# Dynamic Behavior of Twin-Spool Rotor-Bearing System with Pedestal Looseness and Rub Impact
## 2 Modeling
### Bearing Element  
In an aero-engine, the bearing is the core component that connects the rotor to the stator as well as the HP rotor to LP rotor, and it plays a supporting role. According to the different connection objects, they can be divided into ordinary bearings connected with the case and inter-shaft bearings connected with the HP and LP rotor. The inner ring of a ordinary bearing is connected to the rotor shaft and the outer ring is fixed to the bearing housing. The bearing housing is usually not directly connected to the case. Instead, the bearing housing is attached to elastic support (such as squirrel cage elastic support), which is then connected to the case. The inter-shaft bearing's inner ring is linked to the LP rotor shaft, while the outer ring is attached to the HP rotor shaft. In the aero-engine rotor system, the inclusion of inter-shaft bearings can enhance the load-bearing structure and enhance the engine's overall performance. However, this also results in the transmission of vibrations between the HP and LP rotors through the inter-shaft bearings. This leads to a more complex vibration response, which can potentially cause excessive vibrations in certain load conditions. Hence,the twin-spool rotor-bearing model in this paper takes into account the impact of both typical bearings and inter-shaft bearings on the vibration characteristics of the rotor system. During the operation of an aero-engine, the stiffness provided by the bearing fluctuates, but overall, it remains stable and is much greater than the support stiffness provided by the elastic support [27; 28]. According to this characteristic, this paper ignores the motion of the internal structure of the bearings and focuses on the influence of the overall stiffness of the bearings, and it establishes the differential equations of the ordinary bearing element and the inter-shaft bearing element through force analysis and [PERSON]'s second law, respectively. **Ordinary bearing element. ** An ordinary bearing element as shown in Figure 4 is established, and the effects of bearing stiffness/damping, elastic support stiffness/damping, and pedestal mass are considered. In Figure 4, the rotating shaft is affected by the force \(Q_{i}\) from the bearing, and the pedestal is affected by the forces \(Q_{b1},Q_{b2}\) from the bearing and the elastic support at the same time. If it is assumed that the bearing links node \(i\) on the rotating shaft to the case, we can establish the differential equation for the dynamics of the bearing housing using [PERSON]'s second law:  
\[ \begin{split} m_{b}\ddot{V}_{b}&=Q_{bV}=\left[k_{ b}(V_{i}-V_{b})+c_{b}(\dot{V}_{i}-\dot{V}_{b})\right]+\left[k_{s}(-V_{b})+c_{s}(- \dot{V}_{b})\right]\\ m_{b}\ddot{W}_{b}&=Q_{bW}=\left[k_{b}(W_{i}-W_{b})+ c_{b}(\dot{W}_{i}-\dot{W}_{b})\right]+\left[k_{s}(-W_{b})+c_{s}(-\dot{W}_{b}) \right]\end{split}  \tag{8}\]  
where \(Q_{bV}\) and \(Q_{bW}\) represent the external forces acting on the bearing housing in the \(Y\) and \(Z\) directions, respectively. \(V_{i}\) and \(W_{i}\) represent the displacements of the nodes \(i\) on the rotating shaft in the \(Y\) and \(Z\) directions, respectively. Similarly, \(V_{b}\) and \(W_{b}\) represent the displacements of the bearing housing in the \(Y\) and \(Z\) directions, respectively. The variables \(m_{b},k_{b},c_{b}\) represent the mass, stiffness, and damping of the pedestal, respectively. Similarly, \(k_{s}\) and \(c_{s}\) denote the stiffness and damping of the elastic support. The bearing element has an impact on both the pedestal and the \(i\) node on the shaft when the shaft experiences external forces \(Q_{iV},Q_{iW}\) from the bearing in the \(Y,Z\) direction:  
\[ \begin{split} Q_{iV}&=-k_{b}(V_{i}-V_{b})-c_{b}( \dot{V}_{i}-\dot{V}_{b})\\ Q_{iW}&=-k_{b}(W_{i}-W_{b})-c_{b}(\dot{W}_{i}- \dot{W}_{b})\end{split}  \tag{9}\]  
Equations (8) and (9) are written in matrix form to obtain:  
\[