# Modeling of Power Transition in Full Power Shift of Hydromechanical Transmission

Authors: Yong Bao; Zaimin Zhong; Shujun Yang
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

This paper shall explore the dynamics of power transition in full power shift of hydromechanical transmission (HMT) and focus on the ideal target displacement ratio. An arithmetic two-range HMT is taken as the research object. A mathematical model of power transition in full power shift is established, including the hydraulic transmission unit model and the brake torque model during the double brakes overlapping. Aiming at the constant output power of HMT in the shift process, a prediction model of the displacement ratio target value is established, and the prediction method is proposed. By combining theoretical analysis and experimental research, it proves that the power transition model can describe the power transition process. And the prediction method of the displacement ratio target value proposed in this paper can complete the power transition when the double brakes overlap. In the power shift process, the output power can be transmitted normally in full power. The power transition model and the prediction method of the displacement ratio target value can provide theoretical and engineering references for the full power shift of HMT.


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## Body

## 1. Introduction

HMT is a dual power flow transmission consisting of hydraulic transmission unit and mechanical transmission unit [1, 2]. It can realize high-power stepless speed regulation and is suitable for heavy-duty vehicles [3, 4].The theory of design and principle of HMT has been basically mature after years of research. Kress [5] studied the performance of four solutions of HMT. Orshansky and Weseloh [6] and Kireijczyk [7] analyzed the operation of three-range split power HMTs. Ross [8] designed an input-coupled dual mode HMT for heavy-duty trucks. Renius and Resch [9] and Xu et al. [10] designed three-range and six-range HMTs for tractors, respectively, and the performances are good. Liu et al. [11] designed a five-range synthesized power split HMT with planetary gearsets of gear ratio extension, using the modular method. Krauss et al. [12–14] systematically analyzed various HMT schemes and carefully studied the characteristics of HMT. Rossetti and Macor [15, 16] regarded the design of the input-coupled HMT (using three shaft ordinary gear and a simple planetary gear) as an optimization problem and achieved good results. Sun et al. [17–19] designed a power reflux hydromechanical transmission based on a torque converter instead of the traditional hydraulic transmission unit.In the control of HMT, Savaresi et al. [20, 21] designed a controller for a six-range power split continuous variable transmission. Zhu et al. [22, 23] studied the control strategy for a seven-range HMT based on the shifting time. Cheong et al. [24, 25] studied the control of hydromechanical hydraulic hybrid vehicles. Wei et al. [26, 27] pointed out that there are problems such as speed fluctuation, pressure impact, and power interruption during the power shift of the two-range arithmetic HMT (shown in Figure 1) and analyzed the factors affecting the shift quality. In traditional power shift, the target clutch would be engaged after the current clutch is disengaged, which would cause a transitory output power interruption. There is an inertia phase during the shifting process. The pressure building-up process in the traditional power shift process is an uncontrollable passive process. When HMT shifts to the target range, the pressure building-up process just begins, which would cause a transitory output power interruption.Figure 1
Principle of two-range arithmetic HMT.Based on the compressibility of the hydraulic unit, Hu et al. [28] proposed the feasibility of the two-range arithmetic HMT (shown in Figure 1) working in the stage of the dual range. Based on the double brakes overlapping, Yang et al. [29, 30] proposed a full power shift method and analyzed the characteristics. Because of the double brakes overlapping in the full power shift, there is no inertia phase, and only the torque phase is retained. The pressures in the closed hydraulic circuit can be regulated by actively adjusting the displacement ratio. Interchange between two pressure sides and power transition can be completed when the double brakes overlap. Before the current brake is disengaged, the pressure building-up process is completed. But the mathematical model of power transition in full power shift of HMT, which can describe the principle of power transition, has not been established. The pressure building-up process in the full power shift is an active and controllable process. The pressure building-up process is changed from a passively uncontrollable process to an active controllable one. The power transition can only be completed when the displacement ratio is adjusted to the target value. But the prediction method of the displacement ratio target value has not been proposed.This paper shall build the power transition model in full power shift to explore the dynamics of power transition and propose the prediction method of the displacement ratio target value to actively control the pressure building-up process. A full power shift approach of HMT is described in Section2. A mathematical model of power transition in full power shift is established in Section 3, including the hydraulic transmission unit model and the brake torque model during the double brakes overlapping. Aiming at the constant output power of HMT in the shift process, a prediction model of the displacement ratio target value is established in Section 4, and the prediction method is proposed. In Sections 5 and 6, by combining theoretical analysis and experimental research, it proves that the power transition model can describe the power transition process. And the prediction method of the displacement ratio target value proposed in this paper can complete the power transition when the double brakes overlap.The power transition model can describe the principle of power transition, and it can be used to simulate and analyze the dynamics of power transition, which is helpful to understand the principle and develop control strategies. The prediction method of the displacement ratio target value is deployed in the application layer algorithm of the transmission controller, which can actively the power transition process. The power transition model and the prediction method of the displacement ratio target value can provide theoretical and engineering references for the full power shift of HMT.


## 2. Full Power Shift Method of HMT

An arithmetic two-range HMT is taken as the research object. Figure1 shows its principle. The variable-displacement hydraulic component (VDHC) P and the fixed-displacement hydraulic component (FDHC) M form the hydraulic transmission unit. The planetary gearsets K1, K2, and K3, together with the brakes CH and CL, constitute the mechanical transmission unit and confluence unit. ni and no represent, respectively, the input and output speed. The ranges and working parts are shown in Table 1. In the H range, the brake CH is engaged, and the planetary gearset K1 works. In the HM range, the brake CL is engaged and both the planetary gearsets K2 and K3 work.Table 1
Ranges of HMT and working parts.
Range
C
H
C
L
K
1
K
2
K
3
Hydraulic range (H range)
√
√
Hydromechanical range (HM range)
√
√
√A full power shift approach of HMT is proposed based on the double brakes overlapping in [30]. The full power shift is divided into 5 stages: current stage, prior stable stage, power transition stage, poststable stage, and target range. For HMT working at the current range, when the ideal power shift timing comes, the target range brake would be engaged with zero speed difference. Then, HMT comes to the prior stable stage, and its torque and power characteristics remain unchanged. According to [30], regulating properly the displacement ratio can make the high- and low-pressure sides of hydraulic circuit interchange. At the power transition stage, the original low-pressure side pressurizes to the target pressure, and HMT comes to the poststable stage with torque and power characteristics remaining identical with those of the target range. The current range brake is disengaged at zero torque, and HMT enters the target range.In the traditional power shift process, there is a certain time interval between the current range brake disengaging and the target range brake engaging, or there is a short time friction overlap between the two brakes. There is an inertia phase during the shifting process. So, there is the sudden change of speed in the traditional power shift process. But because of the double brakes overlapping in the full power shift, there is no inertia phase, and only the torque phase is retained. So, there is no the sudden change of speed in the full power shift.In traditional power shift process, after the target brake is engaged, the high-pressure side passively completes the pressure building-up process under the action of transmission load torque. The pressure building-up process in the traditional power shift process is an uncontrollable passive process. Hence, there is pressure impact in traditional power shift. When HMT shifts to the target range, the pressure building-up process just begins. So, there is power interruption during the power shift.In the power transition stage of full power shift, the displacement ratio is actively adjusted to the target value. The pressure building-up process in the full power shift is an active and controllable process. Hence, there is no pressure impact in the full power shift. And the pressure building-up process can be accomplished during the double brakes overlapping. Before the current brake is disengaged, the pressure building-up process is completed. In the full power shift process, output power can be transmitted normally in full power. The power transition can be completed during the double brakes overlapping by controlling the high pressure of the target range to the target value actively. In order to improve shift quality, the high pressure of active control needs to be consistent with the pressure of the target range. The displacement ratio target value directly determines the high pressure of active control. The inaccurate prediction value of the displacement ratio will lead to the difference between the high pressure of active control and the pressure of the target range, resulting in shift impact and reducing shift quality. Hence, it is necessary to establish a power transition model and to explore the prediction method of the displacement ratio target value.


## 3. Power Transition Model

According to [29, 30], in the power transition stage, the interchange between two pressure sides of hydraulic transmission unit is completed, and torque can transmit from the current brake to the target brake. Hence, the power transition model should include the hydraulic transmission unit model and the brake torque model.In the full power shift, if the input speed remains unchanged, the speed of VDHC will be constant. The hydraulic transmission unit completes the power transition process through the pressure releasing and building up of the closed hydraulic circuit. Hence, in the modeling of the hydraulic transmission unit, the mass flow model, pressure model, and hydraulic component model that affect the hydraulic circuit are mainly considered. The composition of the hydraulic transmission unit model is shown in Figure2. V1 and V2 are the two closed chambers; np and nm are the rotational speeds of the two hydraulic components; Tp and Tm are the torques of the two hydraulic components; qpi and qpo are the input flow and output flow of VDHC, respectively; qmi and qmo are the input flow and output flow of FDHC, respectively.Figure 2
Composition of the hydraulic transmission unit model.In the hydraulic transmission unit model, the on-way loss of oil flow along pipelines is neglected, and the high- and low-pressure circuits are simplified into two closed chambersV1 and V2. The two hydraulic components are energy conversion devices in the hydraulic system, whose input and output are both mechanical power. Considering the causality between input and output, when the double brakes overlap, the inputs are the speeds of two hydraulic components and the outputs are the torques of two hydraulic components.The schematic diagram of the power transition model is shown in Figure3. In the figure, the speed model and torque model of hydraulic components constitute the models of hydraulic components, and the mass flow model and pressure model of chambers constitute the models of hydraulic pipes.Figure 3
Schematic diagram of the power transition model.


### 3.1. Hydraulic Transmission Unit Model

#### 3.1.1. Principle of Hydraulic Transmission Unit

The hydraulic transmission unit is a volumetric speed regulation system. Its principle is shown in Figure4. The displacement control mechanism is composed of a proportional valve 9 and a servocylinder 10. The displacement is adjusted by the control current. When the HMT works, one side of the closed hydraulic circuit is high pressure and the other side is low pressure. The high- and low-pressure circuits are separated by the two hydraulic components 1 and 2. The one-way valve (5 or 6) on the high-pressure side is closed. If the pressure on the high-pressure side is too high, the relief valve (7 or 8) will open to release pressure to the low pressure system which uses 4 to limit the maximum pressure in this line. The hydraulic reversing valve 11 and the low pressure relief valve 12 ensure that one of the two sides is low pressure. The boost pump 3 replenishes oil to the low-pressure side through the one-way valve (5 or 6), which can cool the oil in the hydraulic system. The low-pressure relief valve 12 overflows the excess oil in the low-pressure circuit to maintain the low pressure. The overflowed oil returns to the tank through the cooling system or directly to the tank (if the tank is equipped with a cooling system). The boost pressure is limited by the boost pressure relief valve 4, which is slightly higher than the low pressure.Figure 4
Principle of hydraulic transmission unit. 1: VDHC, 2: FDHC, 3: boost pump, 4: boost pressure relief valve, 5 and 6: one-way valves, 7 and 8: high-pressure relief valve, 9: proportional valve, 10: servocylinder, 11: hydraulic reversing valve, and 12: low-pressure relief valve.At the beginning of the power transmit stage, the displacement control mechanism is adjusted to reduce the original high pressure gradually, but the oil on the low-pressure side can still be leaked through the low-pressure relief valve. When the original high pressure drops to the same level as the low pressure, the hydraulic reversing valve closes. If the displacement control mechanism is continuously adjusted in the same direction, the original low pressure will start to increase. And the oil on the original high-pressure side will be leaked though the low-pressure relief valve and the original high pressure will become the new low pressure.


#### 3.1.2. Mass Flow Model

According to [29, 30], at the ideal shift timing, the speed difference of to-be-engaged brake is zero. And the speed relationship at the ideal shift timing is as follows:(1)nm=1+k11+k2ni1+k31+k1+k2i2=1+k11+k2npi11+k31+k1+k2i2,where k1, k2, and k3 are the characteristic parameters of planetary gearsets, respectively. np is the speed of VDHC. i1 and i2 are the transmission ratios from the input shaft to the hydraulic transmission unit and the mechanical one, respectively.Before and after the power shift, the functions of two hydraulic components are interchanged and the flow characteristics change. The power flow direction of the hydraulic transmission unit is indicated by “state,” and its value is 0 or 1. When state = 0, the VDHC drives the FDHC. In other words, the outlet of the VDHC and the inlet of the FDHC are connected with the high-pressure side chamber. When state = 1, the FDHC drives the VDHC. In other words, the outlet of FDHC and the inlet of the VDHC are connected with the high-pressure side chamber.The oil leaks from the high-pressure chamber through the two hydraulic components to the low-pressure chamber. According to the law of mass conservation, the mass flow on the high side of each hydraulic component is equal to the sum of the mass flow on the low side and the leakage mass flow. According to Section3.1.1, the mass flow model is established, as shown in equations (2)–(15).(1) State = 0. When state = 0, the input mass flow of VDHC (the theoretical mass flow) is(2)qMpi=ρ2qpi=εVgnpρ260,where ρ2 is the oil density in chamber V2, kg/m3; qpi is the input flow of VDHC, m3/s; ε is the displacement ratio; and Vg is the maximum displacement of VDHC, m3/r.Both the hydraulic components are equivalently simplified to the internal leakage mode. Considering the effects of pipeline pressure, oil viscosity, thermal expansion, and oil bulk modulus, the leakage of hydraulic components is(3)Δq=CsVΔp2πμ1+ΔpEL,where EL is the bulk modulus of oil, MPa; Cs is the leakage coefficient of hydraulic components; V is the displacement of hydraulic components, m3/r; and μ is the dynamic viscosity of oil, N·s/m2.The mass of the oil leaking to the low-pressure chamber is the same before and after expansion. Hence, the mass flow rate of the oil leaking from chamberV1 through the VDHC is(4)ΔqMp=ρ1Δqp=CspVpΔp1ρ12πμ,where Δqp is the leakage flow rate of VDHC, m3/s; ρ1 is the oil density in chamber V1, kg/m3; Csp is the leakage coefficient of VDHC; and Δp1 is the pressure difference between the chambers V1 and V2, MPa.The output mass flow rates of the two hydraulic components are(5)qMpo=qMpi−ΔqMp,(6)qMmo=ρ2qmo=Vgnmρ260,where qmo is the output flow rate of FDHC, m3/s.The mass flow rate of the oil leaking from chamberV1 through the FDHC is(7)ΔqMm=ρ1Δqm=CsmVgΔp1ρ12πμ,where Δqm is the leakage flow rate of FDHC, m3/s and Csm is the leakage coefficient of FDHC.The input mass flow rate of FDHC is(8)qMmi=qMmo+ΔqMm.(2) State = 1. When state = 1, the output mass flow rate of FDHC is(9)qMpo=ρ1qpo=εVgnpρ160.The mass flow rate of the oil leaking from chamberV2 through the VDHC is(10)ΔqMp=ρ2Δqp=CspVpΔp2ρ22πμ.The input mass flow rates of the two hydraulic components are(11)qMpi=qMpo+ΔqMp,(12)qMmi=ρ1qmi=Vgnmρ160.The mass flow rate of the oil leaking from chamberV2 through the FDHC is(13)ΔqMm=ρ2Δqm=CsmVgΔp2ρ22πμ.The output mass flow rate of FDHC is(14)qMmo=qmmi−Δqmm.


#### 3.1.3. Pressure Model of Chambers

The pressure change in the chamber is(15)dpdt=1V∂p∂ρqMi−qMo−ρV˙−∂p∂ρ∂ρ∂TT˙,where qMi and qMo are the input and output mass flow rates of the chamber, respectively, m3/s; V is the volume of chamber, m3; and p is the pressure in chamber, MPa.The shift time is short, so the oil temperature is almost unchanged. The volume change of high- and low-pressure chambers is neglected. Considering the variation of oil bulk modulusEt=ρ∂p/∂ρ and density, equation (16) can be written as(16)dpdt=1VdpdρqMi−qMo=EtρVqMi−qMo.When state = 0, the chamberV1 is the high pressure chamber and the pressure model is(17)dp1dt=1V1dp1dρ1qMpo−qMmi=Et1ρ1V1qMpo−qMmi.When state = 1, the chamberV2 is the high pressure chamber and the pressure model is(18)dp2dt=1V2dp2dρ2qMmo−qMpi=Et2ρ2V2qMmo−qMpi.


#### 3.1.4. Model of Hydraulic Components

(1) Torque Model of Hydraulic Components. When state = 0, the VDHC works as a pump and the FDHC works as a motor. Their torque models are(19)Tp=−εVgΔp2πηmp,Tm=VgΔpηmm2π,where ηmp and ηmm are the mechanical efficiency of the two hydraulic components, respectively.When state = 1, the VDHC works as a motor and the FDHC works as a pump. Their torque models are(20)Tp=−εVgΔpηmp2π,Tm=VgΔp2πηmm.(2) Speed Model of Hydraulic Components. The speed of VDHC is(21)np=nii1.


### 3.2. Torque Model of Brakes

When the double brakes overlap, it is assumed that the input speed is fixed, so the speed of the FDHC and the output speed are constant. That is, the accelerations of the FDHC and planetary carrier ofK1 are 0. Hence, the external torques of the two parts are 0, respectively. The torque relationship of the FDHC can be written as(22)Ts1+Ts2+Tm=0,where Ts1 and Ts2 are the torques of sun gears of K1 and K2, respectively, N·m. In the H range, the torque of sun gear of K2 is Ts2 = 0. And in the HM range, the torque of sun gear of K1 is Ts1 = 0.The torque relationship of planetary carrier ofK1 can be written as(23)Tc1+Tr2+Toi3η3=0,where Tc1 is the torque of planetary carrier of K1, N·m; Tr2 is the torque of ring gear of K2, N·m; To is the load torque, N·m; and η3 is the mechanical efficiency of rear transmission mechanism.The torque relationship of planetary gearsetK1 is(24)Ts1:TH:Tc1ηc1=1:k1:−1+k1,where TH is the torque transmitted by brake CH, N·m, and ηc1 is the mechanical efficiency of the planetary gearset K1.The torque relationship of planetary gearsetK2 is(25)Ts2:Tr2:Tc2ηc2=1:k2:−1+k2,where Tc2 is the torque transmitted by planetary carrier of K2, N·m, and ηc2 is the mechanical efficiency of the planetary gearset K2.According to equations (25)–(28), the torques of sun gears of K1 and K2 and brake CH can be written as(26)Ts1=To/i3η3−k2Tm1+k1ηc1+k2,Ts2=−To/i3η3+1+k1Tmηc11+k1ηc1+k2,TH=k1To/i3η3−k2Tm1+k1ηc1+k2.The torque relationship of planetary gearsetK3 is(27)Ts3:TL:Tc3ηc3=1:k3:−1+k3,where Ts3 is torque of sun gear of K3, N·m; Tc3 is the torque of planetary carrier of K3, N·m; TL is the torque transmitted by brake CL, N·m; and ηc3 is the mechanical efficiency of the planetary gearset K3.The torque theoretical relationship between planetary carriers ofK2 and K3 is(28)Tc2+Tc3=0.The torque transmitted by brakeCL is(29)TL=1+k2k31+k3ηc2ηc3⋅To/i3η3+1+k1Tmηc11+k1ηc1+k2.The power transition model built in Simulink is shown in Figure5.Figure 5
Power transition model in Simulink.


## 3.1. Hydraulic Transmission Unit Model

### 3.1.1. Principle of Hydraulic Transmission Unit

The hydraulic transmission unit is a volumetric speed regulation system. Its principle is shown in Figure4. The displacement control mechanism is composed of a proportional valve 9 and a servocylinder 10. The displacement is adjusted by the control current. When the HMT works, one side of the closed hydraulic circuit is high pressure and the other side is low pressure. The high- and low-pressure circuits are separated by the two hydraulic components 1 and 2. The one-way valve (5 or 6) on the high-pressure side is closed. If the pressure on the high-pressure side is too high, the relief valve (7 or 8) will open to release pressure to the low pressure system which uses 4 to limit the maximum pressure in this line. The hydraulic reversing valve 11 and the low pressure relief valve 12 ensure that one of the two sides is low pressure. The boost pump 3 replenishes oil to the low-pressure side through the one-way valve (5 or 6), which can cool the oil in the hydraulic system. The low-pressure relief valve 12 overflows the excess oil in the low-pressure circuit to maintain the low pressure. The overflowed oil returns to the tank through the cooling system or directly to the tank (if the tank is equipped with a cooling system). The boost pressure is limited by the boost pressure relief valve 4, which is slightly higher than the low pressure.Figure 4
Principle of hydraulic transmission unit. 1: VDHC, 2: FDHC, 3: boost pump, 4: boost pressure relief valve, 5 and 6: one-way valves, 7 and 8: high-pressure relief valve, 9: proportional valve, 10: servocylinder, 11: hydraulic reversing valve, and 12: low-pressure relief valve.At the beginning of the power transmit stage, the displacement control mechanism is adjusted to reduce the original high pressure gradually, but the oil on the low-pressure side can still be leaked through the low-pressure relief valve. When the original high pressure drops to the same level as the low pressure, the hydraulic reversing valve closes. If the displacement control mechanism is continuously adjusted in the same direction, the original low pressure will start to increase. And the oil on the original high-pressure side will be leaked though the low-pressure relief valve and the original high pressure will become the new low pressure.


### 3.1.2. Mass Flow Model

According to [29, 30], at the ideal shift timing, the speed difference of to-be-engaged brake is zero. And the speed relationship at the ideal shift timing is as follows:(1)nm=1+k11+k2ni1+k31+k1+k2i2=1+k11+k2npi11+k31+k1+k2i2,where k1, k2, and k3 are the characteristic parameters of planetary gearsets, respectively. np is the speed of VDHC. i1 and i2 are the transmission ratios from the input shaft to the hydraulic transmission unit and the mechanical one, respectively.Before and after the power shift, the functions of two hydraulic components are interchanged and the flow characteristics change. The power flow direction of the hydraulic transmission unit is indicated by “state,” and its value is 0 or 1. When state = 0, the VDHC drives the FDHC. In other words, the outlet of the VDHC and the inlet of the FDHC are connected with the high-pressure side chamber. When state = 1, the FDHC drives the VDHC. In other words, the outlet of FDHC and the inlet of the VDHC are connected with the high-pressure side chamber.The oil leaks from the high-pressure chamber through the two hydraulic components to the low-pressure chamber. According to the law of mass conservation, the mass flow on the high side of each hydraulic component is equal to the sum of the mass flow on the low side and the leakage mass flow. According to Section3.1.1, the mass flow model is established, as shown in equations (2)–(15).(1) State = 0. When state = 0, the input mass flow of VDHC (the theoretical mass flow) is(2)qMpi=ρ2qpi=εVgnpρ260,where ρ2 is the oil density in chamber V2, kg/m3; qpi is the input flow of VDHC, m3/s; ε is the displacement ratio; and Vg is the maximum displacement of VDHC, m3/r.Both the hydraulic components are equivalently simplified to the internal leakage mode. Considering the effects of pipeline pressure, oil viscosity, thermal expansion, and oil bulk modulus, the leakage of hydraulic components is(3)Δq=CsVΔp2πμ1+ΔpEL,where EL is the bulk modulus of oil, MPa; Cs is the leakage coefficient of hydraulic components; V is the displacement of hydraulic components, m3/r; and μ is the dynamic viscosity of oil, N·s/m2.The mass of the oil leaking to the low-pressure chamber is the same before and after expansion. Hence, the mass flow rate of the oil leaking from chamberV1 through the VDHC is(4)ΔqMp=ρ1Δqp=CspVpΔp1ρ12πμ,where Δqp is the leakage flow rate of VDHC, m3/s; ρ1 is the oil density in chamber V1, kg/m3; Csp is the leakage coefficient of VDHC; and Δp1 is the pressure difference between the chambers V1 and V2, MPa.The output mass flow rates of the two hydraulic components are(5)qMpo=qMpi−ΔqMp,(6)qMmo=ρ2qmo=Vgnmρ260,where qmo is the output flow rate of FDHC, m3/s.The mass flow rate of the oil leaking from chamberV1 through the FDHC is(7)ΔqMm=ρ1Δqm=CsmVgΔp1ρ12πμ,where Δqm is the leakage flow rate of FDHC, m3/s and Csm is the leakage coefficient of FDHC.The input mass flow rate of FDHC is(8)qMmi=qMmo+ΔqMm.(2) State = 1. When state = 1, the output mass flow rate of FDHC is(9)qMpo=ρ1qpo=εVgnpρ160.The mass flow rate of the oil leaking from chamberV2 through the VDHC is(10)ΔqMp=ρ2Δqp=CspVpΔp2ρ22πμ.The input mass flow rates of the two hydraulic components are(11)qMpi=qMpo+ΔqMp,(12)qMmi=ρ1qmi=Vgnmρ160.The mass flow rate of the oil leaking from chamberV2 through the FDHC is(13)ΔqMm=ρ2Δqm=CsmVgΔp2ρ22πμ.The output mass flow rate of FDHC is(14)qMmo=qmmi−Δqmm.


### 3.1.3. Pressure Model of Chambers

The pressure change in the chamber is(15)dpdt=1V∂p∂ρqMi−qMo−ρV˙−∂p∂ρ∂ρ∂TT˙,where qMi and qMo are the input and output mass flow rates of the chamber, respectively, m3/s; V is the volume of chamber, m3; and p is the pressure in chamber, MPa.The shift time is short, so the oil temperature is almost unchanged. The volume change of high- and low-pressure chambers is neglected. Considering the variation of oil bulk modulusEt=ρ∂p/∂ρ and density, equation (16) can be written as(16)dpdt=1VdpdρqMi−qMo=EtρVqMi−qMo.When state = 0, the chamberV1 is the high pressure chamber and the pressure model is(17)dp1dt=1V1dp1dρ1qMpo−qMmi=Et1ρ1V1qMpo−qMmi.When state = 1, the chamberV2 is the high pressure chamber and the pressure model is(18)dp2dt=1V2dp2dρ2qMmo−qMpi=Et2ρ2V2qMmo−qMpi.


### 3.1.4. Model of Hydraulic Components

(1) Torque Model of Hydraulic Components. When state = 0, the VDHC works as a pump and the FDHC works as a motor. Their torque models are(19)Tp=−εVgΔp2πηmp,Tm=VgΔpηmm2π,where ηmp and ηmm are the mechanical efficiency of the two hydraulic components, respectively.When state = 1, the VDHC works as a motor and the FDHC works as a pump. Their torque models are(20)Tp=−εVgΔpηmp2π,Tm=VgΔp2πηmm.(2) Speed Model of Hydraulic Components. The speed of VDHC is(21)np=nii1.


## 3.1.1. Principle of Hydraulic Transmission Unit

The hydraulic transmission unit is a volumetric speed regulation system. Its principle is shown in Figure4. The displacement control mechanism is composed of a proportional valve 9 and a servocylinder 10. The displacement is adjusted by the control current. When the HMT works, one side of the closed hydraulic circuit is high pressure and the other side is low pressure. The high- and low-pressure circuits are separated by the two hydraulic components 1 and 2. The one-way valve (5 or 6) on the high-pressure side is closed. If the pressure on the high-pressure side is too high, the relief valve (7 or 8) will open to release pressure to the low pressure system which uses 4 to limit the maximum pressure in this line. The hydraulic reversing valve 11 and the low pressure relief valve 12 ensure that one of the two sides is low pressure. The boost pump 3 replenishes oil to the low-pressure side through the one-way valve (5 or 6), which can cool the oil in the hydraulic system. The low-pressure relief valve 12 overflows the excess oil in the low-pressure circuit to maintain the low pressure. The overflowed oil returns to the tank through the cooling system or directly to the tank (if the tank is equipped with a cooling system). The boost pressure is limited by the boost pressure relief valve 4, which is slightly higher than the low pressure.Figure 4
Principle of hydraulic transmission unit. 1: VDHC, 2: FDHC, 3: boost pump, 4: boost pressure relief valve, 5 and 6: one-way valves, 7 and 8: high-pressure relief valve, 9: proportional valve, 10: servocylinder, 11: hydraulic reversing valve, and 12: low-pressure relief valve.At the beginning of the power transmit stage, the displacement control mechanism is adjusted to reduce the original high pressure gradually, but the oil on the low-pressure side can still be leaked through the low-pressure relief valve. When the original high pressure drops to the same level as the low pressure, the hydraulic reversing valve closes. If the displacement control mechanism is continuously adjusted in the same direction, the original low pressure will start to increase. And the oil on the original high-pressure side will be leaked though the low-pressure relief valve and the original high pressure will become the new low pressure.


## 3.1.2. Mass Flow Model

According to [29, 30], at the ideal shift timing, the speed difference of to-be-engaged brake is zero. And the speed relationship at the ideal shift timing is as follows:(1)nm=1+k11+k2ni1+k31+k1+k2i2=1+k11+k2npi11+k31+k1+k2i2,where k1, k2, and k3 are the characteristic parameters of planetary gearsets, respectively. np is the speed of VDHC. i1 and i2 are the transmission ratios from the input shaft to the hydraulic transmission unit and the mechanical one, respectively.Before and after the power shift, the functions of two hydraulic components are interchanged and the flow characteristics change. The power flow direction of the hydraulic transmission unit is indicated by “state,” and its value is 0 or 1. When state = 0, the VDHC drives the FDHC. In other words, the outlet of the VDHC and the inlet of the FDHC are connected with the high-pressure side chamber. When state = 1, the FDHC drives the VDHC. In other words, the outlet of FDHC and the inlet of the VDHC are connected with the high-pressure side chamber.The oil leaks from the high-pressure chamber through the two hydraulic components to the low-pressure chamber. According to the law of mass conservation, the mass flow on the high side of each hydraulic component is equal to the sum of the mass flow on the low side and the leakage mass flow. According to Section3.1.1, the mass flow model is established, as shown in equations (2)–(15).(1) State = 0. When state = 0, the input mass flow of VDHC (the theoretical mass flow) is(2)qMpi=ρ2qpi=εVgnpρ260,where ρ2 is the oil density in chamber V2, kg/m3; qpi is the input flow of VDHC, m3/s; ε is the displacement ratio; and Vg is the maximum displacement of VDHC, m3/r.Both the hydraulic components are equivalently simplified to the internal leakage mode. Considering the effects of pipeline pressure, oil viscosity, thermal expansion, and oil bulk modulus, the leakage of hydraulic components is(3)Δq=CsVΔp2πμ1+ΔpEL,where EL is the bulk modulus of oil, MPa; Cs is the leakage coefficient of hydraulic components; V is the displacement of hydraulic components, m3/r; and μ is the dynamic viscosity of oil, N·s/m2.The mass of the oil leaking to the low-pressure chamber is the same before and after expansion. Hence, the mass flow rate of the oil leaking from chamberV1 through the VDHC is(4)ΔqMp=ρ1Δqp=CspVpΔp1ρ12πμ,where Δqp is the leakage flow rate of VDHC, m3/s; ρ1 is the oil density in chamber V1, kg/m3; Csp is the leakage coefficient of VDHC; and Δp1 is the pressure difference between the chambers V1 and V2, MPa.The output mass flow rates of the two hydraulic components are(5)qMpo=qMpi−ΔqMp,(6)qMmo=ρ2qmo=Vgnmρ260,where qmo is the output flow rate of FDHC, m3/s.The mass flow rate of the oil leaking from chamberV1 through the FDHC is(7)ΔqMm=ρ1Δqm=CsmVgΔp1ρ12πμ,where Δqm is the leakage flow rate of FDHC, m3/s and Csm is the leakage coefficient of FDHC.The input mass flow rate of FDHC is(8)qMmi=qMmo+ΔqMm.(2) State = 1. When state = 1, the output mass flow rate of FDHC is(9)qMpo=ρ1qpo=εVgnpρ160.The mass flow rate of the oil leaking from chamberV2 through the VDHC is(10)ΔqMp=ρ2Δqp=CspVpΔp2ρ22πμ.The input mass flow rates of the two hydraulic components are(11)qMpi=qMpo+ΔqMp,(12)qMmi=ρ1qmi=Vgnmρ160.The mass flow rate of the oil leaking from chamberV2 through the FDHC is(13)ΔqMm=ρ2Δqm=CsmVgΔp2ρ22πμ.The output mass flow rate of FDHC is(14)qMmo=qmmi−Δqmm.


## 3.1.3. Pressure Model of Chambers

The pressure change in the chamber is(15)dpdt=1V∂p∂ρqMi−qMo−ρV˙−∂p∂ρ∂ρ∂TT˙,where qMi and qMo are the input and output mass flow rates of the chamber, respectively, m3/s; V is the volume of chamber, m3; and p is the pressure in chamber, MPa.The shift time is short, so the oil temperature is almost unchanged. The volume change of high- and low-pressure chambers is neglected. Considering the variation of oil bulk modulusEt=ρ∂p/∂ρ and density, equation (16) can be written as(16)dpdt=1VdpdρqMi−qMo=EtρVqMi−qMo.When state = 0, the chamberV1 is the high pressure chamber and the pressure model is(17)dp1dt=1V1dp1dρ1qMpo−qMmi=Et1ρ1V1qMpo−qMmi.When state = 1, the chamberV2 is the high pressure chamber and the pressure model is(18)dp2dt=1V2dp2dρ2qMmo−qMpi=Et2ρ2V2qMmo−qMpi.


## 3.1.4. Model of Hydraulic Components

(1) Torque Model of Hydraulic Components. When state = 0, the VDHC works as a pump and the FDHC works as a motor. Their torque models are(19)Tp=−εVgΔp2πηmp,Tm=VgΔpηmm2π,where ηmp and ηmm are the mechanical efficiency of the two hydraulic components, respectively.When state = 1, the VDHC works as a motor and the FDHC works as a pump. Their torque models are(20)Tp=−εVgΔpηmp2π,Tm=VgΔp2πηmm.(2) Speed Model of Hydraulic Components. The speed of VDHC is(21)np=nii1.


## 3.2. Torque Model of Brakes

When the double brakes overlap, it is assumed that the input speed is fixed, so the speed of the FDHC and the output speed are constant. That is, the accelerations of the FDHC and planetary carrier ofK1 are 0. Hence, the external torques of the two parts are 0, respectively. The torque relationship of the FDHC can be written as(22)Ts1+Ts2+Tm=0,where Ts1 and Ts2 are the torques of sun gears of K1 and K2, respectively, N·m. In the H range, the torque of sun gear of K2 is Ts2 = 0. And in the HM range, the torque of sun gear of K1 is Ts1 = 0.The torque relationship of planetary carrier ofK1 can be written as(23)Tc1+Tr2+Toi3η3=0,where Tc1 is the torque of planetary carrier of K1, N·m; Tr2 is the torque of ring gear of K2, N·m; To is the load torque, N·m; and η3 is the mechanical efficiency of rear transmission mechanism.The torque relationship of planetary gearsetK1 is(24)Ts1:TH:Tc1ηc1=1:k1:−1+k1,where TH is the torque transmitted by brake CH, N·m, and ηc1 is the mechanical efficiency of the planetary gearset K1.The torque relationship of planetary gearsetK2 is(25)Ts2:Tr2:Tc2ηc2=1:k2:−1+k2,where Tc2 is the torque transmitted by planetary carrier of K2, N·m, and ηc2 is the mechanical efficiency of the planetary gearset K2.According to equations (25)–(28), the torques of sun gears of K1 and K2 and brake CH can be written as(26)Ts1=To/i3η3−k2Tm1+k1ηc1+k2,Ts2=−To/i3η3+1+k1Tmηc11+k1ηc1+k2,TH=k1To/i3η3−k2Tm1+k1ηc1+k2.The torque relationship of planetary gearsetK3 is(27)Ts3:TL:Tc3ηc3=1:k3:−1+k3,where Ts3 is torque of sun gear of K3, N·m; Tc3 is the torque of planetary carrier of K3, N·m; TL is the torque transmitted by brake CL, N·m; and ηc3 is the mechanical efficiency of the planetary gearset K3.The torque theoretical relationship between planetary carriers ofK2 and K3 is(28)Tc2+Tc3=0.The torque transmitted by brakeCL is(29)TL=1+k2k31+k3ηc2ηc3⋅To/i3η3+1+k1Tmηc11+k1ηc1+k2.The power transition model built in Simulink is shown in Figure5.Figure 5
Power transition model in Simulink.


## 4. Prediction Model of Displacement Ratio Target Value

In order to make the ideal working state of the poststable stage the same as that of the target range, the key state parameters such as the hydraulic pressure, the brake torque, and the speed of the FDHC at the poststable stage should be the same as those in the target range. The speeds of hydraulic components in the current range are the same as those in the target range. However, because of the volumetric efficiency of the hydraulic system, the displacement ratios are different. According to the key state parameters of the current range, two key processes are needed to predict the displacement ratio target value. The first one is to predict the hydraulic pressure in the target range based on the current pressure. The second one is to predict the displacement ratio target value based on the flow continuity equation of the closed hydraulic circuit.


### 4.1. Prediction Model of Target Pressure

In the full power shift, the output power is expected to be transmitted normally in full power. In other words, the aim is to keep the output power unchanged. The hydraulic pressure in the target range is predicted according to the current pressure.According to equations (25)–(28), the torques of the FDHC in the H range and in HM range are(30)TmH=−Toi31+k1η3ηc1,TmHM=Toi3k2η3.The output torque before and after power shift is constant, that is,To/i3η3 is unchanged.Hence, the torque ratio of the FDHC in the HM range and in H range is(31)TmHMTmH=−1+k1ηc1k2.For the HMT shown in Figure2, the relation between characteristic parameters of planetary gearsets is given by(32)1+k1=k2.Hence, the torque ratio of the FDHC can theoretically be written as(33)TmHMTmH=−1.However, it is necessary to consider the influence of mechanical efficiency in practical calculation. The relation in H range can be written as(34)Toi3ηr=−p1h−p2hVg1+k1ηmmηc1,p2h=C,where p1h and p2h are the high and low pressures in the H range, respectively, MPa, and C is the low pressure.The relation in the HM range can be written as(35)Toi3ηr=p1hm−p2hmVgk2ηmm,p1hm=C,where p2hm and p1hm are the high and low pressures in the HM range, respectively, MPa.According to equations (34) and (35), based on the pressures in the H range, the high pressure in the HM range can be predicted by(36)p2hm=p1h−p2h1+k1ηmm2ηc1k2+p1hm=p1h−p2hηmm2ηc1+p1hm.Based on the pressures in the HM range, the high pressure in the H range can be predicted by(37)p1h=p2hm−p1hmk21+k1ηmm2ηc1+p2h=p2hm−p1hmηmm2ηc1+p2h.As can be seen from equations (36) and (37), the pressures in the target range can be predicted according to the pressures in the current range and the mechanical efficiency of the FDHC. The prediction model of target pressure is mainly based on equations (36) and (37) to predict the target pressure. The high pressure has little difference between the H range and HM range. Due to the influence of mechanical efficiency, the high pressure in the H range is slightly higher than that in the HM range.The two efficiency parametersηmm and ηc1 are the most important parameters, which should be calibrated. And only the two parameters need to be calibrated. These two parameters can be calibrated by analyzing the relationship between the output torque and the pressures in the closed hydraulic circuit.


### 4.2. Prediction Model of the Displacement Ratio Target Value

When predicting the displacement ratio target value, the relationship between the volumetric efficiency of the hydraulic system in the two steady state conditions of the prior stable stage and the poststable stage is mainly studied. The high pressure has little difference between the H range and HM range. Therefore, the influence of pressure on viscosity and density can be neglected. At the same time, considering the oil temperature is basically unchanged before and after the power shift, it can be assumed that the oil viscosity remains unchanged during the power shift.According to the flow continuity equation, the outlet flow rate of the VDHC is equal to the inlet flow rate of the FDHC at the ideal shift timing from the H range to HM range:(38)qpo=qmi.That is,(39)εh⟶hm∗Vgnp60−Cspεh⟶hm∗Vgp1h−p2h2πμ=Vgnm60+CsmVgp1h−p2h2πμ,where εh⟶hm∗ is the displacement ratio at the ideal shift timing from the H range to HM range and μ is the dynamic viscosity of oil during the power shift, N·s/m2.The following equation can be obtained:(40)Cspεh⟶hm∗+Csm=2πμεh⟶hm∗np−nm60p1h−p2h=2πμεh⟶hm∗−ε0np60p1h−p2h,where ε0 is the theoretical displacement ratio in the power shift when the leak is ignored. According to [29, 30], ε0 equals to the speed ratio of the two hydraulic components when the double brakes overlap and ε0=nm/np.Considering thatCsp is much smaller than 1 and εh⟶hm∗ is close to ε0, it is assumed that Cspεh⟶hm∗=Cspε0. The following equation can be obtained:(41)Cspεh⟶hm∗+Csm=Cspε0+Csm=2πμεh⟶hm∗−ε0np60p1h−p2h.The inlet flow rate of the VDHC is equal to the outlet flow rate of the FDHC at the ideal shift timing from the HM range to H range:(42)qpi=qmo.That is,(43)εhm⟶h∗Vgnp60+Cspεhm⟶h∗Vgp2hm−p1hm2πμ=Vgnm60−CsmVgp2hm−p1hm2πμ,where εhm⟶h∗ is the displacement ratio at the ideal shift timing from the HM range to H range.The following equation can be obtained:(44)Cspεhm⟶h∗+Csm=2πμnm−εhm⟶h∗np60p2hm−p1hm=2πμε0−εhm⟶h∗np60p2hm−p1hm.Considering that Csp is much smaller than 1 and εhm⟶h∗ is close to ε0, it is assumed that Cspεhm⟶h∗=Cspε0. The following equation can be obtained:(45)Cspεhm⟶h∗+Csm=Cspε0+Csm=2πμε0−εhm⟶h∗np60p2hm−p1hm.The leakage coefficient of hydraulic components and the oil viscosity are unchanged during the power shift. Hence, the following can be obtained from (37) and (41):(46)εh⟶hm∗−ε0ε0−εhm⟶h∗=p1h−p2hp2hm−p1hm.The abovementioned equation can be used to predict the displacement ratio target value. If the high pressure, the displacement ratio in the current range, and the predicted value of high pressure in the target range are known, the displacement ratio target value can be predicted.In engineering applications, equations (36), (37), and (46) are deployed in the application layer algorithm of the transmission controller. The prediction process is divided into two steps. The first step is to predict the target pressure based on the current pressure and the mechanical efficiency of the fixed-displacement hydraulic component. The second step is to predict the displacement ratio target value based on the current displacement ratio, the current pressures, and the target pressures prediction value. During the full power shift process, the displacement ratio of the variable-displacement hydraulic component can be adjusted to the predicted value. Before the current brake is disengaged, the pressure building-up process is completed.


## 4.1. Prediction Model of Target Pressure

In the full power shift, the output power is expected to be transmitted normally in full power. In other words, the aim is to keep the output power unchanged. The hydraulic pressure in the target range is predicted according to the current pressure.According to equations (25)–(28), the torques of the FDHC in the H range and in HM range are(30)TmH=−Toi31+k1η3ηc1,TmHM=Toi3k2η3.The output torque before and after power shift is constant, that is,To/i3η3 is unchanged.Hence, the torque ratio of the FDHC in the HM range and in H range is(31)TmHMTmH=−1+k1ηc1k2.For the HMT shown in Figure2, the relation between characteristic parameters of planetary gearsets is given by(32)1+k1=k2.Hence, the torque ratio of the FDHC can theoretically be written as(33)TmHMTmH=−1.However, it is necessary to consider the influence of mechanical efficiency in practical calculation. The relation in H range can be written as(34)Toi3ηr=−p1h−p2hVg1+k1ηmmηc1,p2h=C,where p1h and p2h are the high and low pressures in the H range, respectively, MPa, and C is the low pressure.The relation in the HM range can be written as(35)Toi3ηr=p1hm−p2hmVgk2ηmm,p1hm=C,where p2hm and p1hm are the high and low pressures in the HM range, respectively, MPa.According to equations (34) and (35), based on the pressures in the H range, the high pressure in the HM range can be predicted by(36)p2hm=p1h−p2h1+k1ηmm2ηc1k2+p1hm=p1h−p2hηmm2ηc1+p1hm.Based on the pressures in the HM range, the high pressure in the H range can be predicted by(37)p1h=p2hm−p1hmk21+k1ηmm2ηc1+p2h=p2hm−p1hmηmm2ηc1+p2h.As can be seen from equations (36) and (37), the pressures in the target range can be predicted according to the pressures in the current range and the mechanical efficiency of the FDHC. The prediction model of target pressure is mainly based on equations (36) and (37) to predict the target pressure. The high pressure has little difference between the H range and HM range. Due to the influence of mechanical efficiency, the high pressure in the H range is slightly higher than that in the HM range.The two efficiency parametersηmm and ηc1 are the most important parameters, which should be calibrated. And only the two parameters need to be calibrated. These two parameters can be calibrated by analyzing the relationship between the output torque and the pressures in the closed hydraulic circuit.


## 4.2. Prediction Model of the Displacement Ratio Target Value

When predicting the displacement ratio target value, the relationship between the volumetric efficiency of the hydraulic system in the two steady state conditions of the prior stable stage and the poststable stage is mainly studied. The high pressure has little difference between the H range and HM range. Therefore, the influence of pressure on viscosity and density can be neglected. At the same time, considering the oil temperature is basically unchanged before and after the power shift, it can be assumed that the oil viscosity remains unchanged during the power shift.According to the flow continuity equation, the outlet flow rate of the VDHC is equal to the inlet flow rate of the FDHC at the ideal shift timing from the H range to HM range:(38)qpo=qmi.That is,(39)εh⟶hm∗Vgnp60−Cspεh⟶hm∗Vgp1h−p2h2πμ=Vgnm60+CsmVgp1h−p2h2πμ,where εh⟶hm∗ is the displacement ratio at the ideal shift timing from the H range to HM range and μ is the dynamic viscosity of oil during the power shift, N·s/m2.The following equation can be obtained:(40)Cspεh⟶hm∗+Csm=2πμεh⟶hm∗np−nm60p1h−p2h=2πμεh⟶hm∗−ε0np60p1h−p2h,where ε0 is the theoretical displacement ratio in the power shift when the leak is ignored. According to [29, 30], ε0 equals to the speed ratio of the two hydraulic components when the double brakes overlap and ε0=nm/np.Considering thatCsp is much smaller than 1 and εh⟶hm∗ is close to ε0, it is assumed that Cspεh⟶hm∗=Cspε0. The following equation can be obtained:(41)Cspεh⟶hm∗+Csm=Cspε0+Csm=2πμεh⟶hm∗−ε0np60p1h−p2h.The inlet flow rate of the VDHC is equal to the outlet flow rate of the FDHC at the ideal shift timing from the HM range to H range:(42)qpi=qmo.That is,(43)εhm⟶h∗Vgnp60+Cspεhm⟶h∗Vgp2hm−p1hm2πμ=Vgnm60−CsmVgp2hm−p1hm2πμ,where εhm⟶h∗ is the displacement ratio at the ideal shift timing from the HM range to H range.The following equation can be obtained:(44)Cspεhm⟶h∗+Csm=2πμnm−εhm⟶h∗np60p2hm−p1hm=2πμε0−εhm⟶h∗np60p2hm−p1hm.Considering that Csp is much smaller than 1 and εhm⟶h∗ is close to ε0, it is assumed that Cspεhm⟶h∗=Cspε0. The following equation can be obtained:(45)Cspεhm⟶h∗+Csm=Cspε0+Csm=2πμε0−εhm⟶h∗np60p2hm−p1hm.The leakage coefficient of hydraulic components and the oil viscosity are unchanged during the power shift. Hence, the following can be obtained from (37) and (41):(46)εh⟶hm∗−ε0ε0−εhm⟶h∗=p1h−p2hp2hm−p1hm.The abovementioned equation can be used to predict the displacement ratio target value. If the high pressure, the displacement ratio in the current range, and the predicted value of high pressure in the target range are known, the displacement ratio target value can be predicted.In engineering applications, equations (36), (37), and (46) are deployed in the application layer algorithm of the transmission controller. The prediction process is divided into two steps. The first step is to predict the target pressure based on the current pressure and the mechanical efficiency of the fixed-displacement hydraulic component. The second step is to predict the displacement ratio target value based on the current displacement ratio, the current pressures, and the target pressures prediction value. During the full power shift process, the displacement ratio of the variable-displacement hydraulic component can be adjusted to the predicted value. Before the current brake is disengaged, the pressure building-up process is completed.


## 5. Simulation Analysis

The input speed of HMT isni = 1000 r/min, and the load torque To = 300 N·m remains unchanged. The low pressure in the closed hydraulic circuit is C = 2.6 MPa. Based on the power transition model established in this paper, the power transition processes are simulated and analyzed. During the simulation, the two brakes are both engaged. From 0 s to 0.3 s, the displacement ratio is the same as the value at ideal shift timing. So, from 0 s to 0.3 s, HMT is at the prior stable stage. At 0.3 s, the displacement ratio changes to the target value, and then it remains unchanged.According to [29, 30], εh⟶hm∗ is the displacement ratio at the ideal shift timing from the H range to HM range and is also the optimal target value of the displacement ratio in the power shift from the HM range to H range. εhm⟶h∗ is the displacement ratio at the ideal shift timing from the HM range to H range and is also the optimal target value of the displacement ratio in the power shift from the H range to HM range.During the power shift from the H range to HM range, the displacement ratio is changed fromεh⟶hm∗ step to εhm⟶h∗ at 0.3 s and then remains unchanged, as shown in Figure 6. In the simulation condition, according to (42), εh⟶hm∗ is equal to 0.92 and εhm⟶h∗ is equal to 0.84. The results are shown in Figure 7.Figure 6
Displacement ratio.Figure 7
Simulation results of the shift from H range to HM range, when the displacement ratio target value isεhm⟶h∗: (a) pressures in closed hydraulic circuit and (b) brake torques.
(a)
(b)As shown in Figure7, during the power shift from the H range to HM range, the displacement ratio is changed from εh⟶hm∗ step to εhm⟶h∗ at 0.3 s, and the original high-pressure side completes the pressure release process at 0.48 s. And the original low-pressure side completes the pressure building-up process at 2.2 s, and the high and low pressures are interchanged. After 2.2 s, the torque of CH is 0 and the load torque transmits from CH to CL. The power transition is completed when the double brakes overlap, and HMT shifts to the poststable stage.During the power shift from the HM range to H range, the displacement ratio is changed fromεh⟶hm∗ step to εhm⟶h∗ at 0.3 s and then remains unchanged, as shown in Figure 8. In the simulation condition, according to (42), εh⟶hm∗ is equal to 0.92 and εhm⟶h∗ is equal to 0.84. The results are shown in Figure 9.Figure 8
Displacement ratio.Figure 9
Simulation results of the shift from HM range to H range, when the displacement ratio target value isεh⟶hm∗: (a) pressures in closed hydraulic circuit and (b) brake torques.
(a)
(b)As shown in Figure9, during the power shift from the HM range to H range, the displacement ratio is changed from εhm⟶h∗ step to εh⟶hm∗ at 0.3 s, and the original high-pressure side completes the pressure release process at 0.42 s. And the original low-pressure side completes the pressure building-up process at 2.5 s, and the high and low pressures are interchanged. After 2.5 s, the torque of CL is 0, and the load torque transmits from CL to CH. The power transition is completed when the double brakes overlap, and HMT shifts to the poststable stage.In summary, the speed ratio of hydraulic components in the target range is the same as that in the current range, but the displacement ratios are different due to the volumetric efficiency of the hydraulic system. Based on the pressures of the hydraulic system in the current range, the pressures in the target range can be predicted, and then the displacement ratio target value can be predicted. If the displacement ratio target value at the poststable stage equals the ideal shift displacement ratio in the target range, the original high pressure at the poststable stage can be maintained atC, the original low pressure at the poststable stage can be equal to the high pressure in the target range, and the torque of the brake to be disengaged is 0. The power transition is completed when the double brakes overlap, and HMT shifts to the poststable stage. The working state of the poststable stage is the same as that of the target range.


## 6. Full Power Shift Test

The full power shift test of the HMT whose configuration is shown in Figure1 is carried out on the test bench, as shown in Figures 10 and 11. It is a typical closed-electric transmission test bench.Figure 10
The schematic diagram of test bench.Figure 11
The photograph of test bench.The input speed remains at 1000 r/min, and the load torque is 300 N·m. The power shifts from the H range to HM range and then shifts from the HM range to H range.The test results of power shift from the H range to HM range are shown in Figure12. The displacement control current is reduced from 549 mA to 512 mA (shown in Figure 12(a)), corresponding to the change of the displacement ratio from 0.92 to 0.83 (shown in Figure 12(b)). The original high pressure in the H range is 8.3 MPa. After the power shift, the test value of the high pressure in the HM range is 6.8 MPa, and the simulation value is 6.7 MPa. And the simulation result of the pressure building-up process is basically consistent with the test result. The speed of the FDHC is basically maintained at 837 r/min, the output speed is basically maintained at 236 r/min, and the output torque is basically maintained at 300 N·m.Figure 12
Test results of the shift from the H range to HM range. (a) Brake pressure and displacement control current. (b) Displacement ratio. (c) Pressure in hydraulic circuit. (d) Speed and output torque.
(a)
(b)
(c)
(d)According to (42), the predicted value of the displacement ratio is 0.84. The actual and theoretical values of the displacement ratio are shown in Figure 12(b). And the predicted value is larger than the actual value by 0.01. The error of the predicted value is acceptable.The test results of power shift from the H range to HM range are shown in Figure13. The displacement control current is increased from 515 mA to 546 mA (shown in Figure 13(a)), corresponding to the change of the displacement ratio from 0.83 to 0.92 (shown in Figure 13(b)). The original high pressure in the HM range is 6.6 MPa. After the power shift, the test value of the high pressure in the H range is 8.0 MPa and the simulation value is 8.1 MPa. And the simulation result of the pressure building-up process is basically consistent with the test result. The speed of the FDHC is basically maintained at 837 r/min, the output speed is basically maintained at 236 r/min, and the output torque is basically maintained at 300 N·m.Figure 13
Test results of the shift from HM range to H range. (a) Brake pressure and displacement control current. (b) Displacement ratio. (c) Pressure in hydraulic circuit. (d) Speed and output torque.
(a)
(b)
(c)
(d)According to (42), the predicted value of the displacement ratio is 0.93. The actual and theoretical values of the displacement ratio are shown in Figure 13(b). And it is larger than the actual value by 0.01. The error of the predicted value is acceptable.According to Figures12(a) and 12(c), the disengaged time of CH is 1.43 s and the completion time of pressure building-up is 1.36 s. According to Figures 13(a) and 13(c), the disengaged time of CL is 1.47 s and the completion time of pressure building-up is 1.42 s. Figures 12(c) and 13(c) show that the interchange between two pressure sides is completed when the double brakes overlap. Figures 12(d) and 13(d) show that the output power can be transmitted normally in full power. The above figures prove the feasibility of the full power shift method and also prove that the power transition model can describe the power transition process.The errors of the predicted values of the displacement ratio shown in Figures12(b) and 13(b) are both 0.01. The prediction process is divided into two steps. The first step is to predict the target pressure based on the current pressure and the mechanical efficiency of the fixed-displacement hydraulic component. The second step is to predict the displacement ratio target value based on the current displacement ratio and the current pressures and the target pressures prediction value. The error of the target pressure predicted in the first step will lead to the error of the displacement ratio predicted value. When deriving (42) in the second step, we assume that Cspεh⟶hm∗=Cspε0 and Cspεhm⟶h∗=Cspε0, which will also cause errors. When the displacement control current is equivalent to the displacement ratio, the hysteresis effect of the solenoid valve is ignored, and the current and the displacement ratio are simplified to a linear relationship. Taking these factors into consideration, the errors of the predicted values are acceptable, and the prediction method of displacement ratio target value is reasonable.


## 7. Conclusions

This paper establishes a power transition model in full power shift of HMT, which is based on the double brakes overlapping. The dynamics of the power transition and the ideal target displacement ratio are studied. The following conclusions could be drawn:(1)
An arithmetic two-range HMT is taken as the research object. Based on the new full power shift method, the power transition model is established, including the hydraulic transmission unit model and the brake torque model during the double brakes overlapping. Under the premise of neglecting the dynamic reversing process of the hydraulic reversing valve, the power transition model can describe the principle of power transition and the interchange between two pressure sides of the hydraulic transmission unit.(2)
Aiming at the constant output power of HMT in the shift process, a prediction model of the displacement ratio target value is established and the prediction method is proposed. The bench test proves that the prediction method of the displacement ratio target value is reasonable, which can make the high-pressure side of the target range complete the pressure building-up process when the double brakes overlap. And the power transition is completed, and the output power can be transmitted normally in full power.



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*Source: 5296713-2020-02-10.xml*