Design and Optimization of Low Impact Shift Control Strategy for Aviation Transmission Power System Based on Response Surface Methodology

Abstract

The utilization of a variable-speed power system significantly improves the forward flight speed and cruising range of the helicopter. Nevertheless, the shock of speed and torque during the shift process brings stability and safety problems that cannot be ignored. Thus, swift and stable shift control is a key issue in the research on aviation power systems. This study focuses on the design and optimization of low-impact shift control strategies for a variable-speed power system, which involves multiple control variables, long adjustment times, and uncontrollable risks due to the nonsteady state. A comprehensive power system model that integrates the engine, a two-speed dual-clutch transmission system, and the main rotor was proposed. By selecting the engine fuel flow, friction clutch hydraulic pressure, and rotor pitch angle as input signals, regression fitting models between the input signals’ starting time points and speed or torque shock were obtained using Response Surface Methodology (RMS). The interaction effect of multiple time series was analyzed, and four kinds of low-impact nonlinear programming multi-objective optimized models for speed or torque are proposed. The results indicate that the P values of the RMS fitting models at upshift and downshift are less than 0.0001 and 0.05, respectively, which are highly significant and can effectively predict the shift dynamic response; under the optimized upshift and downshift control strategy, the speed and torque shock are reduced by 5–10%.

1. Introduction

Currently, the forward flight speeds of conventional civilian and military helicopters are limited to approximately 300–350 km/h, which does not satisfy the design goals and mission requirements of high-speed compound helicopters [1]. With innovation and breakthroughs in power plants and configurations, the horizontal flight speed of fifth-generation helicopters has exceeded 400 km/h [2]. Sikorsky displayed an S-97 Raider helicopter demonstrator for 2021, as shown in Figure 1, demonstrating revolutionary speed, maneuverability, and agility, representing a range of technologies required for future helicopters [3,4]. The novel rotorcraft has two coaxial rotor blades and a single pusher propellor, with a maximum speed exceeding 480 km/h. Its rival, V-280 Valor, features a tilt-rotor design similar to that of V-22 Osprey and is a third-generation tilt-rotor vertical lift helicopter [5]. As shown in Figure 2, it is more efficient and faster in cruise mode. It realized a speed of 555 km/h in a flight test in November 2020 [6] owing to its unique mechanical structure with a variable rotor pull direction.

Variable-speed rotor technology was adopted by the S-97 Raider and V-280 Volar [7]. The effectiveness of this technology in addressing low-speed rotor noise, breaking forward flight speeds, increasing payload and range, and reducing fuel burn and operating costs has been demonstrated by NASA [8,9,10,11]. The variable-speed transmission mechanism increases the complexity of the power system, which is a major reason for the high incidence of V-22 Osprey accidents. Boeing’s A160T hummingbird unmanned helicopter uses a variable-speed rotor [12], which significantly improves the overall aerodynamic performance and fuel economy of the aircraft [13]. However, during the transition of flight, there is a clear safety concern. This is because matching the power output of the turboshaft engine with the aerodynamic characteristics of the main rotor becomes challenging as the rotor speed changes [14].

In existing studies, the collective pitch of the main rotor is generally used as the cross-linking parameter of the flight control system to reduce the interference of the main rotor load on the turboshaft engine [15,16]. To ensure the continuous power output of the turboshaft engine, a sequential torque control method was adopted in [17], and a power turbine speed controller with linear matrix inequality control, and torque feed-forward compensation was designed to track the turbine speed and weaken the torque fluctuation in the process of rotor speed change. In a previous study [18], neural networks were used for rotor demand torque prediction to reduce the power turbine speed overshoot by more than 20%. This mitigated the interference of large changes in the rotor demand torque on the power turbine speed with a faster response and higher robustness.

The aforementioned studies primarily consider a scenario with a constant rotor speed. In such situations, a minor range of rotor speed variations results in limited changes in engine torque demand [19]. However, when there is a broad variation in rotor speed, the torque required by the main rotor and the engine’s output torque must be aligned using variable transmission. The gearshift jerk arising from multi-speed transmission, especially when the motor and clutch reach saturation in a power system, is discussed in ref. [20]. Nevertheless, these studies are predominantly geared toward vehicle systems and are not easily adaptable to variable rotor speeds. Traditional design methods, which ignore the implications of extensive rotor speed variations, fail to align the engine torque output with the dynamic process of variable rotor speed [21]. This makes it challenging to achieve smooth control in systems where the engine, variable speed transmission, and rotor are strongly coupled. Given the torque demand variations due to changing rotor speeds, improving the smoothness of the shifting process becomes a primary focus in the realm of high-speed compound helicopters.

The structure of this article is as follows: First, a brief introduction to the development and application of variable-speed transmission systems in compound helicopters is provided. Subsequently, a universal model of the engine/variable-speed transmission/main-rotor power system is developed. Finally, using XH-59A as a prototype, the Behnken and RSM are used to obtain a regression fitting model between the optimization target and input signal. The control strategy is further optimized using a nonlinear programming model to reduce the speed–torque impact during gear shifting.

2. Integrated Dynamics Model of the Helicopter Engine/Transmission System/Rotor

2.1. Simplified Generic Variable Speed Rotor Helicopter Dynamic Model

A high-speed compound helicopter features a configuration that combines a coaxial counter-rotating rigid dual rotor with a tail propeller. This design was first realized on the Sikorsky XH-59 helicopter in the 1970s [22]. Figure 3 illustrates the layout of the power system, which primarily comprises a dual turboshaft engine, a transmission system, and a twin rotor. With the two sets of rotors rotating coaxially, one above the other, the rotor torque is balanced. Hence, the tail rotor evolves into a propulsive configuration.

As explained in [11,23], the analysis of the turboshaft engine, transmission system, and main rotor subsystem coupling is fundamental for accurately understanding the dynamic characteristics of a power system. A general power system model of a high-speed compound helicopter is developed based on the power system layout shown in Figure 3. To simplify the analysis, a two-stage reduction structure is used instead of the conventional three-stage reduction design, and a single engine is used to replace the dual engine, power P_e, output speed Ω_e, and engine torque T_e. enters the variable speed transmission system through the one-way clutch; the engine speed undergoes a deceleration ratio i_v. Following a secondary deceleration via the parallel wheel train, with a transmission ratio i_p it produces an output torque T_out that drives the rotor to rotate. This results in torsion torque T_MR and lift L_MR. A block diagram illustrating the principle of this power system is shown in Figure 4.

2.2. Turboshaft Engine Modeling

A turboshaft engine is a standard power source for helicopter power systems. It is designed to maintain a consistent output speed during its operation and to swiftly respond to load changes without surpassing its maximum torque limit. When there is a change in the power demand of the main rotor, the fuel injection is adjusted to ensure a consistent rotor speed across various flight conditions [24]. A general gas engine model is adopted to adjust the relationship between the torque T_e and rotational speed Ω_e by using the fuel flow signal Fue. The physical fuel flow signal input specifies the normalized engine torque. The demand power P_e of the engine is a function of the output speed Ω_e, namely g(Ω_e), At steady state, P_e(Ω_e, Fue) = Fue $\times$ g(Ω_e). When the engine speed is in the operating range, the engine power is continuously output, assuming that at Ω_e0, the maximum output power is P_emax, defined w = Ω_e/Ω_e0, g(Ω_e) = P_emax $\times$ p(w), and p(1) = 1, dp(1) = 0, as follows:

$$T_e=\frac{P_{e\max }}{{\Omega }_{\mathrm{e}0}}\times \frac{p(w)}{w}$$

(1)

The function p(w) takes the form of a cubic polynomial, as follows:

$$p(w)=p_1\times w+p_2\times w^2-p_3\times w^3$$

(2)

$$p_1+p_2-p_3=1$$

(3)

$$p_1+2p_2-3p_3=0$$

(4)

The relationship between engine demand torque T_e and output speed Ω_e is shown in Figure 5.

The boundary is defined as: 0 ≤ Ω_emin/Ω_e0 ≤ w ≤ Ω_emax/Ω_e0. At the minimum rotational speed, the engine power p(w_min) ≥ 0, and the following holds.

$$p(w_{\min })=p_1\times w_{\min }+p_2\times w_{\min }{}^2-p_3\times w_{\min }{}^3\ge 0$$

(5)

Similarly, the engine power at the maximum speed must be non-negative: p(w_max) ≥ 0, and the following holds.

$$p(w_{\max })=p_1\times w_{\max }+p_2\times w_{\max }{}^2-p_3\times w_{\max }{}^3\ge 0$$

(6)

2.3. Rotor Dynamic Load Modeling

According to the simplified rotor performance calculation and analysis of Dreier [25,26], and Houghton [26], the main rotor model is established by the blade element analysis theory, and the rotor is divided into several micro-elements along the wing spread direction. The aerodynamic force acting on the rotor blade is decomposed into a two-dimensional profile aerodynamic force and rotor-induced velocity aerodynamic force. Assuming that the angular velocity of the helicopter is zero, the force of each microelement segment is calculated, and the total tension and demand torque of the main rotor blade are obtained via radial and axial integrations.

The aerodynamic diagram of the blade section is shown in Figure 6. Furthermore, the air density is denoted by ρ_a, blade cross-section velocity is denoted by v, blade equivalent chord length is denoted by c_e, rotor pitch is denoted by θ, inflow angle is denoted by α, lift coefficient is denoted by C_L, C_L = a_oα, and drag coefficient is denoted by C_D. Then, the microelement lift dL and resistance dD at the radial distance r_b can be expressed as follows:

$$\mathrm{d}L=\frac{1}{2}{\rho }_a{\widehat{v}}^2{c}_e{C}_L\mathrm{d}{r}_b$$

(7)

$$\mathrm{d}D=\frac{1}{2}{\rho }_a{\widehat{v}}^2{c}_e{C}_D\mathrm{d}{r}_b$$

(8)

The flow speed in the blade section $\widehat{v}$ depends on the main rotor angular velocity ω_MR, air inflow velocity v_c, and rotor-induced velocity v_ind.

$${\widehat{v}}^2={\left({\omega }_{MR}{r}_b\right)}^2+{\left(v_c+v_{ind}\right)}^2$$

(9)

$$\tan \phi =\frac{v_c+v_{ind}}{{\omega }_{MR}{r}_b}$$

(10)

Given that ${\omega }_{MR}r\gg v_c+v_{ind}$, φ denotes a small angle as follows:

$$\phi \approx (v_c+v_{ind})/{\omega }_{MR}{r}_b$$

(11)

$$\alpha =\theta -\phi \approx \theta -(v_c+v_{ind})/{\omega }_{MR}{r}_b$$

(12)

$$\widehat{v}={\omega }_{MR}{r}_b$$

(13)

Substitute Equations (11)–(13) into Equations (7) and (8), as follows:

$$\mathrm{d}L=\frac{1}{2}{\rho }_a{\left({\omega }_{MR}{r}_b\right)}^2{c}_e{a}_0\left(\theta -\frac{v_c+v_{ind}}{{\omega }_{MR}{r}_b}\right)\mathrm{d}{r}_b$$

(14)

$$\mathrm{d}D=\frac{1}{2}{\rho }_a{({\omega }_{MR}{r}_b)}^2{c}_e{C}_D\mathrm{d}{r}_b$$

(15)

For the rotor plane, the blade lift and drag are decomposed in the vertical and horizontal directions, respectively.

$$\mathrm{d}{F}_{MR}=\mathrm{d}L\cos \phi -\mathrm{d}D\sin \phi \approx \mathrm{d}L$$

(16)

$$\mathrm{d}{T}_{MR}=(\mathrm{d}L\sin \phi +\mathrm{d}D\cos \phi )r_b\approx (\mathrm{d}L\phi +\mathrm{d}D)r_b$$

(17)

By integrating (17) in the span direction, the required torque generated in the rotor can be obtained as follows:

$$T_{MR}\approx {\displaystyle \underset{0}{\overset{r}{\int }}\left(\mathrm{d}L\phi +\mathrm{d}D\right)r_b}=\frac{1}{2}{\rho }_a{\displaystyle \underset{0}{\overset{r}{\int }}{\left({\omega }_{MR}{r}_b\right)}^2{c}_e\left[a_0\left(\theta -\frac{v_c+v_{ind}}{{\omega }_{MR}{r}_b}\right)\frac{v_c+v_{ind}}{{\omega }_{MR}{r}_b}+C_D\right]r_b\mathrm{d}{r}_b}$$

(18)

Here, only the effects of the variation in air inflow velocity v_c and main rotor speed ω_MR on the rotor demand torque are considered, and Equation (18) is simplified as follows:

$$T_{MR}\approx {\displaystyle \underset{0}{\overset{r}{\int }}\left(\mathrm{d}L\phi +\mathrm{d}D\right)r_b}=\frac{1}{2}{\rho }_a{c}_e\left[\frac{r^3}{3}{a}_0\theta (v_c+v_{ind}){\omega }_{MR}+\frac{r^4}{4}{C}_D{\omega }_{MR}{}^2\right]-T_A$$

(19)

where $T_A=\frac{1}{2}{\rho }_a{c}_e{\displaystyle \underset{0}{\overset{r}{\int }}{\left(v_c+v_{ind}\right)}^2{a}_0{r}_b\mathrm{d}{r}_b}$.

2.4. 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_v: low gearshift, I_v = i₁; High gearshift, i_v = $i_1{}^{\prime }$. The clutch torque τ_o originates from the one-way clutch, which can be physically modeled as a spring with nonlinear stiffness k_c1. Furthermore, τ_o is determined by external factors, and clutch torque τ_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:

$$i_1=1+\frac{z_{R2}{z}_{P1}}{z_{S1}{z}_{P2}}$$

(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 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:

$$i_1{}^{\prime }=1+\frac{z_{R1}}{z_{S1}}$$

(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:

$$T_{in}=\frac{1}{i_2}{\tau }_o+\frac{1}{i_3}{\tau }_f$$

(22)

$$T_{out}=\frac{i_1}{i_2}{\tau }_o+\frac{i_1{}^{\prime }}{i_3}{\tau }_f$$

(23)

where i₂ denotes the speed ratio between the input shaft and outer ring of the one-way clutch, and i₃ denotes the speed ratio between the input shaft and friction clutch driven end. The torques of the two clutches are calculated as follows:

$${\tau }_o=\left\{\begin{array}{ll}{T}_{out}{i}_2/i_1\hfill & {\dot{\theta }}_{out}={\Omega }_e/i_1\hfill \\ k_{c1}{\theta }_{R1}\hfill & {\Omega }_e/i_1<{\dot{\theta }}_{out}<{\Omega }_e/i_1^{\prime }\hfill \\ 0\hfill & {\dot{\theta }}_{out}={\Omega }_e/i_1^{\prime }\hfill \end{array}\right.$$

(24)

$${\tau }_f=\left\{\begin{array}{ll}0\hfill & {\dot{\theta }}_{out}={\Omega }_e/i_1\hfill \\ F_n{\mu }_d{r}_{e}n\hfill & {\Omega }_e/i_1<{\dot{\theta }}_{out}<{\Omega }_e/i_1^{\prime }\hfill \\ T_{out}{i}_3/i_1^{\prime }\hfill & {\dot{\theta }}_{out}={\Omega }_e/i_1^{\prime }\hfill \end{array}\right.$$

(25)

Specifically, F_n denotes the positive pressure acting on the piston, μ_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].

3. 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.

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. Characteristics of the XH-59A [29,30].

Item	Value	Unit
Number of Blade Nb	3 per rotor (6, total)
Number of rotors	2 (coaxial)
Rotor Radius R	5.49	m
Rotor Solidity σ	0.127
Rotor RPM ωMR
Helicopter	345	rpm
Compound helicopter mode	240	rpm
Drag coefficient cD	0.08
Twist Gradient θtwist	−10	deg
Flap Moment of Inertia IMR	450	kg·m2
Gross weight	4082	kg
Power plants
Thrust	J60-P-3A turbojet engine
Lift	PT6T-3 turboshaft engine

. The basic parameters of the parallel-wheel train and variable-speed transmission system described in Section 2 are listed in

Table 2. Basic parameters of the variable-speed transmission system and parallel wheel train [27].

Item	Value	Unit
Parallel wheel train ip	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 rs	108.75	mm
Planet base circle radius
Wheel rp1	22.5	mm
Pinion rp2	57.35	mm
Ring r1 base circle radius rr1	153.75	mm
Ring r2 base circle radius rr2	185	mm

.

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. Transmission ratio and speed of key components in the power system.

Components	Transmission Ratio	Input Speed (rpm)
Variable speed transmission system	1.712/2.414	6000
Reduction gear train	10	3504.8/2485.5
Rotors	-	350.5/249

. 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 (t_ha, t_hb, t_hc) and downshifting (t_la, t_lb, t_lc) 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. Three level values for upshift starting time points.

Level	Factors
	Fue2	Hyd	Pit2
	tha (s)	thb (s)	thc (s)
−1	15	12	9
0	16.5	12.5	10.5
1	18	13	12

and

Table 5. Three level values for downshift starting time points.

Level	Factors
	Fue2	Hyd	Pit2
	tla (s)	tlb (s)	tlc (s)
−1	45	39	39
0	46.5	40.5	40.5
1	48	42	42

, and the upshifting and shifting strategies are listed in

Table 6. The starting time point under the upshift strategy BBD design (Fue₁: 15 s_, Pit₁: 12 s).

Strategy	Fue2	Hyd	Pit2
	tha (s)	thb (s)	thc (s)
1	16.5	12.5	10.5
2	15	12.5	12
3	16.5	12.5	10.5
4	16.5	13	9
5	18	12.5	9
6	18	12	10.5
7	16.5	12.5	10.5
8	15	12.5	9
9	15	12	10.5
10	16.5	12	9
11	16.5	12.5	10.5
12	16.5	12.5	10.5
13	15	13	10.5
14	16.5	12	12
15	18	12.5	12
16	18	13	10.5
17	16.5	13	12

and

Table 7. The starting time point under downshift strategy BBD design (Fue₁: 45 s_, Pit₁: 42 s).

Strategy	Fue2	Hyd	Pit2
	tla (s)	tlb (s)	tlc (s)
1	48	40.5	42
2	45	40.5	39
3	45	40 (39)	40.5
4	48	40.5	39
5	46.5	42	42
6	46.5	40.5	40.5
7	48	42	40.5
8	46.5	40.5	40.5
9	46.5	39	39
10	46.5	40.5	40.5
11	46.5	41 (39)	42
12	45	42	40.5
13	46.5	42	39
14	45	40.5	42
15	48	40 (39)	40.5
16	46.5	40.5	40.5
17	46.5	40.5	40.5

Note: Downshift strategies 3, 11, and 15 do not converge in the simulation; therefore, the hydraulic pressure relief time is selected, and the original design value in “()” is replaced.

, 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 ΔT_in, rotor speed overshoot value Δω_MR, and transmission system output torque overshoot value ΔT_out. Given that the friction clutch serves as an active control component during the upshift process, the torque overshoot of the friction clutch Δτ_f is also established as a response target. The upshift simulation strategy and its associated response targets are listed in

Table 8. Upshift simulation strategy and response targets.

Strategy	ΔΩe (rad/s)	ΔTin (Nm)	ΔωMR (rad/s)	ΔTout (Nm)	Δτf (Nm)
1	92.30	342.88	8.59	697.52	251.10
2	130.70	483.26	6.39	1028.43	355.33
3	91.75	342.87	8.59	697.52	251.10
4	90.57	226.03	10.6	490.08	162.13
5	83.98	312.1	9.55	656.08	227.05
6	92.14	342.87	8.59	697.52	251.10
7	92.30	342.87	8.59	697.52	251.10
8	84.35	312.1	9.55	656.08	227.05
9	120.17	451.89	7.05	966.76	331.42
10	102.77	383.68	8.49	832.21	279.68
11	91.37	342.86	8.59	697.52	251.10
12	92.14	342.86	8.59	697.52	251.10
13	78.13	225.17	10.21	456.31	162.68
14	158.85	602.29	5.03	1330.07	443.08
15	130.77	483.26	6.39	1028.43	355.37
16	78.13	225.17	10.21	456.31	162.68
17	101.63	376.57	8.01	772.10	276.29

.

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 (t_ha, t_hb, t_hc) of fuel flow Fue₂, hydraulic Hyd, and pitch angle Pit₂.

$$\begin{array}{l}\Delta \Omega \mathrm{e}=2826.91-76.88t_{ha}-336.74t_{hb}+28.44t_{hc}+9.34t_{ha}{t}_{hb}+0.05t_{ha}{t}_{hc}\\ -15.01t_{hb}{t}_{hc}-1.30t_{ha}^2+12.35t_{hb}^2+8.17t_{hc}^2\end{array}$$

(26)

$$\begin{array}{l}\Delta T_{in}=-1811.35-235.59t_{ha}+1063.21t_{hb}+{313.59}_{hc}+36.34t_{ha}{t}_{hb}\\ -22.69t_{hb}{t}_{hc}-6.90t_{ha}^2-64.26t_{hb}^2+31.26t_{hc}^2\end{array}$$

(27)

$$\begin{array}{l}\Delta {\omega }_{MR}=53.65+3.89t_{ha}-16.48t_{hb}+2.82t_{hc}-0.513t_{ha}{t}_{hb}\\ +0.29t_{hb}{t}_{hc}+0.08t_{ha}^2+0.98t_{hb}^2-0.36t_{hc}^2\end{array}$$

(28)

$$\begin{array}{l}\Delta T_{out}=6313.72-651.8t_{ha}+833.43t_{hb}-637.87t_{hc}+89.75t_{ha}{t}_{hb}\\ -71.95t_{hb}{t}_{hc}-14.92t_{ha}^2-78.87t_{hb}^2+79.25t_{hc}^2\end{array}$$

(29)

$$\begin{array}{l}\Delta {\tau }_f=-1739.84-171.09t_{ha}+846.96t_{hb}-233.04t_{hc}+26.77t_{ha}{t}_{hb}\\ +0.004t_{ha}{t}_{hc}-16.41t_{hb}{t}_{hc}-5.16t_{ha}^2-50.07t_{hb}^2+22.98t_{hc}^2\end{array}$$

(30)

Table 9. Significance analysis results of the upshift regression model.

Response Target	F Value	p-Value	R2	R2adj
ΔΩe	28.61	0.0001	0.9735	0.9395
ΔTin	69.28	<0.0001	0.9889	0.9746
ΔωMR	86.58	<0.0001	0.9911	0.9797
ΔTout	52.46	<0.0001	0.9854	0.9666
Δτf	71.21	<0.0001	0.9829	0.9753

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 R² 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 t_hc = 9 s, the changes in t_ha and t_hb have no significant impact on the engine output speed; when t_hb = 12.5 s, the earlier t_ha and t_hc can reduce the overshoot of the engine output speed; when t_ha = 9 s, the earlier t_hb and later t_hc can reduce the overshoot of the engine output speed.

In Figure 9, when t_hc = 10 s, the later t_ha and t_hb can reduce the overshoot of the input torque; when t_hb = 12.5 s, earlier t_ha and later t_hc can reduce the overshoot of input torque; when t_ha = 16.5 s, earlier t_hb and later t_hc can reduce the overshoot of input torque.

In Figure 10, when t_hc = 10.5 s, earlier t_ha and t_hb can reduce the overshoot of rotor speed; when t_hb = 12.5 s, earlier t_ha and later t_hc can reduce the overshoot of rotor speed; when t_ha = 16.5 s, earlier t_hb and later t_hc can reduce the overshoot of rotor speed.

In Figure 11, when t_hc = 10.5 s, earlier t_hc and t_hb can reduce the overshoot of the output torque; when t_hb = 12.5 s, later t_ha and earlier t_hc can reduce overshoot of output torque; when t_ha = 16.5 s, later t_hb and earlier t_hc can reduce the overshoot of the output torque.

In Figure 12, when t_hc = 10.5 s, earlier t_ha and t_hc can reduce clutch torque overshoot; when t_hb = 12.5 s, later t_ha and earlier t_hc can reduce the clutch torque overshoot; when t_ha = 16.5 s, later t_hb and earlier t_hc 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 ΔT_in, rotor speed Δω_MR, and transmission system output torque ΔT_out. Given that the one-way clutch is engaged passively to convey torque during the downshift, the torque overshoot of the one-way clutch Δτ_o is also identified as a response target.

Table 10. Downshift strategy and response objectives.

Strategy	ΔΩe (rad/s)	ΔTin (Nm)	ΔωMR (rad/s)	ΔTout (Nm)	Δτo (Nm)
1	99.41	607.69	8.27	741.44	366.57
2	112.33	386.28	11.63	625.9	330.95
3	82.69	532.76	9.13	647.18	331.56
4	112.53	386.28	11.63	625.9	330.95
5	113.63	471.02	12.06	460.07	265.36
6	145.23	480.11	13.13	453.98	261.86
7	145.98	480.11	13.13	453.98	261.86
8	73.35	431.28	10.35	626.38	331.10
9	89.43	508.26	6.72	675.86	331.53
10	74.04	431.28	7.35	626.39	331.10
11	73.94	482.12	9.56	730.47	363.18
12	145.98	480.11	9.75	433.54	240.93
13	145.98	469.06	13.14	453.94	261.83
14	99.61	607.69	8.2	741.44	366.57
15	82.78	532.76	9.13	647.18	331.56
16	75.1	431.27	10.35	626.38	331.10
17	75.1	431.27	10.36	626.38	331.10

delineates the downshift simulation strategy and its associated response targets.

Regression model between the different response targets during downshift and starting points (t_la, t_lb, t_lc) of fuel flow Fue₂, hydraulic Hyd, and pitch angle Pit₂.

$$\Delta \Omega e=-468.61+24.13t_{lb}-10.16t_{lc}$$

(31)

$$\begin{array}{l}\Delta T_{in}=70489.29-1512.67t_{la}-3095.85t_{lb}+1388.96t_{lc}\\ -49.26t_{lb}{t}_{lc}+16.27t_{la}^2+61.77t_{lb}^2+8.37t_{lc}^2\end{array}$$

(32)

$$\Delta {\omega }_{MR}=-30.83+1.71t_{lb}-0.7t_{lc}$$

(33)

$$\Delta T_{out}=3044.54-100.92t_{lb}+41.14t_{lc}$$

(34)

$$\Delta {\tau }_o=1219.09-37.15t_{lb}+15.061t_{lc}$$

(35)

The significance analysis results of the downshift regression model are shown in

Table 11. Significance Analysis Results of Downshift Regression Model.

Response Target	F Value	p-Value	R2	R2adj
ΔΩe	8.05	0.0047	0.5348	0.4683
ΔTin	20.48	0.0003	0.9634	0.9164
ΔωMR	7.84	0.0052	0.5283	0.4609
ΔTout	12.83	0.0007	0.6471	0.5967
Δτo	10.72	0.0015	0.6050	0.4053

, with p values less than 0.05, indicating that the models are statistically significant. The multiple correlation coefficient R² 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 t_la, t_lb, and t_lc to the 0 level, the response surface graph illustrating the relationship between variations in the other two downshifting time points and response target ΔT_in is obtained, as depicted in Figure 13. It can be observed that when t_lc = 40.5 s, moderate t_la and earlier t_lb can reduce the overshoot of input torque. When t_lb = 40.5 s, moderate t_la and earlier t_lc can reduce the overshoot of input torque; when t_la = 46.5 s, later t_lb and earlier t_lc can reduce the overshoot of input torque.

In Figure 13, it can be observed that at t_lc = 40.5 s, moderate t_la and earlier t_lb can reduce input torque overshoot; when t_lb = 40.5 s, moderate t_la and earlier t_hc can reduce the overshoot of input torque; when t_la = 46.5 s, later t_lb and earlier t_lc 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.

$$\begin{array}{l}\mathrm{Ha}:\min (f_{\mathrm{shift}\_\mathrm{Ha}})=\frac{\Delta {\Omega }_e}{{\Omega }_{e0}}+\frac{\Delta {\omega }_{MR}}{{\omega }_{MRH}}+\frac{\Delta {\tau }_f}{{\tau }_{fH}},\\ \mathrm{Hb}:\min (f_{\mathrm{shift}\_\mathrm{Hb}})=\frac{\Delta T_{in}}{T_{inH}}+\frac{\Delta T_{out}}{T_{outH}}+\frac{\Delta {\tau }_f}{{\tau }_{fH}},\\ \mathrm{s}.\mathrm{t}.\left\{\begin{array}{c}15\le t_{ha}\le 18,\\ 12\le t_{hb}\le 13,\\ 9\le t_{hc}\le 12.\end{array}\right.\end{array}$$

(36)

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 f_{shift_Ha}, t_ha = 15.0 s, t_hb = 13.0 s, t_hc = 10.60 s; At f_{shift_Hb}, t_ha = 18.0 s, t_hb = 13.0 s, t_hc = 9.80 s. Taking t_ha = 15.0 s, t_hb = 12.0 s, t_hc = 9.0 s as the baseline of the upshift strategy, it can be seen from

Table 12. Comparison of the upshift optimization model and baseline performance.

Strategy	ΔΩe (rad/s)	ΔTin (Nm)	ΔωMR (rad/s)	ΔTout (Nm)	Δτf (Nm)
Baseline	104.14	407.66	8.34	885.56	297.31
Model Ha	71.90	209.80	10.56	410.03	151.31
Variation of overshoot Δφ	−5.13%	−8.46%	+8.54%	−10.48%	−9.25%
Model Hb	77.82	215.92	10.73	446.53	155.09
Variation of overshoot Δφ	−4.19%	−8.19%	+9.18%	−9.02%	−9.01%

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.

$$\begin{array}{l}\mathrm{La}:\min (f_{\mathrm{shift}\_\mathrm{La}})=\frac{\Delta {\Omega }_{\mathrm{e}}}{{\Omega }_{\mathrm{e}0}}+\frac{\Delta {\omega }_{MR}}{{\omega }_{MRL}}+\frac{\Delta {\tau }_o}{{\tau }_{oL}},\\ \mathrm{Lb}:\min (f_{\mathrm{shift}\_\mathrm{Lb}})=\frac{\Delta T_{in}}{T_{inL}}+\frac{\Delta T_{out}}{T_{outL}}+\frac{\Delta {\tau }_o}{{\tau }_{oL}},\\ \mathrm{s}.\mathrm{t}.\left\{\begin{array}{c}45\le t_{la}\le 48,\\ 39\le t_{lb}\le 42,\\ 39\le t_{lc}\le 42.\end{array}\right.\end{array}$$

(37)

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 f_{shift_La}, t_la = 46.0 s, t_lb = 39.0 s, t_lc = 42.0 s; At f_{shift_Lb}, t_la = 46.5 s, t_lb = 41.2 s, t_lc = 39.0 s. Furthermore, t_la = 45.0 s, t_lb = 42.0 s, and t_lc = 42.0 s are considered the baseline of the downshift strategy. It can be seen from

Table 13. Comparison of the downshift optimization model and baseline performance.

Strategy	ΔΩe (rad/s)	ΔTin (Nm)	ΔωMR (rad/s)	ΔTout (Nm)	Δτo (Nm)
Baseline	118.05	499.28	11.33	533.78	291.31
Model La	45.66	951.26	6.21	836.54	402.76
Variation of overshoot Δφ	−11.52%	22.80%	13.96%	−6.67%	−4.36%
Model Lb	129.27	370.31	12.08	490.92	275.79
Variation of overshoot Δφ	1.79%	−6.51%	16%	−7.61%	−4.97%

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.

4. Conclusions

(1)

The core components and working principles of the rigid coaxial twin-rotor helicopter power system were introduced, a torque model of the turboshaft engine, and a rigid rotor aerodynamic model were constructed. A composite helicopter engine/transmission system/rotor model was constructed and coupled with a variable-speed transmission system model.

(2)

Using the control time points of the upshift and downshift of fuel flow Fue, hydraulic pressure Hyd, and pitch angle Pit as input variables, 17 upshift/downshift timing strategies were designed using Box–Behnken’s center combination sampling method. Furthermore, RSM was used to obtain regression-fitting models between the overshoot values of engine speed, transmission system input torque, rotor speed, transmission system output torque, friction clutch torque, and one-way clutch torque and input variables. The interaction effects between different upshift and downshift time points were analyzed.

(3)

The upshift optimization models Ha and Hb were designed using nonlinear programming. Compared with model Hb, model Ha has better impact suppression effects in terms of engine speed, input and output torque of the variable speed transmission system, and friction clutch torque. And compared to the baseline upshift strategy, the upshift strategy (t_ha = 15.0 s, t_hb = 13.0 s, t_hc = 10.60 s) optimized by model Ha reduces the engine speed, variable transmission input, output torque, and friction clutch torque overshoot by 5.13%, 8.46%, 10.48%, and 9.25%, respectively.

(4)

The downshift optimization models La and Lb were designed using nonlinear programming. Compared with the optimized model La, the optimized model Lb is slightly weaker in rotor stability control, but more balanced in engine speed, variable transmission system input and output torque and one-way clutch torque impact suppression. And compared to the baseline downshift strategy, the downshift strategy (t_la = 46.5 s, t_lb = 41.2 s, t_lc = 39.0 s) optimized by model La reduces the overshoot of input and output torque of transmission and one-way clutch torque by 6.51%, 7.61%, and 4.97%, respectively.

In this paper, the studied power system primarily focuses on the effect of the time discrepancy in controlling the engine and twin rotor on the system’s dynamic response. In real-world operational conditions, distinct control methods can be applied to various engines to assess the influence of torque output adjustments between the two engines during the shifting process. Additional research can delve into understanding how flexible blades influence the dynamic properties of gear shifting.