## 4 Impacts of Two Uncertainties on Natural Characteristics
### Impacts of Two Uncertainties at Intermediate Support Elements
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**Parameter Uncertainty**} & \multicolumn{3}{c}{**Nonparametric Uncertainty**} \\ \hline \(C_{\omega,\,k}\) & \(n_{\mathcal{C}1}\) & \(n_{\mathcal{C}2}\) & \(n_{\mathcal{C}3}\) & \(n_{\mathcal{C}1}\) & \(n_{\mathcal{C}2}\) & \(n_{\mathcal{C}3}\) \\ \hline
0.05 & 0.010\% & 0.040\% & 0.591\% & 1.292\% & 0.624\% & 2.067\% \\
0.10 & 0.014\% & 0.084\% & 1.154\% & 2.727\% & 1.331\% & 4.135\% \\
0.15 & 0.037\% & 0.143\% & 1.589\% & 3.924\% & 1.816\% & 5.892\% \\
0.20 & 0.035\% & 0.217\% & 3.343\% & 6.268\% & 2.245\% & 9.596\% \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Relative errors of the first three-order critical speeds of the system with two uncertainties at the intermediate support elements. Figure 4: The first three-order critical speeds of the uncertain dual-rotor system for \(C_{\omega,k}\), from 0 to 0.20. (**a**) The first-order critical speed. (**b**) The second-order critical speed. (**c**) The third-order critical speed. of two types of uncertainties on the first three-order vibration modes [30], which can be calculated as follows:  
\[MAC=\frac{\left|[\varphi_{a}]^{\mathsf{T}}[\varphi_{b}]\right|^{2}}{[\varphi_{a}] ^{\mathsf{T}}[\varphi_{a}][\varphi_{b}]^{\mathsf{T}}[\varphi_{b}]} \tag{34}\]  
where \([\varphi_{a}]\) and \([\varphi_{b}]\) represent the vibration mode vectors obtained from the stochastic models (PM and NM) and the MM, respectively. From a mathematical point of view, the MAC represents the cosine value of two vectors in a specific space. The smaller the MAC value, the greater the angle between the two vectors, and it is easier to distinguish the two vectors. In the field of modal analysis, the smaller the MAC value is, the greater the difference between the vibration modes of the stochastic model and the MM, and the greater the impacts of uncertainties on the system's natural characteristics. On the contrary, if the MAC value is closer to one, it means that the vibration modes of the stochastic model and MM are more similar, and the effects of uncertainty are lesser. Figure 5 shows the upper and lower boundaries of the first three-order vibration modes of the system with two uncertainties for \(C_{ov,k}\) = 0.20. In order to show the floating degree of the rotor \(I\) and rotor \(II\) vibration modes more intuitively, the same order vibration modes of the two rotors were placed in one figure. The long lines represent the vibration modes of rotor \(I\), and the short lines represent that of rotor \(II\). The solid blue lines represent the vibration modes calculated with NM, the solid red lines represent those calculated with MCS, and the dotted green line represents those calculated with MM. These descriptions also apply to the following figures. The impacts of two uncertainties on the vibration modes can be easily observed in Figure 6. The MAC values obtained by MCS are almost identical to one, indicating that the parameter uncertainty has little impact on the first three-order vibration modes of two rotors, which means that the vibration modes are insensitive to parameter uncertainty. At the same time, the MAC values of the first three-order vibration modes of rotor \(I\) decreased significantly under nonparametric uncertainty, indicating that the vibration modes of rotor \(I\) are sensitive to nonparametric uncertainty. Figure 5: Upper and lower boundaries of the first three-order vibration modes of two rotors for \(C_{ov,k}\) = 0.20. (**a**) The first-order vibration mode. (**b**) The second-order vibration mode. (**c**) The third-order vibration mode. In short, the natural characteristics of the system are more sensitive to nonparametric uncertainty than to the parameter uncertainty at the intermediate supports.