## 2 Integrated Dynamics Model of the Helicopter Engine/Transmission System/Rotor
### Simulation Parameters of the Helicopter Variable-Speed Power System  
XH-59A is an experimental coaxial composite helicopter developed by Sikorsky Aircraft, which uses a Pratt & Whitney PT6T-3 turboshaft starter to drive the main rotor and two Pratt & Whitney J60-P-3A turbojet engines as auxiliary thrusters. To reduce costs, Sikorsky did not integrate the rotor and transmission systems of an auxiliary propulsion unit. The rotor was powered by a turboshaft engine. The parameters of XH-59A used in this study are listed in Table 1. The basic parameters of the parallel-wheel train and variable-speed transmission system described in Section 2 are listed in Table 2.  
The transmission system between the engine and rotor is divided into two parts: A variable-speed transmission system and a reduction-gear system. The transmission ratios and speed changes in the main components of the power system are listed in Table 3. The total reduction ratios of the helicopter power system in the high and low gears were 17.12 and 24.14, respectively.  
The clutch hydraulic pressure is an important control variable during gear shifting. However, opening the fuel flow affects the output power of the engine, and a change in the pitch angle of the rotor affects the lift and torque of the rotor. According to the path of the power generation transmission action, the engine fuel flow _Fue_, clutch hydraulic pressure _Hyd_, and pitch angle _Pit_ correspond to the input signals. The simulation was set for 60 s, with a preset shift start time at 12 s and a downshift start time at 42 s. The upshift durations for _Fue_, _Hyd_, and _Pit_ were 3 s, 2 s, and 3 s, respectively. The hydraulic pressure curve was an exponential function, and the other variables were linear functions with a downshift duration of 3 s.  
\begin{table}
\begin{tabular}{l c c} \hline \hline
**Item** & **Value** & **Unit** \\ \hline Number of Blade \(N_{b}\) & 3 per rotor (6, total) & \\ Number of rotors & 2 (coaxial) & \\ Rotor Radius \(R\) & 5.49 & m \\ Rotor Solidity \(\sigma\) & 0.127 & \\ Rotor RPM \(\omega_{MR}\) & \\ Helicopter & 345 & rpm \\ Compound helicopter mode & 240 & rpm \\ Drag coefficient \(c_{D}\) & 0.08 & \\ Twist Gradient \(\theta_{twist}\) & \(-\)10 & deg \\ Flap Moment of Inertia \(I_{MR}\) & 450 & kg\(\cdot\)m\({}^{2}\) \\ Gross weight & 4082 & kg \\ Power plants & & \\ \multicolumn{2}{c}{Thrust} & J60-P-3A turbojet engine \\ \multicolumn{2}{c}{Lift} & PT6T-3 turboshaft engine \\ \hline \hline \end{tabular}
\end{table}
Table 1: Characteristics of the XH-59A [29; 30].  
\begin{table}
\begin{tabular}{l c c} \hline \hline
**Item** & **Value** & **Unit** \\ \hline Parallel wheel train \(i_{p}\) & 10 & \\ Variable speed gearbox & & \\ High gear \(i_{1}\) & 1.712 & \\ Low gear \(i_{1}^{\prime}\) & 2.414 & \\ Modulus & 2.5 & mm \\ Sun base circle radius \(r_{s}\) & 108.75 & mm \\ Planet base circle radius & & \\ Wheel \(r_{p1}\) & 22.5 & mm \\ Pinion \(r_{p2}\) & 57.35 & mm \\ Ring r\({}_{1}\) base circle radius \(r_{r1}\) & 153.75 & mm \\ Ring r\({}_{2}\) base circle radius \(r_{r2}\) & 185 & mm \\ \hline \hline \end{tabular}
\end{table}
Table 2: Basic parameters of the variable-speed transmission system and parallel wheel train [27].  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Components** & **Transmission Ratio** & **Input Speed (rpm)** \\ \hline Variable speed & 1.712/2.414 & 6000 \\ transmission system & 10 & 3504.8/2485.5 \\ Reduction gear train & - & 350.5/249 \\ Rotors & - & 350.5/249 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Transmission ratio and speed of key components in the power system.