# 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
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Figure 17: Influence of the parameters of the LuGre model on the dynamic stiffness of the actuator, where (**a**–**f**), respectively, show the effects of \(\sigma_{0}\), \(\sigma_{1}\), \(\sigma_{2}\), \(F_{c}\), \(F_{s}\), and \(V_{s}\). The equivalent stiffness of this nonlinear spring is calculated by the describing function method. The results in Reference [64] show that friction only affects the amplitude of actuator stiffness, which has been modified in this paper according to Figure 17.  
Figure 18 shows the comparison of the frequency-domain and time-domain results. The friction parameters are standard values in Table 3 and keep \(e\) at 0.2\({}^{\circ}\). Due to the nonlinearity of the friction model, as the oscillation amplitude increased, the difference between the time-domain and frequency-domain results became larger, but it did not exceed 4.4%. The oscillation frequency increased slightly as the flight speed increased. It is mentioned in Reference [21] that adding friction makes the frequency of LCO decrease slightly. Figure 19 compares time-domain frequencies in Figures 15b and 18b, which confirms this conclusion. Figure 20 shows the time-domain results, which also confirm that the six parameters had different effects on the system's stability. Figure 19: Comparison of time-domain frequencies for freeplay only and freeplay with friction. Figure 18: Comparison of the results of flutter analysis in time domain and frequency domain with friction, when \(\delta_{0}=1^{\circ}\) and \(e=0.2^{\circ}\), where (**a**) is the flight speed–\(A/e\) curve and (**b**) is the flight speed–oscillation frequency curve. In Figure 20, we can draw a conclusion that \(F_{\text{s}}\) and \(V_{\text{s}}\) have little influence on the aeroelastic characteristics. This corresponds to the fact that they have little effect on dynamic stiffness. As \(F_{\text{s}}\) increases, the flight speed at which the fin begins to oscillate first increases and then decreases slightly, while the oscillation amplitude at the same flight speed is also a function of the time. This is because the time-dependent nature of the flow is not affected by the flow. Figure 20: Influence of the parameters of the LuGre model on the response of the fin deflection angle at different flight speeds with \(\delta_{0}=1^{\circ}\) and \(e=0.2^{\circ}\), where (**a–f**), respectively, show the effects of \(\sigma_{0},\sigma_{1},\sigma_{2}\), \(F_{\text{c}}\), \(F_{\text{s}}\), and \(V_{\text{s}}\). speed remains almost unchanged. As \(V_{\rm s}\) increases, the flight speed at which the fin begins to oscillate also increases slightly, while the oscillation amplitude at the same flight speed is almost unchanged. \(\sigma_{1}\) seems to be the parameter that has the greatest influence on the aeroelastic characteristics in the variable-parameter analysis. It can be seen that increasing \(\sigma_{1}\) can greatly improve the stability of the system. When \(\sigma_{1}\) is 10 times the standard value, the fin deflection angle always converges within the flight speed range of interest. This parameter is difficult to quantify in actual engineering. It can be increased by changing the contact surface geometry, that is, increasing the roughness of the contact surface. More work is needed to quantify and standardize \(\sigma_{1}\) in the future. \(F_{\rm c}\), which stands for Coulomb friction, can be studied as a separate friction model. Compared with what is mentioned in Reference [20]--that increasing the maximum friction torque of Coulomb friction can increase the flutter speed and reduce the amplitude of the pitch response--we discover that the influence of \(F_{\rm c}\) on system stability is complicated. As \(F_{\rm c}\) increases, the speed at which the fin starts to oscillate first increases and then decreases, while the oscillation amplitude at the same flight speed increases. Therefore, \(F_{\rm c}\) is not the larger the better.