# 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
### Mathematical Method of Nonlinear Gas Film Force  
The differential transformation method [28; 29] has significant merits; rapid convergence can be achieved and the computational error can be reduced. The time is discretized in the nonlinear Reynolds equation by the differential transformation method. The gas film pressure \(P\) is a time-dependent function in the time domain, and the differential transform of the \(k\)th derivative of the gas film pressure \(P(t)\) can be expressed as:  
\[P_{t}(k)=\frac{T^{k}}{k!}\left[\frac{\partial^{k}P(t)}{\partial t^{k}}\right] _{t=0} \tag{3}\]  
where \(P_{t}(k)\) denotes the transform function in the transform domain. The differential inverse transform \(P_{t}(k)\) can be written as:  
\[P(t)=\sum_{k=0}^{m}P_{t}(k)\left(\frac{t}{T}\right)^{k}\text{, }0\leq t\leq T \tag{4}\]  
where \(T\) denotes the time gap. While the value \(P(0)\) (\(k=0\)) is known, \(P_{t}\) (\(0\)) can be obtained by Equation (4), and then the other discrete value \(P(k)\) can be obtained in the time interval \(T\)  
Figure 1: Schematic drawing of axial-groove gas bearing: (**a**) 3-axial grooved; (**b**) 2-axial grooved. By using the differential transformation method, the compressible Reynolds equation of the axial-groove gas bearing can be solved, and the transform with respect to the time domain \(\tau\) is made. Introducing \(S(k)=P\otimes P\), \(I(k)=H\otimes H\), and \(J(k)=H\otimes H\otimes H\), \(S(k)\) denotes the differential transform of \(P^{2}\). \(I(k)\) and \(J(k)\) denote the differential transforms of \(H^{2}\) and \(H^{3}\). The differential transformation of Equation (2) can be written as:  
\[ \begin{array}{c}3I\otimes\partial_{\varphi}H\otimes\partial_{\varphi}S+J \otimes\partial_{\varphi}^{2}S+3I\otimes\partial_{\lambda}H\otimes\partial_{ \lambda}S+J\otimes\partial_{\lambda}^{2}S=\\ 2\Lambda\partial_{\varphi}H\otimes P+2\Lambda\partial_{\varphi}P\otimes H+4 \Lambda\partial_{\tau}H\otimes P+4\Lambda\partial_{\tau}P\otimes H\end{array}  \tag{5}\]  
where \(\otimes\) denotes the convolution operator. \[\text{If}\left\{ \begin{array}{c}y(k)=af_{1}(t)\\ y(k)=f_{1}(t)f_{2}(t)\\ y(k)=f_{1}(t)f_{2}(t)f_{3}(t)\end{array} \right.\text{, then }\left\{ \begin{array}{c}Y(k)=a\sum\limits_{m=0}^{k}F_{1}(m)\\ Y(k)=\sum\limits_{m=0}^{k}F_{1}(m)F_{2}(k-m)\\ Y(k)=\sum\limits_{m=0}^{k}F_{1}(k-m)\sum\limits_{n=0}^{m}F_{2}(n)F_{3}(m-n) \end{array} \right.\]  
Equation (5) is discretized in the circumferential and axial directions by the central difference method. The discretized equation can be obtained:  
\[ \begin{array}{c}3\sum\limits_{m=0}^{k}I_{i,j}(k-m)\times\sum\limits_{n=0}^{ m}\Big{[}\Big{(}\frac{S_{i+1,j}(m)-S_{i-1,j}(n)}{2\Delta\varphi}\Big{)}\times( \frac{H_{i+1,j}(m-n)-H_{i-1,j}(m-n)}{2\Delta\varphi})\Big{]}+\\ \sum\limits_{m=0}^{k}\bigg{[}\bigg{(}\frac{S_{i+1,j}(m)-2S_{i,j}(m)+S_{i-1,j}( m)}{(\Delta\varphi)^{2}}\bigg{)}I_{i,j}(k-m)\bigg{]}+\\ 3\sum\limits_{m=0}^{k}I_{i,j}(k-m)\sum\limits_{n=0}^{m}\Big{[}\Big{(}\frac{S_{ i,j+1}(n)-S_{i,j-1}(n)}{2\Delta\lambda}\Big{)}\times\Big{(}\frac{H_{i,j+1}(m-n)-H_{i,j- 1}(m-n)}{2\Delta\lambda}\Big{)}\Big{]}\\ +\sum\limits_{m=0}^{k}\bigg{[}\bigg{(}\frac{S_{i,j+1}(m)-2S_{i,j}(m)+S_{i,j-1}( m)}{(\Delta\lambda)^{2}}\bigg{)}I_{i,j}(k-m)\bigg{]}=2\Lambda\sum\limits_{m=0}^{k} \Big{[}P_{i,j}(m)\Big{(}\frac{H_{i+1,j}(k-m)-H_{i-1,j}(k-m)}{2\Delta\varphi} \Big{)}\Big{]}+\\ 2\Delta\sum\limits_{m=0}^{k}\Big{[}H_{i,j}(m)\Big{(}\frac{P_{i+1,j}(k-m)-P_{i- 1,j}(k-m)}{2\Delta\varphi}\Big{)}\Big{]}+4\Lambda\sum\limits_{m=0}^{k}\Big{[} \Big{(}\frac{m+1}{\Delta\tau}\Big{)}H_{i,j}(k-m)P_{i,j}(m+1)\Big{]}+\\ 4\Lambda\sum\limits_{m=0}^{k}\Big{[}\Big{(}\frac{m+1}{\Delta\tau}\Big{)}P_{i,j }(k-m)H_{i,j}(m+1)\Big{]}\end{array}