# Modeling and Analysis of a Generic Internal Cargo Airdrop System for a Tandem Helicopter
## 2 Internal Cargo Airdrop System
### Linear Dynamics Descriptions  
Using the separation method of the vector mechanics, the forces act on the backward moving cargo are the gravity of itself (\(m_{\text{L}}g\)), the pushing force provided by the two workers (\(\mathbf{F_{P}}\)), and the supporting force (\(N_{\text{B}}\)) and frictional force (\(f_{\text{B}}\)) provided by the helicopter.  
Figure 1: Sketch of the analysis technique of a cargo airdrop system for a tandem helicopter.  
Further using the discretization method, the gravity of the ith discrete section of the cargo in its own body axis system (\(\alpha x_{\text{L}}y_{\text{L}}z_{\text{L}}\)) is written as follows:  
\[\left[\begin{array}{c}\Delta F_{\text{L,v}}^{\text{i}}\\ \Delta F_{\text{L,v}}^{\text{i}}\\ \Delta F_{\text{L,z}}^{\text{i}}\end{array}\right]=\mathbf{T}(\theta_{\text{L}}) \cdot\mathbf{T}(\phi_{\text{B}})\cdot\mathbf{T}(\theta_{\text{B}})\left[\begin{array}[ ]{c}0\\ 0\\ \rho_{\text{L}}^{\text{i}}\cdot\Delta l\cdot g\end{array}\right] \tag{1}\]  
where \(\Delta F_{\text{L,v}}^{\text{i}}\), \(\Delta F_{\text{L,v}}^{\text{i}}\), and \(\Delta F_{\text{L,z}}^{\text{i}}\) are the gravitational components of the ith discrete section of the cargo in the cargo body axis system (\(\alpha x_{\text{L}}y_{\text{L}}z_{\text{L}}\)), \(\mathbf{T}(\cdot)\) the Euler transformation matrix, \(\theta_{\text{B}}\), \(\phi_{\text{B}}\), and \(\psi_{\text{B}}\) the respective pitching, rolling, and yawing attitude angles of the helicopter, \(\rho_{\text{L}}^{\text{i}}\) the linear density of the mass of the ith discrete section of the cargo, \(\Delta l\) the length of each discrete cargo section, and \(g\) the gravitational acceleration.  
In Equation (1), \(\Delta F_{\text{L,z}}^{\text{i}}\) also represents the normal supporting force of the ith discrete section of the cargo provided by the helicopter. Thus, the whole normal supporting force (\(N_{\text{B}}\)) and frictional force of the backward moving cargo (\(f_{\text{B}}\)) can be written as follows:  
\[\left\{\begin{array}{c}N_{\text{B}}=\sum\limits_{\text{i=1}}^{\text{N}} \Delta F_{\text{L,z}}^{\text{i}}\\ f_{\text{B}}=\sum\limits_{\text{i=1}}^{\text{N}}\left(\Delta F_{\text{L,z}}^{ \text{i}}+\left|\Delta F_{\text{L,y}}^{\text{i}}\right|\right)\cdot\mu\end{array}\right. \tag{2}\]  
where N is the total number of the discrete sections of the cargo, and \(\mu\) the equivalent coefficient of the friction between the cargo and the sliding rails. \(N_{\text{B}}\leq 0\) indicates that the cargo has left the helicopter, which can be taken as the first exit condition of the simulation.  
The basic equation of the linear dynamics of the moving cargo in the process of exaction from the helicopter cabin can be deduced as follows:  
\[\dot{V}_{\text{L}}=\left[F_{\text{P}}-\sigma(V_{\text{L}})\cdot f_{\text{B}}- \sum\limits_{\text{i=1}}^{\text{N}}\Delta F_{\text{L,x}}^{\text{i}}\right]/m_ {\text{L}} \tag{3}\]  
where \(V_{\text{L}}\) is the backward moving velocity of the cargo, \(F_{\text{P}}\) the pushing force provided by the workers acting on the cargo, and \(m_{\text{L}}\) the mass of the cargo. \(V_{\text{L}}<0\) indicates that the airdrop operation is failed, which can be taken as the second exit condition of the simulation. The symbolic function \(\sigma(V_{\text{L}})\) can be defined as follows:  
\[\sigma(V_{\text{L}})=\left\{\begin{array}{cc}\frac{V_{\text{L}}}{\left|V_{ \text{L}}\right|},&V_{\text{L}}\
eq 0\\ 1&,&V_{\text{L}}=0\end{array}\right. \tag{4}\]  
Theoretically, each worker can provide a maximum force of 1000 N to push the cargo [38]. Considering some of the possible unexpected factors, however, we assume that each worker provides 80% of the maximum pushing force. In addition, we also assume that when the cargo is going to rotate around the point of tipoff, the two workers withdraw the pushing force immediately. Thus, we can let \(F_{\text{P}}=0\) in this circumstance.