# 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
\[\mathbf{u}_{i\tau}=(\mathbf{K}+a_{0}\mathbf{M})^{-1}\Big{\{}\mathbf{P}_{i\tau}+\mathbf{M}\Big{[}a_ {0}\mathbf{u}_{(i-1)\tau}+a_{2}\dot{\mathbf{u}}_{(i-1)\tau}+a_{3}\ddot{\mathbf{u}}_{(i-1) \tau}\Big{]}\Big{\}} \tag{26}\]  
According to the constrained condition of motion, the \(\ddot{\mathbf{u}}_{i\tau}\), \(\dot{\mathbf{u}}_{i\tau}\), and \(\mathbf{u}_{i\tau}\) are obtained by the Newmark-\(\beta\) method, but according to the dynamic motion equation, the actual acceleration \(\ddot{\mathbf{u}}_{i\tau}\) is obtained by substituting \(\dot{\mathbf{u}}_{i\tau}\) and \(\mathbf{u}_{i\tau}\) into Equation (19). \[\ddot{\mathbf{u}}_{i\tau}=\mathbf{M}^{-1}(\mathbf{P}_{i\tau}-\mathbf{K}\mathbf{u}_{i\tau}) \tag{27}\]  
It can be seen that an increment of acceleration \(\Delta\ddot{\mathbf{u}}_{i\tau}\) at time step \(i\tau\) is produced as Equation (28), which produces a disturbance error at each time step. By substituting \(\ddot{\mathbf{u}}_{i\tau}=\ddot{\mathbf{u}}_{i\tau}-\Delta\ddot{\mathbf{u}}_{i\tau}\) into Equation (27), Equation (29) can be obtained; the increment of acceleration gives rise to an increment of dynamic resultant force \(\Delta\mathbf{P}_{i\tau}=M\Delta\ddot{\mathbf{u}}_{i\tau}\), which is the disturbance increment produced by the increment of acceleration \(\Delta\ddot{\mathbf{u}}_{i\tau}\). \[\Delta\ddot{\ddot{\mathbf{u}}}_{i\tau}=\ddot{\mathbf{u}}_{i\tau}-\ddot{\ddot{\mathbf{u}}}_ {i\tau} \tag{28}\]  
\[\mathbf{u}_{i\tau}=\mathbf{u}_{(i-1)\tau}+\tau\dot{\mathbf{u}}_{(i-1)\tau}+\frac{1}{2}\tau^{2 }\ddot{\mathbf{u}}=\mathbf{u}_{(i-1)\tau}+\tau\dot{\mathbf{u}}_{(i-1)\tau}+\frac{\tau^{2}}{ 2}\Big{[}(1-\delta)\ddot{\mathbf{u}}_{(i-1)\tau}+\delta\ddot{\mathbf{u}}_{i\tau}\Big{]} \tag{20}\]0  
For acquiring the displacements, velocities, and accelerations at time step \(t=i\tau\), which satisfy the constraint condition of motion and dynamic motion equation simultaneously, the increment of dynamic resultant force needs to be eliminated by the following process. According to the motion equation, an increment motion equation is established; \(\Delta\ddot{\mathbf{u}}_{i\tau}\), \(\Delta\ddot{\mathbf{u}}_{i\tau}\) and \(\Delta\mathbf{u}_{i\tau}\) are the displacement increment, velocity increment and acceleration increment caused by \(-\Delta\mathbf{P}_{i\tau}\), respectively, which can be obtained by Equation (30):  
\[\mathbf{u}_{i\tau}=\mathbf{u}_{(i-1)\tau}+\tau\dot{\mathbf{u}}_{(i-1)\tau}+\frac{1}{2}\tau^{2 }\ddot{\mathbf{u}}=\mathbf{u}_{(i-1)\tau}+\tau\dot{\mathbf{u}}_{(i-1)\tau}+\frac{\tau^{2}}{ 2}\Big{[}(1-\delta)\ddot{\mathbf{u}}_{(i-1)\tau}+\delta\ddot{\mathbf{u}}_{i\tau}\Big{]} \tag{20}\]1\[\left\{ \begin{array}{l}\Delta\mathbf{u}_{i\tau}=-\hat{\mathbf{K}}^{-1}\Delta\mathbf{P}_{i\tau} \\ \Delta\dot{\mathbf{u}}_{i\tau}=a_{1}\Delta\mathbf{u}_{i\tau}\\ \Delta\ddot{\mathbf{u}}_{i\tau}=a_{0}\Delta\mathbf{u}_{i\tau}\end{array} \right. \tag{31}\]  
The displacement, velocity, and acceleration can be obtained by adding Equations (29) and (30), which are considering the incremental compensation. \[\left\{ \begin{array}{l}\dot{\mathbf{u}}_{i\tau}{}^{*}=\mathbf{u}_{i\tau}+\Delta\mathbf{u }_{i\tau}\\ \dot{\mathbf{u}}_{i\tau}{}^{*}=\dot{\mathbf{u}}_{i\tau}+\Delta\dot{\mathbf{u}}_{i\tau}\\ \ddot{\mathbf{u}}_{i\tau}{}^{*}=\ddot{\mathbf{u}}_{i\tau}+\Delta\ddot{\mathbf{u}}_{i\tau} \end{array} \right. \tag{32}\]  
The displacement \(\mathbf{u}_{i\tau}{}^{*}\), velocity \(\dot{\mathbf{u}}_{i\tau}{}^{*}\) and acceleration \(\ddot{\mathbf{u}}_{i\tau}{}^{*}\) are the accurate solution of the motion equation, which take the incremental compensation into consideration in Newmark-\(\beta\) method. The solutions conform to the dynamic motion equation, meanwhile, satisfy the motion constraint condition of the Newmark-\(\beta\) method.