# Analysis on Influences of Squeeze Film Damper on Vibrations of Rotor System in Aeroengine
## 2 Dynamic Model of High-Pressure Rotor  
The structure of an aeroengine is simplified, as shown in Figure 2. The engine consists of a high-pressure rotor system and a low-pressure rotor system. In engineering, the high-pressure rotor system shows unstable vibration or bistable vibration caused by SFD. This paper mainly takes the high-pressure rotor system as the research object. The dynamic model of the high-pressure rotor system in Figure 2 is established. As shown in Figure 3, the front fulcrum of the high-pressure rotor is 4# fulcrum, which is composed of the equivalent stiffness of the elastic-supported combined structure and the nonlinear damping coefficient of the squeeze film damper at 4# fulcrum. The equivalent stiffness of the back fulcrum of the high-pressure rotor should be regarded as the series connection between the bearing stiffness of 5# fulcrum and the elastic support stiffness of 6# fulcrum. The series connection stiffness is approximately equal to the stiffness of 6#  
Figure 2: The diagram of the simplified dual-rotor model with four disks and five fulcrums. fulcrum; The damping coefficient of back fulcrum of the high-pressure rotor is provided by the squeeze film damper at 6# fulcrum. The equivalent disk D\({}_{3}\) of the high-pressure compressor has the vibration of \(y_{3}\) (vertical) and \(z_{3}\) (horizontal), and the rotation angles are \(\theta_{y3}\) and \(\theta_{z3}\), respectively. The vibrations corresponding to the equivalent disk D\({}_{4}\) are \(y_{4}\) (vertical), \(z_{4}\) (horizontal), \(\theta_{y4}\), \(\theta_{x4}\). The differential equation of motion of the high-pressure rotor system is as follows:  
\[(m_{3}+m_{4})\ddot{y}_{3}+m_{4}\ddot{\theta}_{z3}a_{34}+q_{b4y}y_{3}+\left(-q_ {b3y}a_{3}+q_{b4y}a_{4}\right)\theta_{z3}+q_{b3y}y_{3}=0 \tag{1}\]  
\[(m_{3}+m_{4})\ddot{z}_{3}-m_{4}\ddot{\theta}_{z3}a_{34}+(q_{b3z}a_{3}-q_{b4z}a _{4})\theta_{z3}+q_{b3z}z_{3}=0 \tag{2}\]  
\[m_{4}\ddot{z}_{3}a_{34}-(m_{4}{a_{34}}^{2}+J_{\Omega}+J_{44})\ddot{\theta}_{y 3}+\Omega_{3}(J_{p3}+J_{p4})\dot{\theta}_{z3}+(q_{b3z}{a_{3}}^{2}+q_{b4z}{a_{4} }^{2}+q_{b4y})\theta_{y3}+z_{3}(q_{b3z}a_{3}-q_{b4z}a_{4})=0 \tag{3}\]  
\[m_{4}\ddot{z}_{3}a_{34}+(m_{4}{a_{34}}^{2}+J_{\Omega}+J_{44})\ddot{\theta}_{z3} -\Omega_{3}(J_{p3}+J_{p4})\dot{\theta}_{y3}+(q_{b3y}{a_{3}}^{2}+q_{b4y}{a_{4} }^{2}+q_{b4y})\theta_{z3}-y_{3}(q_{b3y}a_{3}-q_{b4y}a_{4})=0 \tag{4}\]  
where \(q_{b3y}\), \(q_{b4y}\), \(q_{b3y}\), \(q_{b4y}\) represent the stiffness of the 4# fulcrum and 6# fulcrum in the \(y\) and \(z\) directions, and \(q_{b4a_{2}}\) represents the rotational stiffness of the 6# fulcrum in the \(z\) direction in Figure 3.  
Considering the form of complex variables, and assume:  
\[y=y_{3}+i*z_{3};\ \theta=\theta_{z3}-i*\theta_{y3} \tag{5}\]  
Then the Equations (1)-(4) can be rewritten as:  
\[(m_{3}+m_{4})y_{3}+m_{4}\ddot{\theta}_{z3}a_{34}+q_{b3}y+(q_{b3z}a_{3}-q_{b4z} a_{4})\theta_{z3}+q_{b3y}y=0 \tag{6}\]  
\[m_{4}\ddot{y}_{3}a_{34}+(m_{4}{a_{34}}^{2}+J_{\Omega}+J_{\Omega})\ddot{\theta }-\Omega_{3}(J_{p3}+J_{p4})\dot{\theta}+(q_{b3y}{a_{3}}^{2}+q_{b4y}{a_{4}}^{2}+ q_{b4y}{a_{4}}^{2}+q_{b4y}{a_{4}}^{2})\theta-y(q_{b3y}{a_{3}}-q_{b4y}{a_{4}})=0 \tag{7}\]  
According to the dynamic differential equation of the high-pressure rotor and the stiffness assumption of the high-pressure rotor, the high-pressure rotor system has four degrees of freedom. There is inertia matrix \(\mathbf{M}\), gyro force matrix \(\mathbf{G}\), and stiffness matrix \(\mathbf{K}\), as follows:  
(a) Inertia matrix \(\mathbf{M}\)  
\[\left[ \begin{array}{cccc}m_{3}+m_{4}&0&0&m_{4}a_{34}\\ 0&m_{3}+m_{4}&0&0\\ 0&-m_{4}a_{34}&m_{4}{a_{34}}^{2}+J_{\Omega}+J_{44}&0\\ m_{4}a_{34}&0&0&m_{4}{a_{34}}^{2}+J_{\Omega}+J_{44}\end{array} \right]\]  
Figure 3: Equivalent dynamic model of the high-pressure rotor system. (b) Gyro force matrix \(\mathbf{G}\)  
\[\left[ \begin{array}{cccc}0&0&0&0\\ 0&0&0&0\\ 0&0&0&\Omega_{3}(J_{p3}+J_{p4})\\ 0&0&-\Omega_{3}(J_{p3}+J_{p4})&0\\ \end{array} \right]\]  
(c) Stiffness matrix \(\mathbf{K}\)  
\[\left[