## 1 Introduction  
With the development of rotating machines, the hydrodynamic bearing (HDB) has attracted increasing attention as a critical component in some rotating systems. The HDB provides elastic support to the rotor and reduces wear between journal and bearing shell. However, it can also be a cause of instability in the system. Instabilities induced by an HDB are generally referred to oil whirl and oil whirl [1; 2; 3]. These phenomena are self-excited vibrations, which often result in severe damage to the rotating system. Stability analysis is therefore crucial for maintaining safe operation of the rotor-bearing system. Since the concept of stability threshold was realized, several studies have been conducted to determine the stability threshold speed of a rotating system. The most direct method is to observe the trajectory of the journal at different rotation speeds by experiment, whereby the stability threshold is the speed at which the trajectory starts to diverge [4; 5]. Many theoretical methods have been developed on the basis of this realization. For instance, the Routh-Hurwitz criterion has been introduced to evaluate the stability threshold [6]. With this method, oil-film forces are linearized as functions of eight dynamic coefficients and coupled into the motion equation of the rotor. The stability of the system is then evaluated according to the Routh-Hurwitz stability array by considering coefficients of the characteristic equation. [PERSON] and [PERSON] [6] analyzed the whirl stability of a rotor-bearing system by using the Routh-Hurwitz criterion. They calculated the stability threshold by solving the eigenvalue problem of the rotating system. Since the real part of the eigenvalue represents the growth or decay of vibration, the stability threshold is determined when the real part of the eigenvalue is equal to zero. Based on this approach, [PERSON] and [PERSON] [7] predicted the stability threshold for an actively controlled HDB. [PERSON] et al. [8] discussed different HDB models in detail, including the Infinite Short (IS) model, the Infinite Long (IL) model, and the general finite length model, and then compared the stability threshold based on these models. Using a similar strategy, [PERSON] et al. [9] studied the stability threshold by way of the dynamic coefficients with no coordinate transformation. Another commonly used approach is known as time series analysis, often also referred to as transient simulation [10; 11; 12]. As with the experimental method, in time series analysis, the stability threshold is generally estimated as the rotation speed at which the shaft leaves the steady equilibrium position. [PERSON] et al. [3] numerically calculated the dynamic response of a rotor-bearing system during the run-up and run-down. The stability issue is determined according to the dynamic response, in which the HDB is assumed as short bearing and the bearing forces are considered as nonlinear. [PERSON] et al. [13] investigated both the stability threshold for a rigid rotor-bearing system using a numerical formulation and the impact of different HDB models. Furthermore, other nonlinear-theory-based methods have been employed to investigate the stability problem [14]. For instance,the numerical continuation method has been used to predict the stable and unstable limit cycles and Hopf bifurcation points [15; 16]. The Liapunov direct method [17; 18] or the Floquet and Bifurcation theories [19] have also been applied to analyze the stability threshold of a rotor-bearing system. In previous studies, researchers have assumed that the center line of the journal is parallel to that of the bearing and that the rotor can be considered rigid [8; 13]. Under these simplified conditions, the misalignment effect is neglected. However, in practical situations, the misalignment phenomenon is unavoidable for a number of reasons, such as deflection of the shaft, manufacturing error, or improper installation.