## 1 Introduction  
Rotor systems are widely used in aero-engines, gear transmissions, large steam wheels, and other machines [1]. In some cases, the operating conditions of these systems are pretty serious. Moreover, they will cause significant damage to economic and personal safety if an accident occurs. Therefore, it is essential to carry out a rigorous dynamic characteristics analysis for rotor systems in the design process [2, 3]. So far, many studies on rotordynamics modeling methods have been published. However, there are a lot of uncertainties in the whole life cycle of a device from design, to manufacturing, to operation, to maintenance, which have great impacts on the operation of the device. These uncertainties can then be split into two classes, i.e., aleatoric uncertainty and epistemic uncertainty [4]. The former is mainly caused by changes in operating conditions and errors and is always used to consider the uncertainty of the geometric parameters and physical parameters of the system [5], also known as parameter uncertainty, which has already received great attention and sufficient research in recent years; the latter is induced from the difficulty of accurately modeling the dynamic characteristics of a system [6, 7], known as nonparametric uncertainty, which is mainly introduced by modeling the actual systems abstractly and has become a hot topic in the field of stochastic rotor dynamics. It is the stochastic parametric method that is used to deal with the parametric uncertain problem. In essence, it allows some random variables to be used to describe the uncertain parameters in the dynamic model, and then, one can combine it with the finite element method (FEM) and the Monte Carlo simulation (MCS) to study uncertainties. Nevertheless, it cannot take nonparametric uncertainty induced by model errors and calculation errors into account [8]. For instance, [PERSON] et al. [9, 10] proposed a noninvasive metamodeling method to quantify the parameter uncertainty of a dual-rotor system and calculated the steady-state nonlinear dynamic response of the system under interval uncertainty. [PERSON] et al. [11] proposed the non-probabilistic Chebyshev convex method (CCM) and calculated critical speeds and dynamic responses. The results indicate that the response of the rotor system is no longer deterministic, but rather, it varies in an interval. For nonparametric uncertainty, a nonparametric probabilistic method based on random matrix theory (RMT) has been proposed and widely used, so it will be studied in this paper. [PERSON] [12; 13; 14; 15], who introduced RMT into structural dynamics and developed a large number of fundamental theories, conducted pioneering research in the study of the stochastic dynamics of nonparametric uncertainty systems. [PERSON] et al. [16] introduced the maximum entropy principle (MEP) into the nonparametric modeling method and obtained a nonparametric transfer model of a single rotor system, further analyzing the action of uncertainties on the eigenvalues of the system. [PERSON] et al. [17; 18] not only constructed a nonparametric model of the classic Jeffcott rotor but also studied the effects of nonparametric uncertainty on the critical speeds and unbalanced responses of the rotor with different disk-offsets via numerical simulation. [PERSON] et al. [19] used a nonparametric method to model a high-speed motorized spindle with uncertainty and discussed vibration responses. However, the effects of the two uncertainties were not distinguished. [PERSON] et al. [20] proposed a nonparametric model of a dual-rotor system and investigated the effects of two uncertainties on the critical speeds of the system but did not discuss the impacts on the vibration modes. The above studies show that it is feasible to use the nonparametric modeling method to analyze rotor systems with uncertainty. Nevertheless, most of the investigations focus on the critical speeds and dynamic responses of a single-rotor system.