# Dynamics Investigation on Axial-Groove Gas Bearing-Rotor System with Rod-Fastened Structure
## 2 Kinematic Model and Mathematical Method of Three-Axial-Groove Gas Bearing  
### Kinematic Model  
The axial-groove gas bearing and the coordinate system are shown in Figure 1. Axial grooves are distributed uniformly in the circumferential direction. \(xO_{y}y\) denotes the bearing coordinate system; \(O_{b}\) denotes the bearing center of the axial-groove gas bearing; \(O_{i}\) denotes the journal center; \(\phi\) denotes the angle which is from the negative \(y\) direction to the gas film position along the rotating direction; \(e\) denotes the eccentricity of the gas bearing; \(\omega\) denotes the rotation angular speed; \(R_{b}\) denotes the gas bearing radius; \(c\) denotes the radial clearance; \(\alpha\) denotes the angle of groove width; \(\alpha_{0}\) denotes the angle from leading edge of the pad to the negative \(y\) direction; \(\beta\) denotes the arc bushing angle of the single pad; and \(F_{x}\) and \(F_{y}\) denote the gas film forces of the coordinate system \(xO_{b}y\) in the \(x\) and \(y\) directions.  
In order to obtain the gas film forces \(F_{x}\), \(F_{y}\) of the axial-groove gas bearing, the gas pressure \(p\) needs to be calculated by solving the compressible Reynolds equation. Reynolds equation is the theoretical basis of fluid lubricated bearing research, which depicts the relationship between the gas pressure and gas thickness. Considering a gas journal bearing, the compressible lubrication Reynolds equation is:  
\[\partial_{x}\left(pl^{3}\partial_{x}p\right)+\partial_{z}\left(pl^{3} \partial_{z}p\right)=6\mu U\partial_{x}(ph)+12\mu\partial_{t}(ph) \tag{1}\]  
where \(p\) denotes the gas film pressure in the coordinate system; \(h\) denotes the gas film thickness, for the \(i\)th pad, \(h_{i}=1+e_{i}\text{cos}(\phi-\theta_{i})\); \(z\) denotes the coordinate in the axial direction; \(x\) denotes the angle from the line of centers \(O_{j}\) and \(O_{b}\) to the gas film current position in the circumferential direction; \(\mu\) denotes the average velocity of the gas; and \(U\) denotes the component of velocity in the \(x\) direction.  
The dimensionless procedure is applied to Equation (1) by using the following transformational relations: \(p=P\times Pa\), \(h=H\times c\), \(z=\lambda\times R_{b}\), \(x=\varphi\times R_{b}\), \(\tau=t\times\omega\), \(U=\omega\times R\). A unified equation can be then obtained as follows:  
\[\partial_{\varphi}\left(PH^{3}\partial_{\varphi}P\right)+\partial_{\lambda} \left(PH^{3}\partial_{\lambda}P\right)=\Lambda\partial_{\varphi}(PH)+2\Lambda \partial_{\tau}(PH) \tag{2}\]  
where \(\Lambda=6\mu\omega R^{2}/(Pa^{2})\) and denotes the bearing number of the gas bearing.  
For Equation (2), the boundary conditions are:  
1. The gas film pressures at the both ends of bearing: \(P\left(\varphi,\,B_{b}/2R_{b}\right)=P\left(\varphi,-B_{b}/2R_{b}\right)=1\);
2. The gas film pressures at the leading edge (\(\varphi_{i1}\)) and trailing edge (\(\varphi_{i2}\)) of single pad in the axial direction: \(P\left(\varphi_{1},\,\lambda\right)=P\left(\varphi_{2},\,\lambda\right)=1\).  
By using the differential transformation method, the dimensionless Reynolds equation is solved efficiently.