# High-Fidelity Fin-Actuator System Modeling and Aeroelastic Analysis Considering Friction Effect
## 5 Aeroelastic Analysis Results and Discussion
### Preliminary Flutter Results for No-Freplay Gap (\(e=0\)) and No-Friction  
The fin was released from the initial fin shaft deflection angle of \(1^{\circ}\). Figure 12 shows the time domain response of the fin under aerodynamic force. Comparison has been made between the responses of the fin shaft angle at two flight velocities. Flutter occurred at 765 m/s, and when the flight velocity was lower than 765 m/s, the fin shaft angle response was convergent.  
The flutter results in frequency domain are shown in Figure 13. Eighty-two frequency points were selected from 40 Hz to 120 Hz. The corresponding dynamic stiffness was adopted to obtain the flutter frequency and velocity of the system at each frequency point. Then a line was drawn with slope 1 that indicated the frequencies of abscissa and ordinate were equal. The frequency corresponding to the intersection of the flutter frequency line  
Figure 12: The time domain flutter results of the system without freeplay and friction, where (**a**) is the response of the fin shaft angle at 760 m/s, and (**b**) is at 765 m/s.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Description** & **Symbol** & **Value** & **Unit** \\ \hline Inductance & \(L\) & \(6.53\times 10^{-4}\) & mH \\ Resistance & \(R\) & 1.1 & \(\Omega\) \\ Rotor inertia & \(J_{\mathrm{m}}\) & \(1.094\times 10^{-6}\) & kg \(\cdot\) m\({}^{2}\) \\ Torque coefficient & \(K_{\mathrm{m}}\) & 0.0238 & Nm/A \\ Back EMF coefficient & \(C_{\mathrm{e}}\) & 0.034 & V/ (rad/s) \\ Connection stiffness & \(k_{\mathrm{m}}\) & 1000 & Nm/rad \\ Moment of inertia of gear 1 & \(J_{1}\) & \(2.4\times 10^{-8}\) & kg \(\cdot\) m\({}^{2}\) \\ Radius of gear 1 & \(r_{1}\) & 0.005 & m \\ Moment of inertia of gear 2 & \(J_{2}\) & \(4\times 10^{-6}\) & kg \(\cdot\) m\({}^{2}\) \\ Radius of gear 2 & \(r_{2}\) & 0.0225 & m \\ Meshing stiffness & \(k_{\mathrm{g}}\) & 10\({}^{8}\) & kg \(\cdot\) s\({}^{2}\)/rad \\ Connection stiffness & \(k_{\mathrm{z}}\) & 1000 & Nm/rad \\ Screw inertia & \(I_{\mathrm{gg}}\) & \(3.98\times 10^{-5}\) & kg \(\cdot\) m\({}^{2}\) \\ Radius of screw & \(r_{\mathrm{gg}}\) & 0.006 & m \\ Screw efficiency & \(\eta\) & 0.85 & m \\ Fork length & \(L_{\mathrm{bc}}\) & 0.0285 & m \\ Comprehensive stiffness & \(k_{\mathrm{c}}\) & \(2.27\times 10^{6}\) & kg/s\({}^{2}\) \\ Bristle stiffness & \(\sigma_{0}\) & 300 & Nm/rad \\ Bristle damping & \(\sigma_{1}\) & 2.5 & Nm/(rad/s) \\ Viscous damping & \(\sigma_{2}\) & 0.02 & Nm/(rad/s) \\ Coulomb friction & \(F_{\mathrm{c}}\) & 1.2565 & Nm \\ Maximum static friction force & \(F_{\mathrm{s}}\) & 0.774 & Nm \\ Stribeck velocity & \(V_{\mathrm{s}}\) & 1 & rad/s \\ \hline \hline \end{tabular}
\end{table}
Table 3: Simulation parameters of the actuator. Electromotive force, EMF.  
and the line with slope 1 was the real flutter frequency, which was 55.1077 Hz. The velocity corresponding to the real flutter frequency was the real flutter velocity, which was 769.2596 m/s. The frequency-domain and time-domain results were in good agreement; the discrepancy between which was of the order of 0.5%. This result shows that the calculation method of dynamic stiffness was correct and the flutter result of fin-actuator system calculated using dynamic stiffness was accurate enough.