## 2 Bird Strike and Rotor Dynamic Modeling
### Rotor System Modeling  
The operation state of an aero engine may be affected by the impact force caused by a bird strike. Understanding the response characteristics of the rotor system under bird strike is useful for grasping the aero-engine operation state. The key to studying the dynamic response of an aero-engine rotor system is to develop a rotor dynamics model that can describe the effects of bird strikes on the fan blade. The schematic diagram of the aero-engine low-pressure rotor system is depicted in Figure 6. The low-pressure rotor system is supported by three rolling bearings (#1, #2, and #3), including a 1-stage fan and a 4-stage low-pressure turbine, and the bearing supporting mode is \"0-2-1\". To establish an accurate rotor dynamics model, three coordinate scanning and 3D modeling of the rotor shaft, disk, and other components were carried out: first, BESTSCAN 751 (HOLON 3D, China) 3D scanner and HL-Scan software were used to obtain the point cloud data of the components, and the data was then imported into the software CATIA for constructing 3D models. Combined with the material parameters of each component, the mass distribution and center of mass position of the rotor system were obtained. Based on the obtained data points, the fan, the shaft, and the disk were simplified as several mass elements through the lumped-mass method, and a low-pressure rotor system model of the aero engine was established, as shown in Figure 7.  
Figure 5: Pressure profile on the target of impact under three different impact conditions. Figure 6: Schematic diagram of aero-engine low-pressure rotor structure and its support system. When using the lumped-mass method, the blade and the disk are treated as a mass point, so the bird impact force is considered to the disk, and its actual impact on the blade is similar; the mass of the disk is much larger than the mass of the mass point on the shaft, and the influence of its eccentricity is much larger than the eccentricity of the shaft, so the influence of concentrating the eccentricity on the disk on the calculation results is accepted. As there is no need to consider the connection method of each component in the calculation, considering it as a whole will have no effect on the calculation results; as only the x and y directional displacement of the rotor is considered in this paper, the torque is ignored in the simplification; and as the maximum speed is 3772 r/min and the speed is low, the influence of the gyroscopic force is not obvious, therefore ignoring these two items has little influence on the calculation results. As the focus of this research was the effect of bird strike on the rotor system in the x-y plane of radial vibration, other vibrational effects were not taken into account, and the following assumptions were considered:  
1. The force from the bird strike was taken into account on the disk. 2. The unbalanced mass of the rotor system concentrated on the two rotor disks, and the mass of the rotor system was discretized into the five mass points on the shaft. 3. The materials of the blade and the disk behave according to [PERSON]'s law. 4. The blade, disk, and shaft were considered to be rigidly connected to each other, ignoring the connections between them. 5. The torsional vibration and gyroscopic moment of the system were neglected. Based on [PERSON]'s second law, the equations of motion for the established rotor system were derived as:  
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