## 4 Error Sensitivity Analysis of the Dynamic Performance Based on the PCE Model of the Gear System  
### Global Sensitivity Analysis Method  
Global sensitivity analysis is based on the probability distribution of parameters, and all parameters can change simultaneously during analysis. A quantitative method for global parameter sensitivity analysis was proposed, based on [PERSON]. [PERSON] in 1990. In this method, the interaction between parameters is considered comprehensively when the sensitivity index of a single parameter is analyzed. The main calculation steps are as follows.  
First, all parameters are randomly sampled \(2N\) times to obtain two groups of sample matrices \(\mathbf{A}\) and \(\mathbf{B}\). Then, the sample matrix \(\mathbf{A}_{\mathbf{B}}^{(i)}\) is obtained by replacing the \(i\)th column of matrix \(\mathbf{A}\) with the \(i\)th column of matrix \(\mathbf{B}\) without changing the other columns. Similarly, the sample matrix \(\mathbf{B}_{\mathbf{A}}^{(i)}\) is obtained by replacing the \(i\)th column of matrix \(\mathbf{B}\) with the \(i\)th column of matrix \(\mathbf{A}\) without changing the other columns. Based on the PCE model, the corresponding output response value of the sample matrix is calculated.  
\[\mathbf{A}=\left[\begin{array}{ccccc}x_{1}^{(1)}&x_{2}^{(1)}&\cdots&x_{i}^{(1)}& \cdots&x_{k}^{(1)}\\ x_{1}^{(2)}&x_{2}^{(2)}&\cdots&x_{i}^{(2)}&\cdots&x_{k}^{(2)}\\ \cdots&\cdots&\cdots&\cdots&\cdots\\ x_{1}^{(N-1)}&x_{2}^{(N-1)}&\cdots&x_{i}^{(N-1)}&\cdots&x_{k}^{(N-1)}\\ x_{1}^{(N)}&x_{2}^{(N)}&\cdots&x_{i}^{(N-1)}&\cdots&x_{k}^{(N)}\end{array}\right] \tag{24}\]  
\[\mathbf{B}=\left[\begin{array}{ccccc}x_{k+1}^{(1)}&x_{k+2}^{(1)}&\cdots&x_{k+i}^ {(1)}&\cdots&x_{2k}^{(1)}\\ x_{k+1}^{(2)}&x_{k+2}^{(2)}&\cdots&x_{k+i}^{(2)}&\cdots&x_{2k}^{(2)}\\ \cdots&\cdots&\cdots&\cdots&\cdots\\ x_{k+1}^{(N-1)}&x_{k+2}^{(N-1)}&\cdots&x_{k+i}^{(N-1)}&\cdots&x_{2k}^{(N-1)}\\ x_{k+1}^{(N)}&x_{k+2}^{(N)}&\cdots&x_{k+i}^{(N-1)}&\cdots&x_{2k}^{(N)}\end{array}\right] \tag{25}\]  
Second, calculate the variance of the system output corresponding to the sample matrix \(\mathbf{A}\).  
\[V(Y)=\frac{1}{N}{\sum_{j=1}^{N}{f(\mathbf{A})_{j}^{2}}}-f_{0}^{2} \tag{26}\]  
Calculate the effect of parameter \(x_{i}\) on the variance of the system output.  
\[V_{x_{i}}(E_{x\sim i}(Y|x_{i}))=\frac{1}{N}{\sum_{j=1}^{N}{f(\mathbf{A})_{j}f(\bm {B}_{\mathbf{A}}^{(i)})_{j}}}-f_{0}^{2} \tag{27}\]  
Calculate the effect of parameters other than \(x_{i}\) on the variance of the system output.  
\[V_{x_{i-i}}(E_{x_{i}}(Y|x_{\sim i}))=V(Y)-\frac{1}{N}{\sum_{j=1}^{N}{f(\mathbf{A} )_{j}(f(\mathbf{A})_{j}-f(\mathbf{A}_{\mathbf{B}}^{(i)})_{j})}} \tag{28}\]  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Method** & **MCS** & \multicolumn{3}{c}{**PCE**} \\ \hline
**PCE Order and the Sample Size Needed to** & \multicolumn{3}{c}{**2 (\(\mathbf{M}\) = 15)**} & \multicolumn{3}{c}{**3 (\(\mathbf{M}\) = 35)**} \\ \cline{2-6}
**Determine PCE Coefficient** & & \(\mathbf{N}\) = 2\(\mathbf{M}\) & \(\mathbf{N}\) = 4\(\mathbf{M}\) & \(\mathbf{N}\) = 2\(\mathbf{M}\) & \(\mathbf{N}\) = 4\(\mathbf{M}\) \\ \hline Mean value (N) & 3045.86 & 3198.17 & 3045.86 & 3262.95 & 3042.35 \\ Standard deviation (N) & 594.76 & 538.46 & 482.80 & 943.96 & 606.55 \\ Times calling the black-box system model & 1000 & 30 & 60 & 70 & 140 \\ Computing time (h) & 6 & 0.08 & 0.17 & 0.20 & 0.40 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Comparison of system response results based on different methods.  
Finally, the first and total order indices of variable \(x_{i}\) are calculated.  
\[S_{i}=\frac{V_{x_{i}}(E_{x_{i}}(Y|x_{i}))}{V(Y)} \tag{29}\]  
\[S_{Ti}=1-\frac{V_{x_{i}}(E_{x_{i}}(Y|x_{\sim i}))}{V(Y)} \tag{30}\]