## 3 Results and Discussion
### Multiple Freeplay Nonlinearities
#### 3.2.1 Discrete Scanning Method  
For the convenience of the expression, the function \(f_{\rm DF}()\) is used here to represent the relationship between the equivalent stiffness and the nondimensional LCO amplitude, and the function \(f_{\rm R}()\) is used to represent the relationship between the flutter modal amplitude ratio \(R\) and the stiffnesses \(K_{\rm B}\) and \(K_{\rm T}\) in a linear system. The relevant expressions are as follows:  
\[ \begin{cases}\bar{K}_{\rm B}=f_{\rm DF}(A_{\rm B}/\delta_{\rm B})\\ \bar{K}_{\rm T}=f_{\rm DF}(A_{\rm T}/\delta_{\rm T})\\ R=f_{\rm R}(K_{\rm B},K_{\rm T})\end{cases}  \tag{23}\]  
Figure 7: Observation of the unstable LCO and the application of the harmonic initial condition in time-domain integration (\(K_{\rm B}=5.0\times 10^{3}\) N \(\cdot\) m/rad, \(K_{\rm T}=2.0\times 10^{3}\) N \(\cdot\) m/rad, \(Q=48.0\) kPa). The LCO solutions should satisfy the following equation to achieve alignment of the amplitude of each nonlinear element. \[A_{\rm B}/A_{\rm T}=f_{\rm R}[f_{\rm DE}(A_{\rm B}/\delta_{\rm B}),f_{\rm DF}(A_{ \rm T}/\delta_{\rm T})] \tag{24}\]  
The left side of Equation (24) represents the assumed LCO amplitude ratio, which is a monotonically decreasing function of \(A_{\rm T}\); the right side is the flutter modal amplitude ratio \(R\) of the equivalent linear system, which is a composite function composed of \(f_{\rm R}()\) and \(f_{\rm DF}()\). Solving the LCO characteristics of all-movable fins with both bending and torsion freeplay nonlinearities can be transformed into the following two equivalent problems:  
1. given \(\delta_{\rm B}\) and \(\delta_{\rm T}\), assuming \(A_{\rm B}\), find a suitable \(A_{\rm T}\) to satisfy Equation (24);
2. given \(\delta_{\rm B}\) and \(\delta_{\rm T}\), assuming \(A_{\rm T}\), find a suitable \(A_{\rm B}\) to satisfy Equation (24). Here, we choose the former. According to Equation (14), \(f_{\rm DF}()\) is a monotonically increasing function of \(A_{\rm T}\). For the subject of this study, \(f_{\rm R}()\) is shown in Figure 8. Given different \(A_{\rm B}\) values that correspond to different \(\hat{K}_{\rm B}\) values, the minimum point of \(R\) is roughly on the yellow dotted line. When \(A_{\rm B}\) is fixed, if \(K_{\rm T}\) is smaller than the value on the yellow dotted line, the right side of Equation (24) is a monotonically decreasing function of \(A_{\rm T}\); otherwise, the right side of Equation (24) first decreases and then increases with increasing \(A_{\rm T}\). For the latter case, the LCO amplitude \(A_{\rm T}\) corresponding to the minimum \(R\) is recorded as \(A_{\rm c}\) here. In contrast to previous studies, the ranges of the bending and torsional stiffnesses of the all-movable fin considered in this paper are very large so that its nonlinear aeroelastic characteristics can be comprehensively studied. The torsional stiffness may be relatively low, but no static divergence occurs. In this case, because the function on the right side of Equation (24) may not be monotonous, the previous iterative methods may cause a loss of solutions, and the iteration may not even converge. A more general method is to scan the entire solution interval to find the solutions of Equation (24). However, in the discrete scanning method proposed in this paper, a critical point \(A_{\rm c}\) is used to divide the solution interval into two parts: \(A_{\rm T}\in[A_{\rm min},A_{\rm c}]\cup[A_{\rm c},A_{\rm max}]\), where \(A_{\rm min}\) and \(A_{\rm max}\) are the specified minimum and maximum LCO amplitudes in the torsional DOF, respectively. As shown in Figure 9, when \(A_{\rm T}\in[A_{\rm c},A_{\rm max}]\), the dichotomy method can be directly used because the left and right sides of the equation have different monotonicity. When \(A_{\rm T}\in[A_{\rm min},A_{\rm c}]\), the scanning method is used. Compared with scanning the whole solution interval, this discrete scanning method improves the calculation efficiency while avoiding the loss of solutions.