## 4 Vibroacoustic Analysis for Cabin Interior Noise
### Interior Noise Calculation with Vibroacoustic FE Model  
The HSM, as shown in Figure 14, consists of three glass window parts (one front windshield and two side window glasses of 4 mm thickness), aluminum panels, and sound absorption pads. The absorption layers were constructed to reproduce the effect of the actual interior material in an actual vehicle and ensure that the principal transmission pass to interior noise is only through the side window and front windshield glasses.  
Figure 14: HSM for vibroacoustic computation: (**a**) glass windows and aluminum body panels; (**b**) sound absorption layers; (**c**) wall layers.  
Figure 13: Modal overlap factors of side windows, front windshield, and cavity.  
Figure 15 shows the grid configurations for window glasses and acoustic cabin cavity. Determining the mesh size is the first step in finite element modeling. In general, the wavenumber of the resonant \(k_{mode}\) is calculated such that the resonant frequency mode, which is more than twice the maximum frequency of interest, can be expressed well. In addition, the maximum convective \(k_{c}\), \(max\) and acoustic wave number \(k_{a,max}\) for modeling the wall fluctuating pressure in the turbulent boundary layer are defined as follows:  
\[k_{c,max}=\frac{\omega_{max}}{U_{c}}, \tag{48}\]  
\[k_{a,max}=\frac{\omega_{max}}{c_{0}}, \tag{49}\]  
where \(U_{c}\) and \(c_{0}\) are the convection velocity and the speed of sound, respectively. The mesh size \(\Delta x\) of the finite element model is obtained as follows:  
\[\Delta x=\frac{\pi}{k_{max}}, \tag{50}\]  
where  
\[k_{max}\geq\max(2k_{mode},\;2k_{c,max},\;2k_{a,max}). \tag{51}\]  
When the frequency range of interest is 0-500 Hz, according to Equation (50), the mesh size should be smaller than 7.5 mm for the plate and 88 mm for the cavity. In this case, the grid types (size) of window glasses and acoustic cavity were CTRIA3 (4 mm) and CTETRA (17 mm), respectively.  
The acceleration and force data were measured at the four points on the glass in a frequency range up to 1600 Hz. The glass damping loss factor was estimated using the half-power bandwidth method from the selected resonance peaks. The regression fit was applied to predict the damping loss factor in the frequency range up to 5000 Hz.  
Figure 15: Configuration of computational grids: (**a**) 2D shell elements of glass windows; (**b**) 3D tetrahedral elements of acoustic cavity.  
For the acoustic cabin, the damping loss factor could be determined with the following formula [28]:  
\[\eta_{P}=\frac{13.8}{T_{60}\omega}, \tag{52}\]  
where \(T_{60}\) stands for the reverberation time measured inside a cabin at the frequency \(\omega\). The sound absorption coefficient of the absorption layers could be estimated using the damping loss factor with the following formula:  
\[\alpha=\eta_{P}\frac{8\pi fV}{c_{0}A}. \tag{53}\]  
Then, the inner-wall impedance was calculated using the following equation:  
\[Z=Z_{0}\frac{1+\sqrt{1-\alpha}}{1-\sqrt{1-\alpha}}. \tag{54}\]  
The impedance determined using Equations (52)-(54) was applied as the boundary condition on the cavity wall. Figure 16 shows the applied damping loss factor and inner-wall impedance in the acoustic cavity of HSM.