## 2 Governing Equations and Dynamic Coefficients  
### Reynolds Equation  
The general HDB under consideration and the associated terminology are presented in Figure 1. A thin oil-film is sandwiched between the bearing shell and the rotating journal. Due to the relative motion of the two parts, pressure is generated in the oil-film that allows the HDB to provide support to the journal. Here, the support is commonly referred to as the bearing force. Due to the bearing forces and applied external load, the journal center \(O_{j}\) displays an eccentricity \(e\) with respect to the bearing center \(O_{b}\). The position of the journal is usually determined by two coordinate systems, \((\zeta,\eta)\) and \((x,y)\), of which the former describes the journal center in terms of its eccentricity displacement and angle, and and the latter are the fixed horizontal and vertical coordinates. The pressure of the oil-film is governed by the Reynolds equation derived from the Navier-Stokes equations [44], in which the lubricant oil is assumed to be a laminar, isothermal, and incompressible flow. The Reynolds equation is stated as in [45]  
\[\frac{1}{R^{2}}\frac{\partial}{\partial\Phi}\bigg{(}\frac{h^{3}}{12\mu}\frac{ \partial p}{\partial\Phi}\bigg{)}+\frac{\partial}{\partial z}\bigg{(}\frac{h^ {3}}{12\mu}\frac{\partial p}{\partial z}\bigg{)}=\frac{1}{2}\Omega\frac{ \partial h}{\partial\Phi}+\dot{y}\cos\Phi+\dot{x}\sin\Phi, \tag{1}\]  
where \(p\) represents the oil-film pressure; \(R\) stands for the radius of the bearing; \(\mu\) is the viscosity of the lubricant oil; \(\Phi\) and \(z\) represent the coordinates in circumferential and axial directions, respectively; \(\Omega\) is the rotation speed of the rotor; \(\dot{x}\) and \(\dot{y}\) are the velocity components of the rotating journal; and \(h\) denotes the thickness of the oil-film. According to the generally accepted Reynolds boundary conditions [46], the boundary of the oil-film begins approximately at the maximum film thickness and ends at the position where the pressure equals zero. In general, the thickness of the oil-film in a bearing without considering the misalignment effect is expressed approximately as [20]  
\[h=c+e\cos(\Phi-\Phi_{0}), \tag{2}\]  
in which \(c\) stands for the clearance; \(e\) represents the eccentricity; and \(\Phi_{0}\) denotes the attitude angle, which is used to depict the angle between the direction of the static load and the eccentricity line at equilibrium position. When the misalignment effect is considered, Equation (2) is no longer appropriate for describing the thickness of the oil-film since the thickness in axial direction has changed. The thickness is then given by introducing two misaligned angles as [20; 47; 48; 49; 23]  
\[h=c+e_{0}\text{cos}(\Phi-\Phi_{0})+\text{tan}\gamma\bigg{(}y-\frac{L}{2}\bigg{)} \text{cos}(\Phi-\alpha-\Phi_{0}), \tag{3}\]  
where \(e_{0}\) is the eccentricity at mid-plane, and \(\alpha\) and \(\gamma\) are misaligned angles of rotor in the \(x\)-\(y\) plane and \(y\)-\(z\) planes, respectively, as shown in Figure 2.  
The following dimensionless parameters are introduced to simplify computation  
\[H=\frac{h}{c},\quad\varepsilon=\frac{e}{c},\quad Z=\frac{z}{R},\quad P=\frac{ p}{6\mu\Omega}\Big{(}\frac{c}{R}\Big{)}^{2},\quad\dot{Y}=\frac{\dot{y}}{c \Omega},\quad\dot{X}=\frac{\dot{x}}{c\Omega}, \tag{4}\]  
where \(\varepsilon\) is named as the eccentricity ratio. The reformulated Reynolds equation is rewritten by considering the dimensionless parameters as  
\[\frac{\partial}{\partial\Phi}\bigg{(}H^{3}\frac{\partial P}{\partial\Phi} \bigg{)}+\frac{\partial}{\partial Z}\bigg{(}H^{3}\frac{\partial P}{\partial Z }\bigg{)}=\frac{\partial H}{\partial\Phi}+2\big{(}\dot{Y}\text{cos}\Phi+\dot {X}\text{sin}\Phi\big{)}. \tag{5}\]  
Figure 1: General HDB and associated terminology. The finite difference method (FDM) is adopted to solve the pressure of the lubricant oil. An over-relaxation iteration technique is employed to achieve good convergence of the solution [46].