## 3 Results and Discussion
### Multiple Freeplay Nonlinearities
#### 3.2.1 Discrete Scanning Method
Figure 8: Changes in flutter modal amplitude ratio \(R\) with \(A_{\rm B}\) and \(A_{\rm T}\). Equation (24) can be expressed as:  
\[\frac{A_{\rm B}/\delta_{\rm B}}{A_{\rm T}/\delta_{\rm T}}\frac{\delta_{\rm B}}{ \delta_{\rm T}}=f_{\rm R}(f_{\rm DF}(A_{\rm B}/\delta_{\rm B}),f_{\rm DF}(A_{ \rm T}/\delta_{\rm T})) \tag{25}\]  
where the nondimensional LCO amplitudes \(A_{\rm B}/\delta_{\rm B}\) and \(A_{\rm T}/\delta_{\rm T}\) can be considered as two variables, and the freeplay size ratio \(\delta_{\rm B}/\delta_{\rm T}\) is a parameter. Therefore, in this paper \(\delta_{\rm T}\) is set to 0.1\({}^{\circ}\), and the freeplay size ratio \(\delta_{\rm B}/\delta_{\rm T}\) is changed by using different \(\delta_{\rm B}\) values. The variation of the nondimensional LCO amplitudes with the dynamic pressure is obtained with different freeplay size ratios. It should be noted that although \(\delta_{\rm T}\) here is a definite value, no matter how it changes, as long as \(\delta_{\rm B}/\delta_{\rm T}\) remains unchanged, the solution of the nondimensional LCO amplitudes \(A_{\rm B}/\delta_{\rm B}\) and \(A_{\rm T}/\delta_{\rm T}\) corresponding to the equation will not change; the only thing that changes is the corresponding LCO amplitudes \(A_{\rm B}\) and \(A_{\rm T}\). Therefore, the nonlinear flutter characteristics of the system under different freeplay sizes \(\delta_{\rm B}\) and \(\delta_{\rm T}\) can be obtained by establishing the variation of the nondimensional LCO amplitudes of the system under different freeplay size ratios \(\delta_{\rm B}/\delta_{\rm T}\). Here, \(A_{\rm min}/\delta_{\rm T}\) is set to 1.2 to avoid static aeroelastic divergence, and \(A_{\rm max}/\delta_{\rm T}\) is set to 1000. The calculation error of \(A_{\rm T}/\delta_{\rm T}\) in the dichotomy method is required to be less than \(10^{-6}\). The scanning interval of \(A_{\rm T}\delta_{\rm T}\) is 0.01. When the scanning method is used, the computational domain is divided into \(n\) intervals: \([A_{\rm T}^{i},A_{\rm T}^{i+1}]\) (\(i=1,...,n\)). Then, we successively calculate the values \(F(A_{\rm T}^{i})\). If \(F(A_{\rm T}^{i})\cdot F(A_{\rm T}^{i+1})\leq 0\), then we use the dichotomy method to find the solution of \(F(A_{\rm T})=0\) while \(A_{\rm T}\in[A_{\rm T}^{i},A_{\rm T}^{i+1}]\). 2.2 Case 1: \(K_{\rm B}=5.0\times 10^{3}\) N \(\cdot\) m/rad, \(K_{\rm T}=2.0\times 10^{3}\) N \(\cdot\) m/rad  
In Case 1, a system with only a bending freeplay nonlinearity will exhibit LCO-ii characteristics; with only a torsion freeplay nonlinearity, it will exhibit LCO-i characteristics. Figure 10 shows the LCO solutions obtained by the discrete scanning method at different freeplay size ratios \(\delta_{\rm B}/\delta_{\rm T}\) and the corresponding time-domain bifurcation diagrams. The dynamic pressure range for the time-domain analysis was from 15 kPa to the divergence dynamic pressure. Here, different LCO branches are defined according to the solving interval and the continuity of the LCO solutions with the parameters. Figure 9: The process of solving Equation (24) by discrete scanning method. From the frequency-domain calculation results, we can find that there are two main LCO branches for each \(\dot{\delta}_{\rm B}/\delta_{\rm T}\). Combined with the time-domain results, it can be found that when the freeplay size ratio is small, stable LCO motions appear below the linear flutter dynamic pressure. In addition, irregular motions with limited amplitude are observed in a  
Figure 10: Frequency-domain LCO solutions and corresponding time-domain bifurcation diagrams in bending and torsional DOFs (\(\dot{\delta}_{\rm B}/\delta_{\rm T}\) is 0.01, 0.1, 0.5, 1.0, and 5.0 from top to bottom). certain region of dynamic pressure. With the increase of the freeplay size ratio, the dynamic pressure region of the LCO motions decreases, and the region of the irregular motions increases, but the amplitude tends to decrease.