# High-Fidelity Fin-Actuator System Modeling and Aeroelastic Analysis Considering Friction Effect
## 6 Conclusions
\begin{array}{cc}\mathbf{M_{qq}}&\mathbf{M_{q\delta}}\\ \mathbf{M_{\delta q}}&\mathbf{M_{\delta\delta}}\end{array} \right]\), \(\mathbf{C}=\left[ \begin{array}{cc}\mathbf{C_{qq}}&\mathbf{0}\\ \mathbf{0}&0\end{array} \right]\), \(\mathbf{K}=\left[ \begin{array}{cc}\mathbf{K_{qq}}&\mathbf{0}\\ \mathbf{0}&k_{\delta\delta}(\omega)\end{array} \right]\), \(\mathbf{F}=\left[ \begin{array}{cc}\mathbf{f_{q}}\\ \mathbf{f_{\delta}}\end{array} \right]\), \(\mathbf{M_{qq}}\), \(M_{q\delta^{z}}\), \(M_{\delta\delta}\) are the generalized mass matrix of the elastic modes, the coupling mass vector between the elastic modes and rigid mode, and the rotational inertia of the fin, respectively. \(\mathbf{K_{qq}}\) is the generalized stiffness matrix and corresponds to the elastic modes. \(\mathbf{C_{qq}}\) is the generalized damping matrix and corresponds to the elastic modes. \(\omega\) is the oscillation frequency; \(k_{\delta\delta}(\omega)\) is the supporting stiffness provided by the actuator. The external load is \(\mathbf{F^{\Gamma}}\mathbf{=}\left[\mathbf{f_{q^{\prime}}},\;f_{\delta}\right]\), where \(\mathbf{f_{q}}\) is the generalized force vector corresponding to \(\mathbf{q_{e^{\prime}}}\) and \(f_{\delta}\) is the generalized force corresponding to \(\delta\). The generalized coordinates selected in this way decouples the stiffness of the actuator from the structural stiffness of the fin, according to the inertia-coupling method, making the stiffness matrix \(\mathbf{K}\) a diagonal matrix, which is beneficial to the study of actuator dynamics [65]. \(M\) in Equation (A1) is obtained by Equation (A2) to Equation (A4):  
\[M_{qq}=\Phi_{q}^{\mathrm{T}}M_{s}\Phi_{q}\] (A2)  
\[M_{q\delta}=M_{\delta q}^{\mathrm{T}}=\Phi_{q}^{\mathrm{T}}M_{s}\Phi_{\delta}\] (A3)  
\[M_{\delta\delta}=\Phi_{\delta}^{\mathrm{T}}M_{s}\Phi_{\delta}\] (A4)  
where \(\Phi_{q}\) is the modal matrix of the elastic branch. \(\Phi_{\delta}\) is the modal matrix of the rotational rigid branch. \(M_{s}\) is the discrete mass matrix. Equation (A1) can be written as:  
\[\left[ \begin{array}{cc}M_{qq}&M_{q\delta}\\ M_{\delta q}&M_{\delta\delta}\end{array} \right]\left[ \begin{array}{c}\ddot{q }_{e}\\ \dot{\delta}\end{array} \right]+\left[ \begin{array}{cc}C_{qq}&0\\ 0&0\end{array} \right]\left[ \begin{array}{c}\ddot{q}_{e}\\ \dot{\delta}\end{array} \right]+\left[ \begin{array}{cc}K_{qq}&0\\ 0&k_{\delta\delta}(\omega)\end{array} \right]\left[ \begin{array}{c}q_{e}\\ \dot{\delta}\end{array} \right]=\left[ \begin{array}{c}f_{q}\\ f_{\delta}\end{array} \right]\] (A5)  
The stiffness term can be expanded as:  
\[\left[ \begin{array}{cc}K_{qq}&0\\ 0&k_{\delta\delta}(\omega)\end{array} \right]\left[ \begin{array}{c}q_{e}\\ \dot{\delta}\end{array} \right]=\left[ \begin{array}{cc}K_{qq}&0\\ 0&0\end{array} \right]\left[ \begin{array}{c}q_{e}\\ \dot{\delta}\end{array} \right]+\left[ \begin{array}{cc}0&0\\ 0&k_{\delta\delta}(\omega)\end{array} \right]\left[ \begin{array}{c}q_{e}\\ \dot{\delta}\end{array} \right]\] (A6) \[=\left[ \begin{array}{cc}K_{qq}&0\\ 0&0\end{array} \right]\left[ \begin{array}{c}q_{e}\\ \dot{\delta}\end{array} \right]+\left[ \begin{array}{c}0\\ k_{\delta\delta}(\omega)\delta\end{array} \right]\]  
the torque of the actuator on the fin is:  
\[M_{\mathrm{t}}=-k_{\delta\delta}(\omega)\delta\] (A7)  
therefore, Equation (5) can be written as  
\[\left[ \begin{array}{cc}M_{qq}&M_{q\delta}\\ M_{\delta q}&M_{\delta\delta}\end{array} \right]\left[ \begin{array}{c}\ddot{q }_{e}\\ \dot{\delta}\end{array} \right]_{+}\!\!\left[ \begin{array}{cc}C_{qq}&0\\ 0&0\end{array} \right]\left[ \begin{array}{c}\dot{q}_{e}\\ \dot{\delta}\end{array} \right]_{+}\!\!\left[ \begin{array}{cc}K_{qq}&0\\ 0&0\end{array} \right]\!\left[ \begin{array}{c}q_{e}\\ \dot{\delta}\end{array} \right]\!\!=\!\!\left[ \begin{array}{c}0\\ M_{\mathrm{t}}\end{array} \right]+\left[ \begin{array}{c}f_{q}\\ f_{\delta}\end{array} \right]\] (A8)  
\(F^{\mathrm{T}}\!=\left[f_{q^{\prime}}\;f_{\delta}\right]\) is the aerodynamic force or external exciting force on the fin.