## 4 Global Sensitivity Analysis and Optimisation for Structural Parameters of Coupled Systems
### Optimisation of Structural Parameters of Coupled Systems Based on Sensitivity Analysis  
For submarines and other underwater vehicles, the reduction of acoustic radiated power is the ultimate goal of dynamic structural optimization design, but the actual system's acoustic radiated power function is difficult to obtain, so other relatively easy to obtain vibroacoustic transfer characteristic functions are considered instead of acoustic radiated power as the structural optimization objective function. The basic correspondence between level drop--transmitted power flow--acoustic radiated power has been described through the previous analysis, so in the following, select the transmitted power flow (method 1), level drop (method 2) and transmitted power flow + level drop (method 3) as the objective functions based on the model in Figure 1 (as shown in Table 3), and further discuss the optimal design method for vibration and noise reduction of the coupled system. In Table 3, \(Q_{1}\), \(Q_{2}\) are the root mean square of the power flow and the vibration level drop, respectively. According to the sensitivity analysis in the previous section, set the parameters that have a greater influence on the transmitted power flow (Method 1) and the vibration level drop (Method 2) as the optimisation variables, including the lower vibration isolator spacing \(H_{x}\), the upper unit 1 mass \(m_{1}\), the raft frame mass \(m_{\textit{raft}}\), the upper vibration isolator stiffness \(k_{2}\), the lower vibration isolator stiffness \(k_{R}\), the circular thin shell thickness \(d\) and the vibration isolator loss factor \(\eta\). Table 4 shows the range of values for each parameter variable set for further optimisation calculations. In order to ensure  
\begin{table}
\begin{tabular}{c c c} \hline \hline  & **Method 1** & **Method 2** & **Method 3** \\ \hline Objective functions & \(MinQ=[Q_{1}]\) & \(MinQ=[Q_{2}]\) & \(MinQ=[Q_{1},Q_{2}]\) \\ \hline Binding conditions & \(s.t.\left\{\begin{array}  
structural stability while obtaining a more pronounced optimisation effect, the range of variation of the parameters is set to \(\mu\pm 30\sigma\), using the Latin hypercube sampling method. The values of the parameters that are not set as optimisation variables in the optimisation calculation are still shown in Table 2.  
Adopt the NSGA-2 algorithm for the coupled system model, set the NSGA-2 operation parameters as the population number 100, the iteration number 100, and the crossover frequency 0.7. After the iterative calculation is completed, the results are kept in three valid digits, and the optimized structural parameters of the coupled system are obtained as shown in Table 5 below. According to the optimized structural parameters shown in Table 5, Table 6 shows the comparison of the magnitude of the root mean square of the objective function before and after the optimization for the coupled system dynamics model, and Figure 11 shows the comparison of the spectra of the coupled system before and after the optimization of the vibration level drop, transferred power flow and acoustic radiated power. \begin{table}
\begin{tabular}{c c c c c} \hline
**Names and Symbols** & **Values before** & **Optimisation** & **Optimisation** & **Optimisation** \\  & **Optimisation** & **(Method 1)** & **(Method 2)** & **(Method 3)** \\ \hline transmitted power flow (RMS) & \(1.41\times 10^{-9}\) & \(8.13\times 10^{-10}\) & \(1.25\times 10^{-9}\) & \(9.68\times 10^{-10}\) \\ \hline vibration level dropout (RMS) & \(1.45\times 10^{-4}\) & \(1.75\times 10^{-5}\) & \(1.18\times 10^{-5}\) & \(1.77\times 10^{-5}\) \\ \hline acoustic radiated power (RMS) & \(9.68\times 10^{-13}\) & \(2.95\times 10^{-13}\) & \(4.72\times 10^{-13}\) & \(1.10\times 10^{-13}\) \\ \hline \end{tabular}
\end{table} 
Table 6: Comparison of the objective function before and after optimization for the coupled vibration system.