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.
3.1. 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.
3.2. Control Strategies under Multivariate Time Series
Box–Behnken is an experimental design method used to adjust quadratic functions, which can effectively estimate the first-order and second-order coefficients of fitted models and is commonly used to analyze the nonlinear effects of factors [
31]. The two engine fuel flow valves
Fue1 and
Fue2, friction clutch-driven hydraulic
Hyd, and coaxial dual–rotor pitch angles
Pit1 and
Pit2 are the main influencing factors without further control variable screening. To avoid redundancy in the design of the test points, the upshift control time of fuel flow
Fue1 was set at 15–18 s, and the downshift control time was set at 45–48 s. The upshift control time of pitch angle
Pit1 was set at 12–15 s, and the downshift control time was set at 42–45 s. The starting time points for upshifting (
tha,
thb,
thc) and downshifting (
tla,
tlb,
tlc) of fuel flow
Fue2, hydraulic
Hyd, and pitch angle
Pit2 were considered design factors. The horizontal design of the shifting test points is presented in
Table 4 and
Table 5, and the upshifting and shifting strategies are listed in
Table 6 and
Table 7, respectively.
3.3. Interaction Effects Analysis of Multivariate Time Series in RSM Models
Following the upshift strategy outlined in
Table 6, simulations are conducted. During the upshift, the response targets include the engine speed overshoot value ΔΩ
e, transmission system input torque overshoot value Δ
Tin, rotor speed overshoot value Δ
ωMR, and transmission system output torque overshoot value Δ
Tout. Given that the friction clutch serves as an active control component during the upshift process, the torque overshoot of the friction clutch Δ
τf is also established as a response target. The upshift simulation strategy and its associated response targets are listed in
Table 8.
The data in
Table 8 is processed using Design-Expert v.13. A regression model is established between the various response targets observed during the upshift and starting points (
tha,
thb,
thc) of fuel flow
Fue2, hydraulic
Hyd, and pitch angle
Pit2.
Table 9 presents the significance analysis results of the upshift regression model. With
p values below 0.0001, these results highlight the model’s statistical significance. Both the multiple correlation coefficient R
2 and adjusted determination coefficient R
adj2 are close to 1, illustrating a strong correspondence between the response surface model’s calculations and the existing mathematical model’s simulation results. Consequently, the upshift regression model is suitable for analyzing and forecasting the response traits of various sequential upshifting strategies in variable-speed power systems.
In
Figure 8, when
thc = 9 s, the changes in
tha and
thb have no significant impact on the engine output speed; when
thb = 12.5 s, the earlier
tha and
thc can reduce the overshoot of the engine output speed; when
tha = 9 s, the earlier
thb and later
thc can reduce the overshoot of the engine output speed.
In
Figure 9, when
thc = 10 s, the later
tha and
thb can reduce the overshoot of the input torque; when
thb = 12.5 s, earlier
tha and later
thc can reduce the overshoot of input torque; when
tha = 16.5 s, earlier
thb and later
thc can reduce the overshoot of input torque.
In
Figure 10, when
thc = 10.5 s, earlier
tha and
thb can reduce the overshoot of rotor speed; when
thb = 12.5 s, earlier
tha and later
thc can reduce the overshoot of rotor speed; when
tha = 16.5 s, earlier
thb and later
thc can reduce the overshoot of rotor speed.
In
Figure 11, when
thc = 10.5 s, earlier
thc and
thb can reduce the overshoot of the output torque; when
thb = 12.5 s, later
tha and earlier
thc can reduce overshoot of output torque; when
tha = 16.5 s, later
thb and earlier
thc can reduce the overshoot of the output torque.
In
Figure 12, when
thc = 10.5 s, earlier
tha and
thc can reduce clutch torque overshoot; when
thb = 12.5 s, later
tha and earlier
thc can reduce the clutch torque overshoot; when
tha = 16.5 s, later
thb and earlier
thc can reduce the clutch torque overshoot.
Based on the downshift strategy detailed in
Table 7, simulations are executed. The response targets for the downshift process include overshoot values for engine speed ΔΩ
e, transmission system input torque Δ
Tin, rotor speed Δ
ωMR, and transmission system output torque Δ
Tout. Given that the one-way clutch is engaged passively to convey torque during the downshift, the torque overshoot of the one-way clutch Δ
τo is also identified as a response target.
Table 10 delineates the downshift simulation strategy and its associated response targets.
Regression model between the different response targets during downshift and starting points (
tla,
tlb,
tlc) of fuel flow
Fue2, hydraulic
Hyd, and pitch angle
Pit2.
The significance analysis results of the downshift regression model are shown in
Table 11, with
p values less than 0.05, indicating that the models are statistically significant. The multiple correlation coefficient R
2 is greater than 0.52, and the correction determination coefficient R
adj2 is greater than 0.46, indicating a high degree of similarity between the calculated results of the response surface model and the simulated results of the existing mathematical model. Therefore, the downshift regression model can be used to analyze and predict the response characteristics of downshift strategies in different time series.
Based on the regression model analysis results, the interaction effect between the downshifting time points and the relationship between multiple response targets was determined. When setting
tla,
tlb, and
tlc to the 0 level, the response surface graph illustrating the relationship between variations in the other two downshifting time points and response target Δ
Tin is obtained, as depicted in
Figure 13. It can be observed that when
tlc = 40.5 s, moderate
tla and earlier
tlb can reduce the overshoot of input torque. When
tlb = 40.5 s, moderate
tla and earlier
tlc can reduce the overshoot of input torque; when
tla = 46.5 s, later
tlb and earlier
tlc can reduce the overshoot of input torque.
In
Figure 13, it can be observed that at
tlc = 40.5 s, moderate
tla and earlier
tlb can reduce input torque overshoot; when
tlb = 40.5 s, moderate
tla and earlier
thc can reduce the overshoot of input torque; when
tla = 46.5 s, later
tlb and earlier
tlc can reduce the overshoot of input torque.
3.4. Optimization of a Multi-Objective Low-Impact Control Strategy under Nonlinear Programming
Two upshift quality objective functions were designed for speed and torque impacts. The optimization model Ha considers engine speed stability, rotor speed variation, and friction clutch impact torque as the optimization objectives. Furthermore, the optimization model Hb considers the smoothness of the input and output torques of the variable-speed transmission system and the impact torque of the friction clutch as the optimization objectives.
Solve the optimization model for the upshift strategy (36) to obtain the optimal upshift starting points for fuel flow
Fue2, hydraulic
Hyd, and pitch angle
Pit2, at
fshift_Ha,
tha = 15.0 s,
thb = 13.0 s,
thc = 10.60 s; At
fshift_Hb,
tha = 18.0 s,
thb = 13.0 s,
thc = 9.80 s. Taking
tha = 15.0 s,
thb = 12.0 s,
thc = 9.0 s as the baseline of the upshift strategy, it can be seen from
Table 12 that after optimization, compared with the baseline upshift strategy, the optimized model Ha has a significant decrease in engine speed, variable transmission input, output torque, and friction clutch torque overshoot, with a decrease of 5.13%, 8.46%, 10.48%, and 9.25%, respectively. The rotor speed overshoot increased by 8.54%, and the optimization model Hb showed significant reductions in engine speed, variable transmission input, output torque, and friction clutch torque overshoot, with decreases of 4.19%, 8.19%, 9.02%, and 9.01%, respectively. The rotor speed overshoot increases by 9.18%.
According to
Figure 14, compared with model Hb, model Ha reduces the overshoot of engine speed, transmission system input and output torque, and friction clutch torque by 0.94%, 0.27%, 1.46%, and 0.24% respectively. The optimization model Ha exhibits better performance in suppressing engine speed, input and output torque of the variable-speed transmission system, and torque impact of the friction clutch, and has fewer adverse effects on rotor speed. Therefore, the shift strategy based on model Ha optimization is more suitable as an upshift strategy.
Similarly, the optimization model La considers engine speed stability, rotor speed variation, and one-way clutch impact torque as the optimization objectives. Furthermore, the optimization model Lb considers the smoothness of the input and output torques of the variable-speed transmission system and the impact torque of the one-way clutch as the optimization objectives.
The downshift strategy optimization model (37) was solved to obtain the optimal downshift starting point for fuel flow
Fue2, hydraulic
Hyd, and pitch angle
Pit2: At
fshift_La,
tla = 46.0 s,
tlb = 39.0 s,
tlc = 42.0 s; At
fshift_Lb,
tla = 46.5 s,
tlb = 41.2 s,
tlc = 39.0 s. Furthermore,
tla = 45.0 s,
tlb = 42.0 s, and
tlc = 42.0 s are considered the baseline of the downshift strategy. It can be seen from
Table 13 that compared with the baseline downshift strategy, model La reduces engine speed, variable speed transmission output torque, and one-way clutch torque overshoot by 11.52%, 6.67%, and 4.36%, respectively, and increases variable speed transmission input torque and rotor speed overshoot by 22.8% and 13.96%, respectively. In the optimized model Lb, the overshoot of transmission input and output torque and unidirectional clutch torque are reduced by 6.51%, 7.61%, and 4.97%, respectively, and the overshoot of engine speed and rotor speed is increased by 1.79% and 16%, respectively.
As can be seen from
Figure 15, compared with the La model, the overshoot of input and output torque of the variable speed transmission system and the one-way clutch under the Lb model decreased by 29.31%, 0.94%, and 0.61% respectively. The optimization model Lb offers a more harmonized approach in regulating the engine output speed and input torque of the variable-speed transmission system. It excels in mitigating the effects of the variable-speed transmission output torque and unidirectional clutch torque. Given the contrasting effects of the two optimization models on the performance of the engine, variable-speed transmission, and rotor within the power system, the optimization model Lb emerges as the more fitting choice for a downshift strategy.