Analytical and Experimental Investigation of Nonlinear Dynamic Characteristics of Hydrodynamic Bearings for Oil Film Instability Detection

Abstract

Nonlinear vibration phenomena, such as oil whirl and oil whip, are common indicators of oil film instability in hydrodynamic bearings and are key signs of potential faults in rotating machinery. Excessive vibrations caused by oil film instability can accelerate bearing wear and lead to the failure of the rotating system. This paper presents a model for nonlinear dynamic coefficients, aimed at providing a quantitative approach for monitoring and predicting oil film instability. The impact of operational parameters and perturbation values on both linear and nonlinear stiffness and damping coefficients is investigated. Simulation results and experimental rotor vibration signals demonstrate that the nonlinear dynamic coefficient model effectively characterizes oil film instability and accurately predicts rotor trajectory, while traditional linear models are only applicable under low-speed and small-disturbance conditions. Compared to traditional analytical models and numerical solutions, the nonlinear dynamic coefficients have higher accuracy and efficiency and can reliably identify the onset frequency of oil film instability. This study clarifies the relationship between nonlinear dynamic coefficients and rotor dynamic response, laying a theoretical foundation for the monitoring and prediction of oil film instability.

1. Introduction

Journal bearings, as key components, are widely used in rotating machinery due to their simple structure, strong load capacity, and long service life. As a crucial component of rotor systems, the bearing’s dynamic characteristics directly affect the overall dynamic behaviors and safe operation of the system [1,2,3]. Oil film instability usually occurs under low eccentricity and a high rotational speed [4]. The violent vibration with a sudden increase in amplitude can easily lead to bearing wear and rotor rubbing. Although classical linearization models (such as the short bearing theory) can partially explain the oil film characteristics under steady-state conditions, it is difficult for them to accurately predict the transient instability process caused by the coupling and lag effects of time-varying bearing stiffness and damping. Therefore, accurately determining bearing dynamic coefficients, especially the nonlinear characteristics of oil film instability, is vital for the design and maintenance management of rotor systems [5].

The lubrication characteristics of hydrodynamic bearings were typically studied by directly integrating and solving the Reynolds equation, which incurred high computational costs. To address this issue, researchers introduced the concept of dynamic characteristic parameters. Lune [6] proposed a four-linear-stiffness-coefficient model by using the perturbation method, which has been widely applied. However, the linear dynamic coefficients are inadequate for analyzing and calculating the strongly nonlinear behaviors of hydrodynamic bearings [7]. To address this problem, scholars have initially proposed the concept of nonlinear dynamic characteristics. Czolczynsk [8] determined the nonlinear coefficients of gas bearings via the orbit method. Nevertheless, due to the intrinsic errors associated with the method, its application remains limited in practice. Nishimura et al. [9] explored rotor nonlinear behaviors supported by hydrodynamic bearings using the short bearing theory, considering unbalance and rotor configuration. Choy et al. [10] analytically expressed the nonlinear oil film force as an odd-power series of journal perturbation displacement, based on the short bearing approximation. Chu et al. [11] used a quasi-static nonlinear model to estimate the bearing’s dynamic coefficients. Adiletta et al. [12] applied the long bearing theory to correct approximate theories and analyze the effects of a wide range of parameters on the response of a rigid rotor. However, the short bearing model neglects axial pressure flow, whereas the infinitely long bearing model ignores end leakage effects, so the calculation results of these models all have errors to a certain extent.

The past decade has seen significant improvements in the computational methods for nonlinear dynamic models and nonlinear dynamic characteristic parameters. Andréas and Santiago [13] conducted experiments to evaluate the dynamic response of a large orbit motion with hydrodynamic bearings under a single-frequency excitation force. Meruane et al. [14] proposed a framework that accurately identified dynamic parameters and determined the nonlinear transient response under various operating conditions. Tian et al. [15] developed a finite element model of a TC rotor and calculated the nonlinear bearing forces. Ying et al. [16] compared rotor nonlinear response with and without base excitation at different rotational speeds. Dakel et al. [17] established an innovative Timoshenko beam-based finite element model to analyze dynamic behavior in rigidly supported mechanical systems. Weimin et al. [18] introduced a technique that utilizes partial derivatives to computationally derive the nonlinear dynamic coefficients of journal bearings without prior knowledge of the shaft orbit. Smolík et al. [19] numerically investigated the nonlinear behavior of bearings within the framework of multibody dynamic simulations of turbochargers. Dyk et al. [20] enhanced the analytical representation of oil film nonlinearity by employing a modified polynomial model. Chasalevris et al. [21] considered lemon-bore and partial-arc bearings as two main bearing profiles in the nonlinear impedance analysis. Building on thermohydrodynamic coupling principles, Li [22] formulated a novel computational model that elucidates the nonlinear dynamic behavior of turbocharger rotor–bearing systems. This foundational work has stimulated further investigations into the stability thresholds and complex nonlinear phenomena associated with floating ring bearing configurations under operational conditions [23,24,25].

As rotating machinery advances toward higher speeds and greater precision, the nonlinear behaviors and instability effects induced by oil film forces have gained considerable importance. Dyk and Rendl [26,27,28] introduced a new lubrication groove design and derived analytical expressions for dynamic coefficients, accurately assessing the nonlinear characteristics and stability threshold. The nonlinear force parametrization using the perturbation method paradigm was developed in References [29,30]. EI-Sayed et al. [31,32] conducted a bifurcation analysis using second-order nonlinear coefficients. However, the model showed errors at low Sommerfeld numbers due to the simplification of the oil film forces.

In summary, simplified linear stiffness and damping coefficients cannot evaluate complex nonlinear behaviors. The analytical solution based on the short bearing model is only applicable to full circular bearings with a length-to-diameter ratio far greater than 1, although the convergence speed is fast [33,34]. Numerical solutions for nonlinear oil film forces lead to problems of large computational load and difficult convergence [35]. In addition, although there have been a large number of studies on the nonlinear dynamics of rotor–bearing systems, research on applying nonlinear dynamic coefficients to predict the instability of oil films is still insufficient. Therefore, the main contributions of the proposed method in this paper are summarized as follows:

(1)

The proposed nonlinear dynamic coefficient model enables to evaluation of the nonlinear dynamic characteristics of the rotor–bearing system and solves the limitation of the linear model.

(2)

Compared to traditional analytical models and numerical solutions, the nonlinear stiffness and damping coefficient model significantly saves computation time while accurately expressing the transient nonlinear characteristics of journal bearings.

(3)

The simulation results and experiment signals both show that the nonlinear dynamic coefficient model can effectively characterize oil film instability and accurately predict the rotor trajectory.

The remainder of this paper is organized as follows: Section 2 presents the lubrication model for finite-length hydrodynamic bearings, deriving the analytical expressions for nonlinear stiffness and damping coefficients using the second-order perturbation theory. Section 3 validates the computational results of linear and nonlinear coefficients against the published data. Then, the shaft trajectory, oil whirl, and oil whip phenomena are investigated based on both linear and nonlinear coefficient models. Section 4 verified the theoretical analysis through the experimental displacement signals. Section 5 presents the conclusion of this study.

2. Methodology

2.1. Lubrication Model of Hydrodynamic Bearing

Figure 1 is a schematic diagram of the geometric structure and working principle of the full circular hydrodynamic bearing. $O_b, O_j$ denote the geometric centers of the bearing and journal, respectively; $R$ denotes the bearing radius and $c$ denotes the radial clearance; $e, \theta$ represent the eccentricity and attitude angle, respectively, and $\phi$ denotes the angle that the journal rotates.

Within the classical lubrication theory, the lubricant is assumed to be an incompressible Newtonian fluid undergoing laminar flow (the shear stress has a linear relationship with shear rate). However, under extremely high rotational speed conditions, the Reynolds number may exceed the critical value (Re > 2000), and the turbulence correction factor and inertia effect need to be introduced [36]. Under extremely high pressure and temperature, the characteristics of non-Newtonian fluids—as well as the viscosity–temperature and viscosity–pressure effects of lubricating oil—must be taken into consideration [37].

This paper focuses on the conventional working condition (Re < 2000), so the Newtonian fluid and laminar flow assumptions are adopted. The hydrodynamic within the oil film thickness is mathematically described by the Reynolds equation:

$${\displaystyle \frac{\partial }{\partial x}}({\displaystyle \frac{\rho h^3}{12\eta }}{\displaystyle \frac{\partial p}{\partial x}})+{\displaystyle \frac{\partial }{\partial z}}({\displaystyle \frac{\rho h^3}{12\eta }}{\displaystyle \frac{\partial p}{\partial z}})={\displaystyle \frac{\omega R}{2}}({\displaystyle \frac{\rho \partial h}{\partial x}})+{\displaystyle \frac{\rho \partial h}{\partial t}}$$

(1)

where $x, z$ are coordinate representations in the circumferential and axial directions; $\omega$ is the rotation frequency of the journal bearing; $\eta$ is the lubricant viscosity; and the oil film pressure $p$ is dependent on the oil film thickness $h$, which can be expressed as follows:

$$h(\phi ,z)=c\times (1+\epsilon \times \cos (\phi -\theta ))$$

(2)

We define the dimensionless forms of each variable:

$$H={\displaystyle \frac{h}{c}} \phi ={\displaystyle \frac{x}{R}} Z={\displaystyle \frac{z}{L}}(0\le Z\le 1) P={\displaystyle \frac{p}{p^{\ast }}}\left(p^{\ast }=\eta \omega R^2/c^2\right)$$

(3)

Then, the dimensionless form of Equation (1) can be written as follows:

$${\displaystyle \frac{\partial }{\partial \phi }}(H^3{\displaystyle \frac{\partial P}{\partial \phi }})+({\displaystyle \frac{R}{L}}{)}^2{\displaystyle \frac{\partial }{\partial Z}}(H^3{\displaystyle \frac{\partial P}{\partial Z}})=6p^{\ast }({\displaystyle \frac{\partial H}{\partial \phi }})$$

(4)

$H$ represents the dimensionless fluid film thickness. $P$ represents the dimensionless pressure. $\phi$ and $Z$ represent the dimensionless axial and vertical coordinates, respectively. ${\phi }_1$ denotes the oil film rupture’s start position and ${\phi }_2$ denotes the oil film rupture’s end position. The Reynolds boundary condition is used here with the pressure starting point at $P(\phi ={\phi }_1,Z)=0$ and the pressure endpoint at $P(\phi ={\phi }_2,Z)=0, {\displaystyle {\left.\frac{\partial P}{\partial \phi }\right|}_{\phi =\phi 2}}=0$.

By integrating the oil film pressure distribution over the load-bearing zone of hydrodynamic bearings, the hydrodynamic bearing forces can be mathematically derived. The vertical component $F_x$, axial component $F_y$, and resultant force $F$ from the direct integration of the Reynolds equation are as follows:

$$\left\{\begin{array}{l}{F}_X={\displaystyle {\int }_{-1}^1{\displaystyle {\int }_{{\phi }_1}^{{\phi }_2}P\sin \phi d\phi dZ}}\\ F_X={\displaystyle {\int }_{-1}^1{\displaystyle {\int }_{{\phi }_1}^{{\phi }_2}-P\cos \phi d\phi dZ}}\\ F=\sqrt{F_X^2+F_Y^2}\end{array}\right.$$

(5)

Since the Reynolds equation is an elliptic partial differential equation, obtaining an exact analytical solution is challenging. As a result, the finite difference method is commonly employed to solve the pressure distribution. Figure 2 illustrates that the circumferential direction of the entire oil film region is discretized into m points, and the axial direction is discretized into n points. The difference quotient of adjacent nodes is used to approximate the partial derivative values in different directions. The discrete form of Equation (4) is derived as follows:

$$\begin{array}{l}{\displaystyle \frac{\partial }{\partial \theta }}(H^3{\displaystyle \frac{\partial P}{\partial \theta }})={\displaystyle \frac{H_{i+0.5,j}^3(P_{i+1,j}-P_{i,j})-H_{i-0.5,j}^3(P_{i,j}-P_{i-1,j})}{\Delta {\theta }^2}}\\ ({\displaystyle \frac{R}{L}}{)}^2{\displaystyle \frac{\partial }{\partial Z}}(H^3{\displaystyle \frac{\partial P}{\partial Z}})=({\displaystyle \frac{R}{L}}{)}^2{\displaystyle \frac{H_{i,j+0.5}^3(P_{i,j+1}-P_{i,j})-H_{i,j-0.5}^3(P_{i,j}-P_{i-1,j})}{\Delta Z^2}}\\ 6{\displaystyle \frac{\partial (H)}{\partial \theta }}=6{\displaystyle \frac{H_{i,j}-H_{i-1,j}}{\Delta \theta }}\\ 12{\displaystyle \frac{\partial (H)}{\partial \tau }}=12{\displaystyle \frac{{(H)}^{\tau }-{(H)}^{\tau -1}}{\partial \tau }}\end{array}$$

(6)

Finally, the calculation formula in discrete form is as follows:

$$P_{i,j}={\displaystyle \frac{A{P}_{i-1,j}+B{P}_{i+1,j}+C{P}_{i,j-1}+D{P}_{i,j+1}-F_{i,j}}{E_{i,j}}}$$

(7)

The calculation coefficients $A,B,C,D,E,F$ are, respectively, expressed as follows:

$$\begin{array}{l}A=H_{i-0.5,j}^{3}B=H_{i+0.5,j}^3\\ C=({\displaystyle \frac{D}{L}}\cdot {\displaystyle \frac{\Delta \theta }{\Delta Z}}{)}^2{H}_{i,j-0.5}^{3}D=({\displaystyle \frac{D}{L}}\cdot {\displaystyle \frac{\Delta \theta }{\Delta Z}}{)}^2{H}_{i,j+0.5}^3\\ E=A+B+C+D\\ F=6\Delta \theta (H_{i+0.5}-H_{i+0.5})\end{array}$$

(8)

The coupled equations are solved using a relaxation iteration scheme to accelerate the convergence speed, and the relaxation factor is taken as 1–2, which is applicable for the selection of most lubrication issues [38]. The specific selected value is related to the mesh grid density and boundary conditions. The iterative process terminates when successive approximations satisfy the prescribed convergence tolerance, $\delta ={10}^{-6}$. To determine whether sufficient accuracy has been achieved in each iteration result, the following relative convergence criterion is adopted to determine whether to abort the iteration process:

$$P_{i,j}^k=\beta (P_{i,j}^k-P_{i,j}^{k-1})+P_{i,j}^{k-1}$$

(9)

$${\displaystyle \frac{{\displaystyle \sum _{i=2}^m{\displaystyle \sum _{j=2}^n\left|P_{i,j}^k-P_{i,j}^{k-1}\right|}}}{{\displaystyle \sum _{i=2}^m{\displaystyle \sum _{j=2}^n\left|P_{i,j}^k\right|}}}}\le \delta$$

(10)

2.2. Modeling of Nonlinear Stiffness and Damping Coefficients

The rotor equilibrium position under different operational conditions is determined as the premise for evaluating the lubrication condition. For the hydrodynamic lubrication condition, according to the small perturbation theory [39], the oil film thickness and hydrodynamic pressure are expressed as follows:

$$\begin{array}{l}h=h_0+\Delta x\sin \phi -\Delta y\cos \phi \\ p=p_0+\Delta p=p_0+{\displaystyle \frac{\partial p}{\partial x}}\Delta x+{\displaystyle \frac{\partial p}{\partial y}}\Delta y+{\displaystyle \frac{\partial p}{\partial x^{\prime }}}\Delta x^{\prime }+{\displaystyle \frac{\partial p}{\partial y^{\prime }}}\Delta y^{\prime }\end{array}$$

(11)

Substituting Equation (10) into Equation (4) yields the dimensionless perturbation expression for the Reynolds equation:

$$\begin{array}{l}{\displaystyle \frac{\partial }{\partial \phi }}(H^3{\displaystyle \frac{\partial P_i^{\prime }}{\partial \phi }})+({\displaystyle \frac{R}{L}}{)}^2{\displaystyle \frac{\partial }{\partial Z}}(H^3{\displaystyle \frac{\partial P_i^{\prime }}{\partial Z}})=\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\left\{\begin{array}{l}6\cos \phi -3{\displaystyle \frac{\sin \phi }{H}}{\displaystyle \frac{\partial H}{\partial \theta }}-H^3{\displaystyle \frac{\partial P_0}{\partial \phi }}{\displaystyle \frac{\partial }{\partial \phi }}({\displaystyle \frac{\sin \phi }{\partial \phi }})\hspace{1em}\hspace{1em}{P}_i^{\prime }={\displaystyle \frac{\partial P}{\partial X}}\\ 6\sin \phi -3{\displaystyle \frac{\cos \phi }{H}}{\displaystyle \frac{\partial H}{\partial \phi }}-H^3{\displaystyle \frac{\partial P_0}{\partial \phi }}{\displaystyle \frac{\partial }{\partial \phi }}({\displaystyle \frac{\cos \phi }{\partial \phi }})\hspace{1em}\hspace{1em}{P}_i^{\prime }={\displaystyle \frac{\partial P}{\partial Y}}\\ 12\sin \phi \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{P}_i^{\prime }={\displaystyle \frac{\partial P}{\partial X^{\prime }}}\\ 12\cos \phi \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{P}_i^{\prime }={\displaystyle \frac{\partial P}{\partial Y^{\prime }}}\end{array}\right\}\end{array}$$

(12)

Once the perturbation pressures are calculated, the linearized stiffness and damping coefficients are integrated by the following:

$$\begin{array}{l}\left.\begin{array}{l}{K}_{XX}\\ K_{YX}\end{array}\right\}=-{\displaystyle {\int }_0^1{\displaystyle {\int }_{{\phi }_1}^{{\phi }_2}\frac{\partial P}{\partial X}}}\left\{\begin{array}{l}\sin \phi \\ -\cos \phi \end{array}\right\}d\phi dZ,\left.\begin{array}{l}{K}_{XY}\\ K_{YY}\end{array}\right\}=-{\displaystyle {\int }_0^1{\displaystyle {\int }_{{\phi }_1}^{{\phi }_2}\frac{\partial P}{\partial Y}}}\left\{\begin{array}{l}\sin \phi \\ -\cos \phi \end{array}\right\}d\phi dZ\\ \left.\begin{array}{l}{C}_{XX}\\ C_{YX}\end{array}\right\}=-{\displaystyle {\int }_0^1{\displaystyle {\int }_{{\phi }_1}^{{\phi }_2}\frac{\partial P}{\partial X^{\prime }}}}\left\{\begin{array}{l}\sin \phi \\ -\cos \phi \end{array}\right\}Rd\phi dZ, \left.\begin{array}{r}{C}_{XY}\\ C_{YY}\end{array}\right\}=-{\displaystyle {\int }_0^1{\displaystyle {\int }_{{\phi }_1}^{{\phi }_2}\frac{\partial P}{\partial Y^{\prime }}}}\left\{\begin{array}{l}\sin \phi \\ -\cos \phi \end{array}\right\}Rd\phi dZ\end{array}$$

(13)

The eight stiffness and damping coefficients are calculated by integrating $P_i^{\prime }$ over the oil film region. The relationship between normalized values and real values is converted by $K_b=k_b/W, C_b=c_b*\omega *c/W$, respectively.

The linear coefficients are based on the first-order Taylor expansion. Correspondingly, the solution principle for the nonlinear coefficients is based on applying the second-order Taylor expansion to the rotor’s disturbance displacement and velocity at the equilibrium position.

The expression of the dimensionless pressure distribution P using the second-order expansion for the Taylor series is as follows:

$$\begin{array}{l}P=P_0+P_X\Delta X+P_Y\Delta Y+P_{X^{\prime }}\Delta X^{\prime }+P_{Y^{\prime }}\Delta Y^{\prime }+{\displaystyle \frac{1}{2}}{P}_{XX}\Delta X^2+{\displaystyle \frac{1}{2}}{P}_{YY}\Delta Y^2\\ +P_{XY}\Delta X\Delta Y+P_{YX}\Delta Y\Delta X+P_{X{X}^{\prime }}\Delta X\Delta X^{\prime }+P_{X{Y}^{\prime }}\Delta X\Delta Y^{\prime }++P_{Y{X}^{\prime }}\Delta Y\Delta X^{\prime }{P}_{Y{Y}^{\prime }}\Delta Y\Delta Y^{\prime }\end{array}$$

(14)

Substituting Equation (13) into Equation (4) yields the dimensionless perturbation expression for the Reynolds equation:

$$\begin{array}{l}{\displaystyle \frac{\partial }{\partial \phi }}(H^3{\displaystyle \frac{\partial P}{\partial \phi }})+{\displaystyle \frac{\partial }{\partial Z}}(H^3{\displaystyle \frac{\partial P}{\partial Z}})=\\ {\displaystyle \frac{\partial H_{XX}}{\partial \phi }}-{\displaystyle \frac{\partial }{\partial \phi }}[6H_0^2{H}_X{\displaystyle \frac{\partial P_X}{\partial \phi }}+(6H_0^2{H}_X+3H_0^2{H}_{XX}){\displaystyle \frac{\partial P_0}{\partial \phi }}]\\ -{\displaystyle \frac{\partial }{\partial Z}}[6H_0^2{H}_X{\displaystyle \frac{\partial P_X}{\partial Z}}+(6H_0^2{H}_X+3H_0^2{H}_{XX}){\displaystyle \frac{\partial P_0}{\partial Z}}],\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}P=P_{XX}\\ {\displaystyle \frac{\partial H_{YY}}{\partial \phi }}-{\displaystyle \frac{\partial }{\partial \phi }}[6H_0^2{H}_Y{\displaystyle \frac{\partial P_Y}{\partial \phi }}+(6H_0^2{H}_Y+3H_0^2{H}_{YY}){\displaystyle \frac{\partial P_0}{\partial \phi }}]\\ -{\displaystyle \frac{\partial }{\partial Z}}[6H_0^2{H}_Y{\displaystyle \frac{\partial P_Y}{\partial Z}}+(6H_0^2{H}_Y+3H_0^2{H}_{YY}){\displaystyle \frac{\partial P_0}{\partial Z}}],\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}P=P_{YY}\\ {\displaystyle \frac{\partial H_{XY}}{\partial \phi }}-{\displaystyle \frac{\partial }{\partial \phi }}[3H_0^2{H}_Y{\displaystyle \frac{\partial P_X}{\partial \phi }}+3H_0^2{H}_X{\displaystyle \frac{\partial P_Y}{\partial \phi }}+(6H_0{H}_X{H}_Y+3H_0^2{H}_{YY}){\displaystyle \frac{\partial P_0}{\partial \phi }}]\\ -{\displaystyle \frac{\partial }{\partial Z}}[3H_0^2{H}_Y{\displaystyle \frac{\partial P_X}{\partial Z}}+3H_0^2{H}_X{\displaystyle \frac{\partial P_Y}{\partial Z}}+(6H_0{H}_X{H}_Y+3H_0^2{H}_{YY}){\displaystyle \frac{\partial P_0}{\partial Z}}],\hspace{1em}\hspace{1em}\hspace{1em}P=P_{XY}\\ {\displaystyle \frac{\partial H_{YX}}{\partial \phi }}-{\displaystyle \frac{\partial }{\partial \phi }}[3H_0^2{H}_X{\displaystyle \frac{\partial P_Y}{\partial \phi }}+3H_0^2{H}_Y{\displaystyle \frac{\partial P_X}{\partial \phi }}+(6H_0{H}_Y{H}_X+3H_0^2{H}_{XX}){\displaystyle \frac{\partial P_0}{\partial \phi }}]\\ -{\displaystyle \frac{\partial }{\partial Z}}[3H_0^2{H}_X{\displaystyle \frac{\partial P_Y}{\partial Z}}+3H_0^2{H}_Y{\displaystyle \frac{\partial P_X}{\partial Z}}+(6H_0{H}_Y{H}_X+3H_0^2{H}_{XX}){\displaystyle \frac{\partial P_0}{\partial Z}}],\hspace{1em}\hspace{1em}\hspace{1em}P=P_{YX}\\ 2H_{XX}-{\displaystyle \frac{\partial }{\partial \phi }}(3H_0^2{H}_X{\displaystyle \frac{\partial P_{\dot{X}}}{\partial \phi }})-{\displaystyle \frac{\partial }{\partial Z}}(3H_0^2{H}_X{\displaystyle \frac{\partial P_{\dot{X}}}{\partial Z}}),\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}P=P_{X{X}^{\prime }}\\ 2H_{XX}-{\displaystyle \frac{\partial }{\partial \phi }}(3H_0^2{H}_X{\displaystyle \frac{\partial P_{\dot{Y}}}{\partial \phi }})-{\displaystyle \frac{\partial }{\partial Z}}(3H_0^2{H}_X{\displaystyle \frac{\partial P_{\dot{Y}}}{\partial Z}}),\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}P=P_{X{Y}^{\prime }}\\ 2H_{XY}-{\displaystyle \frac{\partial }{\partial \phi }}(3H_0^2{H}_Y{\displaystyle \frac{\partial P_{\dot{X}}}{\partial \phi }})-{\displaystyle \frac{\partial }{\partial Z}}(3H_0^2{H}_X{\displaystyle \frac{\partial P_{\dot{X}}}{\partial Z}}),\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}P=P_{Y{X}^{\prime }}\\ 2H_{YY}-{\displaystyle \frac{\partial }{\partial \phi }}(3H_0^2{H}_Y{\displaystyle \frac{\partial P_{\dot{Y}}}{\partial \phi }})-{\displaystyle \frac{\partial }{\partial Z}}(3H_0^2{H}_Y{\displaystyle \frac{\partial P_{\dot{Y}}}{\partial Z}}),\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=P_{Y{Y}^{\prime }}\end{array}$$

(15)

Each perturbation pressure $P_{XX},P_{YY},P_{XY},P_{YX},P_{X{X}^{\prime }},P_{X{Y}^{\prime }},P_{Y{X}^{\prime }},P_{Y{Y}^{\prime }}$ corresponds to the solution of each respective perturbation equation.

The nonlinear stiffness and damping coefficients are defined as the second-order approximation, ignoring all perturbation terms higher than the second order. Nonlinear stiffness is the second-order partial derivative of the reaction force with respect to displacement, and damping is the second-order partial derivative of the reaction force with respect to velocity. The detailed terms of each coefficient are as follows:

$$\begin{array}{l}{K}_{XXX}={\displaystyle \frac{{\partial }^2{F}_X}{\partial X^2}},K_{XYY}={\displaystyle \frac{{\partial }^2{F}_X}{\partial Y^2}},K_{XXY}={\displaystyle \frac{{\partial }^2{F}_X}{\partial X\partial Y^{\prime }}},K_{XYX}={\displaystyle \frac{{\partial }^2{F}_X}{\partial X\partial Y^{\prime }}}\\ K_{YYY}={\displaystyle \frac{{\partial }^2{F}_Y}{\partial Y^2}},K_{YXX}={\displaystyle \frac{{\partial }^2{F}_Y}{\partial X^2}},K_{YYX}={\displaystyle \frac{{\partial }^2{F}_Y}{\partial X\partial Y^{\prime }}},K_{YXY}={\displaystyle \frac{{\partial }^2{F}_Y}{\partial X\partial Y^{\prime }}}\\ C_{XXX}={\displaystyle \frac{{\partial }^2{F}_X}{\partial X\partial X^{\prime }}},C_{XXY}={\displaystyle \frac{{\partial }^2{F}_X}{\partial X\partial Y^{\prime }}},C_{XYX}={\displaystyle \frac{{\partial }^2{F}_Y}{\partial X\partial X^{\prime }}},C_{XYY}={\displaystyle \frac{{\partial }^2{F}_X}{\partial Y\partial Y^{\prime }}}\\ C_{YXX}={\displaystyle \frac{{\partial }^2{F}_Y}{\partial X\partial X^{\prime }}},C_{YXY}={\displaystyle \frac{{\partial }^2{F}_Y}{\partial X\partial Y^{\prime }}},C_{YYX}={\displaystyle \frac{{\partial }^2{F}_Y}{\partial Y\partial X^{\prime }}},C_{YYY}={\displaystyle \frac{{\partial }^2{F}_Y}{\partial Y\partial Y^{\prime }}}\end{array}$$

(16)

where $K_{IJO}=k_{ijo}{c}^2/F, C_{IJO}=c_{ijo}{c}^2\omega /F$ are the dimensionless forms of nonlinear coefficients, and $i,j,o$ correspond to $X,Y$, respectively.

The sixteen coefficients are derived by integrating the disturbed oil film pressure distribution:

$$\left\{\begin{array}{l}{K}_{X\zeta \eta }={\displaystyle {\int }_{-1}^1{\displaystyle {\int }_{{\phi }_1}^{{\phi }_2}-P_{\zeta \eta }\sin \phi d\phi dZ}}\\ K_{Y\zeta \eta }={\displaystyle {\int }_{-1}^1{\displaystyle {\int }_{{\phi }_1}^{{\phi }_2}-P_{\zeta \eta }\cos \phi d\phi dZ}}\\ C_{X\zeta \eta }={\displaystyle {\int }_{-1}^1{\displaystyle {\int }_{{\phi }_1}^{{\phi }_2}-P_{\zeta {\eta }^{\prime }}\sin \phi d\phi dZ}}\\ C_{Y\zeta \eta }={\displaystyle {\int }_{-1}^1{\displaystyle {\int }_{{\phi }_1}^{{\phi }_2}-P_{\zeta {\eta }^{\prime }}\sin \phi d\phi dZ}}\end{array}\right.$$

(17)

Substituting Equations (12) and (15) into Equation (17) yields the following expansion of the equivalent force $F_{\zeta }$ for the dimensionless oil film component force expressed by the stiffness and damping coefficient:

$$\begin{array}{l}{F}_{\zeta }=F_{\zeta 0}+K_{\zeta X}\Delta X+K_{\zeta Y}\Delta Y+C_{\zeta Y}\Delta X^{\prime }+C_{\zeta Y}\Delta Y^{\prime }+{\displaystyle \frac{1}{2}}{K}_{\zeta XX}\Delta X^2+{\displaystyle \frac{1}{2}}{K}_{\zeta YY}\Delta Y^2\\ +K_{\zeta XY}\Delta X\Delta Y+C_{\zeta XX}\Delta X\Delta X^{\prime }+C_{\zeta XY}\Delta X\Delta Y^{\prime }+C_{\zeta YX}\Delta Y\Delta X^{\prime }+C_{\zeta YY}\Delta Y\Delta Y^{\prime }\end{array}$$

(18)

2.3. Dynamic Model of Rotor Motion

To study the rotor dynamic response under the linear and nonlinear coefficient model, the dynamic model of the rotor–bearing system was established. As shown in Figure 3b, the equation of motion of the rotor–bearing system can be expressed as follows:

$$\left\{\begin{array}{l}M\ddot{X}=F_x+M{e}_u{\omega }^2\cos (\omega t)\\ M\ddot{Y}=F_y+M{e}_u{\omega }^2\sin (\omega t)-W\end{array}\right.$$

(19)

where $e_u,\omega ,t$ represent the unbalanced eccentricity, rotational speed of the journal, and time, respectively. $F_x,F_y$ denote the bearing forces in the respective directions, obtained using the linear and nonlinear dynamic coefficients, respectively. $W$ denotes the gravity and radial load in the vertical direction.

The rotor motion equation is iteratively solved using the Runge–Kutta numerical integration to determine the rotor’s position and velocity at each time step. A stable rotor vibration response is obtained through a certain number of iterations. The complete flow of the proposed nonlinear stiffness and damping coefficient calculation model, along with its validation method, is shown in Figure 4, with the specific simulation parameters listed in

Table 1. Simulation parameters of the rotor–bearing system.

Bearing Parameter	Specification	Rotor Parameter	Specification
Bearing length L (mm)	20	Shaft radius R1 (mm)	30
Bearing diameter D (mm)	20	Shaft length l (mm)	450
Radial clearance c (μm)	100	Density ρ (kg/m3)	7850
Viscosity η (Pa·s)	0.0135	Load W (N)	500
Sommerfeld number S	0.01~1	Eccentricity $\epsilon$	0.1~0.9

.

3. Simulation Results and Discussion

3.1. Perturbation Analysis

In this section, the effects of both linear and nonlinear parameters on the perturbation force are investigated. The linear and nonlinear coefficients are derived by applying perturbations of varying magnitudes. Three different magnitudes of displacement and velocity perturbations are considered to evaluate the variation in the bearing dynamic coefficients.

Generally speaking, a larger grid number results in higher calculation accuracy and significantly increases calculation time. Therefore, in the numerical calculation process, an appropriate number of grids should be selected to ensure high calculation efficiency and convergence accuracy. Figure 5 shows that the changing trend in the maximum oil film pressure slows down significantly, and the relative error is close to 0.1% when the number of circumferential grids is 440 and the number of axial grids is 200. Thus, a grid size of 440 × 200 is selected for the following numerical calculation.

The first perturbation value is $\Delta X=0.002,\Delta X^{\prime }=0.002,\Delta Y=0.002,\Delta Y^{\prime }=0.002$. The oil film force decreases with the increasing Sommerfeld number when $S<0.1$, while the oil film force increases with the increasing Sommerfeld number when $S>0.1$. The results are displayed in Figure 6, indicating that the evaluated perturbed forces are approximately the same between the linear and nonlinear models at low perturbations.

The second perturbation value is $\Delta X=0.02,\Delta X^{\prime }=0.02,\Delta Y=0.02,\Delta Y^{\prime }=0.02$. The variation trend of the perturbed force with the Sommerfeld number in Figure 7 is the same as that in Figure 6. However, there are small deviations in the absolute force values between the two methods at moderate perturbation levels. The forces based on second-order coefficients are higher than those based on first-order coefficients.

The third perturbation value is $\Delta X=0.2,\Delta X^{\prime }=0.2,\Delta Y=0.2,\Delta Y^{\prime }=0.2$, as illustrated in Figure 8. The absolute value of the perturbation reaction force of the oil film decreases with the Sommerfeld number when $S<0.1$. The perturbation force amplitude presents an increasing progression with operational speed as the Sommerfeld number increases when $S>0.1$, while the linear oil film force has a flat trend. As depicted in Figure 8a,b, the reaction forces evaluated using the two methods are already significantly different at high perturbations. Significant differences are found at both low and high Sommerfeld numbers. The absolute values of the forces based on the second-order coefficients are generally much higher than the absolute values of the forces based on the first-order coefficients when $S<0.1$ and $S>0.6$.

The analysis in this section shows a significant deviation between the forces evaluated based on linear and nonlinear coefficients at large perturbation values.

However, with the increase in the perturbation value, the difference in the reaction force calculated by the linear and nonlinear coefficient models gradually becomes larger with the intensified nonlinear effects of the oil film.

3.2. Results of Nonlinear Stiffness and Damping Coefficients

This section presents a comparison of the calculated linear and nonlinear stiffness-damping coefficients with the results from the published Ref. [29]. Figure 9 illustrates the eight first-order stiffness and damping coefficients. A total of 32 selected nonlinear coefficients were compared, and they were distributed in the following manner. The second-order stiffness and damping coefficients are plotted in Figure 10 and Figure 11. The solid line in the figure represents the results modeled in this paper, while the results of Ref. [29] are represented by the dashed line. $I,J,O$ correspond to $X$ and $Y$, respectively. In contrast, the second-order coefficients reveal significant discrepancies in the cross-coupled stiffness terms.

The simulation results in this paper show good consistency with the literature, regardless of the linear or nonlinear dynamic coefficients. Moreover, the steady-state nonlinear oil film force in Ref. [29] is based on the short bearing theory, which assumes that the axial length of the bearing is much smaller than its diameter (L/D ≪ 1), and the circumferential pressure gradient can be ignored. Therefore, the steady-state force of the finite-length bearing obtained through numerical integration in this paper is more accurate, especially for the second-order nonlinear cross-stiffness coefficients, which require higher convergence accuracy of the oil film force.

The variation trends of the linear and nonlinear dynamic coefficients with different eccentricities are described in Figure 12, Figure 13 and Figure 14. In Figure 12a, the absolute values of the linear stiffness and damping coefficient are high when the eccentricity is $e<0.2$ and $e>0.85$, and tend to zero near the eccentricity $e=0.5$. The absolute values of the nonlinear coefficients far exceed the linearized values when the eccentricity is $e<0.2$ and $e>0.85$, as illustrated in Figure 13 and Figure 14. In particular, the nonlinear coefficients are almost thousands of times higher than the linear results as the eccentricity approaches 0.05 and 0.95.

Under low eccentricity (representing an extremely high rotational speed), stiffness and damping coefficients demonstrate significant variations along the orbital path. Additionally, reduced eccentricity amplifies the nonlinear characteristics of the shaft’s orbital motion relative to high-eccentricity scenarios. Under the condition of high eccentricity (representing an extremely thin oil film), the offset between the shaft and the bearing is larger, and the amplitude of variation in the oil film characteristics increases significantly. The nonlinear dynamic coefficients change significantly, and the bearing’s nonlinear effects become very strong, which is of great significance for the subsequent solution of the dynamic motion equation.

3.3. Results of Nonlinear Motion Trajectory

A comparative evaluation of rotor-dynamic behavior and orbital motion characteristics is conducted in this section. When the journal bearing is subjected only to static loads in the vertical direction, the oil film force generated by the lubricant balances the external load. As the rotational speed increases, the eccentricity of the equilibrium position gradually reduces. Figure 15a illustrates the decrease in eccentricity from 0.95 to 0.17 as the rotational speed increases from 60 rpm to 3600 rpm. Figure 15b shows the polar plot of the bearing’s equilibrium positions at various rotational speeds, where the journal rotates with a small eccentricity near the bearing’s geometric center at 3600 rpm.

The shaft trajectory is calculated in Figure 16 and Figure 17. The rotor motion using linear coefficients only shows the change in the shaft center position with varying speeds at equilibrium positions, as shown in Figure 16a,b, without depicting the oil whirl features at different speeds. In contrast, the orbit calculated using nonlinear coefficients can represent the typical signatures of the oil whirl and whip, i.e., the shaft shows two nested rings or irregular shapes under real operating conditions in Figure 16c,d. When the rotational speed is below 20 Hz (i.e., when the eccentricity $e>0.4$), the shape of the shaft orbit calculated using linear stiffness and damping coefficients is similar to that calculated using nonlinear coefficients, and the corresponding equilibrium position orbit is elliptical. However, the shaft centerline position computed with nonlinear stiffness and damping coefficients is more accurate. When the rotational speed exceeds 20 Hz, the orbit computed with nonlinear coefficients significantly differs from the linear results. The shaft orbit in Figure 17 presents a more complex orbit shape, which is closer to the actual shaft centerline orbit at different speeds under real operating conditions.

The frequency spectra of rotor displacement responses are depicted in Figure 18 and Figure 19. At low rotation speeds of 6 Hz and 14 Hz, the rotor dynamic response contains only the rotational frequency, regardless of whether linear or nonlinear coefficients are used. However, when the rotational speed increases to higher speeds of 30 Hz and 50 Hz, the results of linear coefficients fail to represent sub-synchronous motion, i.e., oil whirl and oil whip, and there is no corresponding half frequency of the rotational speed in the spectrum. In contrast, the results of nonlinear coefficients can display the sub-synchronous whirl component at half the rotational frequency. In particular, the amplitude of sub-synchronous components becomes extremely large when the rotational speed is 50 Hz, which means the oil whip occurs.

Through the analysis in this section, it is evident that under fixed operating conditions, when the rotational speed exceeds a certain value (i.e., when the eccentricity is below a certain value), the rotor motion calculated by the nonlinear model can effectively represent the complex shaft centerline orbit at equilibrium positions corresponding to those speeds. The higher rotational speed implies greater disturbance and strong instability of the oil film. The nonlinear coefficient can predict and characterize the nonlinear characteristics in this case accurately. This capability addresses the complex nonlinear issues in hydrodynamic bearings that linear stiffness and damping coefficient models cannot capture.

It should be noted that the three-order dynamic coefficients have an impact near the critical instability threshold. Under extremely high eccentricity, the third-order coefficient has a significant influence on amplitude variation, and the effective stiffness of the system decreases, resulting in a reduced instability threshold. Moreover, high-order cross-coupling terms may excite subharmonic resonance in strongly nonlinear coupling systems, such as multi-span flexible rotors [40]. The second-order perturbation method adopted in this paper is able to characterize the main nonlinear behaviors, such as oil whirl and oil whip.

4. Experimental Verification

4.1. Test Rig Description

In this section, the simulation results are validated through experimental testing on a double-disk rotor–bearing system, as depicted in Figure 20. The test rig consists of two symmetrically mounted steel disks with a diameter of 12 mm, supported by two hydrodynamic journal bearings. The coupling is connected to a variable-frequency drive motor. High-precision laser sensors used to acquire displacement vibration (with a resolution of 0.1 mm) are fixed on brackets, measuring displacement in both the horizontal and vertical directions relative to the bearings.

The laser detectors transduce optical signals into analog voltages first, which undergo high-speed digitization via a 16-bit data acquisition card to acquire the displacement signals. During the experiment, the synchronous run-up tests are conducted from 1 Hz (60 rpm) to 60 Hz (3600 rpm) in 1 Hz increments, and the sampling rate is set as 10 kHz to ensure the validity of the high-frequency range. Transient stabilization periods of 30 s are enforced before data recording to eliminate transient vibration instability.

4.2. Experimental Data Analysis

The experimental waterfall plots in Figure 21 (horizontal/vertical axes) reveal enhanced spectral complexity relative to numerical predictions, exhibiting multi-order vibration signatures dominated by synchronous (1.0×) and sub-synchronous (0.5×) components alongside higher harmonics. These additional frequency excitations stem from nonlinear dynamic interactions induced by inherent system imperfections, including shaft coupling misalignment effects and bearing clearance-induced parametric instabilities, which are not fully captured in the idealized simulation model. From the waterfall diagram, the variation frequency components of the displacement signals with the increasing rotational speed can be clearly observed.

At very low rotational speeds, the frequency components of the rotor’s motion are primarily the rotational frequency and low-order harmonics. The oil whirl begins at around a rotational speed of 10 Hz. In this range, the rotor’s vortex amplitude is relatively small, and the primary energy remains in the form of synchronous vibrations. The system’s first-order critical rotational speed is approximately 26 Hz. The rotor vibrates intensely, and all frequency components other than the rotational frequency are suppressed. When the rotational speed continuously rises to twice the first-order critical rotational speed, oil whip occurs, and the vibration energy of the oil film instability is the dominant component of the whole system. The phenomenon of oil film instability confirms the calculation result of the nonlinear stiffness and damping coefficients.

Figure 22 and Figure 23 display the detailed time domain waves, the shaft trajectory, and the frequency component distribution. The results at the low rotational speeds of 6 Hz are described in Figure 22a–c, and are characterized by an elliptical orbit with the main frequency component corresponding to the rotational speed. The rotor already begins to exhibit oil whirl at rotational speeds of 15 Hz and 30 Hz, as shown in Figure 22d–f and Figure 23a–c, and the shaft orbit presents two nested circular ring shapes due to the existence of a half-rotational frequency characteristic of oil whirl. When the rotational speed is very high and the eccentricity is small, the shaft trajectories in the experimental results become very complex, as illustrated in Figure 23d–f. Oil whip already occurs when the rotational speed reaches 50 Hz. The experimental results align with the simulation results, showing that linear models are only suitable for low speeds, while nonlinear coefficients could accurately characterize nonlinear dynamic behaviors and predict the occurrence of oil film instability.

5. Conclusions

This study introduces a model for nonlinear dynamic coefficients using the second-order perturbation method. It examines the influence of various operational parameters on linear and nonlinear stiffness and damping coefficients. Finally, it evaluates the nonlinear dynamic response of a rotor–bearing system using both the linear and nonlinear coefficient models. It compares the predicted shaft trajectory and frequency spectrum with the experimental vibration signals. The conclusions of this paper are as follows:

(1)

Compared to the linear coefficients, the nonlinear dynamic coefficients can evaluate the drastic changes in the perturbation reaction force when the Sommerfeld number is very high. The deviation value between the nonlinear and linear coefficient models becomes more significant when the perturbation is large.

(2)

The nonlinear coefficient model is more accurate and can effectively capture complex rotor dynamic characteristics and complex motion trajectories under large perturbations and high speeds. The linear coefficients can only evaluate the rotor motion with small perturbations and low speeds.

(3)

The experimental results confirm that the nonlinear coefficients can predict and assess significant nonlinear phenomena, such as oil whirl and oil whip. Additionally, the nonlinear dynamic coefficients are effective at identifying the onset frequency of oil film instability.

In conclusion, the quantification of nonlinear dynamic coefficients provides an effective theoretical tool for revealing the instability mechanism of oil films in high-speed rotating machinery. Firstly, this approach can guide the optimal design of bearing geometric parameters (such as clearance and lubricant viscosity), thereby suppressing the risk of instability at the source. Moreover, such nonlinear characteristic parameters can be embedded in the rotor dynamics simulation platform to accurately predict transient vibration behavior and provide condition monitoring strategies by capturing fault signatures such as harmonic frequencies. Furthermore, the proposed model lays a theoretical foundation for active control technologies (such as magnetic bearing compensation and adaptive fuel supply systems) as well as for the reliability design of high-end equipment.