## 2 Numerical Simulation Methods  
### Rigid-Flexible Coupling Equation and Numerical Solving Method  
The derivation of the structural dynamics equation and the coupled equation of flight mechanics starts from the Lagrangian equation in quasi-coordinate form:  
\[ \begin{array}{c}\frac{d}{dt}\left(\frac{\partial L}{\partial\sigma_{C}} \right)+\widetilde{\alpha}\frac{\partial L}{\partial\
abla_{e}}-\mathrm{T}_{ \mathrm{G}\partial\sigma_{I}}\frac{\partial L}{\partial R_{I}}=F\\ \frac{d}{dt}\left(\frac{\partial L}{\partial\omega}\right)+\tilde{\
abla}_{c} \frac{\partial L}{\partial\
abla_{e}}+\widetilde{\alpha}\frac{\partial L}{ \partial\omega}-\mathrm{D}^{-\mathrm{T}\frac{\partial L}{\partial\theta}}=M\\ \frac{d}{dt}\left(\frac{\partial L}{\partial\eta}\right)-\frac{\partial L}{ \partial\eta}+\frac{\partial L}{\partial\eta}=Q\end{array}  \tag{1}\]  
where \(L\) presents the difference between kinetic energy and potential energy, \(\Gamma\) represents the internal energy dissipation of the aircraft, \(F\) and \(M\) represent the force and moment represented in the quasi-coordinate system, respectively, \(Q\) represents the generalized force, and (\(\widetilde{\omega}\)) is the cross-product operator representing a skew-symmetry matrix with the following form:  
\[\widetilde{\omega}= \begin{bmatrix}0&-\omega_{3}&\omega_{2}\\ \omega_{3}&0&-\omega_{1}\\ -\omega_{2}&\omega_{1}&0\end{bmatrix}  \tag{2}\]  
where \(\omega=\left[\omega_{1},\ \omega_{2},\ \omega_{3}\right]^{T}\). Figure 1 shows the comparison of the two states before and after the deformation of the aircraft; black is the original state, and red represents the deformed state. Here, \(O_{G}\) represents the inertial frame, where \(O_{b}\) represents the body axis system that is fixed to the aircraft. For any point P on the aircraft, its absolute position in the inertial coordinate system can be expressed as follows:  
\[R=R_{c}+\rho+d \tag{3}\]  
where \(R\) is the position vector of a point on the aircraft in the inertial coordinate system, \(R_{c}\) is the position vector of the origin of the body axis system in the inertial frame, \(\rho\) is the position vector of the point in the body axis system, and \(d\) is the deformation. Assume that the free vibration modes of the body are available; then, for that body undergoing general elastic deformation, the deformation can be described in terms of the mode shapes:  
\[d=\phi\cdot\eta \tag{4}\]  
where \(\eta\) is the generalized coordinate of the mode shape, a dimensionless quantity representing the generalized displacement of the model deformation. \(\phi\) represents the modal shape, which has the following form:  
\[\phi= \begin{bmatrix}x_{1}&x_{2}&\cdots&x_{n}\\ y_{1}&y_{2}&\cdots&y_{n}\\ z_{1}&y_{2}&\cdots&y_{n}\\ \end{bmatrix}  \tag{5}\]  
where \(x_{i}\), \(y_{i}\), and \(z_{i}\) represent the components of the three directions of the \(i^{th}\) structural mode, respectively. According to the definition of the modal superposition method, the velocity vector can be obtained:  
\[\dot{R}=V_{c}-\left(\widetilde{\rho}+\phi\widetilde{\eta}\right)\cdot\omega+ \phi\dot{\eta} \tag{6}\]  
where \(\dot{R}\) is absolute speed in the inertial coordinate system. The kinetic energy of a deformable aircraft can be formulated as follows:  
\[ \begin{array}{ll}T&=\frac{1}{2}\int\
u\,\rho_{m}\dot{R}^{T}\cdot\dot{R}dV\\ &=\frac{1}{2}\int\
u\,\rho_{m}V_{c}{}^{T}\cdot V_{c}dV+\int\
u\,\rho_{m}V_{c}{ }^{T}\left(\widetilde{\rho}+\phi\widetilde{\eta}\right)^{T}dV\cdot\omega\\ &+V_{c}{}^{T}\int\
u\,\rho_{m}\phi dV\cdot\dot{\eta}\\ &+\frac{1}{2}\omega^{T}\int\
u\,\rho_{m}\left(\widetilde{\rho}+\phi \widetilde{\eta}\right)^{T}\left(\widetilde{\rho}+\phi\widetilde{\eta}\right) dV\cdot\omega\\ &+\omega^{T}\int\
u\,\rho_{m}\left(\widetilde{\rho}+\phi\widetilde{\eta} \right)\phi dV\cdot\dot{\eta}\\ &+\frac{1}{2}\dot{\eta}^{T}\int\
u\,\rho_{m}\phi^{T}\phi dV\cdot\dot{\eta}\\ \end{array}  \tag{7}\]  
If the origin of \(O_{b}\) is placed at the undeformed aircraft center of gravity, the term \(\int_{V}\rho_{m}\widetilde{\rho}dV\) is zero.