## 3 Results and Discussion
### Multiple Freeplay Nonlinearities
#### 3.2.1 Discrete Scanning Method
When the freeplay size ratio increases to 0.5, the nondimensional amplitude of the vibration is very small and almost entirely located near the freeplay boundary. After the LCO solutions are obtained, the equivalent stiffness corresponding to the nondimensional LCO amplitude in the bending and torsional DOFs can be calculated. Therefore, each set of LCO nondimensional amplitudes \([A_{\rm B}/\delta_{\rm B},A_{\rm T}/\delta_{\rm T}]\) will correspond to a point \([K_{\rm B},K_{\rm T}]\) in Figure 6. Finally, complete LCO solutions at different \(\delta_{\rm B}/\delta_{\rm T}\) can be drawn in the form of Figure 11. The yellow dotted line in Figure 11 is the same as in Figure 8. For each given \(A_{\rm B}\), it divides the solution domain into upper and lower parts, which correspond to the solution intervals \([A_{\rm c},A_{\rm max}]\) and \([A_{\rm min},A_{\rm c}]\). Finally, the results of branches A and B located on either side of the yellow line are obtained. The LCO solutions of a system with a single torsion or bending freeplay nonlinearity can be seen in the Figure 5 map to the right and the upper dotted lines in Figure 11, respectively. When \(\delta_{\rm B}/\delta_{\rm T}\) decreases, branch A gradually approaches the upper dotted line, and the LCO characteristics of a system with multiple freeplay nonlinearities are more similar to a system with only a torsion freeplay nonlinearity, that is, LCO-i. Similarly, when \(\delta_{\rm B}/\delta_{\rm T}\) increases, the LCO characteristics tend to LCO-ii. Comparing the frequency-domain results at \(\delta_{\rm B}/\delta_{\rm T}=0.01\) in Figures 5a and 10, it can be inferred that when \(\delta_{\rm B}/\delta_{\rm T}\) decreases to 0, the LCO of branch A finally will become the stable LCO branch in Figure 5a, while branch B becomes the unstable LCO branch. As \(\delta_{\rm B}/\delta_{\rm T}\) increases to infinity, the LCO solutions in the bending DOF of branch A will finally become the unstable LCO branch in Figure 5b, while the LCO amplitude of branch B will be reduced to become negligible. In summary, when the difference between the two freeplay sizes is large, the nonlinear aeroelastic characteristics of the system are mainly determined by the freeplay with the larger value. In addition, expressing the LCO solutions in the form of Figure 11 allows more concise and clear observation of the variation rule of the LCO characteristics of the system with the freeplay size ratio. This also contains the variation of the nondimensional LCO amplitude with dynamic pressure. Further, we construct the Poincare section by plotting \(\dot{q}_{\rm B}\) against \(q_{\rm B}\) at time instances when \(q_{\rm T}=\dot{\delta}_{\rm T}\) and \(\dot{q}_{\rm T}>0\). For the Poincare sections to be meaningful, the total simulation time was set to 200 s at each dynamic pressure state, and the last 100 s of data was selected for analysis. In Figure 12, dynamic pressures between 44.0 kPa and 47.6 kPa were chosen for the analysis at \(\dot{\delta}_{\rm B}/\delta_{\rm T}=0.01\). Because the system response decays, the response trajectories do not intersect with the Poincare section when the dynamic pressure is less than 44.2 kPa. A closed curve appears suddenly at 44.4 kPa, and at about 46.2 kPa, this becomes a single point. Figure 13 shows a plot of the response trajectory of the system in the \(q_{\rm T}\)-\(\dot{q}_{\rm T}\) phase plane at 45.0 kPa. The response trajectory of the system is always within a torus. It can be clearly seen that a Hopf bifurcation occurs before 44.4 kPa. The system response changes from decaying motions to quasi-periodic motions somewhere between 44.2 and 44.4 kPa, and it transforms into LCO motions at about 46.2 kPa. From the frequency-domain LCO solutions in Figure 10, we can find two LCO branches that grow from a point around 38 kPa, which have a similar amplitude. Until approximately 46 kPa, the difference in the torsional LCO amplitude increases sharply due to the turning point of branch B.