## 3 Governing Equations
### Dynamic Behavior Analysis of Rotor Using a Linear Model  
According to the simulation of the displacement of the rotor center in the form of limit cycle oscillatory disturbances as shown in Figure 6, the components of the rotor center motion in the direction of the coordinate axes can be expressed as follows [17]:  
\[\overline{X}^{\prime}=Re\Big{(}|\overline{X}^{\prime}|e^{j\gamma\tau}\Big{)} \hskip 56.905512 pt,\hskip 56.905512 pt\overline{Y}^{\prime}=Re\Big{(}|\overline{Y}^{ \prime}|e^{j\gamma\tau}\Big{)} \tag{10}\]  
Within this analytical framework, \(\overline{X}^{\prime}\) and \(\overline{Y}^{\prime}\) characterize the rotor center's oscillatory displacements along coordinate directions referenced to its equilibrium position, with \(|\overline{X}^{\prime}|\)  
Figure 5: Top view of textured surface with boundary and asymptotic lines for simulating cubic, cylindrical, and semi-ellipsoidal textures.  
Figure 6: Closed limit cycle perturbed motion of the rotor center in two-lobe noncircular journal bearing based on the linear dynamic model.  
and \(|\overrightarrow{Y}|\) quantifying their respective magnitudes. The operator Re extracts the real component of complex expressions, where \(\mathfrak{j}\) signifies the imaginary unit (\(\sqrt{-1}\)). The parameter \(\gamma\), known as the whirl frequency ratio, describes the relationship between the angular velocity of the perturbation motion of the shaft center around the static equilibrium point (\(\omega_{P}\)) and the rotor's spin velocity (\(\omega\)). The dynamic pressure fields \(\left(\overrightarrow{P}^{\prime}_{X},\overrightarrow{P}^{\prime}_{Y}\right)\) across the solution domain are obtained through Equation (11), derived by applying partial differentiation to the nondimensional Reynolds equation (Equation 1) with respect to \(\overline{X}^{\prime}\) and \(\overrightarrow{Y}^{\prime}\), followed by evaluation at null perturbation conditions.  
\[\frac{\partial}{\partial\theta}\left[\overline{h}_{0}^{3}\frac{\partial \overline{P}^{\prime}_{I}}{\partial\theta}\right]+\left(\frac{D}{L}\right)^{2 }\frac{\partial}{\partial\overline{z}}\left[\overline{h}_{0}^{3}\frac{ \partial\overline{P}^{\prime}_{I}}{\partial\overline{z}}\right]=\frac{ \partial}{\partial\theta}\left\{\left[\overline{3}\overline{h}_{0}^{2}\frac{ \partial\overline{P}_{0}}{\partial\theta}-6\mu_{rel}\right]f(\theta)\right\}+ \left(\frac{D}{L}\right)^{2}\frac{\partial}{\partial\overline{z}}\left[ \overline{3}\overline{h}_{0}^{2}f(\theta)\frac{\partial\overline{P}_{0}}{ \partial\overline{z}}\right]-12j\gamma\mu_{rel}f(\theta) \tag{11}\]  
The parameters \(\overrightarrow{P}^{\prime}_{I}\) and \(f(\theta)\) in the above equation are:  
\[\overrightarrow{P}^{\prime}_{I}=\overline{P}_{I}+j\gamma\overline{P}_{\hat{I} }\qquad,\qquad I=X,Y \tag{12}\]  
\[f(\theta)=\begin{cases}cos\theta&for\qquad\overline{P}^{\prime}_{X}\\ sin\theta&for\qquad\overline{P}^{\prime}_{X}\end{cases} \tag{13}\]  
In the present study, the finite element numerical method (FEM) based on the Galerkin framework with the isoparametric rectangular elements is used to solve the governing lubrication equation of two-lobe textured journal bearings [14].