## 3 Results and Discussion
### Single Freplay Nonlinearity  
According to the DF method, different LCO amplitudes correspond to different spring stiffnesses. Therefore, the effect of different torsional or bending stiffness on the linear flutter dynamic pressure was investigated first, as shown in Figure 4a and Figure 4b, respectively. According to Figure 4 and Equation (12), the nonlinear aeroelastic characteristics of an all-movable fin with a single freeplay can be quickly and qualitatively judged [21]. The stiffness values given in Table 1 above correspond to state (i) in Figure 4. From Figure 4a we can see that when there is a freeplay nonlinearity in the torsional DOF, as the given nondimensional LCO amplitude \(A_{\rm T}/\delta_{\rm T}\) increases, the bending stiffness \(K_{\rm B}\) does not change, the equivalent torsional stiffness \(\bar{K}_{\rm T}\) increases continuously and tends to the torsional \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Description** & **Value** \\ \hline \(\omega_{1}\) & Modal frequency of mode 1 & 34.3 Hz \\ \(\omega_{2}\) & Modal frequency of mode 2 & 42.6 Hz \\ \(V_{f}\) & Flutter velocity & 921.3 m/s \\ \(Q_{f}\) & Flutter dynamic pressure & 51.8 kPa \\ \(\omega_{f}\) & Flutter frequency & 37.9 Hz \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Linear flutter results of the all-movable fin. Figure 3: Mode shapes of the all-movable fin. Figure 2: Geometrical schematic of the all-movable fin. All dimensions are in millimeters. stiffness of state (i), and the corresponding flutter dynamic pressure first decreases and then increases to the linear flutter dynamic pressure. Similarly, from Figure 4b, we can see that when there is a freeplay nonlinearity in the bending DOF, the flutter dynamic pressure decreases continuously to the linear flutter dynamic pressure with an increase in the given nondimensional LCO amplitude \(A_{\text{B}}/\delta_{\text{B}}\). Figure 5 shows the variation of nondimensional LCO amplitudes \(A_{\text{B}}/\delta_{\text{B}}\) and \(A_{\text{T}}/\delta_{\text{T}}\) with dynamic pressure in state (i); this is consistent with the above qualitative analysis results. In terms of the morphological characteristics of the LCO, for state (i), there are two different types of LCO when the freeplay nonlinearity exists in the torsional or bending DOF, and these are recorded as LCO-i and LCO-ii. It can be seen from Figure 5a that LCO-i contains both stable and unstable LCOs. The system response decays below \(Q_{\text{LCO}}\), which is here referred to as the LCO occurrence boundary. When the dynamic pressure \(Q\) reaches \(Q_{\text{LCO}}\), stable and unstable LCOs exist at the same time, and the system exhibits stable LCO motions in most initial conditions, while in some small initial conditions the system response decays. Then, the amplitude of the stable LCO motions in the nonlinear DOF will increase with increasing dynamic pressure, and this tends to infinity when the dynamic pressure \(Q\) tends to the linear-flutter dynamic pressure \(Q_{\text{LF}}\). When \(Q>Q_{\text{LF}}\), the system response will diverge in most initial conditions, while in some small initial conditions the system response decays. The discussion of initial conditions and the unstable LCOs will be expanded later. In summary, for a system with LCO-i characteristics, LCO may appear below the linear flutter boundary, bringing detrimental effects. Unlike LCO-i, as shown in Figure 5b, LCO-ii only contains unstable LCO. The system response decays when \(Q<Q_{\text{LF}}\). After \(Q\) reaches \(Q_{\text{LF}}\), the response of the system decays or diverges depending on the magnitude of the initial disturbance. The minimum initial condition \(q_{\text{B0}}/\delta_{\text{B}}\) required for divergence is also shown in Figure 5b with all other initial states at zero, and its value decreases as the dynamic pressure increases.