## 4 Global Sensitivity Analysis and Optimisation for Structural Parameters of Coupled Systems
### Global Sensitivity Analysis of Coupled Systems  
Based on the viewpoint of vibration and noise reduction design, take the transmitted power flow, vibration level dropout and acoustic radiated power as the objective functions,respectively, then apply Sobol' sensitivity analysis to compare and analyse the influence weights and cross-influence weights of various structural parameters of the coupled system in Figure 1 on the above three objective functions. This analysis provides a theoretical reference for the optimisation of the vibroacoustic transfer characteristics of similar systems. In order to introduce the Sobol' sensitivity analysis method, first define the interval of variation and the probability distribution of each parameter. Without loss of generality, the distributions of all parameters are set to follow a normal distribution. Since the magnitudes of the different parameters differ, the standard deviation is designed according to the coefficient of variation \(c_{v}=\sigma/\mu\), where \(m\) is the mean and \(s\) is the standard deviation. Let \(c_{u}=0.01\) and use ladin hypercube sampling; Table 2 lists the means, variances and ranges of variation of various parameters. To ensure the accuracy of the calculation and to take into account the length of the calculation, set the sample size of sampling N = 2000. Figure 10 compares the coupled system at the first 5 orders of resonant frequency. The first-order and total sensitivities of the various parameters are listed in Table 2 for the transmitted power flow, vibration level dropout and acoustic radiated power; where a larger first-order sensitivity represents a greater effect of the parameter on the target function, and a larger difference between the first-order sensitivity and the total sensitivity represents a greater effect of the interaction of the parameter with the other parameters on the target function. The resonant frequency points were chosen for sensitivity analysis because they correspond to the peak points of the target function, so that the magnitude of the change in the target function is larger and therefore also has a higher sensitivity value. \begin{table}
\begin{tabular}{c c c c} \hline
**Names and Symbols** & **Mean Value of Parameters (\(\mu\))** & **Variance (\(\sigma\))** & **Range of Variations** \\ \hline Unit quality \(m_{1}\) & 46 & 0.46 & \(\mu\pm 3\sigma\) \\ Unit quality \(m_{2}\) & 70 & 0.7 & \(\mu\pm 3\sigma\) \\ Raft frame quality \(m_{R}\) & 100 & 1 & \(\mu\pm 3\sigma\) \\ Upper vibration isolator stiffness \(k_{1}\) & \(2\times 10^{5}\times(1+0.1\mathrm{i})\) & \((1+0.1\mathrm{i})\times 10^{3}\) & \(\mu\pm 3\sigma\) \\ Upper vibration isolator stiffness \(k_{2}\) & \(2.7\times 10^{5}\times(1+0.1\mathrm{i})\) & \((1+0.1\mathrm{i})\times 10^{3}\) & \(\mu\pm 3\sigma\) \\ Lower vibration isolator stiffness \(k_{2}\) & \(8.6\times 10^{5}\times(1+0.1\mathrm{i})\) & \((1+0.1\mathrm{i})\times 10^{3}\) & \(\mu\pm 3\sigma\) \\ Upper vibration isolator spacing \(h_{1z}\) & 0.2 & 0.002 & \(\mu\pm 3\sigma\) \\ Upper vibration isolator spacing \(h_{1z}\) & 0.2 & 0.002 & \(\mu\pm 3\sigma\) \\ Upper vibration isolator spacing \(h_{2x}\) & 0.3 & 0.003 & \(\mu\pm 3\sigma\) \\ Upper vibration isolator spacing \(h_{2}\) & 0.3 & 0.003 & \(\mu\pm 3\sigma\) \\ Lower vibration isolator spacing \(H_{z}\) & 0.35 & 0.0035 & \(\mu\pm 3\sigma\) \\ Lower vibration isolator spacing \(H_{z}\) & 0.7 & 0.007 & \(\mu\pm 3\sigma\) \\ Cylindrical thin shell thickness \(d\) & 0.02 & 0.0002 & \(\mu\pm 3\sigma\) \\ Vibration isolator loss factor \(\eta\) & 0.1 & 0.001 & \(\mu\pm 3\sigma\) \\ \hline \end{tabular}
\end{table} 
Table 2: Range of values for sensitivity analysis of coupled vibration system parameters.