## 4 Vibroacoustic Analysis for Cabin Interior Noise
### Interior Noise Calculation with Vibroacoustic SE Model
#### 4.3.2 SEA Parameters  
The parameters of SEA include the input power, modal parameter, damping loss factors (DLF), and coupling loss factor (CLF). Parameters are factors needed to describe the total energy of each subsystem and the energy exchange process. The theoretical, experimental, and numerical parameters used in this study are described below in detail. The power supplied to the plate was defined as follows [29],  
\[\langle P\rangle=\int_{A}\langle pv\rangle dA, \tag{56}\]  
where \(p\) is random excitation pressure, \(v\) is the normal velocity of the plate, and \(\langle\rangle\)denotes mean value or expectation. Introducing the cross-power spectral density for pressure and velocity at \(x\), Equation (56) reads as follows:  
\[\langle P\rangle=\frac{1}{2\pi}\int_{A}\int_{-\infty}^{\infty}\int_{A}Y(x,x-, \omega)S_{pp^{\prime}}(\omega)dA^{\prime}d\omega dA, \tag{57}\]  
where \(Y(x,x-,\omega)\) is the admittance between a force at \(x^{\prime}\) and velocity at \(x\), and \(S_{pp^{\prime}}(\omega)\) is the power spectral density of wall pressure at \(x\) and \(x^{\prime}\), respectively. According to Equation (57), the power applied to the plate is expressed by admittance and PSD. This means that, in the case of the distributed pressure field input, the vibration characteristics of the plate should also be considered. Since obtaining admittance for all positions is challenging, the following formula is generally used [30]:  
\[\Pi_{p,in}=\frac{A_{p}\langle p_{TBL}^{2}\rangle}{\pi^{2}f\rho_{p}h_{p}}\left( \frac{U_{c}}{c_{p}}\right),\qquad U_{c}>c_{p}, \tag{58}\]  
\[\Pi_{p,in}=\frac{A_{p}\langle p_{TBL}^{2}\rangle}{2\pi f\rho_{p}h_{p}}\left( \frac{U_{c}}{c_{p}}\right)^{3}\left[\frac{a_{1}}{6}+a_{2}\left(\frac{U_{c}}{2 \pi fL_{s}}\right)^{2}\right],\qquad U_{c}<c_{p}, \tag{59}\]  
where \(\rho_{p}h_{p}\) is the surface density of the plate, \(A_{p}\) is the plate area, \(\langle p_{TBL}^{2}\rangle\) is the spatial distribution of the turbulent boundary layer (TBL) pressure, \(a_{1}\) and \(a_{2}\) are constants depending on the details of the turbulent flow and plate mode shapes, and \(L_{s}\) is the distance between the wavenumber spectra for the TBL pressure and the bending mode shape of the plate. For aerodynamically fast modes, the following formula can be used [19]:  
\[\Pi_{p,in}=\sum_{\omega_{M}\in[\Omega-\Delta,\Omega+\Delta]}\frac{4\pi^{3}}{M _{d}}S_{pp}(k_{M},\omega_{M}),\qquad k_{M}\ll\frac{\omega}{U_{c}}, \tag{60}\]  
Figure 17: Power flow in four subsystems of SEA model for HSM. where \(M_{d}\) is the mass per unit area of the plate, and the subscript \(M\) indicates the number indices of the plate and the cavity, respectively. Then, damping loss factors need to be determined. The two most common methods for experimentally determining the damping loss factor of a subsystem are the decay rate method and the half-power bandwidth methods. The related detailed descriptions are also provided in Section 4.2. Lastly, the coupling loss factors need to be determined. Energy transmission in subsystem \(i\) to \(j\) can be expressed in the form  
\[\Pi_{ij}=\omega\eta_{ij}E_{i}=\omega\eta_{ij}\rho_{i}e_{i}Av_{i}^{2}. \tag{61}\]  
The acoustic power radiated by the plate into the acoustic cavity is then  
\[\Pi_{ij}=\rho_{0}c_{0}A\sigma_{i}v_{i}^{2}, \tag{62}\]  
where \(\rho_{i},\ e_{i},\ \text{and}\ A\) are the density, thickness, and area of the front windshield and side window glass, respectively. By rearranging Equations (61) and (62), the coupling loss factor becomes  
\[\eta_{ij}=\frac{\rho_{0}c_{0}\sigma_{i}}{\omega\rho_{i}e_{i}}, \tag{63}\]  
where \(\sigma_{i}\) is the radiation efficiency of plate. The radiation efficiency is defined as follows:  
\[\sigma=\sum_{i}\frac{W_{i}}{\rho_{0}c_{0}\langle v^{2}\rangle A}, \tag{64}\]  
where \(W_{i}\) is the radiated sound power emitted by the plate, and \(\left\langle v^{2}\right\rangle\) is the spatially averaged RMS value of velocity across the surface. Estimation of radiation efficiency can be obtained experimentally, theoretically, and analytically.