# 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  
### Dynamic Motion of the Rod-Fastened Rotor-Gas Bearing System  
The rod-fastened rotor-gas bearing system is depicted in Figure 3. The rod-fastened rotor consists of shafts, disks, and gas bearings. Two rigid disks are bolted by rods, which are circumferentially distributed on the disk. The mass and damping of the rod are ignored when it is equivalent to a nonlinear stiffness spring structure. The rod-fastened rotor is supported by two- and three-axial-groove gas bearings, and the mass of the axial-groove gas bearing is not considered. In Figure 3a, \(O_{1}\) and \(O_{2}\) denote the journal centers at the left- and right-bearing stations, respectively; \(m_{1}\) and \(m_{2}\) denote the lumped masses of the left and right segments, respectively; \(l_{1}\) and \(l_{2}\) denote the lengths of the left and right segments, respectively; \(O_{d1}\) and \(O_{d2}\) denote the centers of the rigid disks, respectively; and \(m_{d1}\) and \(m_{d2}\) denote the masses of two rigid disks, respectively. Figure 3b shows the circumferential distribution of the rods; \(R_{d}\) denotes the radius of rigid disks. Figure 2: The calculation flow chart of the differential transform method. The dynamic motion model of the rod-fastened rotor system with two- and three-axial-groove gas bearings support follows:  
\[\mathbf{M}\ddot{\mathbf{u}}+\mathbf{K}\mathbf{u}=\mathbf{P} \tag{14}\]  
where \(\mathbf{u}=[x_{1},\,y_{1},\,x_{2},\,y_{2},\,x_{d1},\,y_{d1},\,x_{d2},\,y_{d2}]^{T}\); \(\mathbf{M}\) denotes the mass matrix of the dynamic motion; \(\mathbf{K}\) denotes the stiffness matrix; and \(\mathbf{P}\) denotes the dynamic resultant force matrix. \[\mathbf{M}=\left[ \begin{array}{cccccccc}m_{1}&0&0&0&0&0&0&0\\ 0&m_{1}&0&0&0&0&0&0\\ 0&0&m_{2}&0&0&0&0\\ 0&0&0&m_{2}&0&0&0&0\\ 0&0&0&0&m_{d1}&0&0\\ 0&0&0&0&0&m_{d1}&0&0\\ 0&0&0&0&0&m_{d2}&0\\ 0&0&0&0&0&0&m_{d2}\end{array} \right] \tag{15}\]  
\[\mathbf{K}=\left[ \begin{array}{cccccccc}k_{1}&0&0&0&-k_{1}&0&0&0\\ 0&k_{1}&0&0&0&-k_{1}&0&0\\ 0&0&k_{2}&0&0&0&-k_{2}&0\\ 0&0&0&k_{2}&0&0&0&-k_{2}\\ -k_{1}&0&0&0&k_{1}-k_{b}&0&k_{b}&0\\ 0&-k_{1}&0&0&0&k_{1}-k_{b}&0&k_{b}\\ 0&0&-k_{2}&0&-k_{b}&0&k_{2}+k_{b}&0\\ 0&0&0&-k_{2}&0&-k_{b}&0&k_{2}+k_{b}\end{array} \right] \tag{16}\]  
Figure 3: The schematic drawing of the rod-fastened rotor system supported by two- and three-axial-groove gas bearings: (**a**) schematic drawing of the rod-fastened rotor; (**b**) the circumferential distribution of rods. \[\mathbf{P}=\left[ \begin{array}{c}-f_{x1}\\ -f_{y1}+m_{1}g\\ -f_{x2}\\ -f_{y2}+m_{2}g\\ f_{rx1}+m_{d1}e_{d1}\omega^{2}\cos(\omega t)+m_{d1}e_{d1}\omega^{2}\sin(\omega t) \\ f_{ry1}+m_{d1}e_{d1}\omega^{2}\cos(\omega t)-m_{d1}e_{d1}\omega^{2}\sin(\omega t )+m_{d1}g\\ f_{rx2}+m_{d2}e_{d2}\omega^{2}\cos(\omega t)+m_{d2}e_{d2}\omega^{2}\sin(\omega t )\\ f_{ry2}+m_{d2}e_{d2}\omega^{2}\cos(\omega t)-m_{d2}e_{d2}\omega^{2}\sin(\omega t )+m_{d2}g\\ \end{array} \right] \tag{17}\]  
where \(f_{x}\) and \(f_{y}\) denote the gas film forces of the axial-groove gas bearing, respectively; \(e_{d1}\) and \(e_{d1}\) are the mass eccentricities of the rigid disks, respectively; \(x_{d1}\) and \(y_{d1}\) denote the displacements of disk 1 in the \(x\) and \(y\) directions, respectively; \(x_{d2}\) and \(y_{d2}\) denote the displacements of disk 2 in the \(x\) and \(y\) directions, respectively; \(k_{1}\) and \(k_{2}\) denote the bending stiffnesses of the left and right segments, respectively; \(k_{b}\) denotes the bending stiffness of the bolted rod; \(f_{rx1}\) and \(f_{ry1}\) denote the restoring forces of the disk 1, respectively; \(f_{rx2}\) and \(f_{ry2}\) denote the restoring forces of the disk 2, respectively; and \(k_{r}\) denotes the restoring stiffness. \[\left\{ \begin{array}{c}f_{rx1}=k_{r}(x_{d2}-x_{d1})^{3}\\ f_{ry1}=k_{r}(y_{d2}-y_{d1})^{3}\\ \end{array} \right.,\ \left\{ \begin{array}{c}f_{rx2}=-k_{r}(x_{d2}-x_{d1})^{3}\\ f_{ry2}=-k_{r}(y_{d2}-y_{d1})^{3}\\ \end{array} \right.