## 3 Hybrid FE-SEA Modeling Parameter Identification Algorithm
### The Algorithm of a Case Analysis
#### 3.6.2 System Response Analysis  
The FE-SEA hybrid method and the improved FE-SEA method were used to solve the subterms of Equation (73), and the inverse matrix results of the energy density of the subsystem are shown in Figures 9-11. When plate \(1\) was used for the excitation, as shown in Figure 9a, an ill-posed matrix appeared when solving the inverse matrix of the energy density through the mixed FE-SEA method, and a singular value appeared when solving the inverse matrix of the energy density. The principal-element iteration of the ill-conditioned matrix parameters was carried out by the inverse operation of the energy density matrix. Thus, the above problems were solved. In the whole calculation frequency band, no ill-posed matrix appeared, and the fitting degree with respect to the reference solution was high. Figure 9b-d are the inverse matrix distributions obtained by the reference solution method, the proposed method, and the mixed FE-SEA method, respectively. It can be seen from the figure that the inverse matrix obtained by this method was reasonably distributed and had no ill-posed singular matrix values.  
When plate \(2\) was used as the excitation surface, as shown in Figure 10a, the ill-posed matrix appeared when solving the inverse matrix of the energy density by the mixed FE-SEA method. Singular solutions appeared in the inverse matrix of the ratio of subsystem energy to subsystem modal density. There was no ill-posed matrix in the whole calculated frequency band, and the improved method had a high degree of fit with the reference solution. Figure 10b-d show the inverse matrix distribution obtained by the reference solution method, the proposed method, and the mixed FE-SEA method, respectively. It can be seen from the distribution results that the distribution of the inverse matrix solution  
\begin{table}
\begin{tabular}{p{42.7 pt} p{42.7 pt} p{42.7 pt} p{42.7 pt} p{42.7 pt} p{42.7 pt} p{42.7 pt}} \hline \hline \multirow{2}{*}{**Name**} & **Density (kg/m\({}^{3}\))** & **Poisson’s Ratio** & **Elasticity Modulus (GPa)** & **Damping Loss Factor** & **Steel Plate Size (mm)** & **Thickness of Steel Plate (mm)** \\ \hline steel plate & 7800 & 0.3 & 200 & 0.01 & \(1000\times 1000\) & \(1\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Mixed-model material parameters.  
Figure 8: Parameter test system.  
obtained by the method in this paper was reasonable and that no ill-posed matrix appeared. Similarly, as shown in Figure 11, the conclusion regarding system excitation plate 3 was basically consistent with that for excitation plates 1 and 2. The parameter iteration of the ill-conditioned matrix can solve the inverse problem of energy density in the solution process of the mixed FE-SEA. Compared with reference [21], the method for solving the energy matrix of the mixed FE-SEA method was extended. The disadvantages of neglecting the inverse matrix of the energy density of the process parameters in the existing calculation methods are given. A new method for solving the hybrid model is presented which has good research significance for further solving the effective coupling loss factor of the system.