# Dynamics Test Research of Bounce of Carrier-Based Aircraft Arresting Hook
## 3 Test Key Technology
### Simulation of Cushtioning Performance of Longitudinal Damper of Arresting Hook  
In the test the longitudinal damper of arresting hook is used, and the rebound diagram of the arresting hook is shown in Figure 10.  
In the figure, \(AB\) is the hook, \(PQ\) is the longitudinal buffer, \(P\) is the hinge point of the two, \(B\),\(Q\) are connection points between the hook bar and the fuselage. The length and angle of each segment are shown in the figure, \(\theta\) is the angle between the axis of the arresting hook and the vertical direction, \(\theta_{1}\) is the angle between the line of hook bar and fuselage and the vertical direction. The length of \(BQ\) is \(a\)  
Figure 10: Rebound diagram of the arresting hook.  
Figure 9: Design rendering and material object of disc.  
and the length of \(BP\) is \(b\), and the length of \(PQ\) is \(s\). \(F_{s}\) is the cushioning force of longitudinal buffer acting on \(P\), and \(m_{l}S\) is the gravity of the arresting hook acting on centroid \(C\).  
From the geometric relations in the graph, we can get the following conclusions:  
\[\left\{\begin{array}{c}s=\sqrt{a^{2}+b^{2}+2 ab\cos(\theta+\theta_{1})}\\ \dot{s}=\frac{-ab\sin(\theta+\theta_{1})}{\sqrt{a^{2}+b^{2}+2 ab\cos\cos( \theta+\theta_{1})}}\dot{\theta}\\ \Delta s=\sqrt{a^{2}+b^{2}+2 ab\cos(\theta_{0}+\theta_{1})}-\sqrt{a^{2}+b^{2}+ 2 ab\cos(\theta+\theta_{1})}\end{array}\right., \tag{3}\]  
where \(\Delta s\) is stroke of longitudinal damper, \(\theta_{0}\) is initial value of angle between arresting hook axis and vertical direction, \(\dot{\theta}\) is angular velocity of arresting hook rotating with the center of \(B\).  
According to the theorem of moment of momentum, it can be concluded that  
\[\left(J_{C}+m_{l}d_{1}^{2}\right)\ddot{\theta}=-m_{l}gd_{1}\sin\theta-F_{s}b \sin\theta. \tag{4}\]  
The principle of the longitudinal damper of the arresting hook is similar to that of the landing gear buffer. Here, it is a single buffer type, and its force \(F_{s}\) is mainly composed of air spring force and oil damping force. The internal structure of longitudinal buffer is shown in Figure 11.  
For single cavity buffer, the calculation formula of air spring force is  
\[F_{a}=A_{a}\bigg{[}P_{0}\bigg{(}\frac{V_{0}}{V_{0}-A_{a}\Delta s}\bigg{)}^{n} -P_{atm}\bigg{]}, \tag{5}\]  
where \(A_{a}\) is the compressed air area, \(P_{0}\) is the initial filling pressure of the buffer, \(V_{0}\) is the initial inflation volume of the buffer, \(n\) is the gas polytropic index, \(P_{atm}\) is the local atmospheric pressure, \(\Delta\)\(s\) is the damper stroke.  
The calculation formula of oil damping force is as follows:  
\[F_{h}=\frac{\rho_{oil}A_{Zh}^{3}\dot{s}}{2C_{d}^{2}A_{Zd}^{2}}\dot{s}|_{J}, \tag{6}\]  
where \(\rho_{oil}\) is density of the oil, \(A_{Zh}\) is the effective oil pressure area of the arresting hook damper, \(A_{Zd}\) is area of the damper oil hole, \(C_{d}\) is contraction coefficient of the oil hole.  
The final bouncing height of hook head is  
\[h_{A}=l\sin\beta-l\cos\theta-vt\sin\alpha_{l}, \tag{7}\]  
where \(\beta\) is the deck angle of aircraft when touching the deck, \(\alpha_{l}\) is angle of the glide path of aircraft.  
The heading displacement of hook head is  
\[x_{A}=l\cos\beta-l\sin\theta+vt\cos\alpha_{l}. \tag{8}\]