Optimization of Shifting Quality for Hydrostatic Power-Split Transmission with Single Standard Planetary Gear Set

: To improve the driving comfort of continuously variable transmission (CVT) tractors, the shifting quality of hydrostatic power-split transmission with a standard planetary gear set was optimized. Firstly, the powertrain of the CVT and two shift strategies, direct-shift and bridge-shift, were introduced; then, a dynamic model of tractor shifting was constructed, and the models of key components such as wet clutches and proportional pressure valves were experimentally veriﬁed. Finally, the control parameters of the above two shifting strategies were optimized, and the acceleration impact and sliding energy loss caused by them were compared. The results showed the following: the minimum peak acceleration of the bridge-shift method was 0.385807 m/s 2 ; the energy consumption of the bridge-shift method was signiﬁcantly lower than that of the direct-shift method; the sliding friction work of clutches decreased by 14.92% and 75.84%, respectively, while their power loss decreased by 22.82% and 74.48%, respectively.


Introduction
The working conditions of tractors are complex, as they require more gears to meet different operational needs, but this also leads to complex transmission structures and difficult gear selection.Tractors with continuously variable transmission (CVT) can effectively solve the above problems.At present, there are three common forms of tractor CVT [1][2][3]: hydrostatic transmission, steel belt transmission, and hydrostatic power-split transmission.Among them, the energy consumption of hydrostatic transmission is very high, and the torque transmitted by steel belt transmission is very limited, while hydrostatic power-split transmission has both a high efficiency and a large load driving capacity [4][5][6].Since the release of the first CVT tractor "926 Vario" by Fent in 1996, hydrostatic power-split transmissions have gradually been applied to various pieces agricultural machinery [7].Afterwards, transmission manufacturers began to introduce various concepts of tractor CVTs [8,9], such as the Eccom produced by ZF and the Autopowr produced by John Deere.The above-mentioned transmissions all adopt multi-range technology to achieve continuous speed adjustment [10][11][12], so it is necessary to conduct research to improve the quality of the shift [13][14][15], including changing the gear ratio, the clutch engagement time, and the displacement ratio of the swash plate axial piston units.For tractors with heavy load operations such as plowing as their main operating conditions, while paying attention to their riding comfort we also need to consider issues such as power interruption and clutch damage, all of which are important criteria for evaluating the shifting quality.Typically, cascading multiple planetary gears together to form a compound planetary gear set can improve the shifting quality through speed synchronization [10].This is currently the mainstream strategy for CVT design, as its shifting logic is very simple.In this research field, Bao et al. [16] constructed a clutch control system based on solenoid directional valves and optimized the clutch pressure, the flow, and the displacement ratio of the pump to the motor to improve the shifting quality of the power-split CVT.Chen et al. [17] proposed a simulation model of a similar hydraulic control system, making it possible to further study the shifting dynamics of CVT tractors through computer simulation.Iqbal et al. [18] conducted similar work.Wang et al. [19] analyzed the reliability of clutch control systems based on on-off logic and discussed the possible influence of hydraulic system failures on the shifting quality.To further improve the shifting quality of this type of transmission, it is necessary to use proportional pressure valves to accurately control the clutch action.For example, Xiang et al. [20] proposed a control strategy for dual-clutch transmissions that can maintain the sliding in the torque phase to improve the shifting quality.Li and Görges [21] conducted similar work.Li et al. [22] used a PID controller to track the pressure of the clutch to ensure the repeatability of proportional pressure control.Although a compound planetary gear set can improve the shifting smoothness through speed synchronization, its structure needs to fully consider support and load balance issues when applied, which brings difficulties to its design, manufacturing, and assembly.In contrast, using a single standard planetary gear to merge the power is simple and cost effective.Currently, some companies such as Hofer have shown great interest in this new concept of transmission.However, the shift logic of the transmission is complex, requiring the simultaneous adjustment of multiple wet clutches and swash plate axial piston units during shifting, which is much more difficult to control than traditional power-split CVTs.To improve the shifting quality of this cost-effective power-split CVT with a single planetary gear set and promote its application in tractors, a new strategy called the bridge-shift method is proposed in this study.

Powertrain
The hydrostatic power-split transmission proposed in this study has two ranges, HM 1 and HM 2 , in the forward direction, which can achieve a stepless speed regulation of the tractor within a range of 0-30 km/h.The principle of the transmission is shown in Figure 1.The engine power is divided into two parts on the input shaft, with part of the power being transferred to the sun gear of the planetary gear set through the swash plate axial piston units and the rest of the power entering the ring gear of the planetary gear set through the gear train.The above two parts of power are marked as the hydraulic circuit power and mechanical circuit power, respectively.The transmission ratio of the mechanical circuit is fixed, so the output speed of the transmission only depends on the displacement ratio of the pump to the motor, which is numerically equal to the actual displacement of the pump divided by the rated displacement of the motor.Since the displacement of the pump changes in two directions with the inclination angle of its swash plate, the displacement ratio ranges from −1 to +1 ("+" indicates that the speed direction of the pump and motor is the same, while "−" indicates the opposite).In each range, the displacement ratio of −1 corresponds to the lowest speed of the tractor, while the displacement ratio of +1 corresponds to the highest speed of the tractor.Before starting the tractor, the transmission control unit (TCU) needs to adjust the displacement ratio of the pump to the motor to −1 (i.e., the displacement ratio corresponding to the minimum CVT output speed of the range HM1), engage clutches C1 and C3, and separate clutches CR, C2, and C4.Then, the TCU slowly engages the clutch CF to bring the tractor to its minimum operating speed.
After starting, the transmission operates in the range HM1.As the displacement ratio changes in the direction of "−1→+1", the tractor speed continuously increases.Once the tractor reaches its predetermined speed, the TCU separates clutches C1 and C3, engages clutches C2 and C4, and reversely adjusts the displacement ratio of the pump to the motor to achieve the equal-speed shifting of the transmission, thereby switching the working range of the transmission from HM1 to HM2.The speed adjustment process of the ranges HM1 and HM2 is completely the same and will not be repeated here.
When clutches CR, C1, and C4 are engaged and clutches CF, C2, and C3 are separated, the transmission operates in the reverse range HMR.The speed of the tractor in this range covers two directions, and the displacement ratio corresponding to its zero speed is approximately −0.9.When the displacement ratio changes from −0.9 to +1, the tractor can achieve a stepless speed regulation within the range of 0-16 km/h in the reverse direction.
The clutch schedule of this transmission is shown in Table 1.

Control Strategies
The transmission involves the separation or engagement of four clutches during shifting, and the action timing of each clutch will have a significant impact on the shifting process.For example, after the separation of clutches C1 and C3, the speed of the tractor will continuously decrease under the action of the load.If the engagement of clutches C2 and C4 is slow, the driving and driven plates of the clutch will be in a continuous sliding Before starting the tractor, the transmission control unit (TCU) needs to adjust the displacement ratio of the pump to the motor to −1 (i.e., the displacement ratio corresponding to the minimum CVT output speed of the range HM 1 ), engage clutches C 1 and C 3 , and separate clutches C R , C 2 , and C 4 .Then, the TCU slowly engages the clutch C F to bring the tractor to its minimum operating speed.
After starting, the transmission operates in the range HM 1 .As the displacement ratio changes in the direction of "−1→+1", the tractor speed continuously increases.Once the tractor reaches its predetermined speed, the TCU separates clutches C 1 and C 3 , engages clutches C 2 and C 4 , and reversely adjusts the displacement ratio of the pump to the motor to achieve the equal-speed shifting of the transmission, thereby switching the working range of the transmission from HM 1 to HM 2 .The speed adjustment process of the ranges HM 1 and HM 2 is completely the same and will not be repeated here.
When clutches C R , C 1 , and C 4 are engaged and clutches C F , C 2 , and C 3 are separated, the transmission operates in the reverse range HM R .The speed of the tractor in this range covers two directions, and the displacement ratio corresponding to its zero speed is approximately −0.9.When the displacement ratio changes from −0.9 to +1, the tractor can achieve a stepless speed regulation within the range of 0-16 km/h in the reverse direction.
The clutch schedule of this transmission is shown in Table 1.

Control Strategies
The transmission involves the separation or engagement of four clutches during shifting, and the action timing of each clutch will have a significant impact on the shifting process.For example, after the separation of clutches C 1 and C 3 , the speed of the tractor will continuously decrease under the action of the load.If the engagement of clutches C 2 and C 4 is slow, the driving and driven plates of the clutch will be in a continuous sliding state, which will burden the cooling system and shorten the service life of the clutch.In severe cases, it can also directly cause power interruption.On the contrary, if the engagement of clutches C 2 and C 4 is very fast, the speed difference between the driving and driven plates of the clutch is quickly eliminated, which will cause severe speed oscillations and significantly reduce the riding comfort of the tractor.The actual shifting process is very complex, which requires precise control of the actions of each clutch in the time domain.To solve this problem, two shifting strategies are proposed: the direct-shift and bridge-shift strategies.
Taking the switching of range HM 1 to range HM 2 as an example, the former separates clutches C 1 and C 3 while directly engaging clutches C 2 and C 4 , while the latter needs to insert a transitional state of C 2 and C 3 engagement during the aforementioned process, as shown in Figure 2.
Agriculture 2023, 13, x FOR PEER REVIEW 4 of 19 state, which will burden the cooling system and shorten the service life of the clutch.In severe cases, it can also directly cause power interruption.On the contrary, if the engagement of clutches C2 and C4 is very fast, the speed difference between the driving and driven plates of the clutch is quickly eliminated, which will cause severe speed oscillations and significantly reduce the riding comfort of the tractor.The actual shifting process is very complex, which requires precise control of the actions of each clutch in the time domain.To solve this problem, two shifting strategies are proposed: the direct-shift and bridge-shift strategies.
Taking the switching of range HM1 to range HM2 as an example, the former separates clutches C1 and C3 while directly engaging clutches C2 and C4, while the latter needs to insert a transitional state of C2 and C3 engagement during the aforementioned process, as shown in Figure 2.

Modeling of the Swash Plate Axial Piston Units
The swash plate axial piston units are the core speed-regulating components of the CVT, consisting of a variable-displacement pump and a fixed-displacement motor.Its pressure, torque, flow, and speed meet the following equations: where Tp and Tm are the theoretical torques of the pump shaft and motor shaft, respectively, N•m; ΔPp and ΔPm are the pressure differences between the inlet and outlet of the pump and motor, respectively, MPa; Qp and Qm are the theoretical flows of the pump and motor, respectively, L/min; np and nm are the rotation speeds of the pump shaft and motor

Modeling of the Swash Plate Axial Piston Units
The swash plate axial piston units are the core speed-regulating components of the CVT, consisting of a variable-displacement pump and a fixed-displacement motor.Its pressure, torque, flow, and speed meet the following equations: where T p and T m are the theoretical torques of the pump shaft and motor shaft, respectively, N•m; ∆P p and ∆P m are the pressure differences between the inlet and outlet of the pump and motor, respectively, MPa; Q p and Q m are the theoretical flows of the pump and motor, respectively, L/min; n p and n m are the rotation speeds of the pump shaft and motor shaft, respectively, r/s; V p and V m are the rated displacements of the pump and motor, respectively, cm 3 /r; and e is the displacement ratio of the pump to the motor.
In actual systems, it is necessary to consider the torque loss and flow loss caused by mechanical friction and oil leakage: T mr = T m η mm (6) where T pr and T mr are the real torques of the pump shaft and motor shaft, respectively, N•m; η mp and η mm are the mechanical efficiencies of the pump and motor, respectively; Q pr and Q mr are the real flows of the pump and motor, respectively, L/min; and η vp and η vm are the volume efficiencies of the pump and motor, respectively.

Modeling of the Power-Shift System
The power-shift system consists of wet clutches and a corresponding hydraulic circuit.The frictional torque that the clutch can transmit is: where T c is the frictional torque transmitted in the clutch plates, N; µ is the coulomb friction coefficient; F n is the normal force acting on the clutch plates, N; n p is the number of clutch contact faces; r o and r i are the outside radius and inside radius of the friction plates, respectively, mm; R n is the relative velocity, r/min; and d v is the rotary stick velocity threshold, r/min.The normal force F n is determined by the combination of the oil pressure, centrifugal force, and spring force: where P c is the oil pressure, MPa; A c is the effective area of the piston, mm 2 ; F c is the centrifugal force, N; k c is the stiffness of the spring, N/mm; and x ci and ∆x c are the initial compression and relative displacement of the spring, respectively, mm.The rotary hydraulic cylinder is a typical coupling element used in multi-plate wet clutches in the transmission.Its rotational speed is high, so its oil chamber is subjected to centrifugal acceleration.The structure of the clutch hydraulic cylinder is shown in Figure 3, and the centrifugal force acting on its piston is calculated as follows: where ρ is the bulk density of the hydraulic oil, kg•m 3 ; ω is the angular velocity, r/min; r p and r l are the outside radius and inside radius of the fluid volume acting on the piston, respectively, mm; and r r is the inside radius of the piston, mm.A proportional pressure valve is used to control the working pressure of the clutch, consisting of a proportional electromagnet and a three-way spool valve, as shown in Fig- ure 4. When the electromagnetic force increases, the valve spool moves to the right to increase the opening of the outlet port, causing the output pressure to increase.When the electromagnetic force decreases, the valve spool moves to the left, causing the output pressure to decrease.On this basis, when the output pressure increases, the valve spool moves left to allow excess oil to flow back to the tank through the oil return port, thereby reducing the output pressure.When the output pressure decreases, the piston moves right to increase the output pressure.According to its working principle, the force equation of the valve spool is expressed as: where As is the effective area of the valve spool, mm 2 ; pout is the output pressure of the valve, MPa; Fjet is the jet force, N; ks is the stiffness of the spring, N/mm; xsi and Δxs are the initial compression of the spring and the displacement of the valve spool, respectively, mm; and Fg is the electromagnetic force, N. The jet force is calculated using the following equation: where Cq is the flow coefficient; ds is the equivalent diameter of the valve spool, mm; Δpv is the pressure difference between the inlet and outlet of the valve, MPa; and αjet is the jet angle, rad.The displacement of the valve spool obtained by simultaneous Equations ( 12) and ( 13) is as follows: A proportional pressure valve is used to control the working pressure of the clutch, consisting of a proportional electromagnet and a three-way spool valve, as shown in Figure 4.When the electromagnetic force increases, the valve spool moves to the right to increase the opening of the outlet port, causing the output pressure to increase.When the electromagnetic force decreases, the valve spool moves to the left, causing the output pressure to decrease.On this basis, when the output pressure increases, the valve spool moves left to allow excess oil to flow back to the tank through the oil return port, thereby reducing the output pressure.When the output pressure decreases, the piston moves right to increase the output pressure.According to its working principle, the force equation of the valve spool is expressed as: where A s is the effective area of the valve spool, mm 2 ; p out is the output pressure of the valve, MPa; F jet is the jet force, N; k s is the stiffness of the spring, N/mm; x si and ∆x s are the initial compression of the spring and the displacement of the valve spool, respectively, mm; and F g is the electromagnetic force, N. A proportional pressure valve is used to control the working pressure of the clutch, consisting of a proportional electromagnet and a three-way spool valve, as shown in Fig- ure 4. When the electromagnetic force increases, the valve spool moves to the right to increase the opening of the outlet port, causing the output pressure to increase.When the electromagnetic force decreases, the valve spool moves to the left, causing the output pressure to decrease.On this basis, when the output pressure increases, the valve spool moves left to allow excess oil to flow back to the tank through the oil return port, thereby reducing the output pressure.When the output pressure decreases, the piston moves right to increase the output pressure.According to its working principle, the force equation of the valve spool is expressed as: where As is the effective area of the valve spool, mm 2 ; pout is the output pressure of the valve, MPa; Fjet is the jet force, N; ks is the stiffness of the spring, N/mm; xsi and Δxs are the initial compression of the spring and the displacement of the valve spool, respectively, mm; and Fg is the electromagnetic force, N. The jet force is calculated using the following equation: where Cq is the flow coefficient; ds is the equivalent diameter of the valve spool, mm; Δpv is the pressure difference between the inlet and outlet of the valve, MPa; and αjet is the jet angle, rad.The displacement of the valve spool obtained by simultaneous Equations ( 12) and ( 13) is as follows: The jet force is calculated using the following equation: where C q is the flow coefficient; d s is the equivalent diameter of the valve spool, mm; ∆p v is the pressure difference between the inlet and outlet of the valve, MPa; and α jet is the jet angle, rad.
The displacement of the valve spool obtained by simultaneous Equations ( 12) and ( 13) is as follows: According to the above analysis, the output pressure of the proportional valve depends on the electromagnetic force F g , which is controlled by a current signal.To clarify the corresponding relationship between the input current and output pressure, a signal generator is used to calibrate the valve, and the results are shown in Figure 5.It can be seen that within the current range of 4-20 mA, the input current of the proportional valve exhibits a clear linear relationship with oil pressure.Therefore, we used the calibration data to construct an electromagnetic model of the valve.(14) According to the above analysis, the output pressure of the proportional valve depends on the electromagnetic force Fg, which is controlled by a current signal.To clarify the corresponding relationship between the input current and output pressure, a signal generator is used to calibrate the valve, and the results are shown in Figure 5.It can be seen that within the current range of 4-20 mA, the input current of the proportional valve exhibits a clear linear relationship with oil pressure.Therefore, we used the calibration data to construct an electromagnetic model of the valve.The hydraulic circuit constructed based on clutches and proportional valves is the core of the power-shift system and requires independent experimental verification of its mathematical model.We closed the outlet of the proportional valve before the experiment, and the PLC controlled its AD module to output a step signal corresponding to the rated pressure of the clutch.Note that the output signal of the AD module used was the voltage, which needed to be converted into a 4-20 mA current through a converter module to control the proportional valve.At the same time, the Labview program controlled the data acquisition card (NI USB-6009) to capture the output pressure of the proportional valve feedback from the sensor.The input signal of the simulation model was consistent with the experiment, that is, the input current was modulated from 0 to the maximum value in the experiment in a very short time to observe the pressure response of the model.The simulation and measurement results of the step response of the proportional pressure valve are shown in Figure 6.From the figure, it can be seen that the simulation results of the mathematical model constructed in this study were highly consistent with the experimental results and could meet the needs of subsequent dynamic analysis.
We connected the model of the proportional valve with the model of the wet clutch, further constructed the model of the power-shift system and conducted an experimental verification of it.The simulation and measurement results are shown in Figure 7.The figure shows that under the same input signal of the proportional valve, the clutch pressure response of the simulation model was basically consistent with the experimental results, thus proving the reliability of the constructed model.
The models of the proportional valve and wet clutch were relatively complex; for some models not covered in this article, please refer to the AMESim manual.The key parameters used in the simulation calculations are shown in Tables 2 and 3.The hydraulic circuit constructed based on clutches and proportional valves is the core of the power-shift system and requires independent experimental verification of its mathematical model.We closed the outlet of the proportional valve before the experiment, and the PLC controlled its AD module to output a step signal corresponding to the rated pressure of the clutch.Note that the output signal of the AD module used was the voltage, which needed to be converted into a 4-20 mA current through a converter module to control the proportional valve.At the same time, the Labview program controlled the data acquisition card (NI USB-6009) to capture the output pressure of the proportional valve feedback from the sensor.The input signal of the simulation model was consistent with the experiment, that is, the input current was modulated from 0 to the maximum value in the experiment in a very short time to observe the pressure response of the model.The simulation and measurement results of the step response of the proportional pressure valve are shown in Figure 6.From the figure, it can be seen that the simulation results of the mathematical model constructed in this study were highly consistent with the experimental results and could meet the needs of subsequent dynamic analysis.
We connected the model of the proportional valve with the model of the wet clutch, further constructed the model of the power-shift system and conducted an experimental verification of it.The simulation and measurement results are shown in Figure 7.The figure shows that under the same input signal of the proportional valve, the clutch pressure response of the simulation model was basically consistent with the experimental results, thus proving the reliability of the constructed model.
The models of the proportional valve and wet clutch were relatively complex; for some models not covered in this article, please refer to the AMESim manual.The key parameters used in the simulation calculations are shown in Tables 2 and 3.

Modeling of Gears and Shafts
The torque and speed of the two meshing gears satisfy the following equations: T 2 = i 12 T 1 (16) where i 12 is the transmission ratio of the gear pairs; n 1 and n 2 are the speeds of the two gears, r/min; and T 1 and T 2 are the torques of the two gears, N•m.The speed and torque between the three basic components of the planetary gear, the sun gear, the ring gear, and the carrier satisfy the following equations: T s : T r : where n s , n r , and n c are the speeds of the sun gear, ring gear, and carrier, respectively, r/min; T s , T r , and T c are the torques of the sun gear, ring gear, and carrier, respectively, N•m; and k is the standing ratio of the standard planetary gear.In this study, the moment of inertia of each component is calculated by the SolidWorks 2016 software and is equivalent to the transmission shaft.Its influence on the torque of each shaft is as follows: where T a and T 0 are the actual torque and theoretical torque of the shaft, respectively, N•m; J is the moment of inertia, kg•m 2 ; ω is the angular velocity of the shaft, rad/s; and t is the time, s.

Modeling of Tractor
Based on the above equations, a shift dynamics model of the entire continuously variable transmission tractor was constructed using AMESim, as shown in Figure 8.

Modeling of Gears and Shafts
The torque and speed of the two meshing gears satisfy the following equations: 2 =  12  1 (16) where i12 is the transmission ratio of the gear pairs; n1 and n2 are the speeds of the two gears, r/min; and T1 and T2 are the torques of the two gears, N•m.The speed and torque between the three basic components of the planetary gear, the sun gear, the ring gear, and the carrier satisfy the following equations: :   :   = 1: : (1 + ) where ns, nr, and nc are the speeds of the sun gear, ring gear, and carrier, respectively, r/min; Ts, Tr, and Tc are the torques of the sun gear, ring gear, and carrier, respectively, N•m; and k is the standing ratio of the standard planetary gear.In this study, the moment of inertia of each component is calculated by the Solid-Works 2016 software and is equivalent to the transmission shaft.Its influence on the torque of each shaft is as follows: where Ta and T0 are the actual torque and theoretical torque of the shaft, respectively, N•m; J is the moment of inertia, kg•m 2 ; ω is the angular velocity of the shaft, rad/s; and t is the time, s.

Modeling of Tractor
Based on the above equations, a shift dynamics model of the entire continuously variable transmission tractor was constructed using AMESim, as shown in Figure 8.

Evaluation Indicators
The speed drop is defined as the difference between the output speed of the transmission before the shift and the lowest output speed during the shift: where δ 1 is the speed drop, r/min; ω 1 is the output speed before the shift, r/min; and ω 2 is the lowest output speed during the shift, r/min.According to Duncan and Wegscheid [23], the peak acceleration of a tractor in the longitudinal direction can well reflect its driving comfort during shifting.Therefore, this study took the peak acceleration as one of the indicators for evaluating the shifting quality of the CVT, and its expression is as follows: where δ 2 is the peak acceleration of the tractor during the shift, m/s 2 .When the clutch is engaged, a large amount of heat will be generated due to friction, and in severe cases, it may burn out the clutch.The power loss during the aforementioned process is as follows: where δ 3 is the maximum power loss during the shift, kW; T c is the friction torque, N•m; and ∆ω is the difference in the angular speed of the clutch driving and driven disc, r/min.On this basis, sliding friction work is defined as the integral of the power loss over time: where δ 4 is the sliding friction work, kJ; and t 1 and t 2 are the start and end times of the shift, s.

Determination of Shift Points
The process of direct shifting relatively simple, with clutches C 1 and C 3 being separate while clutches C 2 and C 4 engage.During the shift process, the displacement ratio is synchronously adjusted, and its initial and final values need to meet the following relationship: where i HM1 and i HM2 are the transmission ratios in HM 1 and HM 2 , respectively; and i x is the transmission ratio of the gear pair g x .
From the perspective of transmission efficiency and energy consumption, the authors have demonstrated in previous research that the optimal shift point for this CVT in the range HM 1 is e = 1.Based on the above equations, the displacement ratio after the shift was calculated to be e = −0.8034.

Optimization of Shifting Quality
The pressure control signal of the proportional valve based on the direct-shift method is shown in Figure 9.If we define the pressure relief time of proportional valves 1 and 3 as t s1 (i.e., reference time, 10 s), then ∆T is the start time of the pressure rise for proportional valves 2 and 4 relative to t s1 .T 4 and T 2 are the times corresponding to the two inflection points in the pressure rise curve of proportional valve 2, respectively.T 1 and T 5 are the times corresponding to the two inflection points in the pressure rise curve of proportional valve 4, respectively.K 2 and K 1 are the percentages of the input signals corresponding to the second inflection point in the pressure rise curve of proportional valves 2 and 4, respectively.The starting time of T 1 and T 4 is the same, and the starting time and duration of the reverse change in the displacement ratio are T s and T d , respectively.On this basis, we adopted an orthogonal experiment with nine factors and four levels to optimize the above control parameters.The schedule of the experiment is shown in Table 4, and the results are shown in Tables 5 and 6.Considering that the peak acceleration directly affects the driving comfort, we only optimized the parameters for this indicator when designing the orthogonal experiments, but we considered other indicators together when analyzing the results.

Optimization of Shifting Quality
The pressure control signal of the proportional valve based on the direct-shift method is shown in Figure 9.If we define the pressure relief time of proportional valves 1 and 3 as ts1 (i.e., reference time, 10 s), then ΔT is the start time of the pressure rise for proportional valves 2 and 4 relative to ts1.T4 and T2 are the times corresponding to the two inflection points in the pressure rise curve of proportional valve 2, respectively.T1 and T5 are the times corresponding to the two inflection points in the pressure rise curve of proportional valve 4, respectively.K2 and K1 are the percentages of the input signals corresponding to the second inflection point in the pressure rise curve of proportional valves 2 and 4, respectively.The starting time of T1 and T4 is the same, and the starting time and duration of the reverse change in the displacement ratio are Ts and Td, respectively.On this basis, we adopted an orthogonal experiment with nine factors and four levels to optimize the above control parameters.The schedule of the experiment is shown in Table 4, and the results are shown in Tables 5 and 6.Considering that the peak acceleration directly affects the driving comfort, we only optimized the parameters for this indicator when designing the orthogonal experiments, but we considered other indicators together when analyzing the results.According to the results of the orthogonal range analysis, when switching from HM1 to HM2, the degree of influence of each factor on the direct-shift method was ranked as follows: the reverse starting point Ts, the time difference ΔT, time T2, current K1, current K2, time T5, time T4, time T1, and the reverse duration Td.The best combination of factors was A3B4C3D1E1F4G3H3I1.By substituting the optimized parameters into the simulation model, the various indicators for the direct-shift method were obtained as follows: the speed drop was 30.67 r/min (no power interruption), the peak acceleration was 0.384535 m/s 2 , the power loss of clutch C2 was 22.7383 kW, the sliding friction work of clutch C2 was 8.0752 kJ, the power loss of clutch C4 was 18.1166 kW, and the sliding friction work of clutch C4 was 2.3906 kJ.According to the results of the orthogonal range analysis, when switching from HM 1 to HM 2 , the degree of influence of each factor on the direct-shift method was ranked as follows: the reverse starting point T s , the time difference ∆T, time T 2 , current K 1 , current K 2 , time T 5 , time T 4 , time T 1 , and the reverse duration T d .The best combination of factors was A 3 B 4 C 3 D 1 E 1 F 4 G 3 H 3 I 1 .By substituting the optimized parameters into the simulation model, the various indicators for the direct-shift method were obtained as follows: the speed drop was 30.67 r/min (no power interruption), the peak acceleration was 0.384535 m/s 2 , the power loss of clutch C 2 was 22.7383 kW, the sliding friction work of clutch C 2 was 8.0752 kJ, the power loss of clutch C 4 was 18.1166 kW, and the sliding friction work of clutch C 4 was 2.3906 kJ.

Determination of Shift Points
The process of bridge shifting is divided into two stages.In the first stage, the transmission shifts from the low-speed range HM 1 to the transition range, where clutch C 1 separates while clutch C 2 engages.In the second stage, the transmission shifts from the transition range to the high-speed range HM 2 , where clutch C 3 separates and clutch C 4 engages.During the shift process, the displacement ratio is synchronously adjusted, and its initial and final values are the same as those of the direct-shift method.However, the displacement ratio of the transition range will be used as an optimization variable, which will be discussed later.

Optimization of Shifting Quality
The pressure control signal of the proportional valve based on the bridge-shift method is shown in Figure 10.The factors T 1 , T 2 , K 1 , T 4 , T 5 , and K 2 in the bridge-shift method are the same as those specified in the direct-shift method.We define the pressure relief times for proportional valves 1 and 3 as t s1 (10 s) and t s2 (10 s + ∆T 3 ), respectively; then, ∆T 1 is the start time of the pressure increase for proportional valve 2 relative to t s1 .∆T 2 is the start time of the pressure increase for proportional valve 4 relative to t s2 .The starting time of the reverse change in displacement ratio is T s .The duration of the two stages of the reverse change in the displacement ratio are T d1 and T d2 , respectively.The displacement ratio of the transition range is e t .On this basis, we adopted an orthogonal experiment with thirteen factors and three levels to optimize the above control parameters.The schedule of the experiment is shown in Table 7, and the results are shown in Tables 8 and 9.

Determination of Shift Points
The process of bridge shifting is divided into two stages.In the first stage, the transmission shifts from the low-speed range HM1 to the transition range, where clutch C1 separates while clutch C2 engages.In the second stage, the transmission shifts from the transition range to the high-speed range HM2, where clutch C3 separates and clutch C4 engages.During the shift process, the displacement ratio is synchronously adjusted, and its initial and final values are the same as those of the direct-shift method.However, the displacement ratio of the transition range will be used as an optimization variable, which will be discussed later.

Optimization of Shifting Quality
The pressure control signal of the proportional valve based on the bridge-shift method is shown in Figure 10.The factors T1, T2, K1, T4, T5, and K2 in the bridge-shift method are the same as those specified in the direct-shift method.We define the pressure relief times for proportional valves 1 and 3 as ts1 (10 s) and ts2 (10 s + ΔT3), respectively; then, ΔT1 is the start time of the pressure increase for proportional valve 2 relative to ts1.ΔT2 is the start time of the pressure increase for proportional valve 4 relative to ts2.The starting time of the reverse change in displacement ratio is Ts.The duration of the two stages of the reverse change in the displacement ratio are Td1 and Td2, respectively.The displacement ratio of the transition range is et.On this basis, we adopted an orthogonal experiment with thirteen factors and three levels to optimize the above control parameters.The schedule of the experiment is shown in Table 7, and the results are shown in Tables 8 and 9.According to the results of the orthogonal range analysis, when switching from HM 1 to HM 2 , the degree of influence of each factor with the bridge-shift method was ranked as follows: the reverse starting point T s , time T 1 , time T 4 , time difference ∆T 3 , current K 1 , time difference ∆T 1 , time T 2 , the reverse duration T d1 , the reverse duration T d2 , current K 2 , time difference ∆T 2 , displacement ratio e t , and time T 5 .The optimum level combination was A 3 B 3 C 2 D 3 E 1 F 1 G 1 H 2 I 1 J 3 K 3 L 1 M 1 , and after substituting the parameters into the simulation model, the various indicators for the bridge-shift method were obtained as follows: the speed drop was 16.035 r/min (no power interruption), the peak acceleration was 0.385807 m/s 2 , the power loss of clutch C 2 was 17.5495 kW, the sliding friction work of clutch C 2 was 6.8700 kJ, the power loss of clutch C 4 was 4.6241 kW, and the sliding friction work of clutch C 4 was 0.5775 kJ.
The shifting results under the two control strategies are shown in Figure 11.Compared to the direct-shift method, the shifting quality of the tractor based on the bridge-shift method was greatly improved: the speed drop was reduced by 47.72%, the peak acceleration was increased by 0.33% (which can be ignored), the power loss of clutch C 2 was reduced by 22.82%, the sliding friction work of clutch C 2 was reduced by 14.92%, the power loss of clutch C 4 was reduced by 74.48%, and the sliding friction work of clutch C 4 was reduced by 75.84%.model, the various indicators for the bridge-shift method were obtained as follows: the speed drop was 16.035 r/min (no power interruption), the peak acceleration was 0.385807 m/s 2 , the power loss of clutch C2 was 17.5495 kW, the sliding friction work of clutch C2 was 6.8700 kJ, the power loss of clutch C4 was 4.6241 kW, and the sliding friction work of clutch C4 was 0.5775 kJ.
The shifting results under the two control strategies are shown in Figure 11.Compared to the direct-shift method, the shifting quality of the tractor based on the bridgeshift method was greatly improved: the speed drop was reduced by 47.72%, the peak acceleration was increased by 0.33% (which can be ignored), the power loss of clutch C2 was reduced by 22.82%, the sliding friction work of clutch C2 was reduced by 14.92%, the power loss of clutch C4 was reduced by 74.48%, and the sliding friction work of clutch C4 was reduced by 75.84%.

Discussion
Due to the involvement of multiple clutch actions, the generation of parasitic power is inevitable.To determine the direction of the power flow, we observed the power loss of all the clutches during shifting, as shown in Figure 12a,b.The figure shows that in the direct-shift method, parasitic power mainly flowed back along clutch C3, while in the bridge-shift method, parasitic power mainly flowed back along clutch C1.However, compared to the power flowing in the forward direction in clutches C2 and C4, the parasitic power generated under both shifting strategies was not significant, so the energy loss of clutches C1 and C3 was not discussed in this study.

Discussion
Due to the involvement of multiple clutch actions, the generation of parasitic power is inevitable.To determine the direction of the power flow, we observed the power loss of all the clutches during shifting, as shown in Figure 12a,b.The figure shows that in the direct-shift method, parasitic power mainly flowed back along clutch C 3 , while in the bridge-shift method, parasitic power mainly flowed back along clutch C 1 .However, compared to the power flowing in the forward direction in clutches C 2 and C 4 , the parasitic power generated under both shifting strategies was not significant, so the energy loss of clutches C 1 and C 3 was not discussed in this study.
Tractors can operate within a larger range of loads, so it was necessary to further analyze the shifting quality of the tractor under different tractive forces, as shown in Figure 13.Tractors can operate within a larger range of loads, so it was necessary to further analyze the shifting quality of the tractor under different tractive forces, as shown in Figure 13.
From Figure 13a,b, it can be seen that when the tractive force was less than 20,000 N, the speed drop and peak acceleration both rapidly decreased with the increase in the load.When the tractive force was greater than 20,000 N, the speed drop no longer changed significantly, while the peak acceleration still decreased slightly with the increase in the load.Figure 13c shows that the power loss of clutch C4 decreased with increasing the load, while the variation law of clutch C2 was opposite.However, the above law was very insignificant, especially since the variation in clutch C2 with the load was very small.Figure 13d shows that the sliding friction work of clutch C2 increased with increasing the load, while clutch C4 showed the same law but was not significant.
In response to the above laws, we provide the following explanation: the speed impact and energy losses were mutually affected and formed a causal relationship, that is, the clutch absorbed the speed and acceleration impact of the transmission system through its sliding process.Therefore, the load reduced the speed impact, and its cost was the severe sliding of the clutch friction plates and high energy losses.
In addition, by comparing the response of the two shifting strategies to the load, it was found that bridge-shift method had a significantly lower speed drop, power loss, and sliding friction work than the direct-shift method, except for its peak acceleration, which was comparable to that of the direct-shift method.It should be further emphasized that the tractor did not experience power interruption in all the simulation results.Therefore, the bridge-shift method proposed in this study is widely applicable to various load conditions of tractors.

Conclusions
This study conducted a shift dynamics analysis of a hydrostatic power-split tractor transmission with a single standard planetary gear set.Two power-shift strategies were proposed and compared, and the conclusions obtained are as follows: (1) The degree of influence of each factor with the direct-shift method is ranked as follows: the reverse starting point Ts, the time difference ΔT, time T2, current K1, current K2, time T5, time T4, time T1, and the reverse duration Td.The best combination of factors is A3B4C3D1E1F4G3H3I1.From Figure 13a,b, it can be seen that when the tractive force was less than 20,000 N, the speed drop and peak acceleration both rapidly decreased with the increase in the load.

Figure 1 .
Figure 1.Transmission scheme of hydrostatic power-split CVT.Note: the symbol g represents the gear pair, and the symbol C represents the wet clutch.

Figure 1 .
Figure 1.Transmission scheme of hydrostatic power-split CVT.Note: the symbol g represents the gear pair, and the symbol C represents the wet clutch.

Figure 3 .
Figure 3. Schematic diagram of the clutch hydraulic cylinder.

Figure 4 .
Figure 4. Schematic diagram of the proportional pressure valve.

Figure 3 .
Figure 3. Schematic diagram of the clutch hydraulic cylinder.

Figure 3 .
Figure 3. Schematic diagram of the clutch hydraulic cylinder.

Figure 4 .
Figure 4. Schematic diagram of the proportional pressure valve.

Figure 4 .
Figure 4. Schematic diagram of the proportional pressure valve.

Figure 5 .
Figure 5. Calibration of the proportional pressure valve.(a) Hydraulic system used for calibration testing.(b) Calibration results of the proportional valve.

Figure 5 .
Figure 5. Calibration of the proportional pressure valve.(a) Hydraulic system used for calibration testing.(b) Calibration results of the proportional valve.

Figure 6 .Figure 7 .
Figure 6.Simulation and measurement results of step response of proportional pressure valve.

Figure 6 .
Figure 6.Simulation and measurement results of step response of proportional pressure valve.

Figure 6 .Figure 7 .
Figure 6.Simulation and measurement results of step response of proportional pressure valve.

Figure 7 .
Figure 7. Experimental verification of the simulation model of the shift hydraulic system.(a) Simulation model of shift hydraulic system.(b) Simulation and measurement results of the response of clutch pressure to control signals.Note: the relief valve was designed for rapid pressure relief and was consistent with the actual hydraulic circuit.

Figure 8 .
Figure 8. Shifting dynamics model of continuously variable transmission tractor.Figure 8. Shifting dynamics model of continuously variable transmission tractor.

Figure 8 .
Figure 8. Shifting dynamics model of continuously variable transmission tractor.Figure 8. Shifting dynamics model of continuously variable transmission tractor.

Figure 9 .
Figure 9. Pressure control signal of the clutch under direct-shift strategy.Note: considering that the rated pressure of the proportional valve and the clutch are not consistent, the input signal has been redefined here, with the maximum signal corresponding to the rated pressure of the clutch.

Figure 9 .
Figure 9. Pressure control signal of the clutch under direct-shift strategy.Note: considering that the rated pressure of the proportional valve and the clutch are not consistent, the input signal has been redefined here, with the maximum signal corresponding to the rated pressure of the clutch.

Figure 10 .
Figure 10.Pressure control signal of the clutch under bridge-shift strategy.Note: considering that the rated pressure of the proportional valve and the clutch are not consistent, the input signal has been redefined here, with the maximum signal corresponding to the rated pressure of the clutch.

Figure 10 .
Figure 10.Pressure control signal of the clutch under bridge-shift strategy.Note: considering that the rated pressure of the proportional valve and the clutch are not consistent, the input signal has been redefined here, with the maximum signal corresponding to the rated pressure of the clutch.

Figure 11 .
Figure 11.Comparison of shifting quality under different control strategies.(a) Output speed of the transmission during shift.(b) Acceleration of the tractor during shift.(c) Power loss of the clutches during shift.(d) Sliding friction work of the clutches during shift.

Figure 11 .
Figure 11.Comparison of shifting quality under different control strategies.(a) Output speed of the transmission during shift.(b) Acceleration of the tractor during shift.(c) Power loss of the clutches during shift.(d) Sliding friction work of the clutches during shift.

Figure 12 .
Figure 12.Power loss of clutches.(a) Power loss in direct-shift method.(b) Power loss in bridgeshift method.

Figure 12 .Figure 13 .
Figure 12.Power loss of clutches.(a) Power loss in direct-shift method.(b) Power loss in bridge-shift method.Agriculture 2023, 13, x FOR PEER REVIEW 17 of 19

Figure 13 .
Figure 13.Shifting quality under different tractive forces.(a) Speed drop of the transmission during shifting.(b) Peak acceleration of the tractor during shifting.(c) Power loss of the clutches during shifting.(d) Sliding friction work of the clutches during shifting.Note: The clutch still loses energy under no load because the model takes into account the effects of cab air resistance and tire rolling resistance.Moreover, inertial loads can also cause energy losses.

Table 1 .
Clutch schedule of the hydrostatic power-split transmission.

Table 1 .
Clutch schedule of the hydrostatic power-split transmission.

Table 2 .
Configuration parameters of proportional valves.

Table 3 .
Configuration parameters of wet clutches.

Table 2 .
Configuration parameters of proportional valves.

Table 3 .
Configuration parameters of wet clutches.

Table 2 .
Configuration parameters of proportional valves.

Table 3 .
Configuration parameters of wet clutches.

Table 4 .
Factors and levels used for direct-shift optimization.

Table 5 .
Orthogonal simulation sequence for direct-shift optimization.

Table 6 .
Range analysis of orthogonal optimization for direct-shift strategy.

Table 7 .
Factors and levels used for bridge-shift optimization.

Table 8 .
Orthogonal simulation sequence for bridge-shift optimization.

Table 9 .
Range analysis of orthogonal optimization for bridge-shift strategy.