## 3 Results and Discussion
### Single Freplay Nonlinearity
In summary, the divergence boundary of a nonlinear system with LCO-ii characteristics is not lower than the flutter boundary of the linear system, indicating that the freeplay nonlinearity will instead bring favorable effects. Figure 4: Linear flutter dynamic pressure with the variation of torsional or bending stiffness. (I: monotonically increasing; II: monotonically decreasing). In fact, the minimum point of the dynamic pressure in Figure 4 divides the entire calculation region into two parts, that is, I and II. The position of the support stiffness in the nonlinear DOF determines the LCO characteristics of the fin. In region I, the system exhibits LCO-i characteristics, while in region II, the system exhibits LCO-ii characteristics. In addition, we can calculate the variation of the linear flutter dynamic pressure with different bending and torsional stiffnesses, as shown in Figure 6. In Figure 6, along the horizontal direction, the bending stiffness of the fin changes, while along the vertical direction, the torsional stiffness changes. The blue arrow indicates that the linear flutter dynamic pressure decreases with increasing stiffness, while the magenta arrow indicates that it increases. The results shown in Figure 4a,b finally map to the vertical and horizontal lines through point (i) in Figure 6, respectively. For example, the vertical line through point (i) (marked as (i)-V) reflects that the fin with single torsion freeplay nonlinearity (marked as Ndot = T) exhibits LCO-i characteristics, corresponding to the results of Figures 4a and 5a. Moreover, the horizontal line through the point (i) (marked as (i)-H) reflects that the fin with single bending freeplay nonlinearity (marked as Ndot = B) exhibits LCO-ii characteristics, corresponding to the results of Figures 4b and 5b. Similarly, it can be found that the LCO characteristics of the fin in state (ii) are completely opposite to those of state (i). Furthermore, the tangent points of the pressure contour lines and the vertical or horizontal lines can be connected into two orange straight lines; the slopes of these lines are recorded here as \(\xi=\xi_{\rm T}\) and \(\xi=\xi_{\rm B}\), where \(\xi\) represents the ratio of torsional and bending stiffness. Another understanding of \(\xi_{\rm B}\) and \(\xi_{\rm T}\) is that if the bending stiffness \(K_{\rm B}\) is constant, when \(K_{\rm T}\) satisfies \(K_{\rm T}/K_{\rm B}=\xi_{\rm T}\), the flutter dynamic pressure \(Q_{f}\) will achieve its minimum value; similarly, if the torsional stiffness \(K_{\rm T}\) is constant, \(Q_{f}\) achieves the minimum value when \(K_{\rm B}\) satisfies \(K_{\rm T}/K_{\rm B}=\xi_{\rm B}\). When there is a torsion freeplay nonlinearity, \(\xi=\xi_{\rm T}\) divides the entire computational domain into two parts, where \(\xi\leq\xi_{\rm T}\) corresponds to LCO-ii, and \(\xi>\xi_{\rm T}\) corresponds to LCO-i. When there is a bending freeplay nonlinearity, \(\xi=\xi_{\rm B}\) divides the entire computational domain into two parts, where \(\xi<\xi_{\rm B}\) corresponds to LCO-i, and \(\xi\geq\xi_{\rm B}\) corresponds to LCO-ii. Therefore, if the bending and torsional stiffnesses of the system are located between the lines \(\xi=\xi_{\rm T}\) and \(\xi=\xi_{\rm B}\), that is, \(\xi_{\rm T}<\xi<\xi_{\rm B}\), the system exhibits LCO-i characteristics no matter whether there is a bending or torsion freeplay nonlinearity. Furthermore, from the results of the calculation, it can be seen that changing the bending and torsional stiffness does not change the values of \(\xi_{\rm B}\) and \(\xi_{\rm T}\); therefore, the LCO characteristics of the system can be changed by changing the ratio of bending and torsional stiffnesses. Figure 5: Nonlinear aeroelastic characteristics of an all-movable fin with torsion or bending freeplay (\(K_{\rm B}=5.0\times 10^{3}\) N \(\cdot\) m/rad, \(K_{\rm T}=2.0\times 10^{3}\) N \(\cdot\) m/rad). Figure 5 also shows the results of stable and unstable LCOs obtained from time-domain simulations.