# A Novel Inverse Simulation Method of Helicopter Maneuvering Flight
## 4 Case Studies and Results
### Case I: Pull-Up Maneuver
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Time Points (TPs)** & \(\mathbf{t}\), \(\mathbf{s}\) & \(S_{\rm P}\), \(\mathbf{m}\) & \(V_{\rm P}\), \(\mathbf{\rm km}/\mathbf{h}\) & \(\dot{\Psi}_{\rm P}\), \(\mathbf{\rm km}/(\mathbf{h}\cdot\mathbf{s})\) \\ \hline
1 & 0 & 0 & 284 & 0 \\
2 & 2 & - & 284 & 0 \\
3 & 5.34 & - & 227 & \(-18.52\) \\
4 & 9.3 & - & 175 & \(-5\) \\ \hline Time Points (TPs) & \(t\), s & \(\theta_{\rm PE}\), deg & \(\dot{\theta}_{\rm PE}\), deg/s & \(\dot{\theta}_{\rm PE}\), deg/s\({}^{2}\) \\ \hline
1 & 0 & 0 & 0 & 0 \\
2 & 2 & - & - & - \\
3 & 5.34 & - & - & - \\
4 & 9.3 & 35 & 0 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Boundary conditions for the pull-up maneuver. Figure 5: UH\(-\)60A (**a**) normal load factor, (**b**) rotor thrust, and (**c**) pitch angle, during the UTTAS pull-up maneuver. Figure 6: UH\(-60\)A (**a**) collective and pedal controls, (**b**) longitudinal and lateral cyclic controls, and (**c**) stabilator angle, during the UTTAS pull-up maneuver. The normal load factor, pitch angle, and rotor thrust in the longitudinal channel are compared with the flight test data. Because the effects of the unsteady aerodynamics and the dynamic stall of the main rotor in the helicopter flight dynamics model have not been considered in this case study, the present peak normal load factor in Figure 5a only reaches 1.88 g (slightly lower than the flight data of 2.12 g), although the peak trimming rotor thrust in Figure 5b matches the flight data. The present peak pitch angle in Figure 5c is lower than the flight data, but these show a similar overall trend. In general, our inverse simulation results are credible, indicating that the degeneration method is reasonable and effective. Figure 6 shows the time histories of the cockpit controls and the stabilator angle during the UTTAS pull-up maneuver. To realize a pull-up maneuver, two primary controls (i.e., longitudinal cyclic and collective sticks) are required. In detail, the longitudinal cyclic control stick is moved backward, and the collective control stick is put down, to enter the maneuver. Then, the longitudinal cyclic control stick is moved forward, and the collective control stick is raised, to recover from the maneuver. In addition, the stabilator of the helicopter (i.e., the flexible horizontal tail) is controlled automatically by the flight control system, and depends on the flight velocity; and the collective controls the stabilator angle changes, from 1.0 deg to approximately 7.0 deg, during the whole pull-up maneuver, to provide an additional pitch moment for the helicopter. Note that the increase in the stabilator angle generates a pitching-down moment, while the decrease in the stabilator angle generates a pitching-up moment. Thus, the evident phenomenon is that the minimum value reaches approximately \(-1.0\) deg when entering the maneuver (in this case, the helicopter needs a pitching-up moment, to quickly increase the pitch attitude), and the angle needs to be gradually increased when recovering from the maneuver (in this case, the helicopter needs a pitching-down moment, to restrain the increasing trend of the pitch attitude). Our method and the inverse simulation results illustrate this phenomenon well.