## 2 Aeroelastic Model and Analysis Method  
### Structural Model  
Figure 1 shows a diagram of the typical all-movable fin in a supersonic flow that is analyzed in this paper. A right-handed reference coordinate system is coincident with the fin root. The \(x\)-axis is along the free-stream direction, and the \(y\)-axis is along the fin span. The flexibility of the fin root support is modeled by two springs that constrain its bending (B) and torsional (T) motions, and \(K_{\text{B}}\) and \(K_{\text{T}}\) denote their linear spring stiffnesses. Using the Branch Mode method [34], the linear fin can be divided into the following two branches: (1) a rigid fin with elastic supports at the root; (2) an elastic fin with rigid supports at the root. The bending and torsional modes of branch 1 and some elastic modes of branch 2 are retained for analysis. The displacement response in the \(z\) direction of the fin at position \((x,y)\) can be written as follows:  
\[z(x,y,t)=\Phi_{\text{B}}(x,y)q_{\text{B}}(t)+\Phi_{\text{T}}(x,y)q_{\text{T}}(t )+\sum_{j=1}^{n}\Phi_{\text{Ej}}(x,y)q_{\text{Ej}}(t) \tag{1}\]  
where \(\Phi_{\text{B}}\) and \(\Phi_{\text{T}}\) represent the mode shape functions of the bending and torsional modes, \(\Phi_{\text{Ej}}\) represents the \(j\)th elastic mode shape function, \(q_{\text{B}}\), \(q_{\text{T}}\) and \(q_{\text{Ej}}\) are the corresponding  
Figure 1: Simplified model of the all-movable fin examined in this paper. generalized coordinates, \(n\) is the number of modes retained in branch 2. By means of the Lagrange equations, the motion equations of the linear fin can be derived as  
(2)  
where \(I_{xx}\), \(I_{yy}\), and \(I_{xy}\) denote the mass moments of inertia about the corresponding axes in the reference coordinates, \(\mathbf{RE}\) is the coupling mass term of branches 1 and 2, \(\mathbf{M_{E}}\) and \(\mathbf{K_{E}}\) are the generalized mass and stiffness matrix of branch 2, \(\mathbf{q_{E}}\) is the corresponding generalized modal coordinate vector, \(Q_{\text{B}}\) and \(Q_{\text{T}}\) are the generalized aerodynamic forces, \(\mathbf{Q_{E}}\) is the generalized aerodynamic force vector. Generally, for a small aspect ratio supersonic fin, the structure of the fin is more rigid than the attachment part [35], which means the modal frequency of branch 2 is much higher than that of branch 1. So the consideration of branch 2 has little influence on the aeroelastic characteristics of the fin [36], and it can be approximated as a rigid body. Equation (2) can be reduced to the following form:  
\[ \begin{bmatrix}I_{xx}&-I_{xy}\\ -I_{xy}&I_{yy}\end{bmatrix}\begin{bmatrix}\dot{q}_{\text{B}}\\ \dot{q}_{\text{T}}\end{bmatrix} + \begin{bmatrix}K_{\text{B}}\\ &K_{\text{T}}\end{bmatrix}\begin{bmatrix}q_{\text{B}}\\ \dot{q}_{\text{T}}\end{bmatrix} = \begin{bmatrix}Q_{\text{B}}\\ Q_{\text{T}}\end{bmatrix}  \tag{3}\]  
For an all-movable fin with symmetric freeplay in the bending and torsional directions, the nonlinear elastic restoring moments \(M_{\text{B}}\) and \(M_{\text{T}}\) can be described by  
\[ \begin{split} M_{\text{B}}(q_{\text{B}})=\begin{cases}K_{\text{B} }(q_{\text{B}}-\delta_{\text{B}})&q_{\text{B}}>\delta_{\text{B}}\\ 0&-\delta_{\text{B}}\leq q_{\text{B}}\leq\delta_{\text{B}}\\ K_{\text{B}}(q_{\text{B}}+\delta_{\text{B}})&q_{\text{B}}<-\delta_{\text{B}} \end{cases}\\ M_{\text{T}}(q_{\text{T}})=\begin{cases}K_{\text{T}}(q_{\text{T}}-\delta_{ \text{T}})&q_{\text{T}}>\delta_{\text{T}}\\ 0&-\delta_{\text{T}}\leq q_{\text{T}}\leq\delta_{\text{T}}\\ K_{\text{T}}(q_{\text{T}}+\delta_{\text{T}})&q_{\text{T}}<-\delta_{\text{T}} \end{cases}\end{split}  \tag{4}\]  
where \(\delta_{\text{B}}\) and \(\delta_{\text{T}}\) represent the unilateral sizes of the freeplay in the bending and torsional directions, respectively. Thus, the motion equations of the nonlinear fin can be written in the following form. \[ \begin{bmatrix}I_{xx}&-I_{xy}\\ -I_{xy}&I_{yy}\end{bmatrix}\begin{bmatrix}\dot{q}_{\text{B}}\\ \dot{q}_{\text{T}}\end{bmatrix} + \begin{bmatrix}M_{\text{B}}\\ M_{\text{T}}\end{bmatrix} =