## 4 Global Sensitivity Analysis and Optimisation for Structural Parameters of Coupled Systems  
### Sensitivity Theory  
Complex dynamical systems have many influencing parameters and the acoustic characteristics of the coupled system vary for different values and combinations of various parameters. The influence of various structural parameters on the isolation effect in vibration isolation systems has been studied extensively in the past, but mainly by means of the single factor method. Sensitivity analysis focuses on identifying the importance of the influence of different structural parameters on the target function and can show the degree of interaction between different structural parameters. The core idea of [PERSON]' sensitivity analysis is to decompose the function Y into a sum of subterms in the case where the input parameter domain \(k\) is a \(k\) dimensional unit cell  
\[Y=f_{0}+\sum_{i=1}^{k}f_{i}(x_{i})+\sum_{ \begin{array}{c}i,j=1\\ i<j\end{array} }^{k}f_{i,j}(x_{i},x_{j})+\cdots+f_{1,2,\ldots,k}(x_{1},x_{2}, \ldots,x_{k}) \tag{25}\]  
where \(x_{i}\) is the \(i\)th parameter, \(f_{0}\) is a constant, and the remaining subterms have zero integration over any variable they contain  
\[\int_{0}^{1}f_{i_{1},i_{2},\ldots,i_{k}}(x_{i_{1}},x_{i_{2}},\ldots,x_{i_{k}} )dx_{i_{k}}=0 \tag{26}\]  
Figure 9: Ratio between vibration level dropout and transmitted power flow. Sub-terms are orthogonal to each other, so if (\(i_{1}\), \(i_{2}\),..., \(i_{s}\)) \(\
eq\) (\(j_{1}\), \(j_{2}\),..., \(j_{l}\)), then  
\[\int_{R}f_{i_{1},i_{2},...,i_{s}}\cdot f_{j_{1},i_{2},...,j_{l}}dX=0 \tag{27}\]  
Due to the above relationship, the decomposition of Equation (25) is unique and the subterms can be found by multiple integration  
\[\left\{ \begin{gathered} f_{0}=\int_{R^{i}}f(X)\mathrm{d}X\\ f_{0}+f_{i}(x_{i})=\int_{0}^{1}\cdots\int_{0}^{1}f(X)\prod\limits_{l \to i}\mathrm{d}x_{l}\\ f_{0}+f_{i}(x_{i})+\cdots+f_{j}(x_{j})+f_{i,j}\big{(}x_{i},x_{j} \big{)}=\int_{0}^{1}\cdots\int_{0}^{1}f(X)\prod\limits_{l\to i,j}\mathrm{d}x_{ l}\end{gathered} \right. \tag{28}\]  
Assuming that \(f(X)\) squared is productable, square both sides of Equation (25) and integrate over the entire domain of definition yields  
\[\int_{R^{i}}f^{2}(X)\mathrm{d}X-f_{0}^{2}=\sum\limits_{ \begin{subarray}{c}s=1 \\ 1\leq i_{1}<i_{2}<...<i_{s}\leq k\end{subarray} }^{k}\int_{0}^{1}\cdots\int_{0} ^{1}f_{i_{1},i_{2},...,i_{s}}^{2}\big{(}x_{i_{1}},x_{i_{2}},\ldots,x_{i_{s}} \big{)}\mathrm{d}x_{i_{1}}\mathrm{d}x_{i_{2}}\cdots\mathrm{d}x_{i_{s}} \tag{29}\]  
From this, define the total variance of \(f(X)\)  
\[D=\int_{R^{i}}f^{2}(X)\mathrm{d}X-f_{0}^{2} \tag{30}\]  
and the bias variance of \(f_{i_{1},i_{2},...,i_{s}}\big{(}x_{i_{1}},x_{i_{2}},\ldots,x_{i_{s}}\big{)}\)  
\[D_{i_{1},i_{2},...,i_{s}}=\int_{0}^{1}\cdots\int_{0}^{1}f_{i_{1},i_{2},...,i_{ s}}^{2}\big{(}x_{i_{1}},x_{i_{2}},\ldots,x_{i_{s}}\big{)}\mathrm{d}x_{i_{1}} \mathrm{d}x_{i_{2}}\cdots\mathrm{d}x_{i_{s}} \tag{31}\]  
\[D=\sum\limits_{i=1}^{k}D_{i}+\sum\limits_{ \begin{subarray}{c}i_{j}=1\\ i<j\end{subarray} }^{k}D_{i,j}+\cdots+D_{1,2,...,k} \tag{32}\]  
The global sensitivity \(S_{i_{1},i_{2},...,i_{s}}\) can therefore be expressed as  
\[S_{i_{1},i_{2},...,i_{s}}=D_{i_{1},i_{2},...,i_{s}}/D,\ (1\leq i_{1}<i_{2}< \ldots<i\leq k) \tag{33}\]  
\(S_{i}\) is the first-order sensitivity coefficient of the factor \(x_{i}\) and represents the main effect of \(x_{i}\) on the output (the effect of \(x_{i}\) on \(f(X)\)); \(S_{i,j}\) (\(i\
eq j\)) is the second-order sensitivity coefficient and represents the cross-effect of the two parameters \(x_{i}\), \(x_{j}\) (the sum of the effects of \(x_{i}\), \(x_{j}\) alone cannot directly represent the changes caused by them), and  
\[\sum\limits_{i=1}^{k}S_{i}+\sum\limits_{ \begin{subarray}{c}i_{j}=1\\ i<j\end{subarray}