# Modeling and Analysis of a Generic Internal Cargo Airdrop System for a Tandem Helicopter
## 2 Internal Cargo Airdrop System
### Forces and Moments Acting on the Helicopter  
We mainly discuss the forces and moments of the worker-cargo system acting on the helicopter according to the principle of the biomechanical analysis of fundamental human movements [38]. Using the separation method of vector mechanics, the frictional forces of the worker-cargo system act on the helicopter are written as follows:  
\[\left\{\begin{array}{l}F_{\rm P}^{\prime}=F_{\rm P}\\ f_{\rm L}=f_{\rm B}\end{array}\right. \tag{12}\]where \(F_{\rm P}^{\prime}\) is the frictional force provided by the workers acting on the helicopter, and \(f_{\rm L}\) the frictional force provided by the cargo acting on the helicopter.  
Two types of the flight operation are considered, including the helicopter flight operation carrying an internal cargo and the helicopter flight operation in the process of the cargo airdrop.  
1. In the first type of the flight operation, the cargo is stationary relative to the helicopter. Thus, we can let \(F_{P}=0\) in this circumstance. The gravity of the ith discrete section of the cargo acting on the helicopter in the helicopter body axis system (\(\alpha x_{\rm B}y_{\rm B}z_{\rm B}\)) can be written as follows: \[\left[\begin{array}{c}\Delta X_{\rm L,\ B}^{\rm i}\\ \Delta Y_{\rm L,\ B}^{\rm i}\\ \Delta Z_{\rm L,\ B}^{\rm i}\end{array}\right]=T(-\theta_{\rm L})\left[ \begin{array}{c}\Delta F_{\rm L,\ x}^{\rm i}\\ \Delta F_{\rm L,\ y}^{\rm i}\\ \Delta F_{\rm L,\ z}^{\rm i}\end{array}\right]\] (13)
2. In the second type of the flight operation, the cargo is moving backward and airdropping. Thus, the forces of the ith discrete section of the cargo acting on the helicopter in the helicopter body axis system (\(\alpha x_{\rm B}y_{\rm B}z_{\rm B}\)) can be written as follows: \[\left[\begin{array}{c}\Delta X_{\rm L,\ B}^{\rm i}\\ \Delta Y_{\rm L,\ B}^{\rm i}\\ \Delta Z_{\rm L,\ B}^{\rm i}\end{array}\right]=T(-\theta_{\rm L})\left[ \begin{array}{c}-\sigma(V_{\rm L})\cdot f_{\rm L}^{\rm i}\\ \Delta F_{\rm L,\ y}^{\rm i}\\ \Delta F_{\rm L,\ z}^{\rm i}\end{array}\right]\] (14)  
Thus, the resultant force vector of the cargo acting on the helicopter in the above two types of the flight operations can be written as follows:  
\[\mathbf{F}_{\rm L}=\sum\limits_{\rm i=1}^{\rm N}\left[\begin{array}{c}\Delta X_ {\rm L,\ B}^{\rm i}\\ \Delta Y_{\rm L,\ B}^{\rm i}\\ \Delta Z_{\rm L,\ B}^{\rm i}\end{array}\right]+\mathbf{T}(\phi_{\rm B})\cdot\mathbf{T}( \theta_{\rm B})\left[\begin{array}{c}0\\ 0\\ m_{\rm P}\cdot g\end{array}\right]+\left[\begin{array}{c}F_{\rm P}^{\prime}\\ 0\\ 0\end{array}\right] \tag{15}\]  
where \(m_{\rm P}\) is the mass of the workers.  
The resultant moment vector of the cargo acting on the helicopter in the above two types of the flight operations is written as follows:  
\[M_{\rm L}=\sum\limits_{\rm i=1}^{\rm N}\left\{\left[\begin{array}{c}-S_{ \rm i}(t)\\ 0\\ h\end{array}\right]\times\left[\begin{array}{c}\Delta X_{\rm L,\ B}^{\rm i}\\ \Delta Y_{\rm L,\ B}^{\rm i}\\ \Delta Z_{\rm L,\ B}^{\rm i}\end{array}\right]\right\}+\left[\begin{array}{c} -S(t)\\ 0\\ h\end{array}\right]\times\left\{T(\phi_{\rm B})\cdot T(\theta_{\rm B})\left[ \begin{array}{c}0\\ 0\\ m_{\rm P}\cdot g\end{array}\right]+\left[\begin{array}{c}F_{\rm P}^{\prime}\\ 0\\ 0\end{array}\right]\right\} \tag{16}\]  
where \(h\) is the vertical distance between the center gravity of the helicopter and its cabin floor in the helicopter body axis system (\(\alpha x_{\rm B}y_{\rm B}z_{\rm B}\)). The terms of the right hand of the equation represent the moment provided by the cargo and the workers, respectively.