## 4 Results
### Application of the Monte Carlo Method
#### 4.1.1 Number of Samples  
The accuracy of the technique depends on the number of samples used for the input values. Increasing the number of samples, the accuracy of the results will be higher, but the computational cost of the calculation will also be increased. Thus, an agreement must be reached between accuracy and computational cost.  
In order to quantify the accuracy of the method, the Monte Carlo probable error [35] was employed. According to Equation (41), the standard deviation of the output mean (\(\sigma_{\text{M}}\)) can be calculated in the function of the standard deviation of all the output values (\(\sigma\)) and the number of samples.  
\[\sigma_{\text{M}}=\frac{\sigma}{\sqrt{N}} \tag{41}\]  
As a criterion, a maximum error of \(\pm 0.5\%\) was established, with a confidence rate of 99.7%, which is equivalent to three standard deviations. Thus, simulations were performed for different values of \(N\), and Table 2 shows the standard deviation obtained for the vibration and the torque in each case.  
Therefore, among the studied cases, 5000 samples would need to be employed in order to fulfil the criterion, as a maximum error of 0.465% and 0.351% would be committed in the 99.7% of the cases for the vibration and the torque, respectively.  
Additionally, the sensitivity indices obtained for these cases are shown in Tables 3-6:  
\begin{table}
\begin{tabular}{c c c c c} \hline  & \(N=200\) & \(N=1000\) & \(N=5000\) & \(N=10,000\) \\ \hline Vibration & 0.773\% & 0.346\% & 0.155\% & 0.110\% \\ \hline Torque & 0.583\% & 0.261\% & 0.117\% & 0.082\% \\ \hline \end{tabular}
\end{table}
Table 2: Standard deviations of the mean for different numbers of samples.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline  & \(D_{\mathbf{i}}\) & \(h_{\mathbf{sy}}\) & \(h_{\mathbf{m}}\) & \(g\) & \(L_{\mathbf{s}}\) & Total \\ \hline Vibration & \(S_{i}\) & 0.718 & 0.173 & 0.005 & 0.123 & 0.012 & 1.031 \\ \(S_{i}^{\text{tot}}\) & 0.760 & 0.133 & 0.009 & 0.111 & 0.006 & 1.019 \\ \hline Torque & \(S_{i}\) & 0.476 & 0.001 & 0.006 & 0.015 & 0.456 & 0.954 \\ \(S_{i}^{\text{tot}}\) & 0.469 & 0.000 & 0.017 & 0.016 & 0.473 & 0.976 \\ \hline \end{tabular}
\end{table}
Table 3: Sensitivity indices—200 samples.  
Generally, quite similar values are obtained for all the cases. However, some illogical results are obtained for low values of \(N\). For instance, when using 200 samples, the first-order indices of several parameters are higher than their total-effect indices; the sum of the first-order indices is higher than 1 for the vibration, and the sum of total-effect indices is lower than 1 for the torque. All these results are incorrect according to the definition of the sensitivity indices stated in Section 3.2, and some slight incoherences are still observed for 1000 samples.  
For 5000 and 10,000 samples, all the values of the sensitivity indices are logical. Therefore, according both to the Monte Carlo probable error criterion and the definition of the sensitivity indices, a value of 5000 was selected as the number of samples for the analysis, in order to reach an agreement between accuracy and computational cost.  
Another conclusion that can be obtained is that the effect of the interactions between the input parameters is negligible, as all the total-effect indices are almost equal to their corresponding first-order indices. Therefore, only the total-effect indices were considered when analysing the results of the sensitivity analysis.