# Dynamic Behavior of Twin-Spool Rotor-Bearing System with Pedestal Looseness and Rub Impact
## 2 Modeling
### Twin-Spool Rotor-Bearing Model of a Typical Turbofan  
Based on the actual structure of the PW6000 engine (Figure 11) and applying the rotor system modeling methodology described in the previous section, the corresponding simplified dynamics model is developed in this section. The rotor structure of the PW6000 engine consists of a fan, a four-stage LP compressor, a six-stage HP compressor, a one-stage HP turbine, and a three-stage LP turbine; three bearings are located at the LP compressor, the HP compressor, and the LP turbine; and an inter-shaft bearing, which connects the LP rotor to the HP rotor, is located at the HP  
Figure 11: Cross-section of the PW6000 engine. Figure 10: Schematic of the forces on the disk during the rub impact failure. turbine. According to the data published by Pratt and Whitney (2018), the PW6000 has a total length of about 2.75 m and a total mass of about 2289 kg. In this paper, it assumes that the PW6000 has a total rotor system length of about 2.4 m and a rotor system mass of about 400 kg. Using the location of the disk and bearings as nodes, the HP and LP shafts are divided into a number of elements, as shown in Figure 12. The location information of each node is shown in Table 1.  
After node division, the material and geometry of each element need to be determined to calculate the element matrix. The fan, compressor and turbine are simplified as homogeneous discs with a certain thickness, whose geometry and materials ([PERSON] et al., 2018) are shown in Table 2. In addition, let the bearings 1\(\sim\)3 have a mass of 5 kg, a support stiffness of \(10^{8}\) N/m, and damping of 300 N \(\cdot\) s/m; let the inter-shaft bearing have a stiffness of \(10^{9}\) N/m and damping of 3000 N \(\cdot\) s/m. The element matrix is calculated based on the material and geometrical dimensions of each component and assembled to obtain the differential equations for the twin-spool rotor-bearing system with 70 DOFs:  
\[[\mathbf{M}]\ddot{\boldsymbol{q}}+\big{(}[\mathbf{C}]-\dot{\Phi}[\mathbf{G}] \big{)}\dot{\boldsymbol{q}}+[\mathbf{K}]\boldsymbol{q}-\ddot{\Phi}[\mathbf{N}] \boldsymbol{q}=\boldsymbol{Q} \tag{21}\]  
where  
\[\boldsymbol{q}=\big{\{}V_{1}\quad W_{1}\quad B_{1}\quad\Gamma_{1}\quad V_{2} \quad W_{2}\quad B_{2}\quad\Gamma_{2}\quad\ldots\quad V_{17}\quad W_{17}\quad V _{18}\quad W_{18}\quad V_{19}\quad W_{19}\big{\}}^{\mathrm{T}}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Node** & **Location** & **Distance from the Left End (m)** & **Components** & **Node Association** \\ \hline
1 & LP Shaft & 0 & Fan & / \\
2\(\sim\)5 & LP Shaft & 0.4670, 0.5557, 0.6443, 0.7330 & LP Compressor & 2 to 17 \\
6 & LP Shaft & 1.9600 & / & 6 to 16 \\
7\(\sim\)9 & LP Shaft & 2.1000, 2.2500,2.4000 & LP Turbine & 9 to 18 \\
10\(\sim\)15 & HP Shaft & 0, 0.0934, 0.1868, 0.2802, 03736 & HP Compressor & 10 to 19 \\
16 & HP Shaft & 0.9600 & HP Turbine & 16 to 6 \\
17\(\sim\)18 & Bearing & 0.4670,2.4000 & / & 17 to 2, 18 to 9 \\
19 & Bearing & 0 & / & 19 to 10 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Node information for PW6000 twin-spool rotor system. Figure 12: Nodal division for PW6000 twin-spool rotor-bearing system. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Component** & **Material** & **Length (m)** & **Inner and Outer Radius (m)** & **Density (kg/m)** & **Elastic Modulus (GPa)** & **Poisson’s Ratio** & **Eccentricity (m)** \\ \hline LP Shaft & GH4169 ([PERSON] et al., 2018) & 2.4 & 0.015, 0.035 & 8240 & 150 (700 \({}^{\circ}\)C) & 0.3 & / \\ HP Shaft & GH4169 ([PERSON] et al., 2018) & 0.96 & 0.04, 0.056 & 8240 & 150 (700 \({}^{\circ}\)C) & 0.3 & / \\ Fan & TC4 & 0.0005 & 0, 0.7 & 4510 & / & / & \(8\times 10^{-5}\) \\ LP Compressor & V78 & 0.01 & 0, 0.3 & 4430 & / & / & \(5\times 10^{-5}\) \\ HP Compressor & V78 & 0.015 & 0, 0.2 & 4430 & / & / & \(4\times 10^{-5}\) \\ LP Turbine & N07718 & 0.01 & 0, 0.35 & 8190 & / & / & \(5\times 10^{-5}\) \\ HP Turbine & N07718 & 0.02 & 0, 0.25 & 8190 & / & / & \(5\times 10^{-5}\) \\ \hline \hline \end{tabular}
\end{table}