# A Novel Inverse Simulation Method of Helicopter Maneuvering Flight
## 4 Case Studies and Results
### Case I: Pull-Up Maneuver  
To validate our novel inverse simulation method, a case study of a helicopter flying a dynamic pull-up maneuver (in the longitudinal/vertical channel) was conducted. This maneuver was based on the Utility Tactical Transport Aerial System (UTTAS) maneuver of the original UH\(-\)60A design specification [30]. It was an instantaneous down-to-up turning maneuver in the vertical plane, and was designated by the Counter 11029 flight from the UH\(-\)60A flight test database, which is a terrain-avoidance maneuver in the longitudinal/vertical channel, and is initiated from a high-speed steady flight by pitching-up the helicopter for a rapid gain in altitude [31]. A sketch of the pull-up maneuver is shown in Figure 2. The maneuver lasted for approximately 9.3 s, reaching a maximum load factor of 2.12 g.  
The relevant parameters of the UH\(-\)60A helicopter for the inverse simulation are shown in Table 1. By using rotor vortex, blade element, and rotor momentum theories [32], the conventional flight dynamics of the UH\(-\)60A helicopter can be developed (see [33; 34; 35]). In this model, the fuselage, horizontal, and vertical tail aerodynamics modeling is mainly based on wind tunnel data from NASA [36]. Tail rotor aerodynamics modeling is based on rotor momentum theory. The main rotor aerodynamics modeling is based on rotor vortex and element theories. Figure 2: Sketch of the pull-up maneuver. The pull-up maneuver only involves the elements in the longitudinal channel; i.e., \(V_{\rm P}\) and \(\theta_{\rm PE}\) in the transitional vector \(\mathbf{\eta}(t)\). Thus, these directional elements \(\psi_{\rm PE}\) and \(\dot{\psi}_{\rm BP}\) can be set as zero in the whole time histories. The boundary conditions of the time histories \(V_{\rm P}\) and \(\theta_{\rm PE}\) are shown in Table 2. In detail, four time points of \(V_{\rm P}\) and two time points of \(\theta_{\rm PE}\) are used to prescribe, according to the nature of this pull-up maneuver. Here, because the transitional element \(V_{\rm P}\) is a rate variable, the displacement variable \(S_{\rm P}\) is also used to follow the definition of the polynomial scheme, i.e., Equation (8). The boundary value of \(S_{\rm P}\) of time point 1 (i.e., the start) can be set as zero, and its values at the other time points are not necessary, because they can be inherited from the previous calculation stage. To illustrate the numerical stability of the degenerated method, three simulation time steps (i.e., \(\Delta T=0.02\) s, \(\Delta T=0.05\) s, and \(\Delta T=0.10\) s) are adopted. The primary results \begin{table}
\begin{tabular}{l c} \hline \hline
**Main Rotor** & SC 1095 \\ Number of Blades & 4 \\ Radius, m & 8.178 \\ Blade Chord, m & 0.5273 \\ Rotational Speed, rad/s & 27 \\ Longitudinal Shaft Tilt, deg & \(-3\) \\ Linear Blade Twist, rad/m & \(-18\) \\ Lock Number & 8.1936 \\ Flapping Hinge Offset, m & 0.381 \\ Mass Moment, kg-m & 385.66 \\ Inertia Moment, kg-m\({}^{2}\) & 2050.81 \\ \hline
**Fuselage** & \\ \hline Gross Weight, kg & 7876.18 \\ Roll Inertia, kg-m\({}^{2}\) & 6316.8 \\ Pitch Inertia, kg-m\({}^{2}\) & 52,215.6 \\ Yaw Inertia, kg-m\({}^{2}\) & 49,889.0 \\ Product of Inertia, kg-m\({}^{2}\) & 2551.7 \\ \hline
**Tail Rotor** & SC 1095 \\ Number of Blades & 4 \\ Radius, m & 1.6764 \\ Blade Chord, m & 0.2469 \\ Rotational Speed, rad/s & 124.62 \\ Linear Blade Twist, rad/m & \(-18\) \\ Lock Number & 3.3783 \\ Installation Angle, deg & 70 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: UH\(-60\)A helicopter parameters used for the inverse simulation.