# High-Fidelity Fin-Actuator System Modeling and Aeroelastic Analysis Considering Friction Effect
## 6 Conclusions  
In this article, a high-fidelity model of a fin-actuator system with a typical configuration was established based on modularization, in which a friction sheet was modeled utilizing the LuGre model, which can better reflect the friction characteristics. The whole modeling method can be used for dynamic analysis and aeroelastic analysis. The aeroelastic characteristics of the system were analyzed using frequency-domain and time-domain methods. The results of the two methods were highly consistent. The time-domain method can deal with highly nonlinear situations, while the frequency-domain method based on the DF method is suitable for weak nonlinear systems. Initial conditions, freeplay, and friction will all affect the stability of the nonlinear system. As the initial deflection angle \(\delta_{0}\) increases, the gain of the dynamic stiffness will be larger, but the increase was not proportional, and the increase of the phase was minor. The oscillation occurred at smaller flight speed, and the oscillation amplitude \(A\) became larger. Different \(\delta_{0}\) have little effect on the ratio of \(A\) to the half width of freeplay \(e\) at the same flight speed. Dynamic stiffness of the actuator with a smaller freeplay gap will reach a larger value, while the phase was not affected much by freeplay gap. The oscillation amplitude increased as the width of freeplay increased. Friction affects the amplitude and phase of actuator dynamic stiffness greatly. Adding friction makes the frequency of LCO decrease slightly. The LuGre friction model has six parameters. Friction sheets that do not use lubricant, \(\sigma_{2}\), which reflects the viscosity of the lubricant, were not the focus of this study. \(F_{\text{s}}\) and \(V_{\text{s}}\) have little influence on the aeroelastic characteristics, which corresponds to the fact that they have little effect on dynamic stiffness. \(\sigma_{1}\) is the parameter that has the greatest influence on the aeroelastic characteristics. Increasing \(\sigma_{1}\) can greatly improve the stability of the system. However, \(\sigma_{1}\) is difficult to quantify in actual engineering. The influence of \(F_{\text{c}}\) on system stability is complicated. As \(F_{\text{c}}\) increased, the speed at which the fin started to oscillate first increased and then decreased, while the oscillation amplitude at the same flight speed increased. Therefore, \(F_{\text{c}}\) was not better when larger. An optimal \(F_{\text{c}}\) can be obtained by appropriately designing the lubrication, pressure, and the geometry of the contact surface between the friction sheet and fork, so that the system is stable and friction is reduced. The smaller the \(\sigma_{0}\), the greater the flight speed at which the fin starts to oscillate, which was the desired effect. However, once the oscillation started in this case, the amplitude increased sharply. A single factor may have limitations. The synergistic effect of multiple parameters can suppress the LCO in the low-speed range, and it is also effective in increasing the flutter velocity as well as reducing the oscillation amplitude. Data curation, formal analysis, investigation, software, validation, visualization, and writing (original draft) were conducted by [PERSON] Conceptualization, methodology, and project administration were conducted by [PERSON] Writing (review and editing) was conducted by both [PERSON] and [PERSON]. Resources and supervision were conducted by both [PERSON] and [PERSON]. All authors have read and agreed to the published version of the manuscript. This research received no external funding. This appendix provides the matrices and vectors of the fin model, unsteady aerodynamics, and state-space form of the fin model in Section 3.1, Section 3.2, Section 3.3. The dynamic equation of the fin structure is [47]:  
\[\mathbf{M\ddot{x}}\mathbf{+}\mathbf{C\dot{x}}\mathbf{+}\mathbf{K}\mathbf{x}\mathbf{=}\mathbf{F} \tag{1}\]  
where \(\mathbf{x}=\left[ \begin{array}{c}\mathbf{q}_{\text{e}}\\ \delta\end{array} \right]\), \(\mathbf{M}=\left[