## 1 Introduction  
Research on the asymmetric rotor system is one branch of rotor dynamics. They widely exist in industrial plants, such as two-pole generator rotor [1] and cracked rotor [2; 3]. The unequal principal second moments of area of the shaft section can make the dynamics of the asymmetric rotor systems different from the traditional rotor systems with circle shaft sections. In the previous studies focusing on the asymmetric rotor systems, the properties of the rotor system and the boundary condition are deterministic [4; 5; 6]. However, uncertainties are inevitable in reality, which can lead to uncertain dynamics. To gain a deep understanding of the asymmetric rotor system, uncertainties are considered in this paper. In 1988, [PERSON] and [PERSON] [7] presented the matrix expressions of beam elements in the rotating frame, which promoted the development of the research on the asymmetric rotor system. [PERSON] and [PERSON] [8] established the finite element model in the rotating frame; their model considered the effects of rotary inertial, gyroscopic moment, transverse shear deformation, and internal damping. They used this model to analyze the whirl speed and forced vibration. [PERSON] et al. [9] studied the dynamics of the asymmetric rotor system in the fixed frame. The matrix expressions of beam elements were presented in their study. The influences of the asymmetry of the disk and shaft on the forced responses were studied. Their study also considered the general asymmetric rotor systems. However, with the demand of capability for modeling large-scale asymmetric rotor systems without much simplification,the modeling techniques based on beam elements were gradually replaced by a three-dimensional finite element model (3D FEM). [PERSON] and [PERSON] [PERSON] and [PERSON] (2005) modeled the asymmetric rotor by using 3D finite elements. The transient response due to unbalance alone and gravity alone were investigated. The gravity critical speed was reported in their study. [PERSON] et al. [PERSON] _et al._ (2008) established the 3D FEM of the rotary part and stationary part in the rotating frame and fixed frame, respectively. The global equations of motion were obtained by coupling the two parts in a different frame. To accelerate the stability and steady-state analysis, component mode synthesis (CMS) was used to reduce the degrees of freedom (DOFs). [PERSON] et al. [PERSON] _et al._ (2010) used a free-interface CMS to establish a reduced-order model (ROM). Thus, the DOFs of anisotropic rotor-bearing systems were reduced. The influences of four dimensionless parameters describing the anisotropy of bearing on the unbalance and gravity responses were analyzed and discussed. [PERSON] et al. [PERSON] _et al._ (2010) experimentally and numerically studied the whirl speed of an asymmetric rotor system. In their numerical analysis, 3D FEM combined with model order reduction using modal shapes was employed. Then, their model order reduction technique was employed by [PERSON] et al. [PERSON] _et al._ (2010) to study the instability of the asymmetric rotor system with uncertainties. [PERSON] et al. [PERSON] _et al._ (2010) proposed a method that can reduce the interface DOFs when multiple substructures were used. This method was able to solve the steady-state response of large-scale asymmetric rotor systems. The above studies indicated that valuable results concerning the stability and steady-state response of asymmetric rotors have been obtained by deterministic study, where the properties of rotor system and boundary conditions were deterministic. However, uncertainties were inevitable in reality, which could result in uncertain problems. The uncertain rotor system attracted the attention of researchers. The Monte Carlo method was one of the most straightforward methods to handle uncertain problems. [PERSON] _et al._ (2010) employed the Monte Carlo method to study the bounds of the spin speed of unstable regions in an asymmetric rotor system.