# Launch Bar Dynamics Character Analysis of Carrier-Based Aircraft Catapult Launch
## 2 Mathematical Model of Catapult Launch
### Mathematical Model of the Launch Bar
#### 2.2.2 The Mathematical Model of the Launch Bar after the Launch Bar Automatically Disengages  
At the end of the catapult, the launch bar pops out of the shuttle, as presented in Figure 3.  
Figure 3: The system of the launch bar, nose gear, and shuttle. The equation of the cg position of the launch bar is as follows:  
\[\left\{ \begin{array}{l}x_{cg}=x_{O}+K_{launch}L_{launch}\sin\theta_{r}\\ z_{cg}=z_{O}+K_{launch}L_{launch}\cos\theta_{r}\end{array} \right., \tag{4}\]  
where \(\theta_{r}\) is the angle between the launch bar and nose gear. Because the pivot point coordinate system is parallel to the carrier-body coordinate system, the equation of the momentum theorem for the launch bar is given by  
\[M_{r}-M_{g}-M_{a}-M_{t}=\dot{J}\ddot{\theta}_{r}, \tag{5}\]  
where \(M_{r}\) is the restoring moment of the launch bar, \(M_{g}\) is the moment of gravity of the launch bar, \(M_{a}\) is the moment of the force of inertia due to the longitudinal acceleration of the pivot point, \(M_{t}\) is the moment of the force of inertia due to the nose landing gear's sudden extension, \(J\) is the moment of inertia of the launch bar, and \(\ddot{\theta}_{r}\) is the angular acceleration of the launch bar. The equation of the angle between the launch bar and nose gear is as follows:  
\[\theta_{r}=\theta_{r0}+\Delta\theta_{r}, \tag{6}\]  
where \(\theta_{r0}\) is the initial angle between the launch bar and nose gear, and \(\Delta\theta_{r}\) is the change value of the angle between the launch bar and nose gear. The equation of the restoring moment of the launch bar is as follows:  
\[M_{r}=k_{r}, \tag{7}\]  
where \(k_{r}\) is the restoring moment of the launch bar, which is a constant in this article. The equation of the moment of gravity of the launch bar is as follows:  
\[M_{g}=m_{launch}k_{launch}L_{launch}\sin\theta_{r}, \tag{8}\]  
where \(m_{launch}\) is the mass of the launch bar, and \(g\) is the acceleration due to gravity. The equation of the moment of the force of inertia due to the longitudinal acceleration of the pivot point is as follows:  
\[M_{a}=m_{launch}ak_{launch}L_{launch}\cos\theta_{r}, \tag{9}\]  
where \(a\) is longitudinal acceleration of the aircraft. The equation of the moment of the force of inertia due to the nose landing gear's sudden extension is as follows:  
\[M_{t}=m_{launch}a_{2}K_{launch}L_{launch}\sin\theta_{r}, \tag{10}\]  
where \(a_{2}\) is the vertical acceleration of the elastic support mass. The vertical acceleration of the elastic support mass can be calculated as follows:  
\[a_{2}=\left\{ \begin{array}{ll}\frac{m_{2}g-F_{n}-f_{n}-F_{h}-Y_{n}}{m_{2}}& \dot{u}\geq 0\\ \frac{m_{2}g-F_{n}+f_{3}+F_{h}-Y_{n}}{m_{2}}&\dot{u}<0\end{array} \right., \tag{11}\]  
where \(m_{2}\) is the elastic support mass of the nose gear, \(F_{n}\) is the air spring force of the nose gear, \(f_{3}\) is the strut friction force of the nose gear, \(F_{h}\) is the hydraulic damping force of the nose gear, \(Y_{n}\) is the equivalent aerodynamic lift acting on the nose gear, and \(\dot{u}\) is the gear stroking velocity of the nose gear. The air spring force can be calculated as follows:  
\[F_{n}=A_{a}\bigg{[}P_{0}\bigg{(}\frac{V_{0}}{V_{0}-A_{a}\cdot u}\bigg{)}^{ \
u}-P_{atm}\bigg{]}, \tag{12}\]where \(A_{a}\) is the effective area of pressure in the air chamber, \(P_{0}\) is the initial pressure of air in the chamber, \(V_{0}\) is the initial volume of air in the chamber, \(u\) is the stroke of the damper, \(\gamma\) is gas polytropic index which varies between 1.1 and 1.4, and \(P_{atm}\) is the air pressure. The equation for calculating the hydraulic damping force is shown below.