## 4 Result Analysis
### Optimization of Patch Location on Damping Loss Factor  
In this section, a numerical analysis is carried out to study the optimal patch location for a sandwich plate with partial constraining damping in four different boundary conditions (CFFF, CFCF, SFSF CGSF) and the first four order damping loss factors are taken into account. There are four classical boundary conditions (F-free, S-simply supported, C-clamped and G-guided) in this paper. The basic parameters of MPGA are shown in Table 5. The geometry and material properties are as follows: \(L=W=0.4\) m, \(L_{P}=W_{P}=0.08\) m, \(h_{c}=h_{b}=h_{d}=0.005\) m, \(\rho_{b}=\rho_{c}=7800\) kg/m\({}^{3}\), \(\rho_{d}=2000\) kg/m\({}^{3}\), \(E_{c}=E_{b}\) = 207 GPa, \(G_{d}=4\) MPa, and \(\beta=0.38\). Then the simulation results can be obtained according to the above conditions and all calculations are done by using MATLAB software. For CFFF, it can be clearly seen that the optimal location and vibration modes of the first four order are shown in Figure 6. The blue square patch represents the constrained damping layer, and the numbers in parentheses represent the coordinates of the lower left corner of the patch. The mode shapes in Figure 6a-d correspond to the location of the patch. In fact, the location of the patch has little effect on the mode shapes by comparison. Figure 6 indicates that the optimal locations of each order are far from the fixed constraints. In practice, far away from the fixed constraints, there is a greater dynamic response under external forces. Therefore, it is a reasonable optimization result. The first four loss factors of a sandwich plate with \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Order**} & **Theoretical Calculation** & **Experiment Result (\(E\))** & **Error** \\  & **(T) (\%)** & **(\%)** & **\(\mid\)T-E 1/E (\%)** \\ \hline
1 & 0.196 & 0.188 & 4.23 \\
2 & 0.139 & 0.132 & 5.30 \\
3 & 0.087 & 0.092 & 5.43 \\
4 & 0.060 & 0.056 & 7.14 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The first four order damping loss factor of the sandwich plate obtained by the presented method and experiment. \begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Parameters.**} & **Value** \\ \hline Number of individuals & 40 \\ Precision of variables & 20 \\ Population size & 10 \\ Cross probability & 0.7 \\ Mutation probability & 0.05 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: The parameters of multi population genetic algorithm (MPGA). partially constrained layer damping after optimizing the location of the patch are shown in Table 6. In Table 6, bold numbers indicate the results of the loss factor optimization for the corresponding order. With the increase in order, the loss factor decreases gradually and the loss factor of the optimized target order is greater than others. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Figure number** & **1 st Order** & **2 nd Order** & **3 rd Order** & **4 th Order** \\ \hline Figure 6a & **0.04077** & 0.02864 & 0.01239 & 0.0011 \\ Figure 6b & 0.04069 & **0.03095** & 0.01353 & 0.0008 \\ Figure 6c & 0.04028 & 0.02753 & **0.01492** & 0.0125 \\ Figure 6d & 0.03946 & 0.02704 & 0.01451 & **0.0169** \\ \hline \hline \end{tabular}
\end{table} 
Table 6: The first four loss factors after optimizing the location of the patch for CFFF. Figure 6: Optimal location of the patch for CFFF. (**a**) optimization for first order loss factor; (**b**) optimization for second order loss factor; (**c**) optimization for third order loss factor; (**d**) optimization for fourth order loss factor. Further, the same optimization is carried out for the other three boundary conditions (CFCF, SFSF CGSF) and all optimization results are shown in Table 7. For the sake of simplicity, only the coordinates of the patch and the target order loss factor are displayed after optimization. Then, the influence of the boundary conditions on the optimization of the damping loss factor can be analyzed, based on the simulation results.