# Modeling, Simulation, and Cruise Characteristics of Wingtip-Jointed Composite Aircraft
## 2 Dynamic Modeling of Wingtip-Jointed Composite Aircraft
### Derivation of Dynamic and Kinematic Equations of the Composite System
where the Lagrange function is described as  
\[L=\sum_{k=1}^{n}\{\frac{1}{2}m_{k}(V_{kx}^{2}+V_{ky}^{2}+V_{kz}^{2})+\frac{1}{2}l_ {kx}\varphi_{k}^{2}+mgz_{k}\}. \tag{15}\]  
Thus, Equation (14) can be expanded as:  
\[\dot{\varphi}_{12}=\frac{l_{x}\dot{p}+(ml^{2}+\frac{1}{2}l_{x})\dot{p}_{12}+ml( \tilde{z}_{1e}\cos\varphi_{12}-\tilde{y}_{1e}\sin\varphi_{12})+\frac{1}{2}mgl}{ ml(\dot{z}_{1e}\sin\varphi_{12}+\dot{y}_{1e}\cos\varphi_{12})}, \tag{16}\]  
where \(Z_{1e}\) and \(y_{1e}\) represent the center of gravity of the aircraft on the right side. The zero-order variables in the composite body co-ordinate system are \((\alpha_{c},y_{c},z_{c},\varphi_{c},\dot{\alpha}_{c},\dot{\
u}_{12},\varphi_{12})\). The final forms for the multibody dynamic equation and the kinematic equation are derived as Equations (17) and (18), respectively:  
\[\left\{ \begin{array}{l}X-mg\sin\theta+T\cos\varphi_{T}=m(\dot{v}_{x}+\omega_ {y}v_{2}-\omega_{z}v_{y})+2\omega_{y}\dot{S}_{z}-2\omega_{z}\dot{S}_{y}+\dot{ \omega}_{y}S_{z}-\dot{\omega}_{z}\dot{S}_{y}+\omega_{y}\Big{(}\omega_{x}S_{y} -\omega_{y}S_{z}\Big{)}\\ -\omega_{z}(\omega_{z}S_{x}-\omega_{x}S_{z})+\tilde{S}_{x}\\ Y+mg\cos\theta\sin\varphi=m(\dot{v}_{y}+\omega_{z}v_{x}-\omega_{x}v_{z})+2 \omega_{z}\dot{S}_{x}-2\omega_{x}\dot{S}_{z}+\dot{\omega}_{z}S_{x}-\dot{ \omega}_{x}S_{z}+\omega_{z}\Big{(}\omega_{y}S_{z}-\omega_{z}S_{y}\Big{)}\\ -\omega_{x}\Big{(}\omega_{x}S_{y}-\omega_{y}S_{x}\Big{)}+\tilde{S}_{y}\\ Z+mg\cos\theta\cos\varphi-T\sin\varphi_{T}=m(\dot{v}_{z}+\omega_{x}v_{y}-\omega _{y}v_{x})+2\omega_{x}\dot{S}_{y}-2\omega_{y}\dot{S}_{x}+\dot{\omega}_{x}S_{y }-\dot{\omega}_{y}S_{x}\\ +\omega_{x}(\omega_{z}S_{x}-\omega_{x}S_{z})-\omega_{y}\Big{(}\omega_{y}S_{z}- \omega_{z}S_{y}\Big{)}+\tilde{S}_{z}\\ +\omega_{x}\Big{(}-l_{xy}\omega_{x}+l_{y}\omega_{y}-l_{zx}\omega_{z})-\omega_ {z}(S_{y}v_{z}-S_{y}v_{z})+l_{x}\omega_{x}-l_{xy}\omega_{y}-l_{zx}\omega_{z} \\ +\omega_{y}\Big{(}-l_{x}\omega_{x}-l_{yz}\omega_{y}+l_{z}\omega_{z}\Big{)}- \omega_{z}(l_{xy}\omega_{x}+l_{y}\omega_{y}-l_{zx}\omega_{z})\\ m=S_{z}\dot{v}_{x}-S_{z}\dot{v}_{z}+\dot{\omega}_{z}\Big{(}S_{y}v_{z}-S_{z}v_{y }\Big{)}-\omega_{x}\Big{(}S_{z}v_{y}-S_{y}v_{z}\Big{)}-l_{xy}\omega_{x}+\dot{ l}_{y}\omega_{y}-\dot{l}_{yz}\omega_{z}-l_{xy}\dot{\omega}_{z}\\ +\omega_{z}\Big{(}l_{x}\omega_{x}-l_{xy}\omega_{y}-l_{zx}\omega_{z}\Big{)}- \omega_{x}\Big{(}-l_{xz}\omega_{x}-l_{yz}\omega_{y}+l_{z}\omega_{z}\Big{)}\\ n=S_{z}\dot{v}_{y}-S_{y}\dot{v}_{x}+\omega_{x}(S_{z}v_{x}-S_{x}v_{z})-\omega_ {y}\Big{(}S_{y}v_{z}-S_{z}v_{y}\Big{)}-l_{zx}\omega_{x}-l_{yz}\omega_{y}+\dot{ l}_{z}\omega_{z}-l_{xz}\dot{\omega}_{x}-l_{yz}\dot{\omega}_{y}+l_{z}\dot{ \omega}_{z}\\ +\omega_{x}\Big{(}-l_{xy}\omega_{x}+l_{y}\omega_{y}-l_{yz}\omega_{z}\Big{)}+ \omega_{y}\Big{(}l_{x}\omega_{x}-l_{xy}\omega_{y}-l_{zx}\omega_{z}\Big{)}\\ \end{array} \right. \tag{17}\]  
\[\dot{\varphi}=p+q\sin\varphi\tan\theta+r\cos\varphi\tan\theta \tag{18}\]  
\[\dot{\theta}=q\cos\varphi-r\sin\varphi\qquad,\]  
\[\dot{\psi}=q\sin\varphi\sec\theta+r\cos\varphi\sec\theta\]  
\[\dot{\varphi}_{12}=\frac{l_{x}\dot{p}+(ml^{2}+\frac{1}{2}l_{x})\dot{p}_{12}+ml (\tilde{z}_{1e}\cos\varphi_{12}-\tilde{y}_{1e}\sin\varphi_{12})+\frac{1}{2}mgl }{m(\dot{z}_{1e}\sin\varphi_{12}+\dot{y}_{1e}\cos\varphi_{12})}.\]  
Equations (17) and (18) show that there are additional terms in the multibody dynamic equations, caused by the mass center shift in the composite flight process. Thus, there is a significant difference in the non-linear equations between composite aircraft with jointed wingtips and single conventional configuration aircraft. In Equation (17), \(S_{x}\), \(S_{y}\), and \(S_{z}\) represent the additional forces produced by mass center shift in composite flight, while \(\dot{S}_{z}\) and \(\ddot{S}_{z}\) represent the additional forces produced by the velocity and acceleration, respectively, of the aircraft's mass center shift.