## 4 Impacts of Two Uncertainties on Natural Characteristics
### Impacts of Two Uncertainties at Intermediate Support Elements  
In this section, the effects of parameter uncertainty and nonparametric uncertainty on the natural characteristics of the dual-rotor system are studied via numerical simulations. As shown in Figure 2, the ninth element is the intermediate support element connecting two rotors, and its uncertainties will greatly affect the whole transfer model. Therefore, the uncertainties at the intermediate support elements have to be analyzed under the same level of uncertainty so that one can compare the contribution of two types of uncertainties in the system. Due to the significant amount of calculation, only 100 numerical simulations utilizing the nonparametric model (NM) were finished. In comparison, 100 MCS were executed based on the parameter model (PM) proposed in Reference [29]. Furthermore, the results calculated via the mean model (MM) are provided. In actual engineering applications, engineers often focus only on the low-order modes, because they have large amplitude, large equivalent mass, and are the most destructive. Therefore, only the first three-order modes of the system are analyzed in this paper. Figure 4 shows the first three-order critical speeds of the system with a different \(C_{\varpi,k}\). \(C_{\varpi}\) is the coefficient of variation, which is equivalent to \(\delta\) in Equation (19). It is often set as 0.1 in practice [11]. In this paper, \(C_{\varpi}\) is set as 0-0.20 (0, 0.05, 0.10, 0.15, and 0.20) to study the impacts of two types of uncertainties on natural characteristics. In order to evaluate the impacts of uncertainties on the natural characteristics of the system, relative errors were introduced to characterize the changes in the first three-order critical speeds. It can be seen that the two types of uncertainties at the intermediary support element have similar impacts on the first three-order critical speeds from Table 2; that is, the relative errors of the critical speeds will increase with two uncertainties. At the same time, there are some differences. On the one hand, at the same level of uncertainty, the errors caused by nonparametric  
Figure 3: Flow chart for calculating the natural characteristics of the system. uncertainty are significantly larger than those caused by parameter uncertainty, indicating that the first three-order critical speeds are more sensitive to nonparametric uncertainty. [PERSON] et al. [29] reached a similar conclusion when conducting a quantitative study on the uncertainty of complex aircraft. On the other hand, at the same level of uncertainty, the impacts will increase with the orders of critical speed, indicating that the high-order critical speeds are more sensitive to the parameter uncertainty than the low-order critical speeds. At the same level of nonparametric uncertainty, this rule cannot be established. The errors of \(n_{\mathcal{C}2}\) are the smallest, followed by \(n_{\mathcal{C}1}\), and \(n_{\mathcal{C}3}\) is the largest, which indicates that the impacts of nonparametric uncertainty on different-order critical speeds are different. In short, the critical speeds of the system will vary with its own orders and the type of uncertainty. The critical speeds reflect the speed range at which the rotor system resonates, but the vibration modes reflect the vibration form of the system, which is also an important natural characteristic of the rotor system. Therefore, it is necessary to analyze the uncertainty of the vibration modes. The modal assurance criteria (MAC) are used to analyze the effects