## 2 Bird Strike and Rotor Dynamic Modeling
### Rotor System Modeling
\begin{array}{l}M_{R1}\ddot{x}_{R1}+C_{R}\dot{x}_{R1}+K_{r}(x_{R1}-x_ {B1})=M_{R1}e_{R1}\omega^{2}\cos\omega t+B_{x}\\ M_{R1}\ddot{y}_{R1}+C_{R}\dot{y}_{R1}+K_{r}(y_{R1}-y_{B1})=M_{R1}e_{R1}\omega ^{2}\sin\omega t+B_{y}-M_{R1}g\\ M_{R2}\ddot{x}_{R2}+C_{R}\dot{x}_{R1}+K_{r}(x_{R2}-x_{B2})+K_{r}(x_{R2}-x_{R3} )=M_{R2}e_{R2}\omega^{2}\cos\omega t\\ M_{R2}\ddot{y}_{R2}+C_{R}\dot{y}_{R2}+K_{r}(y_{R2}-y_{B2})+K_{r}(y_{R2}-y_{B2} )=M_{R1}e_{R1}\omega^{2}\sin\omega t-M_{R2}g\\ M_{B1}\ddot{x}_{R1}+C_{B}\dot{x}_{R1}+K_{r}(x_{B1}-x_{R1})+K_{r}(x_{B1}-x_{B2} )=F_{B1x}\\ M_{B1}\ddot{y}_{B1}+C_{B1}\dot{y}_{B1}+K_{r}(y_{R2}-y_{B2})+K_{r}(y_{R2}-y_{B2} )=F_{B1y}-M_{B1}g\\ M_{B2}\ddot{x}_{B2}+C_{B}\dot{x}_{B2}+K_{r}(x_{B2}-x_{B1})+K_{r}(x_{B2}-x_{R2} )=F_{B2x}\\ M_{B2}\ddot{y}_{B2}+C_{B}\dot{y}_{B2}+K_{r}(y_{B2}-y_{B1})+K_{r}(y_{B2}-y_{B2} )=F_{B2y}-M_{B2}g\\ M_{B3}\ddot{x}_{B3}+C_{B}\dot{x}_{R3}+K_{r}(x_{B3}-x_{R2})=F_{B3x}\\ M_{B3}\ddot{y}_{B3}+C_{B}\dot{y}_{B3}+K_{r}(y_{B3}-y_{R2})=F_{B3y}-M_{B1}g\\ \end{array} \right. \tag{4}\]  
where \(M_{R1}\) is the mass of Disk 1 of the rotor, \(M_{R2}\) is the mass of Disk 2, and \(M_{B1}\), \(M_{B2}\), \(M_{B3}\) are the masses of the concentrated points at the three rolling bearings, respectively. \(C_{R}\) and \(C_{B}\) are the damping coefficients of the disk and ball bearings, \(K_{r}\) is the stiffness of the shaft, supposing that the stiffness of each section of the rotor is identical. \(e_{R1}\) and \(e_{R2}\) are the eccentricities of Disk 1 and Disk 2, respectively. \(B_{x}\) and \(B_{y}\) are the decomposition of bird impact forces into the x and y directions, respectively. \(F_{Bix}\) and \(F_{Biy}\) (\(i=1,\,2,\,3\)) are the  
Figure 7: The dynamical model of rotor system. bearing support forces in the x and y directions. \(\omega\) is the rotational speed of the rotor in rad/s. As this paper mainly studies the impact of bird strike on the fan rotor system, the bearings are regarded as an elastic system. The bearing stiffness in the rotor system was considered to be constant (\(K_{b}\)). The equations relating \(F_{Bix}\) and \(F_{Biy}\) (\(i=1,\,2,\,3\)), stiffness, and displacements could be given by:  
\[\left\{ \begin{array}{l}F_{Bix}=-K_{b}x_{bi}\\ F_{Biy}=-K_{b}y_{bi}\end{array} \right. \tag{5}\]  
From the above equations, the bearing support force is proportional to the displacement, and the negative sign indicates that the reaction force points in the opposite direction of the displacement. In our previous study [31], we used the same 3D scan method to build a '0-2-1' rotor test setup using the kinetic similarity principle (see Figure 8). The vibration response of the rotor system at 1500 rpm was measured using a test rig and is shown in Figure 9. At a speed of 1500 r/min, the vibration waveform of the system shows obvious peaks at the peaks and valleys. The fundamental frequency amplitude is the main frequency of the rotor system, and its amplitude is much larger than other high frequencies. The numerical method is used to calculate the rotor response at 1500 r/min, as shown in Figure 9, and it can be found that some of the results are the same, differing in units due to different model simplification treatments, which verifies the accuracy of the proposed rotor system model to a certain extent. Figure 8: The rotor test rig. Figure 9: _Cont._