# Research on the Performance of an Aircraft Landing
## 3 Results of Dynamic and Static Analyses
### Bifurcation Analysis Results  
The equilibrium points of this mechanism system were determined using the numerical continuation algorithm in MATLAB (Mathworks, Natick, US). All the parameters are shown in Table 1 and the results are displayed in Figure 5.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameters** & **Values** & **Parameters** & **Values** & **Parameters** & **Values** \\ \hline \(l_{1}\) (mm) & 40 & \(a_{1}\) (mm) & 12.5 & \(\alpha_{1}\) (degree) & 2.5 \\ \(l_{2}\) (mm) & 41 & \(a_{2}\) (mm) & 22 & \(a_{2}\) (degree) & 18 \\ \(l_{3}\) (mm) & 54 & \(b_{1}\) (mm) & 21 & \(\beta_{1}\) (degree) & 32 \\ \(m_{1}\) (kg) & 0.1 & \(b_{2}\) (mm) & 45 & \(\beta_{2}\) (degree) & 82.5 \\ \(m_{2}\) (kg) & 0.1 & \(c_{1}\) (mm) & 21 & \(\gamma_{1}\) (degree) & 37.8 \\ \(m_{3}\) (kg) & 0.2 & \(c_{2}\) (mm) & 21.6 & \(\gamma_{2}\) (degree) & 42.2 \\ \(m\) (mm) & 62 & \(n\) (mm) & 50 & & \\ \hline \hline \end{tabular}
\end{table}
Table 1: Each parameter of hook lock.  
Figure 4: Pressure curves from hook lock performance test.  
In Figure 5, the black line represents the locking process, where \(F_{Unlock}=0\), and the dark gray line represents the unlocking process, where M = 0, namely, \(M_{\omega}=0\). Here, the unlocking process is studied first without considering the influence of gravity on the landing gear temporarily. Detailed discussions will be given in the following section. Meanwhile, the solid line represents the stable equilibrium point, and the dotted line represents the unstable equilibrium point. If the system undergoes a small disturbance near a stable equilibrium point, it will converge to the stable point. Moreover, if the system suffers a small disturbance near an unstable equilibrium point, it will immediately jump to a stable equilibrium point.  
Combined with the schedules of the hook lock mechanism in Figure 6, the locking and unlocking process can be inferred from the equilibrium solution curves. In Figure 5, position 1 is the stable equilibrium point of the unlocked state. When the hook lock begins to lock, the upper lock moves from position 1 to position 2 as \(M\) slowly increases. Position 2 is the bifurcation point of the locking process (black solid dot). As \(M\) continues to increase, the mechanism quickly jumps to position 3, where the lock enters the locked state. It is thus clear that the locking impact moment corresponding to position 3 is precisely the critical impact force in the experimental test. The value is \(M_{Impact}=1311\) N \(\cdot\) mm, with a 2.7% error compared with the test result, showing a high goodness of fit. Subsequently, as \(M\) decreases, the system moves to position 4 instead of returning by the original path. This suggests that position 4 is the stable equilibrium point of the locked state. The value of \(\theta_{1}\) corresponding to this position is denoted by \(\theta_{1}^{c}\) representing the locking ability of the hook lock mechanism. When the hook lock begins to unlock, the upper lock moves from position 4 to position 5 in the locked state as \(F_{Unlock}\) slowly increases. Position 6 is the bifurcation point of the unlocking process (black solid dot), at which the lock mechanism jumps rapidly to position 6, so that the hook lock reaches the locked state. Then, as \(F_{Unlock}\) decreases, the system moves back to position 1. Notably, each component of the lock will suffer interference in positions 1 and 4. Actually, in the abrupt locking and unlocking processes, the lock will be limited by a stop block to prevent interference.  
Figure 5: Numerical continuation results of a typical locking and unlocking process.  
Figure 6: Kinematic sketches of the hook lock mechanism.