## 4 Impacts of Two Uncertainties on Natural Characteristics
### Impacts of Two Uncertainties at Disk-Shaft Elements  
In an actual dual-rotor system, the mass will also have a significantly large effect on the natural characteristics. In addition, it can be found from Equation (4) that the mass of elements is needed to construct the transfer matrices, so the mass parameter uncertainty will definitely affect the natural characteristics of the system. Therefore, this section will focus on the impacts of mass parameter uncertainty and nonparametric uncertainty at the 10 th element on the natural characteristics of the dual-rotor system. Figure 7 and Table 3 show the first three-order critical speeds of the system and their relative errors for \(C_{ox,k}=0\)-\(0.20\), respectively. For \(n_{c1}\) and \(n_{c2}\), the results are similar to those in Section 4.1; i.e., the critical speeds of NM envelope those of PM. However, the results have changed for \(n_{c3}\): the results of PM envelop those of NM, which indicates that the \(n_{c3}\) of this system is more sensitive to mass parameter uncertainty. [PERSON] [20] drew a similar conclusion in studying the effects of uncertainties on the critical speeds of the rotating system. Figure 6: MAC values of the first three-order vibration modes of an uncertain dual-rotor system for \(C_{ox,k}=0\)–\(0.20\). (**a**) MAC values of the first-order vibration mode. (**b**) MAC values of the second-order vibration mode. (**c**) MAC values of the third-order vibration mode. As shown in Figure 8, under the impacts of mass parameter uncertainty and nonparametric uncertainty at the disk-shaft element, the first three-order vibration modes change; that is, they are no longer fixed in shape, instead constantly varying within a certain interval. It should be noted that both the second-order and third-order vibration modes display a phenomenon where the vibration modes under parameter uncertainty envelope those under nonparametric uncertainty, which is different from Figure 5, indicating that the mass uncertainty of the disk-shaft elements and the stiffness uncertainty of the intermediate supports have different impacts on the vibration modes. The MAC variation in Figure 9 also confirms the existence of this phenomenon. It may be due to the large mass of the 10 th element, resulting in greater contributions from mass parameter uncertainty than the corresponding nonparametric uncertainty. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**Parameter Uncertainty**} & \multicolumn{3}{c}{**Nonparametric Uncertainty**} \\ \hline \(C_{ov,\,m}\) & \(n_{c1}\) & \(n_{c2}\) & \(n_{c3}\) & \(n_{c1}\) & \(n_{c2}\) & \(n_{c3}\) \\ \hline
0.05 & 0.048\% & 0.157\% & 0.050\% & 1.486\% & 0.413\% & 0.100\% \\
0.10 & 0.096\% & 0.283\% & 0.135\% & 4.025\% & 1.002\% & 0.189\% \\
0.15 & 0.168\% & 0.363\% & 0.181\% & 5.319\% & 1.385\% & 0.340\% \\
0.20 & 0.240\% & 0.659\% & 0.236\% & 8.554\% & 1.831\% & 0.391\% \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Relative errors of the first three-order critical speeds of a system with two uncertainties at the disk-shaft element. Figure 7: The first three-order critical speeds of the uncertain dual-rotor system for \(C_{ov,m}\), from 0 to 0.20. (**a**) The first-order critical speed. (**b**) The second-order critical speed. (**c**) The third-order critical speed. Figure 8: Upper and lower boundaries of the first three-order vibration modes of two rotors for \(C_{ov,m}=0.20\). (**a**) The first-order vibration mode. (**b**) The second-order vibration mode. (**c**) The third-order vibration mode. Figure 9: MAC values of the first three-order vibration modes of an uncertain dual-rotor system for \(C_{ov,m}=0\)–\(0.20\). (**a**) MAC values of the first-order vibration mode. (**b**) MAC values of the second-order vibration mode. (**c**) MAC values of the third-order vibration mode. The above analyses reveal that the mass parameter uncertainty, stiffness parameter uncertainty, and the corresponding nonparametric uncertainty will alter the natural characteristics of the system.