# 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
\begin{array}{l}F_{x}=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}S_{y}+\omega_{y}\Big{(}\omega_{x}S_{y}-\omega_{y}S_{x}\Big{)}-\omega_{z}( \omega_{z}S_{x}-\omega_{z}S_{z})+\ddot{S}_{x}\\ F_{y}=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}_{z}S_{z}+\omega_{z} \Big{(}\omega_{y}S_{z}-\omega_{z}S_{y}\Big{)}-\omega_{x}\Big{(}\omega_{y}S_{y} -\omega_{y}S_{x}\Big{)}+\ddot{S}_{y}\\ F_{z}=m(\dot{v}_{z}+\omega_{x}v_{y}-\omega_{y}v_{x})+2\omega_{z}\dot{S}_{y}-2 \omega_{y}\dot{S}_{x}+\dot{\omega}_{x}\dot{S}_{y}-\dot{\omega}_{y}S_{x}+\omega _{x}(\omega_{z}S_{x}-\omega_{z}S_{z})-\omega_{y}\Big{(}\omega_{y}S_{z}-\omega_ {z}S_{y}\Big{)}+\ddot{S}_{z}\\ M_{x}=S_{y}\ddot{v}_{z}-S_{z}\dot{v}_{y}+\omega_{y}\Big{(}S_{x}v_{y}-S_{y}v_{x} \Big{)}-\omega_{z}(S_{z}v_{x}-S_{x}v_{z})+I_{x}\omega_{x}-I_{y}\omega_{y}-I_{z }\omega_{z}+I_{z}\alpha_{x}-I_{zy}\omega_{y}-I_{z}\alpha_{z}\\ +\omega_{y}\Big{(}-I_{x}\omega_{x}-I_{y}\omega_{x}+I_{z}\alpha_{z}\Big{)}- \omega_{z}(-I_{y}\omega_{x}+I_{y}\omega_{y}-I_{y}\omega_{z}\omega_{z})\\ M_{y}=S_{z}\ddot{v}_{x}-S_{z}\dot{v}_{z}+\omega_{z}\Big{(}S_{y}v_{z}-S_{z}v_{y} \Big{)}-\omega_{x}\Big{(}S_{x}v_{y}-S_{y}v_{x}\Big{)}-I_{zy}\alpha_{x}+\dot{I} _{y}\omega_{y}-\dot{y}_{z}\alpha_{z}-I_{xy}\omega_{z}\\ +\omega_{x}\Big{(}I_{x}\omega_{x}-I_{y}\omega_{x}-I_{z}\alpha_{z}\Big{)}-\omega _{x}\Big{(}I_{x}\omega_{x}-I_{z}\alpha_{x}-I_{y}\omega_{y}+I_{z}\alpha_{z}\\ M_{z}=S_{x}\ddot{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{)}-I_{z}\alpha_{x}-I_{yz}\alpha_{y} +\dot{I}_{z}\alpha_{z}-I_{zx}\alpha_{x}-I_{yz}\alpha_{y}+I_{z}\alpha_{z}\\ +\omega_{x}\Big{(}-I_{xy}\omega_{x}+I_{y}\omega_{y}-I_{z}\alpha_{z}\Big{)}+ \omega_{y}\Big{(}I_{x}\omega_{x}-I_{y}\omega_{y}-I_{z}\alpha_{z}\Big{)}\end{array} \right. \tag{9}\]  
Meanwhile, the overall kinematic equations of the composite system can be derived as  
\[\left[ \begin{array}{c}\dot{x}_{\rm e}\\ \dot{y}_{e}\\ \dot{z}_{e}\end{array} \right]=L_{ef}(\varphi,\theta,\psi)\left[ \begin{array}[] {c}u\\ v\\ w\end{array} \right],\left[ \begin{array}{c}\dot{\varphi}=p+q\sin\varphi\tan \theta+r\cos\varphi\tan\theta\\ \dot{\theta}=q\cos\varphi-r\sin\varphi\\ \dot{\psi}=q\sin\varphi\sec\theta+r\cos\varphi\sec\theta\end{array} \right.. \tag{10}\]  
In the composite flight process, internal relative rotational movement exists on the additional 7 th DOF, compared with a regular single aircraft's flight process. As shown in Figure 5, \(\varphi_{12}\) is defined as the relative roll angle between the aircraft on the right side (subscript 1) and the one on the left side (subscript 2). According to the moment of momentum theorem, the relationship between the motion and moment of momentum of the composite system can be expressed as:  
\[\dot{\varphi}_{12}=-\frac{l_{1}}{I_{O/1}}+\frac{l_{2}}{I_{O/2}}, \tag{11}\]  
which can be derived as:  
\[I_{O_{f}1}I_{O/2}\dot{\varphi}_{12}=l_{2}I_{O/1}-l_{1}I_{O/2}. \tag{12}\]  
The geometric movement state and motion relationship can be derived as:  
\[\left\{ \begin{array}{c}\varphi_{12}=\varphi_{2}-\varphi_{1}\\ V_{x2}=V_{x1}\\ V_{y2}=V_{y1}-\dot{\varphi}_{12}I\sin\varphi_{12}\\ V_{z2}=V_{z1}-\dot{\varphi}_{12}I\cos\varphi_{12}\end{array} \right. \tag{13}\]  
Based on the second-order Lagrange equation  
\[\frac{d}{dt}[\frac{\partial L}{\partial\dot{\varphi}_{12}}]-\frac{\partial L}{ \partial\dot{\varphi}_{12}}=0, \tag{14}\]  
Figure 5: Internal relative movement of the multibody system.