## 3 Hybrid FE-SEA Modeling Parameter Identification Algorithm
### Derivation of the Energy Response of Random Subsystems of Three-Coupled Systems  
#### 3.2.1 Random Subsystem Energy Response of Three-Coupled Systems  
The power-flow model of the three-coupled system is shown in Figure 3. The external input power of the random subsystem is taken into account, and the power flow between the random subsystem is ignored. In the figure, the solid line represents the deterministic boundary, while the dashed line represents the random boundary. The random subsystem \(j\) (\(j=1\), \(K=2\)) satisfies the following equation:  
\[P_{in,dir}^{(j)}+P_{in,j}=P_{out,rev}^{(j)}+P_{diss}^{(j)}=P_{out,rev}^{(1)}+ P_{diss}^{(1)} \tag{33}\]  
where \(P_{in,dir}^{(j)}=P_{in,dir}^{(1)}\) represents the average direct field power received by the random subsystem \(j=1\), \(P_{in,j}\) represents the direct input power received by the random subsystem \(j=1\), \(P_{out,rev}^{(j)}\) represents the output power of the reverberation field to the random subsystem \(j=1\), and \(P_{diss}^{(j)}\) represents the power consumed by the random subsystem \(j=1\). By expanding the above Equation (29), the displacement response of the deterministic subsystem is composed of two parts. One part is the response \(q_{d}\) generated by excitation, and the other part is the response \(\sum\limits_{k=1}^{N}q^{(k)}\) generated by the reverberation field excitation of the other random subsystems. Then, the displacement response \(q\) is expressed as [9]:  
\[q=D_{tot}^{-1}\Bigg{(}f_{ext}+\sum\limits_{k=1}^{N}f_{rev}^{(k)}\Bigg{)}=q_{d}+ \sum\limits_{k=1}^{N}q^{(k)}=q_{d}+\Big{(}q^{(1)}+q^{(2)}\Big{)} \tag{34}\]  
\[q_{d}=D_{tot}^{-1}f_{ext} \tag{35}\]  
\[q^{(k)}=q^{(2)}=D_{tot}^{-1}f_{rev}^{(2)} \tag{36}\]  
The direct field power flow \(P_{in,dir}^{(j)}\) received by a random subsystem (\(j=1\)) is expressed as:  
\[ \begin{array}{l}P_{in,dir}^{(j)}=E\Big{[}\frac{\omega}{2}\text{Im}\Big{\{}q^ {H}D_{dir}^{(j)}q\Big{\}}\Big{]}=\frac{\omega}{2}\text{Im}\Big{\{}\sum\limits _{r}\sum\limits_{s}\Big{(}D_{dir,js}^{(j)}E[q_{r}q_{s}^{*}]\Big{)}\Big{\}}= \frac{\omega}{2}\sum\limits_{rs}\Big{[}\text{Im}\Big{\{}D_{dir,js}^{(j)} \Big{\}}\Big{]}S_{qq,rs}=\\ \frac{\omega}{2}\text{Im}\Big{\{}\sum\limits_{r}\sum\limits_{s}\Big{(}D_{dir,1s}^{(1)}E[q_{r}q_{s}^{*}]\Big{)}\Big{\}}=\frac{\omega}{2}\sum\limits_{rs} \Big{[}\text{Im}\Big{\{}D_{dir,1s}^{(1)}\Big{\}}\Big{]}S_{qq,rs}\end{array}  \tag{37}\]  
where \(r\) and \(s\) represent the degrees of freedom of a random subsystem (\(j=1\)) at a deterministic boundary. Substitute Equation (34) into Equation (37):  
\[ \begin{array}{l}P_{in,dir}^{(j)}=\frac{\omega}{2}\sum\limits_{rs}\Big{[} \text{Im}\Big{\{}D_{dir,js}^{(j)}\Big{\}}\Big{(}D_{tot}^{-1}S_{ff}^{ext}D_{ tot}^{-1H}\Big{)}_{rs}\Big{]}+\frac{\omega}{2}\sum\limits_{rs}\sum\limits_{ rs}\Big{[}\text{Im}\Big{\{}D_{dir,js}^{(j)}\Big{\}}\Big{(}\frac{4E_{s}}{ \omega\pi n_{2}}\Big{)}\Big{(}D_{tot}^{-1}\Big{\{}D_{dir}^{(k)}\Big{\}}D_{tot }^{-1H}\Big{)}_{rs}\Big{]}=\\ \frac{\omega}{2}\sum\limits_{rs}\Big{[}\text{Im}\Big{\{}D_{dir,1s}^{(1)}\Big{\}} \Big{(}D_{tot}^{-1}S_{ff}^{ext}D_{tot}^{-1H}\Big{)}_{rs}\Big{]}+\frac{\omega}{2 }\sum\limits_{rs}\Big{[}\text{Im}\Big{\{}D_{dir,1s}^{(1)}\Big{\}}\Big{(} \frac{4E_{s}}{\omega\pi n_{2}}\Big{)}\Big{(}D_{tot}^{-1}\Big{\{}D_{dir}^{(2)} \Big{\}}D_{tot}^{-1H}\Big{)}_{rs}\Big{]}\end{array}  \tag{38}\]  
where the first term on the right of the equation is the power of the excitation force of the deterministic subsystem acting on the system. The second term on the right of the equation is the power of the reverberation load on the system generated by other random subsystems.