# High-Fidelity Fin-Actuator System Modeling and Aeroelastic Analysis Considering Friction Effect
## 5 Aeroelastic Analysis Results and Discussion
### Aeroelastic Characteristics of the System with Freeplay and Friction  
The LuGre model is represented by several nonlinear functional equations, including six parameters in which \(\sigma_{0}\) and \(\sigma_{1}\) are two dynamic parameters, and \(\sigma_{2}\), \(F_{\rm c}\), \(F_{\rm s}\), and \(V_{\rm s}\) are four static parameters. Analyzing the effect of friction on the flutter characteristics is challenging since coupling exists among the six parameters [57,58]. However, it is undoubtedly an effective way to analyze the aeroelastic characteristics with the help of the concept of dynamic stiffness and the physical meaning of the parameters. \(\sigma_{0}\) and \(\sigma_{1}\) are related to material properties. If a lubricant exists, \(\sigma_{2}\) indicates the viscous properties of the lubricant. For dry friction, \(\sigma_{2}\) is pretty small and close to zero [26]. The pressure influences on \(F_{\rm s}\), \(F_{\rm c}\), and \(V_{\rm s}\)[59]. The variable-parameter analysis is conducted to evaluate the influence of each parameter. Table 4 summarizes the range of friction parameters and the factors that parameter depends principally upon. As can be seen, lubricant is an important factor, but there is no lubricant added to the friction sheet of the actuator generally. Since the LuGre parameters depend on the material properties and working conditions, when the parameters are changed for stability analysis, they cannot be divorced from reality. Here, the upper and lower bounds of the parameters were derived from the literature [26,41,60,61,62]. For comparison, the standard value of each friction parameter in Table 3 was multiplied by factors of 0.1 and 10, after which the parameter values were still in the range of Table 4.  
Figure 17 shows the impact of the variation of six friction parameters on the actuator dynamic stiffness when \(\delta_{0}\) is \(1^{\circ}\) and \(e\) is \(0.2^{\circ}\). It was discovered that changing \(\sigma_{2}\), \(F_{\rm s}\), or \(V_{\rm s}\) had little effect on the dynamic stiffness, while changes in \(\sigma_{0}\), \(\sigma_{1}\), and \(F_{\rm c}\) had a greater effect. It can be observed that, generally, friction had a huge influence on both gain and phase of \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Description**} & \multirow{2}{*}{**Symbol**} & **PARAMETER** & \multirow{2}{*}{**Unit**} & **Parameter Depends** \\  & & **RANGE** & & **Principally upon**[26,59,63] \\ \hline Bristle stiffness & \(\sigma_{0}\) & 10.23–6.5 \(\times\) 10\({}^{4}\) & Nm/rad & Material properties \\ Bristle damping & \(\sigma_{1}\) & 2.719 \(\times\) 10\({}^{-3}\)–45.2 & Nm/(rad/s) & Contact geometry and lubricant \\ Viscous damping & \(\sigma_{2}\) & 9.216 \(\times\) 10\({}^{-6}\)–1.819 & Nm/(rad/s) & \multirow{2}{*}{Lubricant} \\ Coulomb friction & \(F_{\rm c}\) & 1.895 \(\times\) 10\({}^{-3}\)–2646.856 & Nm & \\ Maximum static friction & \(F_{\rm s}\) & 3.124 \(\times\) 10\({}^{-3}\)–8.558 & Nm & Boundary lubrication and pressure \\ Stribeck velocity & \(V_{\rm s}\) & 6.109 \(\times\) 10\({}^{-2}\)–88.1 & rad/s & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Approximate ranges and influence factors for the parameters of the LuGre model. Figure 16: The influence of \(e\) with \(\delta_{0}=1^{\circ}\), where (**a**) is the influence of \(e\) on the dynamic stiffness of the actuator and (**b**) is the influence of \(e\) on the response of the fin deflection angle at different flight speeds. actuator dynamic stiffness. This is because the friction model used in this paper is complex and highly nonlinear. Friction has been oversimplified in some literatures. The actuator in Reference [64] was equivalent to a spring with friction nonlinearity, and the relationship between displacement and restoring force is given by:  
\[F(x)=\left\{ \begin{array}{cc}Kx+F_{f}&at&x>0\\ Kx-F_{f}&at&x<0\end{array} \right.