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 whip [
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. Abdel and Nasar [
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, Rho and Kim [
7] predicted the stability threshold for an actively controlled HDB. Dyk 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, Huang 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. Castro 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. Smolík 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. 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. Sun and Changlin [
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. Zhang 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. Ebrat et al. [
22] and Xu 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, Mukherjee [
26] and Rao [
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 Ahmend and El-Shafei [
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. Uncertainty quantification using MC simulation would, in this case, become computationally very expensive. To achieve better computational efficiency, various non-sampling-based methods, such as generalized polynomial chaos (gPC) expansion, have received much attention in recent years [
38,
39,
40]. In the gPC expansion method, the uncertain parameters are represented by a series of orthogonal polynomials with unknown coefficients [
41,
42]. Recently, Garoli et al. [
43] demonstrated the application of the gPC expansion in a stochastic dynamic response analysis of a nonlinear rotor-bearing system, in which the nonlinear bearing forces are obtained using the IS model.
The literature review reveals that researchers have, for the most part, conducted deterministic stability analyses of rotor-bearing systems. However, a few studies are available that investigate the issue of uncertain stability on the basis of a simplified HDB model using a sampling-based method, such as MC simulation, but without considering the misalignment effect. This study reports an uncertain stability analysis for a rotor system supported by finite-length HDBs while taking into account the misalignment effect. We develop a stability analysis framework guided by the Routh–Hurwitz criterion. The characteristics of the HDB used for stability analysis are evaluated by solving the Reynolds equation numerically. Collocation-based gPC expansion is employed to determine the impact of uncertainties.
The paper is organized as follows: The next section introduces the theoretical background on characteristics of the general finite-length HDB. The process of determining the stability threshold is presented in
Section 3, and the uncertain stability analysis based on the gPC expansion is presented in
Section 4.
Section 5 presents the numerical results of a rotor-bearing system under uncertainty. The paper’s conclusion then follows.
6. Conclusions
This paper has presented a stability analysis of a rotor-bearing system taking into consideration the misalignment effect and uncertainty in a finite-length HDB. Determination of the stability threshold is guided by the Routh–Hurwitz method. The gPC expansion is used to quantify the impact of uncertainty. The characteristics of the HDB used to determine the stability threshold are calculated by numerical solving of the governing equations. The accuracy of the results is verified by comparison with those published in previous papers. The misalignment effect is investigated by introducing two misaligned angles in the expression of thickness of the oil-film. The results show that misalignment has a remarkable effect on the characteristics of the HDB and the run-up curve. The impact of misalignment on the borderline is negligible for a small eccentricity ratio, but evident for a larger eccentricity ratio. Moreover, the stability threshold increases as the misaligned angles increase. To study the influence of uncertainties on the stability threshold, clearance and viscosity are assumed as random parameters with predefined distribution. Using the collocation method, the gPC expansion is constructed by solving the responses at a series of specific collocation points using the deterministic code. MC simulation is implemented as a validation. The results show that the stability threshold becomes uncertain since the random input parameters can affect the static and dynamic characteristics of the system. By comparing the evaluated statistical properties of random outputs, it is apparent that gPC expansion is able to approximate the random parameters with reasonable accuracy. The results also illustrate that gPC expansion is more efficient, since it requires far fewer sample realizations.
In this study, the bearing force is considered linear and an analytical rotor is employed for the stability analysis. These are the main limitations of this study, which might limit its significance for direct industrial application. However, according to the previous studies, it is suggested that the developed framework is suitable to understand the performance of the rotor-bearing system. In addition, the interdisciplinary approach of combining stability analysis, misalignment effect, and uncertainty analysis using a gPC approach builds a bridge between these subjects. In the future, the nonlinear behavior of physical quantities and a complex rotor model will be simulated and investigated experimentally to close the gap between the purely numerical approach and a real physical model.