# Dynamics Investigation on Axial-Groove Gas Bearing-Rotor System with Rod-Fastened Structure
## 3 Dynamic Model and Method of the Rod-Fastened Rotor-Gas Bearing System
### Newmark-\(\beta\) Method Based on Disturbance Compensation  
For solving the dynamic motion model of the rod-fastened rotor-gas bearing system efficiently, the modified Newmark-\(\beta\) method is employed by considering the disturbance compensation. The Newmark-\(\beta\) method is based on the average constant acceleration theory; that is, the acceleration is assumed to be a constant between \(\tilde{u}_{(i-1)\tau}\) and \(\tilde{u}_{i\tau}\). The integration constants of Newmark-\(\beta\) method are given in Table 1.  
For the equation of motion, the dynamic response equation at time step \(t=i\tau\) is expressed as follows:  
\[\mathbf{M}\tilde{u}_{i\tau}+\mathbf{K}\mathbf{u}_{i\tau}=\mathbf{P}_{i\tau} \tag{19}\]  
where \(\tilde{u}_{i\tau}\) and \(\mathbf{u}_{i\tau}\) denote the acceleration and displacement at time step \(i\tau\). By the average constant acceleration theory, the average constant acceleration \(\vec{\mathbf{u}}\) can be obtained as \(\tilde{\mathbf{u}}=\tilde{\mathbf{u}}_{(i-1)\tau}+\delta(\tilde{u}_{i\tau}-\tilde{u}_ {(i-1)\tau})\), which is between previous time step \((i-1)\tau\) and time step \(i\tau\) \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Value or Formula** \\ \hline \(\alpha\) & 0.25 \\ \(\delta\), \(\delta=2\alpha\) & 0.5 \\ \(a_{0}\) & \(1/\alpha\tau^{2}\) \\ \(a_{1}\) & \(\delta/\alpha\tau_{2}\) \\ \(a_{2}\) & \(1/\alpha\tau\) \\ \(a_{3}\) & \(1/2\alpha-1\) \\ \(a_{4}\) & \(\delta/\alpha-1\) \\ \(a_{5}\) & \((1/2\alpha-1)\) \\ \(a_{6}\) & \((1-\delta)\tau\) \\ \(a_{7}\) & \(\delta\tau\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Integration constant parameters of Newmark-\(\beta\) method. \(\ddot{\mathbf{u}}_{(i-1)\tau}\) denotes the acceleration at previous time step \((i-1)\tau\). \(\dot{\mathbf{u}}_{i\tau}\) denotes the acceleration at time \(i\tau\); \(\dot{\mathbf{u}}_{i\tau}\) and \(\mathbf{u}_{i\tau}\) can be written as:  
\[\mathbf{u}_{i\tau}=\mathbf{u}_{(i-1)\tau}+\tau\dot{\mathbf{u}}_{(i-1)\tau}+\frac{1}{2}\tau^{2 }\ddot{\mathbf{u}}=\mathbf{u}_{(i-1)\tau}+\tau\dot{\mathbf{u}}_{(i-1)\tau}+\frac{\tau^{2}}{ 2}\Big{[}(1-\delta)\ddot{\mathbf{u}}_{(i-1)\tau}+\delta\ddot{\mathbf{u}}_{i\tau}\Big{]} \tag{20}\]  
\[\ddot{\mathbf{u}}_{i\tau}=\dot{\mathbf{u}}_{(i-1)\tau}+\tau\ddot{\mathbf{u}}=\dot{\mathbf{u}}_{ (i-1)\tau}+\tau(1-\delta)\ddot{\mathbf{u}}_{(i-1)\tau}+\tau\delta\ddot{\mathbf{u}}_{i \tau}=\dot{\mathbf{u}}_{(i-1)\tau}+a_{\theta}\ddot{\mathbf{u}}_{(i-1)\tau}+a_{\gamma} \ddot{\mathbf{u}}_{i\tau} \tag{21}\]  
where \(\mathbf{u}_{(i-1)\tau}\) and \(\dot{\mathbf{u}}_{(i-1)\tau}\) are the velocity and displacement at the previous time step \((i-1)\tau\). According to Equation (20), \(\ddot{\mathbf{u}}_{i\tau}\) can be written as:  
\[\ddot{\mathbf{u}}_{i\tau}=\frac{1}{\alpha\tau^{2}}(\mathbf{u}_{i\tau}-\mathbf{u}_{(i-1) \tau})-\frac{1}{\alpha\tau}\dot{\mathbf{u}}_{(i-1)\tau}-(\frac{1}{2\alpha}-1) \ddot{\mathbf{u}}_{(i-1)\tau}=a_{1}(\mathbf{u}_{i\tau}-\mathbf{u}_{(i-1)\tau})-a_{4}\dot{ \mathbf{u}}_{(i-1)\tau}-a_{5}\ddot{\mathbf{u}}_{(i-1)\tau} \tag{22}\]  
By substituting Equations (21) and (22) into Equation (19), the equivalent motion Equation (23) can be obtained:  
\[\dot{\mathbf{K}}\mathbf{u}_{i\tau}=\hat{\mathbf{P}}_{i\tau} \tag{23}\]  
where \(\hat{\mathbf{K}}\) denotes the equivalent stiffness matrix, which does not change with time; \(\hat{\mathbf{P}}_{i\tau}\) denotes the equivalent dynamic resultant force matrix which is relevant to time. \[\hat{\mathbf{K}}=\mathbf{K}+a_{0}\mathbf{M} \tag{24}\]  
\[\hat{\mathbf{P}}_{i\tau}=\mathbf{P}_{i\tau}+\mathbf{M}\Big{[}a_{0}\mathbf{u}_{(i-1)\tau}+a_{ 2}\dot{\mathbf{u}}_{(i-1)\tau}+a_{3}\ddot{\mathbf{u}}_{(i-1)\tau}\Big{]} \tag{25}\]  
The displacement \(\mathbf{u}_{i\tau}\) at time \(i\tau\) can be obtained by solving Equation (23).