## 1 Introduction
Further, it is observed from the governing equation of the lubricated oil-film pressure that the pressure is determined by the thickness of the oil-film, while the thickness changes in the presence of the misalignment effect. Under this recognition, the misalignment effect is gradually taken into consideration. [PERSON] and [PERSON] [20] studied the steady characteristics of an HDB considering the misalignment caused by deflection of the shaft. Herein, the misalignment effect is described by introducing two misaligned angles. The results show that the maximum pressure and the oil-film forces apparently vary with the misaligned angles. [PERSON] et al. [21] investigated the relationship between the load-carrying capacity and the misalignment angle for a misaligned water-lubricated HBD under different bearing parameters. [PERSON] et al. [22] and [PERSON] et al. [23] studied the influence of misalignment on the dynamic characteristics of an HDB. It can be concluded from these explorations that the characteristics of an HDB and the stability of the rotor-bearing system are affected by misalignment. Therefore, it is essential to consider the misalignment effect in a stability analysis of a rotor-bearing system. When the effect of misalignment is considered, there exist additional moments in the rotor-bearing system. In order to describe these moments, some researchers introduced the moments coefficients for a misaligned system [24; 25]. However, after evaluating the moments' coefficients, [PERSON] [26] and [PERSON] [27] pointed out that the moments' coefficients are generally insignificant and have little impact on the dynamic behavior of journal bearings. Further, the research of [PERSON] and [PERSON] [28] demonstrated that the moments' coefficients are much smaller than the force coefficients. Based on this evaluation, the moments' coefficients are neglected in the following research [29]. Therefore, the stability analysis for a misaligned rotor-bearing system in this study only considers the conventional eight dynamic coefficients. In addition, most previous research focused on deterministic analysis and ignored the inherent uncertainties in rotor-bearing systems caused by such factors as variations in wear and operating conditions. In such situations, the characteristics of the HDB, the dynamics response, and the stability of the rotor-bearing system all become uncertain due to the random nature of the input parameters. Hence, the effect of random parameters should also be taken into account to ensure the reliability of the system. In recent years, some uncertain quantification methods are introduced to evaluate the uncertainty in the rotor systems, such as interval methods [30; 31], fuzzy models [32], probabilistic techniques [33; 34], and hybrid approaches [35]. The selection of these methods depends on the available information of the uncertain system. In this study, it is assumed that the statistical information of the system has been obtained, which corresponds to the probabilistic approach and assumes aleatoric uncertainties. In this condition, several approaches have been developed to quantify the uncertainty, including sampling- and non-sampling-based methods. Monte-Carlo (MC) simulation is a conventional sampling-based method that addresses the stochastic problem. Generally, MC simulation can quantify the effect of uncertainty on dynamic response with considerable accuracy. However, this method converges only slowly and is thus rather inefficient, particularly for complex dynamic systems. In stability analysis, the computational costs of uncertainty quantification using MC simulation are acceptable when adopting simplified HDB models, such as IS and IL. Since these simplified models allow the analytical solution to be obtained for the characteristics of HDB, they can compensate for the high computational cost of MC simulation. Some related studies can be found in [36; 37]. The behavior of the general finite-length HDB is governed by the Reynolds equation and is usually solved numerically.