## 2 Structure and Model
### Mathematical Model  
To determine the pressure distribution in the gas film, the Reynolds equation is the basic governing equation, which is expressed as follows [24]:  
\[\frac{\partial}{\partial x}\bigg{(}ph^{3}\frac{\partial p}{\partial x}\bigg{)}+ \frac{\partial}{\partial z}\bigg{(}ph^{3}\frac{\partial p}{\partial z}\bigg{)}= 6\mu U_{0}\frac{\partial}{\partial x}(ph)+12\mu\frac{\partial(\text{ph})}{ \partial t} \tag{1}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Structural Parameters** & **Value** & **Unit** \\ \hline bearing length & 40 & mm \\ bearing radius & 20 & mm \\ film clearance & 20 & \(\mu\)m \\ groove depth & 20 & \(\mu\)m \\ groove angle & 60 & \(\circ\) \\ number of grooves & 4 & / \\ groove circumferential ratio & 0.5 & / \\ groove axial ratio & 0.5 & / \\ eccentricity ratio & 0.5 & / \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Structural parameters of herringbone grooved journal bearing. Figure 1: Schematic diagram of herringbone grooved journal bearing. (**a**) Three-dimensional structure; (**b**) expanded view; (**c**) cylindrical coordinate system. To simplify the calculation, it is assumed that the gas is isothermal ideal gas, and the gas rarefaction effect is ignored. The Reynolds number is calculated as follows [25]:  
\[\text{Re}=\frac{\rho r\omega c_{0}}{\mu} \tag{2}\]  
where \(\rho\) represents density, \(r\) represents radius, \(\omega\) represents rotational speed, \(c_{0}\) represents gas film clearance, and \(\mu\) represents dynamic viscosity. When the ambient pressure is 1.5 MPa and the rotational speed is 80,000 rpm, the Reynolds number of the gas film is only 152, and this flow pattern can be regarded as laminar flow. According to the coordinate system shown in Figure 1c, Equation (1) can be expressed as:  
\[\frac{1}{r^{2}}\frac{\partial}{\partial\theta}\bigg{(}ph^{3}\frac{\partial p}{ \partial\theta}\bigg{)}+\frac{\partial}{\partial z}\bigg{(}ph^{3}\frac{ \partial p}{\partial z}\bigg{)}=6\mu\omega\frac{\partial}{\partial\theta}(ph)+ 12\mu\frac{\partial(\text{ph})}{\partial t} \tag{3}\]  
The following dimensionless quantities are introduced:  
\[ \begin{array}{l}P=p/p_{a};\ x=r\theta;H=h/c_{0};\ H_{1}=h_{g}/c_{0};\\ Z=z/r;\ T=\upsilon t;\ \gamma=\frac{\upsilon}{\omega};\Lambda=\big{(}6\mu \omega r^{2}\big{)}/\big{(}p_{a}c_{0}^{2}\big{)}\end{array}  \tag{4}\]  
where \(c_{0}\) represents bearing clearance, \(\omega\) represents rotor speed, \(\upsilon\) represents whirl frequency, \(\mu\) represents gas dynamic viscosity, \(p_{a}\) represents ambient pressure, and \(\Lambda\) represents the bearing number. The dimensionless Reynolds equation is obtained as follows:  
\[\frac{\partial}{\partial\theta}\bigg{(}PH^{3}\frac{\partial P}{\partial\theta }\bigg{)}+\frac{\partial}{\partial Z}\bigg{(}PH^{3}\frac{\partial P}{\partial Z }\bigg{)}=\Lambda\frac{\partial(PH)}{\partial\theta}+2\Lambda\gamma\frac{ \partial(PH)}{\partial T} \tag{5}\]  
To solve the above equation, the linear perturbation method is used. This method was proposed by [PERSON] et al. and has been widely used and demonstrated to be very effective [26; 27]. Therefore, the linear perturbation method is used to analyze the characteristics of the journal bearings. Since the trajectory of the axis approximates an ellipse shape, a simple harmonic model is adopted. Dimensionless disturbances include the following:  
\[ \begin{array}{l}\text{Displacement disturbance:}\qquad\begin{cases}\Delta X =\frac{\Delta X}{c_{0}}=|\Delta X|\cdot e^{it}\\ \Delta Y=\frac{\Delta y}{c_{0}}=|\Delta Y|\cdot e^{it}\end{cases}\end{array}  \tag{6}\]  
\[ \begin{array}{l}\text{Velocity disturbance:}\qquad\begin{cases}\Delta\dot{X} =\frac{\partial\Delta X}{\partial T}=i\cdot\Delta X\\ \Delta\dot{Y}=\frac{\partial\Delta Y}{\partial T}=i\cdot\Delta Y\end{cases}\end{array}  \tag{7}\]  
\[ \begin{array}{l}\text{Acceleration disturbance:}\qquad\begin{cases}\Delta \ddot{X}=\frac{\partial\Delta\dot{X}}{\partial T}=-\Delta X\\ \Delta\ddot{Y}=\frac{\partial\Delta Y}{\partial T}=-\Delta Y\end{cases}\end{array}