# Dynamic Behavior of Twin-Spool Rotor-Bearing System with Pedestal Looseness and Rub Impact
## 2 Modeling
### Shaft Element  
In order to reduce the weight, the shaft in the aero-engine usually is a hollow shaft with a drum structure [26], and the whole rotor shaft is mainly subjected to transverse loads (loads in the \(OYZ\)). Therefore, beam models are commonly used in engineering to simulate rotor shafts in aero-engine rotor systems. Considering the bending, shearing, gyroscopic effect, and transient vibration of the shaft, the dynamic differential equations of the shaft element are obtained by the theory of the Timoshenko beams, [PERSON]'s equations, and the finite element method. Any cross-section of the rotary axis in the model produces only translational displacements \((V,W)\) in the \(Y,Z\)directions and rotational displacements \((\Phi,B,\Gamma)\) around \(z^{\prime},y^{\prime\prime},x\), as shown in Figure 3.  
The matrix form of the differential equations for the shaft element is:  
\[([\mathbf{\mathrm{M}}_{\mathrm{T}_{\mathrm{S}}}]+[\mathbf{\mathrm{M}}  
elastic modulus,second axial moment of area, and shear modulus, and \(A_{s}\) is the effective shear area. And the expression for each element in the matrix is shown below:  
\[ \begin{array}{l}M_{T1}=\frac{13}{35}+\frac{7}{10}\varphi_{s}+\frac{1}{3}\varphi _{s}^{2}&M_{T2}=\left(\frac{1}{105}+\frac{1}{60}\varphi_{s}+\frac{1}{120} \varphi_{s}^{2}\right)l^{2}\\ M_{T3}=\frac{9}{70}+\frac{3}{10}\varphi_{s}+\frac{1}{6}\varphi_{s}^{2}&M_{T4}= \left(\frac{11}{210}+\frac{1}{120}\varphi_{s}+\frac{1}{24}\varphi_{s}^{2} \right)l\\ M_{T5}=\left(\frac{13}{420}+\frac{3}{40}\varphi_{s}+\frac{1}{24}\varphi_{s}^{2} \right)l&M_{T6}=-\left(\frac{1}{140}+\frac{1}{60}\varphi_{s}+\frac{1}{120} \varphi_{s}^{2}\right)l^{2}\\ \end{array} \]  
Mass matrix \([\mathbf{M}_{Rs}]\):  
\[[\mathbf{M}_{Rs}]=k_{MRs}\left[ \begin{array}{cccccccc}M_{R1}&0&0&M_{R4}&-M_{ R1}&0&0&M_{R4}\\ 0&M_{R1}&M_{R4}&0&0&-M_{R1}&-M_{R4}&0\\ 0&-M_{R4}&M_{R2}&0&0&M_{R4}&M_{R3}&0\\ M_{R4}&0&0&M_{R2}&-M_{R4}&0&0&M_{R3}\\ -M_{R1}&0&0&-M_{R4}&M_{R1}&0&0&-M_{R4}\\ 0&-M_{R1}&M_{R4}&0&0&M_{R1}&M_{R4}&0\\ 0&-M_{R4}&M_{R3}&0&0&M_{R4}&M_{R2}&0\\ M_{R4}&0&0&M_{R3}&-M_{R4}&0&0&M_{R2}\\ \end{array} \right] \tag{4}\]  
\(k_{MRs}\) is constant:  
\[k_{MRs}=\frac{I_{p}}{l(1+\varphi_{s})^{2}}=\frac{\rho_{l}I}{IA(1+\varphi_{s}) ^{2}}=\frac{\rho I}{l(1+\varphi_{s})^{2}}\]  
where \(I\) denotes the cross-section moment of inertia, and \(I_{p}\) denotes the polar moment of inertia. And the expression for each element in the matrix is shown below:  
\[ \begin{array}{l}M_{R1}=\frac{6}{5}&M_{R2}=\left(\frac{2}{15}+\frac{1}{6} \varphi_{s}+\frac{1}{3}\varphi_{s}^{2}\right)l^{2}\\ M_{R3}=\left(-\frac{1}{30}-\frac{1}{6}\varphi_{s}+\frac{1}{6}\varphi_{s}^{2} \right)l^{2}&M_{R4}=\left(\frac{1}{10}-\frac{1}{2}\varphi_{s}\right)l\\ \end{array} \]  
Transient matrix \([\mathbf{N}_{s}]\):  
\[[\mathbf{N}_{s}]=k_{Ns}\left[\begin{array}{cccccccc}0&-N_{s1}&N_{s2}&0&0&N_{ s1}&N_{s2}&0\\ 0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0\\ 0&-N_{s2}&N_{s4}&0&0&N_{s2}&-N_{s3}&0\\ 0&N_{s1}&-N_{s2}&0&0&-N_{s1}&-N_{s2}&0\\ 0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0\\ 0&-N_{s2}&-N_{s3}&0&0&N_{s2}&N_{s4}&0\\ \end{\[\text{Gyroscopic matrix}\left[\mathbf{G}_{s}\right]=\left(\left[\mathbf{N}_{s}\right]- \left[\mathbf{N}_{s}\right]^{\text{T}}\right)\text{:} \tag{6}\]  
\(k_{Gs}=k_{Ns}\) is constant. And the expression for each element in the matrix is shown below:  
\[ \begin{array}{l}G_{s1}=36\hskip 28.452756 ptG_{s2}=3l-15l\varphi_{s}\\ G_{s3}=l^{2}+5l^{2}\varphi_{s}-5l^{2}\varphi_{s}\hskip 28.452756 ptG_{s4}=4l^{2}+5l^{2 }\varphi_{s}+10l^{2}\varphi_{s}\end{array} \]  
\(\text{Stiffness matrix}\)\(\left[\mathbf{K}_{s}\right]\):  
\(k_{Ks}\) is constant:  
\[k_{Ks}=\frac{EI}{l^{3}(1+\varphi_{s})}\]  
The expression for each element in the matrix is shown below:  
\[ \begin{array}{l}K_{s1}=12\hskip 28.452756 ptK_{s2}=l^{2}(\varphi_{s}+4)\\ K_{s3}=l^{2}(2-\varphi_{s})\hskip 28.452756 ptK_{s4}=6l\end{array} \]  
In this subsection, we have developed the differential equations that describe the dynamics of the rotating shaft element using the Timoshenko beam theory.