# Dynamic Behavior of Twin-Spool Rotor-Bearing System with Pedestal Looseness and Rub Impact
## 2 Modeling
### Bearing Element
\begin{split}\begin{cases}\mathbf{Q}_{i}\\ \mathbf{0}\end{cases}=\begin{bmatrix}-\mathbf{K}_{in}&\mathbf{K}_{in}\\ \mathbf{K}_{in}&-\mathbf{K}_{in2}\end{bmatrix}\begin{cases}V_{i}\\ V_{b}\\ V_{b}\end{cases}+\begin{bmatrix}-\mathbf{C}_{in}&\mathbf{C}_{in}\\ \mathbf{C}_{in}&-\mathbf{C}_{in2}\end{bmatrix}\begin{cases}\dot{V}_{i}\\ \dot{W}_{i}\\ \dot{V}_{b}\\ \dot{W}_{b}\end{cases}-\begin{bmatrix}\mathbf{0}&\mathbf{0}\\ \mathbf{0}&\mathbf{M}_{in}\end{cases}\begin{cases}\dot{V}_{i}\\ \dot{W}_{i}\\ \dot{V}_{b}\\ \dot{W}_{b}\end{cases}\end{cases}\end{split}  \tag{10}\]  
Figure 4: Bearing element. (**a**) Interaction between shaft and pedestal. (**b**) Schematic of bearing element. where:  
\[\mathbf{K}_{in}= \begin{bmatrix}k_{b}&0\\ 0&k_{b}\end{bmatrix} ,\ \mathbf{C}_{in}= \begin{bmatrix}c_{b}&0\\ 0&c_{b}\end{bmatrix} ,\ \mathbf{M}_{in}= \begin{bmatrix}m_{b}&0\\ 0&m_{b}\end{bmatrix} ,\ \mathbf{Q}_{i}= \begin{Bmatrix}Q_{iV}\\ Q_{iW}\end{Bmatrix} \]  
\[\mathbf{K}_{in2}= \begin{bmatrix}k_{b}+k_{s}&0\\ 0&k_{b}+k_{s}\end{bmatrix} ,\ \mathbf{C}_{in2}= \begin{bmatrix}c_{b}+c_{s}&0\\ 0&c_{b}+c_{s}\end{bmatrix} \]  
The bearing element is incorporated into the dynamic differential equation of the complete model as an external force term within the assembly. To simplify, the equation mentioned above is enlarged into a \(six\)-dimensional equation. The dynamic equation of the ordinary bearing in the full model is expressed as follows:  
\[[\mathbf{M}_{b1}]\ddot{q}_{b1}+[\mathbf{C}_{b1}]\dot{q}_{b1}+[\mathbf{K}_{b1 }]\dot{q}_{b1}=0 \tag{11}\]  
\[[\mathbf{K}_{b1}]= \begin{bmatrix}\mathbf{K}_{b1}^{11}&\mathbf{K}_{b1}^{12}\\ \mathbf{K}_{b1}^{21}&\mathbf{K}_{b1}^{22}\end{bmatrix} = \begin{bmatrix} \mathbf{K}_{in}&\mathbf{0}^{{}^{\prime}}&-\mathbf{K}_{in}\\ \mathbf{-\ddot{K}}_{in}&\mathbf{0}^{{}^{\prime}}&\mathbf{0}^{{}^{\prime}}& \mathbf{0}^{{}^{\prime}}\\ \mathbf{-\ddot{K}}_{in}&\mathbf{0}^{{}^{\prime}}&\mathbf{\ddot{K}}_{in2}^{{}^{ \prime}}\end{bmatrix} ,\ \mathbf{q}_{b1}=\left\{V_{i}\quad W_{i}\quad B_{i}\quad\Gamma_{i}\quad V_{b }\quad W_{b}\right\}^{\mathsf{T}}\]  
where the dimensions of the submatrices \(\mathbf{0},\mathbf{M}_{in},\mathbf{C}_{in},\mathbf{K}_{in},\mathbf{C}_{in2}, \mathbf{K}_{in2}\) are \(2\times 2\) and the subscript \(i\) indicates that the bearing element is connected to the node \(i\) on the rotating shaft. **Inter-Shaft Bearing. ** An inter-shaft bearing element is built as shown in Figure 5, connecting the \(i\) node on the HP rotor shaft with the \(j\) node on the LP rotor shaft, taking into account the effect of bearing stiffness/damping. From Figure 5a, it can be seen that the force generated by the inter-shaft bearing is applied to both the HP rotor shaft and the LP rotor shaft, and the magnitude of the force is \(Q_{i},Q_{j},\) and \(Q_{i}=-Q_{j}\). The force on the rotor shaft in the \(Y,Z\) direction is:  
\[ \begin{cases}Q_{iV}=k_{t}(V_{j}-V_{i})+c_{t}(\dot{V}_{j}-\dot{V}_{i})\\ Q_{iW}=k_{t}(W_{j}-W_{i})+c_{t}(\dot{W}_{j}-\dot{W}_{i})\end{cases} \quad \begin{cases}Q_ {jV}=-Q_{iV}\\ Q_{jW}=-Q_{iW}\end{cases}  \tag{12}\]  
where \(Q_{iV},Q_{iW},Q_{jV},Q_{jW}\) denote the force acting on node \(i,j\) in the \(Y,Z\) direction, \(V_{i},W_{i},V_{j},W_{j}\) denote the displacement of node \(i,j\) in the \(Y,Z\) direction, and \(k_{t},c_{t}\) denote the bearing stiffness and damping. Figure 5: Inter-shaft bearing element. (**a**) Interaction between HP shaft and LP shaft. (**b**) Schematic of inter-shaft bearing element. Organizing the above equation into matrix form gives:  
\[ \begin{cases}Q_{iV}\\ Q_{iW}\\ Q_{jV}\\ Q_{jW}\end{cases} =\left[ \begin{array}{cc}-\mathbf{K}_{im}&\mathbf{K}_{im}\\ \mathbf{K}_{im}&-\mathbf{K}_{im}\\ Q_{jW}\end{array} \right] \begin{cases}V_{i}\\ W_{i}\\ W_{j}\end{cases} +\left[ \begin{array}{cc}-\mathbf{C}_{im}&\mathbf{C}_{im}\\ \mathbf{C}_{im}&-\mathbf{C}_{im}\end{array} \right]