## 2 Numerical Methods and Target Model
### Vibro-Acoustics Model
#### 2.3.2 Statistical Energy Analysis  
Statistical energy analysis (SEA) is a theory proposed by [PERSON] et al. (2019) in the 1970s under the assumption that multiple resonance modes participate in the system response. The SEA is a response analysis method that calculates the average response of the modes existing within an arbitrary frequency band from the difference in the power balance between coupled subsystems. Figure 2 shows the power flow relationship between two coupled subsystems. Each subsystem can receive, dissipate, or exchange power with other subsystems to which it is connected, where \(\Pi_{1,\ in}\) and \(\Pi_{2,\ in}\) are the power entering each subsystem, \(\Pi_{1,\ diss}\) and \(\Pi_{2,\ diss}\) are the power dissipated in each subsystem, and \(\Pi_{12}\) and \(\Pi_{21}\) are the power moving between subsystems, as shown in Figure 2(2019).  
In many practical problems, a structural system consists of a large number of multi-modal subsystems, which are physically connected to one or more other subsystems. The entire system is modeled as the sum of the modes of each subsystem that stores power. In SEA, power transfer between subsystems is performed between resonant systems having a resonant frequency in the frequency band of interest and is expressed as follows:  
\[\Pi_{ij}=\omega\big{(}\eta_{ij}E_{i}-\eta_{ji}E_{j}\big{)}, \tag{36}\]  
\[N_{i}\eta_{ij}=N_{j}\eta_{ji}, \tag{37}\]  
where \(N_{i}\) is the number of vibration modes existing in the frequency band of interest of the subsystem, \(E_{i}\) is the total energy of \(N_{i}\) vibration modes in the subsystem, \(\omega\) is the center frequency, and \(\eta_{ij}\) is the coupling loss factor. The power dissipated in subsystem \(i\) is given by  
\[\Pi_{i,diss}=\omega\eta_{i}E_{i}, \tag{38}\]  
where \(\eta_{i}\) is the damping loss factor. The power balance equation for the \(k\) subsystems can be derived as  
Figure 2: Power flow in two subsystems of SEA model.  
\[\Pi_{i,in}=\Pi_{i,diss}+\sum_{i\
eq j}^{k}\Pi_{ij}-\sum_{i\
eq j}^{k}\Pi_{ji}. \tag{39}\]  
Substituting Equations (36) and (38) into Equation (39) gives the following equation:  
\[\Pi_{i,in}=\omega\eta_{i}E_{i}+\omega\sum_{i\
eq j}^{k}\bigl{(}\eta_{ij}E_{i}- \eta_{ji}E_{j}\bigr{)}. \tag{40}\]  
The second term on the right side of Equation (40) is the net transmission loss energy and can be expressed with the number of resonance modes \(N_{i}\) of the subsystem \(i\) and the reciprocal relationship in the form  
\[\Pi_{i,in}=\omega\eta_{i}E_{i}+\omega\sum_{i\
eq j}^{k}N_{i} \Bigl{(}\eta_{ij}\frac{E_{i}}{N_{i}}-\eta_{ji}\frac{E_{j}}{N_{i}}\Bigr{)} \tag{41}\] \[=\omega\eta_{i}E_{i}+\omega\sum_{i\
eq j}^{k}\eta_{ij}N_{i} \Bigl{(}\frac{E_{i}}{N_{i}}-\frac{E_{j}}{N_{j}}\Bigr{)}.\]  
In Equation (41), \(E_{i}/N_{i}\) is the average modal energy of the subsystem. The power balance equation for a multiple-degrees-of-freedom vibration system with a large number of natural vibration modes, as looked at above, can be rewritten as a determinant as follows:  
\[\{\Pi\}=\omega[C]\{E\}, \tag{42}\] \[\left[\begin{array}{c}\Pi_{1}\\ \Pi_{2}\\ \vdots\\ \Pi_{k}\end{array}\right]=\omega\begin{bmatrix}\Bigl{(}\eta_{1}+\sum_{i\
eq 1 }^{k}\eta_{1i}\Bigr{)}N_{1}&-\eta_{12}N_{1}&\cdots&-\eta_{1k}N_{1}\\ -\eta_{21}N_{2}&\Bigl{(}\eta_{2}+\sum_{i\
eq 1}^{k}\eta_{2i}\Bigr{)}N_{2}& \cdots&-\eta_{2k}N_{2}\\ \vdots&\vdots&\vdots&\vdots\\ -\eta_{k1}N_{k}&\cdots&\cdots&\Bigl{(}\eta_{k}+\sum_{i\
eq 1}^{k}\eta_{k} \Bigr{)}N_{k}\end{bmatrix}\left[\begin{array}{c}\frac{E_{1}}{N_{1}}\\ \frac{E_{2}}{N_{2}}\\ \vdots\\ \frac{E_{2}}{N_{2}}\end{array}\right], \tag{43}\]  
where \(\{\Pi\}\) and \([C]\) are the input power vector and coupling loss coefficient matrices, respectively, and \(\{E\}\) is the average modal energy vector of the subsystem that is unknown in the equation.