in this study is presented in Fig. 1. Studs that connect two surface panels were modeled as a parallel combination of mass and a spring. For this model, sound propagates in three regions, and region 3 is assumed to be extended to infinity; i.e. no reflection wave exists in this region. If the sound is incident on the panel at an angle θ 1 to the normal direction of the surface, the vibrations of surface panels are governed by the following expressions [7]: where m pi is the surface density of the panel, M is the line density of stud, L is the spacing of the stud, ρ i represents the density of the medium, D i is the bending stiffness of a plate, W i is the normal displacement of the panel, i represents the velocity potential functions of the sound field and K t is the stiffness of the spring used to model the stud. The incident and reflected wave velocity potentials of each sound fields are expressed as follows: where A 1 and A 2 represent the amplitudes of incident and reflected waves in medium 1, C is the amplitude of the transmitted wave, B 1 , and B 2 are the amplitudes of two-directional waves in medium 2, and k n and θ n are the corresponding wave number and the sound propagation angle in each medium n, respectively. The normal displacements W i of each panel are expressed as follows: If only the thin plate is considered, the continuity of the normal velocity should be satisfied, which is required for the plate and medium to remain in contact. Therefore, we have the following expressions: We considered the medium in regions 1 and 3 to be air and that in region 2 to be the sound-absorbing porous material. In this study, the sound-absorbing porous material was equivalent to a fluid with complex wave number k 2 , dynamic density ρ b and complex sound velocity c 2 . The sound velocity and density of air were denoted as c 0 and ρ 0 ., respectively. Substituting (3)- (7) into (1), (2), (8) and (9), we obtain the following expressions: where k x (= k sin θ ) and k y (= k cos θ ) are the components of the wave number in the x-and y-directions. Here, k = ω/c is the wave number for sound propagation in air. According to Snell's law, the y-direction wave number component k 2y in porous material can be expressed as follows: According to Delany and Bazley [17], for fibrous materials with porosity close to 1, the complex wave number k 2 and velocity c 2 of the equivalent fluid can be expressed as follows: where f is frequency and X denotes the flow resistivity of the porous material. Delany and Bazley suggested that empirical formulas should satisfy the following expression: If the medium in region 2 is air, then the complex wave number k 2 and velocity c 2 are replaced by k and c of air. Equations (10)- (15) can be combined to obtain the relationship between the amplitude of incident and transmitted waves as follows: where a 11 , a 12 , a 21 and a 22 are expressed as follows: If the random incidence situation is considered, then according to the derivation of Fahy [18], the transmission coefficient can be obtained by calculating the average of all incident angles and expressed as follows: The transmission loss (TL) of the panel can be calculated by the following expression: TL = −10 log 10 (T av ) dB.

STIFFNESS OF THE STUD AND VERIFICATION OF THE APPROACH
The equivalent stiffness of stud considerably affects the sound insulation of a double-leaf panel and should be determined first. In this study, panel structures measured by Halliwell et al. [11] were used to examine the mass-stiffness model of studs. Figure 2 depicts the structure and dimension measured in Halliwell's experiments. The surface panels comprised a 16-mm gypsum board and non-load-bearing (C channel steel) steel studs screwed to surface panels. Stud separation distances of 406 and 610 mm and panel spacings (or depth of C channel steel) of 65 and 90 mm are considered. The material properties are listed in Table 1. Referring to Wang' model [7], Table 1 Material properties of the double-leaf panels measured in [11].

Young's modulus of panel (N/m 2 )
Poisson's ratio of panel

Figure 3
Comparison of the analytical transmission loss for the stiffness obtained using Hooke's law to that of the experiment [11].
the equivalent stiffness of C channel steel based on Hooke's law is expressed as follows: where E is Young's modulus, and t 0 and l are the width and depth of C channel steel, respectively. The estimated K t was ∼7 × 10 10 (N/m 2 ). The theoretical and experimental transmission losses are illustrated in Fig. 3 for the case of panel spacing and stud distance of 65 and 406 mm, respectively. The analytical transmission loss was similar to that of a single panel and considerably deviated from experimental results. The stiffness derived from Hooke's law appears too strong for the present approach. Therefore, the equivalent stiffness of studs should be modified in the model. Poblet-Puig [19] performed a vibration test on studs to measure the compliance (which is defined as the inverse of stiffness) distribution. Vigran [20] also derived the compliance curve of C channel steel studs. Davy et al. [21] used sound insulation values derived by Halliwell et al. [11] to calculate the compliance of studs inversely. Figure 4 shows the compliances of steel stud obtained by different researchers, where Max and Min represent the collected maximum and minimum compliance values by Davy et al. and were depicted in fig. 4 in [21]. Results indicated that the compliance of steel stud was frequency dependent, and its reciprocal was considerably smaller than stiffness obtained using Hooke's law. To obtain a reasonable compliance value, the mean value at each frequency of the figure was inserted in the model to analyze transmission loss. Results were compared with Halliwell's experimental results, which revealed a large discrepancy in values. Finally, a lower compliance value, that is, the blue line in Fig. 4, was selected as the equivalent stiffness in our model. The computed transmission losses of four experimental results of different stud distances and panel spacings are depicted in Fig. 5. Although some discrepancy still persisted, results were consistent.
The panel spacing with the sound absorption material was also investigated. The required properties of mineral fibers considered in cases (a) and (b) of Fig. 5 are listed in Table 2. A comparison of the results of the presented method ( Fig. 6) with that of Halliwell's measurements [11] revealed that the result was consistent. The accuracy of the current method was verified. Therefore, the compliance to be selected in the model was proven and can be used in a future study.

FACTOR S AFFECTING SOUND TR ANSMISSION LOSS 4.1 Panel thickness
The panel spacing (90 mm) and stud distance (610 mm) were fixed, and only the thickness of panels was varied. Three gypsum boards with surface panel thicknesses of 9, 16 and 20 mm were considered. The transmission loss is depicted in Fig. 7. The sound insulation performance increased considerably with an increase in panel thickness, and the coincidence frequency decreased with an increase in panel thickness. Because of the  Table 2 Properties of porous fiber material [11].

Mineral fiber Density (kg/m 3 ) F l o w r e s i s t i v i t y ( N / m 4 s) Thickness (mm)
Case 1 36.7 11 400 65 Case 2 32. 6 12 700 90 increase in the surface density in the frequency range of 250-1500 Hz, transmission loss reduced by ∼10 dB when thickness increased from 9 to 20 mm.

Surface panel materials
Only two widely used surface panel materials, namely gypsum and calcium silicate boards, were analyzed. The thickness, panel spacing and stud distance of analyzed double-leaf panels were 16, 90 and 406 mm, respectively. The density, Young's modulus and Poisson's ratio of these two materials are listed in Table 3.
The calculated transmission losses of these boards are depicted in Fig. 8. Because the density of the calcium silicate board was larger than that of the gypsum board, results revealed increased transmission loss in the mass control region apparently.

Panel spacing (or depth of C channel steel)
In this section, the influence of panel spacing on sound transmission loss is discussed. The gypsum board panel with a

Material Density (kg/m 3 ) Young's modulus (N/m 2 ) Poisson's ratio
Gypsum board 690 2.5 × 10 9 0.3 Calcium silicate board 1200 9 × 10 9 0.1  reveal that the sound insulation capacity increases with an increase in the panel spacing. However, at frequencies higher than the coincidence frequency, the effect of panel spacing was negligible.

Stud distance
The surface panels of a double-leaf panel are connected with studs. Therefore, the stud's distribution considerably affects the strength of the double-leaf panel and sound insulation property. The same gypsum double-leaf panel with a thickness of 16 mm and panel spacing of 90 mm was analyzed for three stud distances of 300, 600 and 900 mm. Results are depicted in Fig. 10.
Although the difference was not obvious, the transmission loss decreased as the stud distance decreased, because the closer the studs, the stronger the connection of panels. Thus, sound propagation through the connection stud was easier and consequently provided less sound insulation.    Table 4. Simulation results are illustrated in Fig. 12, which revealed that porous material with higher density and flow resistivity provided a superior sound insulation performance.

CONCLUSIONS
Equivalent stiffness and mass were used to model the stud of a double-leaf panel. A comparison of computed transmission loss with experimental measurements revealed that estimated stiffness from Hooke's law was unsuitable for the proposed model. A frequency-dependent and low-stiffness curve was selected in this study. Calculated results showed that an increase in the thickness of the surface panel increased sound insulation in the mass control frequency region and reduced the coincidence frequency. In addition, for the same thickness, using higher density surface panels also provided similar results. The analysis of various panel spacings (i.e. depth of channel steel) showed that increasing the panel spacing considerably increased the transmission loss for frequencies less than the coincidence frequency. However, increasing the connection of panels reduced the sound insulation slightly. Finally, the effect of the sound-absorbing materials was investigated. Sound-absorbing materials inserted into the space between the two surfaces increased transmission loss considerably. Higher density and flow resistivity porous sound-absorbing material exhibited better performance.