## 2 Integrated Dynamics Model of the Helicopter Engine/Transmission System/Rotor
### Variable Speed Gearbox Dynamic Modeling  
The variable-speed transmission system uses a double-row double planetary gear train--wet friction clutch--one-way clutch configuration [27], as shown in Figure 7. The variable speed gearbox reduction ratio \(i_{p}\): low gearshift, \(I_{v}=i_{1}\); High gearshift, \(i_{v}=i_{1}\). The clutch torque \(\tau_{o}\) originates from the one-way clutch, which can be physically modeled as a spring with nonlinear stiffness \(k_{c1}\). Furthermore, \(\tau_{o}\) is determined by external factors, and clutch torque \(\tau_{f}\) is determined by the state of the friction clutch.  
High-speed gear: The friction clutch engages, ring R2 locks up, and ring R1 rotates in the same direction as the input shaft. The one-way clutch is in an overrunning state, and the transmission ratio is:  
\[i_{1}=1+\frac{z_{R2}z_{P1}}{z_{S1}z_{P2}} \tag{20}\]  
where \(z\) denotes the number of gear teeth, and the subscript indicates the gear train parts. Low-speed gear: The friction clutch is disengaged, ring R2 rotates freely, and ring R1  
Figure 7: Double row double planetary structure with multiple clutches.  
produces a tendency to rotate in the opposite direction to the input shaft, resulting in the one-way clutch locking up, and the transmission ratio is:  
\[{i_{1}}^{\prime}=1+\frac{z_{R1}}{z_{S1}} \tag{21}\]  
According to the kinematics of the compound planetary gear train, the following relationship exists between the input and output torques and the multiple clutch torques:  
\[T_{in}=\frac{1}{i_{2}}\tau_{o}+\frac{1}{i_{3}}\tau_{f} \tag{22}\]  
\[T_{out}=\frac{i_{1}}{i_{2}}\tau_{o}+\frac{{i_{1}}^{\prime}}{i_{3}}\tau_{f} \tag{23}\]  
where \(i_{2}\) denotes the speed ratio between the input shaft and outer ring of the one-way clutch, and \(i_{3}\) denotes the speed ratio between the input shaft and friction clutch driven end. The torques of the two clutches are calculated as follows:  
\[\tau_{o}=\begin{cases}T_{out}i_{2}/i_{1}&\dot{\theta}_{out}=\Omega_{e}/i_{1}\\ k_{c1}\theta_{R1}&\Omega_{e}/i_{1}<\dot{\theta}_{out}<\Omega_{e}/i_{1}^{\prime} \\ 0&\dot{\theta}_{out}=\Omega_{e}/i_{1}^{\prime}\end{cases} \tag{24}\]  
\[\tau_{f}=\begin{cases}0&\dot{\theta}_{out}=\Omega_{e}/i_{1}\\ F_{n}\mu_{d}r_{e}n&\Omega_{e}/i_{1}<\dot{\theta}_{out}<\Omega_{e}/i_{1}^{\prime} \\ T_{out}i_{3}/i_{1}^{\prime}&\dot{\theta}_{out}=\Omega_{e}/i_{1}^{\prime}\end{cases} \tag{25}\]  
Specifically, \(F_{n}\) denotes the positive pressure acting on the piston, \(\mu_{d}\) denotes the coefficient of dynamic friction of the clutch, \(r_{e}\) denotes the equivalent radius of the friction plate, and \(n\) denotes the number of friction surfaces. The friction torque model of the wet hydraulic clutch and the nonlinear stiffness model of the one-way clutch are obtained from the reference [28].  
Optimization of Low Impact Shift Control Strategy for Dual-Engine Dual-Rotor Variable Speed Power System  
Given the pronounced nonlinearity between these input variables and the dynamic characteristics involved with the dual-engine, dual-rotor variable-speed transmission dynamic model--the RSM is employed. RSM transforms the multi-variable, strongly coupled dynamic model into a more manageable and analyzable mathematical model.