# Dynamic Behavior of Twin-Spool Rotor-Bearing System with Pedestal Looseness and Rub Impact
## 2 Modeling
### Assembly  
Based on the dynamic differential equations of each element and the geometric structure of the dual-rotor system, this subsection describes the method of assembling the elements by taking the twin-spool rotor system shown in Figure 6 as an example. It should be noted that the structure of the twin-spool rotor system in Figure 6 is different from that of a real aero-engine, and it is only a case study to illustrate the methodology of assembly. According to the geometry of the rotor system, nodes are set up at critical locations, such as the rotor, the bearing location, and the center of the rotor shaft. Nodes \(1\)\(\sim\)\(5\) are located at the LP rotating shaft, nodes \(6\)\(\sim\)\(10\) are located at the HP rotating shaft, and nodes \(11,12\) are located at the bearing housing connected to nodes \(2,10\), respectively. In addition, the inter-shaft bearing connects nodes \(4\) and \(9\) on the HP and LP shafts, and the disk element is located at node \(7\).  
In this case, nodes \(1\)\(\sim\)\(10\) are located in the rotating shaft, each node corresponds to \(4\) degrees of freedom (DOF) \(V_{i},W_{i},B_{i}\), and \(\Gamma_{i}\); nodes \(11,12\) are located in the bearing housing, each node corresponds to \(2\)-DOF \(V_{i},W_{i}\), where the subscript \(i\) denotes the node number. The overall matrix with \(44\times 44\) dimensions is shown in Figure 7. The small squares in the diagram represent the elements in the matrix, and a blank square means that the position is \(0\). The blocks with different colors represent different elements: blue  
Figure 6: Case A: node division of a twin-spool rotor used to illustrate overall matrix assembly.  
for the shaft element, yellow for bearing 1, brown for bearing 2, hollow pink for the disk element, and red for the inter-shaft bearing. The overlapping color blocks mean that the overlapping elements are superimposed. The following describes the method of grouping each element separately.  
First, the rotating shaft element is assembled. Nodes 1\(\sim\)5 divide the LP shaft into four segments, and nodes 6\(\sim\)10 divide the HP shaft into four segments. Each segment is a rotating shaft element. The mass, stiffness, gyroscopic, and transient matrices of each shaft element are partitioned according to the DOF of node. Take the stiffness matrix as an example:  
\[[\mathbf{K}_{\mathrm{s}}]=\begin{bmatrix}\mathbf{K}_{1}^{11}&\mathbf{K}_{1}^ {12}\\ \mathbf{K}_{\mathrm{s}}^{21}&\mathbf{K}_{\mathrm{s}}^  
The overall differential equations of the twin-spool rotor-bearing system are obtained:  
\[[\mathbf{M}]\bar{\boldsymbol{q}}+([\mathbf{C}]-[\mathbf{G}])\boldsymbol{q}+[ \mathbf{K}]\boldsymbol{q}-[\mathbf{N}]\boldsymbol{q}=\boldsymbol{Q} \tag{15}\]  
\[\boldsymbol{q}=\big{\{}V_{1}\quad W_{1}\quad B_{1}\quad\Gamma_{1}\quad V_{2} \quad W_{2}\quad B_{2}\quad\Gamma_{2}\quad\ldots\quad V_{11}\quad W_{11}\quad V _{12}\quad W_{12}\big{\}}^{\mathrm{T}}\]  
where \([\mathbf{M}],[\mathbf{C}],[\mathbf{G}],[\mathbf{K}]\), and \([\mathbf{N}]\) are the overall mass, damping, gyroscopic, stiffness, and transient matrices,\(\boldsymbol{Q}\) is the external force vector, and \(\boldsymbol{q}\) is the system displacement, respectively. It should be noted that the rotational speeds of the HP and LP rotors are usually not the same, and when the rotational speed ratio is \(r=\frac{\Phi_{\mathrm{HP}}}{\Phi_{\mathrm{LP}}}\) (\(\dot{\Phi}_{\mathrm{LP}},\dot{\Phi}_{\mathrm{HP}}\) are the HP and LP rotor speeds, respectively), the LP part of the matrix \([\mathbf{G}]\) needs to be multiplied by the LP shaft speed \(\Phi\) and the HP part by the HP shaft speed \(r\Phi\). The LP part of the matrix \([\mathbf{N}]\) needs to be multiplied by the LP shaft acceleration \(\dot{\Phi}\) and the HP part needs to be multiplied by the HP shaft acceleration \(r\dot{\Phi}\).