# 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  
In the flying process of wingtip-jointed composite aircraft, the relative movement between each aircraft can cause a shift of the mass center of the multibody system. As shown in Figure 4, composite body co-ordinate system (\(o_{fy}x_{fy}y_{fy}z_{fb}\)) was chosen, where the origin of this co-ordinate system is located at the wingtip joint. A composite air co-ordinate system (\(Ox_{fa}y_{fa}z_{fa}\)) and path coordinate system (\(Ox_{fy}y_{fa}z_{fb}\)) are defined in the corresponding method. The ground co-ordinate system is described as \(o_{g}x_{g}y_{g}z_{g}\). According to multibody dynamics theory, the velocities of the two aircraft in the multibody system can be described as:  
\[\overset{\rightarrow}{v}_{i}=\overset{\rightarrow}{v}_{0}+\overset{ \rightarrow}{r}_{OGi}+\overset{\rightarrow}{a}_{i}\times\overset{ \rightarrow}{r}_{OGi},\ i=1,2. \tag{1}\]  
The momentum \(\overset{\rightarrow}{p}\) and the moment of momentum \(\overset{\rightarrow}{L}\) can be described as:  
\[ \begin{array}{l}\overset{\rightarrow}{p}=\sum\limits_{i}^{n}m_{i}\bigg{(} \overset{\rightarrow}{v}_{i}+\overset{\rightarrow}{r}_{OGi}+\overset{ \rightarrow}{a}_{i}\times\overset{\rightarrow}{r}_{OGi}\bigg{)},\ i=1,2.\\ \overset{\rightarrow}{L}=\sum\limits_{i}^{n}\bigg{(}\overset{\rightarrow}{r}_ {OGi}\times m\overset{\rightarrow}{v}_{i}+\overset{\rightarrow}{I}_{CIi} \cdot\left(\overset{\rightarrow}{a}+\overset{\rightarrow}{a}_{i}\right) \bigg{)},\ i=1,2.\end{array}  \tag{2}\]  
The augmented body method [13] introduced the definition of the augmented body mass M, the augmented body rotation tensor \(\overset{\leftrightarrow}{I}_{c}\), and the augment body static moment \(\overset{\rightarrow}{S}^{x}\). The relative equations are defined, respectively, as follows:  
\[M=\sum\limits_{i=1}^{k}(m_{i}), \tag{3}\]  
\[\overset{\leftrightarrow}{I}_{C}^{\ast}=\iint_{V}\rho\bigg{(}\overset{ \rightarrow}{r}^{2}\overset{\leftrightarrow}{1}-\overset{\rightarrow}{r} \overset{\rightarrow}{r}\bigg{)}dV=\left[ \begin{array}{cccc}I_{x}^{\ast}&-I _{xy}^{\ast}&-I_{zx}^{\ast}\\ -I_{xy}^{\ast}&I_{y}^{\ast}&-I_{yz}^{\ast}\\ -I_{zx}^{\ast}&-I_{yz}^{\ast}&I_{z}^{\ast}\end{array} \right] \tag{4}\]  
Figure 4: Wingtip-jointed aircraft system body co-ordinates. Figure 3: Motion analysis of the composite aircraft multibody system. \[\overrightarrow{S}^{*}=\int\overrightarrow{r}\times dm=S_{x}\overrightarrow{i}+S_{y }\overrightarrow{j}+S_{z}\overrightarrow{k}. \tag{5}\]  
The numbers of augmented body parameters in different cases are shown in Table 1. The variable double composite aircraft numbers change obviously, compared with those for the single aircraft case. The relative roll angle, \(\varphi_{12}\), can determine the corresponding numerical value of the augmented body parameters. Based on Equations (2)-(5), the equations of \(\overrightarrow{p}\) and \(\overrightarrow{L}\) become  
\[ \begin{array}{l}\overrightarrow{p}=M\cdot\overrightarrow{v}+\overset{ \cdot\cdot}{S}^{*}+\overset{\cdot\cdot}{\overrightarrow{\omega}}\times \overset{\rightarrow}{S}\\ \overrightarrow{L}=\overrightarrow{S}\times\overset{\rightarrow}{v}+ \overset{\cdot\cdot\cdot}{I}_{O}\cdot\overset{\rightarrow}{\omega}\end{array}  \tag{6}\]  
The origin of the composite body co-ordinate system is not always located at the mass center of the whole multibody system, according to the momentum theorem:  
\[ \begin{array}{l}\overrightarrow{F}=\overrightarrow{p}\\ \overrightarrow{M}=\overset{\cdot}{L}+\overrightarrow{r}\times m\overrightarrow {v}=\overset{\cdot}{L}+\overrightarrow{v}\times\overset{\cdot}{S}\end{array}  \tag{7}\]  
Equation (7) can be further derived as:  
\[\left\{