Research ArticleResearch on Transmission Characteristics of HydromechanicalContinuously Variable Transmission of Tractor

Junyan Wang12 Changgao Xia 1 Xin Fan13 and Junyu Cai1

1School of Automotive and Traffic Engineering Jiangsu University Zhenjiang 212013 China2School of Transportation Zhenjiang College Zhenjiang 212028 China3School of Automobile and Traffic Engineering Jiangsu University of Technology Changzhou 213001 China

Correspondence should be addressed to Changgao Xia xiacgujseducn

Received 9 January 2020 Accepted 13 June 2020 Published 12 August 2020

Academic Editor Giorgio Besagni

Copyright copy 2020 JunyanWang et al )is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

)is paper proposes a new transmission scheme of hydromechanical continuously variable transmission (HMCVT) for tractors)e HMCVT has 4 working ranges in each of the front and rear directions )e speed characteristic and the torque characteristicof HMCVTare theoretically derived On the basis of HMCVTpower flow direction the Lrfkoes formula is used to calculate thetransmission efficiency )en the image analysis method is used to study the influence of parameters on the transmissionefficiency of HMCVT and the main influencing factors are found )e results of theoretical derivation demonstrate that bycoordinating control of the HST displacement ratio and the engagement conditions of shifting clutches the stepless speedregulation of HMCVT at the tractor speed of 0ndash50 kmh can be realized )e proposed HMCVT has the ability to continuouslytransmit and change torque over all working ranges )e overall transmission efficiency of HMCVT is at a high level To verify thetheoretical derivation Amesim simulation software is used for the modeling and simulation of HMCVT )e simulation resultsare consistent with the theoretical analysis results )erefore the HMCVTproposed in this paper has the advantages of compactstructure and high transmission efficiency and it is suitable for matching tractors

1 Introduction

In addition to transportation themain function of tractors isto drive agricultural machinery needed for agriculturalproduction which needs the transmission of tractors toprovide constantly changing speed and torque to adapt tofrequent external load changes [1]

In this context the use of a continuously variabletransmission (CVT) for tractors has become a trend )emost common type of CVTs is the hydraulic static trans-mission (HST) [2] it mainly includes hydraulic pumpshydraulic motors and control mechanism HST has goodperformance at low speed and can easily switch betweenpositive rotation and negative rotation However itstransmission efficiency is much lower than that of geartransmission which leads to the fact that it is rarely used intractors

In order to overcome the above shortcomings the HST isconnected in parallel with the mechanical components toform a hydromechanical continuously variable transmission(HMCVT) [3ndash6] HMCVT transmits only part of the powerthrough the HST the remaining power is transmittedthrough the mechanical components so the efficiency ismuch higher than HST HMCVT can realize stepless speedregulation through the HST while relying on mechanicalcomponents to achieve high efficiency transmission Inaddition the HMCVT can control the engine to operate onan optimal power curve or optimal economic curve thusachieving optimum power performance or optimum fueleconomy for tractors [7ndash9]

)e research on power split continuously variabletransmission technology began in the early 20th century inthe world However limited by the level of hydraulic systemmanufacturing this type of transmission was not used in

HindawiMathematical Problems in EngineeringVolume 2020 Article ID 6978329 14 pageshttpsdoiorg10115520206978329

armored and heavy construction vehicles until the late 1960s[10] Literature [8] first proposes a transmission scheme forsingle-row planetary gears with two hydromechanicalworking ranges In each of the two working ranges themaximum speed of the tractor can reach 32 kmh and 50 kmh respectively )en according to the market demand onlyone hydromechanical working range transmission scheme isproposed which can match the 75 kw engine )e forwarddriving speed is from 0 to 40 kmh and the reverse drivingspeed is from 0 to 25 kmh [11] )e transmission developedby Steyr company has three planetary gear rows controlledby 4 clutches and 2 brakes )is transmission can achievespeed adjustments from 0 to 50 kmh [12] )e authors of[13 14] study a HMCVT that has four planetary gear rowsand can achieve speed adjustment from 0 to 40 kmh )eauthor of [15] proposes a transmission scheme in which asingle-row planetary gear is connected in series with amechanical stepped transmission )is transmission has 6ranges in the forward direction and 3 ranges in the reversedirection

In general the more the transmission ranges the greaterthe expansion of the hydraulic power [10] but it will lead tofrequent switching during the ranges and reducing thetransmission performance)e single-row planetary schemehelps to reduce the size of the transmission but the hydraulicmotor needs to be reversed quickly at the moment ofswitching which needs high performance of the hydraulicmotor

)is paper will propose a transmission scheme for a new4-range HMCVT suitable for wheeled tractors and thenconduct a detailed theoretical analysis on the transmissioncharacteristics (speed characteristic torque characteristicand efficiency characteristic) of HMCVT and find out theimportant factors that affect the efficiency of HMCVT fi-nally the Amesim simulation software is used to verify thetransmission scheme

2 Transmission Scheme of HMCVT

)e structure of HMCVTproposed in this paper is shown inFigure 1 It includes the input shaft pump-motor hydraulicsystem planetary gear mechanism gear pairs shiftingclutches and directional clutches )e hydraulic path ofHMCVT is the HST and its mechanical path is the Rav-igneaux planetary gear mechanism (PGM) )e hydraulicpath and the mechanical path are connected in parallel toform a closed loop and then are connected in series with therange switching mechanism In the PGM when the outputcomponent is the carrier c it is an ordinary PGM when theoutput component is the small sun gear s2 then it is aRavigneaux PGM In addition part of the engine power isthrough the HST into the PGM and the HST realizes thestepless transmission

)ere are 6 clutches in the HMCVT which are 4 shiftingclutches and 2 directional clutches Different clutches canrealize 4 working ranges in the front or rear directions Inorder to ensure that engine power can be transferred from

the input shaft to the output shaft one shifting clutch andone directional clutch must be engaged )e engagementconditions of clutches in each range of HMCVT are shownin Table 1

3 Theoretical Analysis on TransmissionCharacteristics of HMCVT

31 Speed Characteristic

311 Ranges HM1 and HM3 )e power transmissionroutes of ranges HM1 andHM3 are basically the same exceptfor the difference between gear pairs i7 and i8 Under thesetwo conditions the power is input by the large sun gear s1and the ring gear r and output by the carrier c

If ne denotes the engine speed the variable pump inputshaft speed np is

np ne

i1i2( 1113857 (1)

Ignoring the effect of the volumetric efficiency of thevolumetric speed control loop the relationship between themotor output shaft speed nm the variable pump input shaftspeed np and the pump-motor displacement ratio ε is

nm εnp εne

i1i2 (2)

where ε QpQmminus1le εle 1 Qp is the displacement of thevariable pump and Qm is the displacement of the quanti-tative motor

)e input speed of the large sun gear in PGM ns1is

ns1 nm

minusi3( 1113857

εne

minusi1i2i3 (3)

)e input speed of the ring gear nr equals ne Accordingto the PGM speed characteristic formula [16] the rela-tionship between the input speed of large sun gear ns1

theinput speed of ring gear nr the output speed of carrier ncand the PGM standing transmission ratio minusk1 is satisfied

ns1minus k1nr minus 1 minus k1( 1113857nc 0 (4)

where minusk1 ZrZs1gt 1 Zr is the number of teeth of the ring

gear and Zs1is the number of teeth of the large sun gear

)erefore the output speed of the carrier nc is

nc minusk1i1i2i3 minus ε1 minus k1( 1113857i1i2i3

ne (5)

)e intermediate shaft speed nz is

nz

minusk1i1i2i3 minus ε1 minus k1( 1113857i1i2i3i4i8

ne HM1( 1113857

minusk1i1i2i3 minus ε1 minus k1( 1113857i1i2i3i4i7

ne HM3( 1113857

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(6)

Finally the output shaft speed of HMCVT no is

2 Mathematical Problems in Engineering

no

minusk1i1i2i3 minus ε1 minus k1( 1113857i1i2i3i4i8i11i12

ne HM1( 1113857

minusk1i1i2i3 minus ε1 minus k1( 1113857i1i2i3i4i7i11i12

ne HM3( 1113857

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(7)

)e transmission ratio of HMCVT ig is

ig ne

no

1 minus k1( 1113857i1i2i3i4i8i11i12

minusk1i1i2i3 minus ε HM1( 1113857

1 minus k1( 1113857i1i2i3i4i7i11i12

minusk1i1i2i3 minus εHM3( 1113857

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(8)

312 Ranges HM2 and HM4 )e power transmissionroutes of ranges HM2 and HM4 are the same except for the

difference between gear pairs i6 and i9 Under these twoconditions the power is input by the large sun gear s1 andthe ring gear r and output by the small sun gear s2

)e same as the previous case the input speed of thelarge sun gear ns1

is ns1 minusεne(i1i2i3) the input speed of the

ring gear nr equals ne According to the PGM speed char-acteristic formula (16) the relationship between the inputspeed of large sun gear ns1

the input speed of ring gear nr theoutput speed of the small sun gear ns2

and the PGM standingtransmission ratios k2 and minusk3 is satisfied

1 minus k3

k2 minus k3nr minus

1 minus k2

k2 minus k3ns1

minus ns2 0 (9)

)erefore the output speed of the small sun gear ns2is

ns2

1 minus k3( 1113857i1i2i3 + 1 minus k2( 1113857εk2 minus k3( 1113857i1i2i3

ne (10)

Table 1 Engagement conditions of clutches of HMCVT

Range Directional clutch Shifting clutchCF CR C1 C2 C3 C4

Forward

First + +Second + +)ird + +Fourth + +

Reverse

First + +Second + +)ird + +Fourth + +

Engine

Tei1

i2

p1

r

p2

S1

S2c

i5

i4

i6

i7

i9

i8

i3

i12

i11

i10

ne

Tp

npTm

nm

Tz

nz

To

no

C1C3

C2C4

Pump Motor

CR CF

Figure 1 )e structure of HMCVT

Mathematical Problems in Engineering 3

where k2 Zs2Zr gt 0 minusk3 Zs2

Zs1gt 0 and Zr Zs1

Zs2are

the numbers of teeth of ring gear large sun gear and smallsun gear respectively

)e intermediate shaft speed nz is

nz

1 minus k3( 1113857i1i2i3 + 1 minus k2( 1113857εk2 minus k3( 1113857i1i2i3i5i9

ne HM2( 1113857

1 minus k3( 1113857i1i2i3 + 1 minus k2( 1113857εk2 minus k3( 1113857i1i2i3i5i6

ne HM4( 1113857

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(11)

Finally the output shaft speed of HMCVT no is

no

1 minus k3( 1113857i1i2i3 + 1 minus k2( 1113857εk2 minus k3( 1113857i1i2i3i5i9i11i12

ne HM2( 1113857

1 minus k3( 1113857i1i2i3 + 1 minus k2( 1113857εk2 minus k3( 1113857i1i2i3i5i6i11i12

ne HM4( 1113857

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(12)

)e transmission ratio of HMCVT ig is

ig ne

no

k2 minus k3( 1113857i1i2i3i5i9i11i121 minus k3( 1113857i1i2i3 + 1 minus k2( 1113857ε

HM2( 1113857

k2 minus k3( 1113857i1i2i3i5i6i11i12

1 minus k3( 1113857i1i2i3 + 1 minus k2( 1113857ε HM4( 1113857

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(13)

32 Determination of Parameters of HMCVT )e ratedspeed of the engine matching with the HMCVT is 2200 rmin the minimum speed of the tractor is 4 kmh themaximum speed is 50 kmh the main reducer ratio of therear axle is 37 the reduction ratio of the wheel side is 56 thetotal transmission ratio of the rear axle is 2072 and the drivewheel radius is 08579m

)e range of the transmission ratio in the drive train isdetermined by parameters such as the speed of the tractorthe rated speed of the engine and the drive wheel radius Sothe transmission ratio of each range of HMCVT is cal-culated as follows

igj 0377nerd

v (14)

where j 1 2 3 4 ne is the rated speed of the engine rd is thedrive wheel radius v is the theoretical driving speed of thetractor

)e HMCVT has 4 working ranges so the common ratioof the transmission ratios is

φ

igmax

igmin

4

1113971

(15)

)e HMCVT adopts a proportional transmission so ineach working range the transmission ratio has the followingrelationship

igjmax

igjmin φ (16)

When HMCVT is switching between adjacent rangesthe conditions must be met at the changing point thetransmission ratios are equal and the pump-motor dis-placement ratios are equal

ig1min(minus1) ig2min(minus1)

ig2min(1) ig3min(1)

ig3min(minus1) ig4min(minus1)

ig4min(1) imin

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(17)

)e gear ratio of the connecting gears between theengine and the hydraulic variable pump needs to be met

i1i2 genemax

npmax (18)

According to the previous content if i10 i11 and i12 areset to 1 then the relationship between the various param-eters of HMCVT can be obtained

k1i1i2i3 minus32727

1 minus k3 k1 k2 minus 1( 1113857

i5i6 08969k1

1 + k1( 1113857

i4i7 minus16865k1

1 minus k1( 1113857

i4i8 minus59615k1

1 minus k1( 1113857

i5i9 31702k1

1 + k1( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(19)

)ese parameters are all expressions about the PGMstanding transmission ratio minusk1 Once the value of minusk1 isdetermined the values of these parameters will also bedetermined which will be described later

33 Torque Characteristic

331 Ranges HM1 and HM3 Ignoring the power losses ifTo denotes the load torque of output shaft the intermediateshaft torque Tz is

Tz To

i11i12 (20)

and then the output torque of carrier in PGM Tc is

Tc

Tz

i4i8

To

i4i8i11i12 HM1( 1113857

Tz

i4i7

To

i4i7i11i12 HM3( 1113857

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(21)

4 Mathematical Problems in Engineering

According to the PGM torque characteristic formula(16) the relationship between the input torque of large sungear Ts1

the input torque of ring gear Tr the output torqueof carrierTc and the PGM standing transmission ratio minusk1 issatisfied

Ts1

Tr

minusk1 minus

Tc

1 minus k1 (22)

where minusk1 ZrZs1gt 1 Zr is the number of teeth of the ring

gear and Zs1is the number of teeth of the large sun gear

)erefore the large sun gear torque Ts1is

Ts1 minus

Tc

1 minus k1

minusTo

i4i8i11i12 1 minus k1( 1113857 HM1( 1113857

minusTo

i4i7i11i12 1 minus k1( 1113857 HM3( 1113857

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(23)

)en the input torque of ring gear Tr is

Tr minusk1Ts1

k1To

i4i8i11i12 1 minus k1( 1113857 HM1( 1113857

k1To

i4i7i11i12 1 minus k1( 1113857 HM3( 1113857

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(24)

)us the motor shaft load torque Tm is

Tm Ts1

minusi3

To

i3i4i8i11i12 1 minus k1( 1113857 HM1( 1113857

To

i3i4i7i11i12 1 minus k1( 1113857 HM3( 1113857

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(25)

)e pump shaft torque Tp is

Tp εTm

εTo

i3i4i8i11i12 1 minus k1( 1113857 HM1( 1113857

εTo

i3i4i7i11i12 1 minus k1( 1113857 HM3( 1113857

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(26)

)e engine output torque Te is

Te Tr +Tp

i1i2

k1i1i2i3 + ε( 1113857To

i1i2i3i4i8i11i12 1 minus k1( 1113857 HM1( 1113857

k1i1i2i3 + ε( 1113857To

i1i2i3i4i7i11i12 1 minus k1( 1113857 HM3( 1113857

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(27)

Finally the ratio of the output torque of HMCVT to theinput torque of the engine μ is

μ To

Te

1 minus k1( i1i2i3i4i8i11i12

ε + k1i1i2i3 HM1( 1113857

1 minus k1( 1113857i1i2i3i4i7i11i12ε + k1i1i2i3

HM3( 1113857

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(28)

332 Ranges HM2 and HM4 )e same as the previous caseTo denotes the load torque of output shaft and the inter-mediate shaft torque Tz is Tz To(i11i12) )en the outputtorque of small sun gear in PGM Ts2

is

Ts2

Tz

i5i9

To

i5i9i11i12 HM2( 1113857

Tz

i5i6

To

i5i6i11i12 HM4( 1113857

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(29)

)e external torque algebra sum of PGM is zero soTs1

+ Tr + Ts2 0 According to the PGM torque character-

istic formula (16) the relationship between the input torque oflarge sun gear Ts1

the output torque of small sun gear Ts2 and

the PGM standing transmission ratios k2 and minusk3 is satisfied

Ts1

k2 minus 1k3 minus k2

Ts2

k2 minus 1( 1113857To

k3 minus k2( 1113857i5i9i11i12 HM2( 1113857

k2 minus 1( 1113857To

k3 minus k2( 1113857i5i6i11i12 HM4( 1113857

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(30)

)e engine output torque Te is

Te

1 minus k2( 1113857ε + 1 minus k3( 1113857i1i2i3

k3 minus k2( 1113857i1i2i3i5i9i11i12To HM2( 1113857

1 minus k2( 1113857ε + 1 minus k3( 1113857i1i2i3

k3 minus k2( 1113857i1i2i3i5i6i11i12To HM4( 1113857

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(31)

Finally the ratio of the output torque of HMCVT to theinput torque of the engine μ is

μ To

Te

k3 minus k2( 1113857i1i2i3i5i9i11i12

1 minus k2( 1113857ε + 1 minus k3( 1113857i1i2i3 HM2( 1113857

k3 minus k2( 1113857i1i2i3i5i6i11i12

1 minus k2( 1113857ε + 1 minus k3( 1113857i1i2i3 HM4( 1113857

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(32)

34 Circulating Power Flow Calculation

341 Ranges HM1 and HM3 )e speed ratio of the powertransmitting from the engine to the carrier ice is

ice nc

ne

minusk1i1i2i3 minus ε1 minus k1( 1113857i1i2i3

irce + i

s1ce (33)

)e speed ratios of the power transmitting from theengine to the large sun gear and ring gear ireis1e are

ire 1

is1e minusε

i1i2i3

(34)

In the PGM the speed ratios between the input elements(large sun gear s1 and ring gear r) and the output element(carrier c) are

Mathematical Problems in Engineering 5

ircs1

nc

ns1

1

1 minus k1

is1cr

nc

nr

minusk1

1 minus k1

(35)

)en we can get

irce i

rcs1

is1e minusε

i1i2i3 1 minus k1( 1113857

is1ce i

s1crire minus

k1

1 minus k1

(36)

Next two situations will be discussed as follows [17]

(1) When εlt 0 is1ce gt 0 irce gt 0 and i

s1cei

rce gt 0 the PGM has

no circulating power PL 0(2) When εgt 0 i

s1ce gt 0 irce lt 0 and i

s1cei

rce lt 0 the PGM has

circulating power substituting equation (19) fork1i1i2i3 in equation (33) ice is

ice 32727 minus ε

32727 + i1i2i3gt 0 (37)

Because ofPe gt 0 Pr (is1ceice)Pe gt 0 Ps1

(irceice)Pe lt 0the output power P is

P Pe minusPc Pr + Ps1 Pr minus Ps1

11138681113868111386811138681113868

11138681113868111386811138681113868 (38)

)en

Pr P + Ps1

11138681113868111386811138681113868

11138681113868111386811138681113868 P + PL

11138681113868111386811138681113868111386811138681113868 (39)

)e circulating power PL is

PL Ps1

irce

ice

Pe (40)

Assume that the closed power factor KL is

KL PL

11138681113868111386811138681113868111386811138681113868

Pe

irce

ice

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

εk1i1i2i3 + ε

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 |ϕ(ε)| (41)

where ϕ(ε) εk1i1i2i3 + ε and the derivative of ϕ(ε) isdϕ(ε)dε k1i1i2i3(k1i1i2i3 + ε)2 lt 0 so function ϕ(ε) is amonotonically decreasing function )en KL(0) 0KL(1) 044 and the maximum value of the circulatingpower is 044Pe

342 Ranges HM2 and HM4 )e speed ratio of the powertransmitting from the engine to the small sun gear is2e is

is2e ns2

ne

1 minus k3( 1113857i1i2i3 + 1 minus k2( 1113857ε

k2 minus k3( 1113857i1i2i3 i

rs2e + i

s1s2e (42)

)e speed ratios of the power transmitting from theengine to the large sun gear and ring gear ireis1e are

ire 1

is1e minusε

i1i2i3

(43)

In the PGM the speed ratios between the input elements(large sun gear s1 and ring gear r) and the output element(small sun gear s2) are

irs2s1

k2 minus 1k2 minus k3

is1s2r

1 minus k3

k2 minus k3

(44)

)en we can get

irs2e i

rs2s1

is1e minusk2 minus 1( 1113857ε

i1i2i3 k2 minus k3( 1113857

is1s2e i

s1s2rire

1 minus k3

k2 minus k3

(45)

Next two situations will be discussed as follows [17]

(1) When εlt 0 is1s2e gt 0 irs2e lt 0 and i

s1s2ei

rs2e lt 0 the PGM

has circulating power substituting equation (19) fork2 k3 i1i2i3 in equation (42) is2e is

is2e 32727 + ε

32727 minus 32727 minusk1( 1113857gt 0 (46)

Because of Pe gt 0 Pr is1ceicePe gt 0 Ps1

irceicePe lt 0the output power P is

P Pe minusPs2 Pr + Ps1

Pr minus Ps1

11138681113868111386811138681113868

11138681113868111386811138681113868 (47)

)en

Pr P + Ps1

11138681113868111386811138681113868

11138681113868111386811138681113868 P + PL

11138681113868111386811138681113868111386811138681113868 (48)

)e circulating power PL is

PL Ps1 irs2e

is2e

Pe (49)

Assume that the closed power factor KL is

KL PL

11138681113868111386811138681113868111386811138681113868

Pe

irs2e

is2e

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

minusεk1i1i2i3 minus ε

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 |ϕ(ε)| (50)

where ϕ(ε) minusεk1i1i2i3 minus ε and the derivative ofϕ(ε) is dϕ(ε)dε minusk1i1i2i3(k1i1i2i3 minus ε)2 gt 0 sofunction ϕ(ε) is a monotonically increasing function)en KL(0) 0 KL(1) 044 and the maximumvalue of the circulating power is 044Pe

(2) When εgt 0 is1s2e gt 0 irs2e gt 0 and i

s1s2ei

rs2e gt 0 the PGM

has no circulating power PL 0

35 Hydraulic Split Power Calculation Hydraulic power-split ratio ρ PHPe Ps1

Pe hydraulic power is input tothe PGM through the large sun gear s1

6 Mathematical Problems in Engineering

351 Ranges HM1 and HM3

ρ Ps1

Pe

irce

ice

ε

k1i1i2i3 + ε ϕ(ε) (51)

and as can be obtained from the above function ϕ(ε) is amonotonically decreasing function so

ρ(minus1) 0234gt 0

ρ(0) 0

ρ(1) minus044lt 0

⎧⎪⎪⎨

⎪⎪⎩(52)

352 Ranges HM2 and HM4

ρ Ps1

Pe

irs2e

is2e

minusε

k1i1i2i3 minus ε ϕ(ε) (53)

As can be obtained from the above function ϕ(ε) is amonotonically increasing function so

ρ(minus1) minus044lt 0

ρ(0) 0

ρ(1) 0234gt 0

⎧⎪⎪⎨

⎪⎪⎩(54)

According to the theoretical analysis on the circulatingpower flow and the hydraulic split power the power flowdirection of HMCVT in each working range is shown inFigure 2 )is result lays the foundation for the analysis onthe efficiency characteristic of HMCVT

36 Efficiency Characteristic )e Lrfkoes formula (17) isused to calculate the transmission efficiency of HMCVT

η minus1113957μig

minusToTe

neno

(55)

where 1113957μ is the ratio of output torque to input torque ofHMCVTconsidering power losses and ig is the transmissionratio of HMCVT

)e transmission efficiency calculation process of eachworking range of HMCVT is shown in Tables 2 and 3Among them the calculation processes of the first range andthe third range are listed in Table 2 and the calculationprocesses of the second range and the fourth range are listedin Table 3 In Tables 2 and 3 i1 minus i12 are the gear ratios of thegear pairs η1 minus η12 are the transmission efficiency of the gearpairs ηH is the hydraulic system transmission efficiency andηs1r ηs1s2

and ηrs2are the transmission efficiencies of the

large sun gear to the ring gear the large sun gear to the smallsun gear and the ring gear to the small sun gear in the PGMrespectively

4 Analysis on the Influence of Parameters onthe Transmission Efficiency of HMCVT

)e results of the transmission efficiency in each workingrange of HMCVT are as follows

ηHM1

ηs1r minus k11113872 1113873η4η8η11η12 ε + k1i1i2i3( 1113857

1 minus k1( 1113857 η1η2η3ηHηs1rε + k1i1i2i31113872 1113873 (εgt 0)

ηs1r minus k11113872 1113873η1η2η3ηHη4η8η11η12 ε + k1i1i2i3( 1113857

1 minus k1( 1113857 i1i2i3η1η2η3ηHk1 + εηs1r1113872 1113873 (εle 0)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

ηHM2

k3ηs1s2minus k2ηrs2

1113872 1113873η5η9η11η12 1 minus k3( 1113857i1i2i3 + 1 minus k2( 1113857ε1113858 1113859

k3 minus k2( 1113857 1 minus k3ηs1s21113872 1113873i1i2i3 minus η1η2η3ηHε k2ηrs2

minus 11113872 11138731113960 1113961 (εlt 0)

k3ηs1s2minus k2ηrs2

1113872 1113873η1η2η3η5η9η11η12ηH 1 minus k3( 1113857i1i2i3 + 1 minus k2( 1113857ε1113858 1113859

k3 minus k2( 1113857 1 minus k3ηs1s21113872 1113873i1i2i3η1η2η3ηH minus ε k2ηrs2

minus 11113872 11138731113960 1113961 (εge 0)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

ηHM3

ηs1r minus k11113872 1113873η4η7η11η12 ε + k1i1i2i3( 1113857

1 minus k1( 1113857 η1η2η3ηHηs1rε + k1i1i2i31113872 1113873 (εgt 0)

ηs1r minus k11113872 η1η2η3ηHη4η7η11η12 ε + k1i1i2i3( 1113857

1 minus k1( 1113857 i1i2i3η1η2η3ηHk1 + εηs1r1113872 1113873 (εle 0)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

ηHM4

k3ηs1s2minus k2ηrs2

1113872 1113873η5η6η11η12 1 minus k3( 1113857i1i2i3 + 1 minus k2( 1113857ε1113858 1113859

k3 minus k2( 1113857 1 minus k3ηs1s21113872 1113873i1i2i3 minus η1η2η3ηHε k2ηrs2

minus 11113872 11138731113960 1113961 (εlt 0)

k3ηs1s2minus k2ηrs2

1113872 1113873η1η2η3η5η6η11η12ηH 1 minus k3( 1113857i1i2i3 + 1 minus k2( 1113857ε1113858 1113859

k3 minus k2( 1113857 1 minus k3ηs1s21113872 1113873i1i2i3η1η2η3ηH minus ε k2ηrs2

minus 11113872 11138731113960 1113961 (εge 0)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(56)

Mathematical Problems in Engineering 7

When HMCVT is working in HM1 or HM3 range if thetransmission efficiencies of the gear pairs i9 and i8 are equal(η7 η8) the transmission efficiencies of HMCVT in HM1and HM3 ranges are equal

When HMCVT is working in HM2 or HM4 range if thetransmission efficiencies of the gear pairs i9 and i6 are equal(η9 η6) the transmission efficiencies of HMCVT in HM2and HM4 ranges are also equal

Due to the complexity of the above results it is necessaryto evaluate the influence of displacement ratio ε standingtransmission ratio of PGM minusk1 gear ratio i1i2i3 and hy-draulic system efficiency ηH on the transmission efficiency of

HMCVT by means of image analysis method We assumethat η1 minus η12 are 098 ηs1r is 098 ηs1s2

and ηrs2are 097 and

k1i1i2i3 is minus32727 )e evaluation results are shown inFigure 3

According to the analysis on Figure 3 the following rulesare obtained

In each range of HMCVT when the displacement ratio εis 0 the transmission efficiency reaches the maximum valueNow the hydraulic path (HST) does not transmit power andthe power is completely transmitted by the mechanical pathWhen the displacement ratio ε is minus1 or 1 the transmissionefficiency reaches the minimum value and the hydraulic

Table 2 Equation of efficiency calculation for ranges HM1 and HM3

ElementsTorque equation considering power losses

εgt 0 εlt 0HM1 HM3 HM1 HM3

Engine Te Tr + η1η2Tpi1i2 Te Tr + Tpi1i2η1η2Pump Tp εηHTm Tp εTmηH

Motor Tm η3Ts1(minusi3) Tm Ts1

(minusη3i3)PGM Ts1

+ Tr + Tc 0PGM Tr minusk1Ts1

ηs1r

Intermediate shaft Tc Tzi4i8η4η8 Tc Tzi4i7η4η7 Tc Tzi4i8η4η8 Tc Tzi4i7η4η7Output shaft Tz To(i11i12η11η12)Torque ratio 1113957μ ToTe

Efficiency η minus1113957μig

144 max

100 44 max

44 max

Engine

r

S1

S2c

100

(a)

766 min

100

100

234 max

234 max

Engine

r

S1

S2c

(b)

766 min

100 234 max

234 max

Engine

r

S1

S2c

100

(c)

144 max

100

100

44 max

44 max

Engine

r

S1

S2c

(d)

Figure 2 Power flow direction of the PGM of HMCVT (a) Range HM1HM3 εgt 0 power-circulation (b) Range HM1HM3 εlt 0 power-split (c) Range HM2HM4 εgt 0 power-split (d) Range HM2HM4 εlt 0 power-circulation

8 Mathematical Problems in Engineering

Table 3 Equation of efficiency calculation for ranges HM2 and HM4

ElementsTorque equation considering power losses

εgt 0 εlt 0HM2 HM4 HM2 HM4

Engine Te Tr + Tpi1i2η1η2 Te Tr + η1η2Tpi1i2Pump Tp εTmηH Tp εηHTm

Motor Tm Ts1(minusη3i3) Tm η3Ts1

(minusi3)

PGM Ts1+ Tr + Ts2

0PGM Tr 1 minus k3ηs1s2

k3ηs1s2minus k2ηrs2

Ts2Intermediate shaft Ts2

Tzi5i9η5η9 Ts2 Tzi5i6η5η6 Ts2

Tzi5i9η5η9 Ts2 Tzi5i6η5η6

Output shaft Tz To(i11i12η11η12)Torque ratio 1113957μ ToTe

Efficiency η minus1113957μig

ndashk1 = 25172ndashk1 = 29130

ndashk1 = 17805ndashk1 = 21875

095

09

085

08

075

07ndash1 ndash08 ndash06 ndash04 ndash02 0 02 04 06 08 1

Displacement ratio

ηH = 081

Effici

ency

(a)

ndashk1 = 25172ndashk1 = 29130

ndashk1 = 17805ndashk1 = 21875

ηH = 081

095

09

085

08

075

07ndash1 ndash08 ndash06 ndash04 ndash02 0 02 04 06 08 1

Displacement ratio

Effici

ency

(b)

ηH = 081ηH = 090

ηH = 064ηH = 072

095

09

085

08

075

07ndash1 ndash08 ndash06 ndash04 ndash02 0 02 04 06 08 1

Displacement ratio

Effici

ency

ndashk1 = 21875

(c)

ηH = 081ηH = 090

ηH = 064ηH = 072

095

09

085

08

075

07ndash1 ndash08 ndash06 ndash04 ndash02 0 02 04 06 08 1

Displacement ratio

Effici

ency

ndashk1 = 21875

(d)

Figure 3 Continued

Mathematical Problems in Engineering 9

path transmits more power at this time resulting in a loweroverall efficiency of HMCVT

In each range of HMCVT the transmission efficiencyvalue in the power circulation phase is lower than thetransmission efficiency value in the power split phase whichindicates that the power circulation will cause the trans-mission efficiency of HMCVT to decrease

)e larger the value of the PGM standing transmissionratio minusk1 the higher the transmission efficiency of HMCVTbut the influence on improving the transmission efficiency is

not obvious )e PGM standing transmission ratio minusk1ranges from 15 to 3 )e gear ratio i1i2i3 and the PGMstanding transmission ratio minusk1 are in an inversely pro-portional relationship the smaller the value of the gear ratioi1i2i3 the higher the transmission efficiency of HMCVT andthe influence on improving the transmission efficiency isalso not obvious )e gear ratio i1i2i3 ranges from 11 to 22

)e greater the value of HSTefficiency ηH the higher theoverall transmission efficiency of HMCVT )e influence ofηH on improving transmission efficiency of HMCVT issignificant

i1i2i3 = 13001i1i2i3 = 11234

i1i2i3 = 18381i1i2i3 = 14961

ηH = 081

095

09

085

08

075

07ndash1 ndash08 ndash06 ndash04 ndash02 0 02 04 06 08 1

Displacement ratio

Effici

ency

(e)

i1i2i3 = 13001i1i2i3 = 11234

i1i2i3 = 18381i1i2i3 = 14961

ηH = 081

095

09

085

08

075

07ndash1 ndash08 ndash06 ndash04 ndash02 0 02 04 06 08 1

Displacement ratio

Effici

ency

(f )

Figure 3)e influence of parameters on the transmission efficiency of HMCVT (a))e influence of minusk1 on HM1 andHM3 ranges (b))einfluence of minusk1 on HM2 and HM4 ranges (c) )e influence of ηH on HM1 and HM3 ranges (d) )e influence of ηH on HM2 and HM4ranges (e) )e influence of i1i2i3 on HM1 and HM3 ranges (f ) )e influence of i1i2i3 on HM2 and HM4 ranges

9

8

7

6

5

4

3

2

1

0

Tran

smiss

ion

ratio

ndash15 ndash1 ndash05 0 05 1 15Displacement ratio

HM1HM2

HM3HM4

Figure 4 Relationship between transmission ratio and displace-ment ratio

60

50

40

30

20

10

0

Trac

tor s

peed

(km

h)

ndash15 ndash1 ndash05 0 05 1 15Displacement ratio

HM1HM2

HM3HM4

Figure 5 )e curve of the tractor speed

10 Mathematical Problems in Engineering

5 Analysis on the Transmission CharacteristicsResults of HMCVT

According to the previous content the value of the PGMstanding transmission ratio minusk1 is set to 21875 and the HSTefficiency value ηH is set to 081 then the other parameters ofHMCVT are k2 02714 minusk3 05938 i1 i2 1 i3 15i4 i5 1 i6 166 i7 116 i8 410 i9 584 andi10 i11 i12 1

)e relationship between the transmission ratio anddisplacement ratio is shown in Figure 4 )e curve of thetractor speed is shown in Figure 5 In the first range thedisplacement ratio changes from 1 to minus1 the transmissionratio decreases from 862 to 459 and the tractor speedincreases from 0 to 750 kmh In the second range thedisplacement ratio changes from minus1 to 1 the transmissionratio decreases from 459 to 243 and the tractor speedincreases from 750 kmh to 1412 kmh In the third rangethe displacement ratio changes from 1 to minus1 the trans-mission ratio decreases from 243 to 130 and the tractorspeed increases from 1412 kmh to 2644 kmh In thefourth range the displacement ratio changes from minus1 to 1the transmission ratio decreases from 130 to 069 and thetractor speed increases from 2644 kmh to 50 kmh )eresults show that the stepless speed regulation of HMCVTin the tractor speed range of 0ndash50 kmh can be realized bythe coordinated control of the HST displacement ratio andthe clutches

)e ratios of the output shaft torque to the input shafttorque in each range of HMCVT are shown in Figure 6(a))e continuous change of torque can be realized in the entireworking ranges )e ratios of the motor shaft torque to theoutput shaft torque in each range of HMCVT are shown inFigure 6(b) )e torque ratio remains constant in each range

and varies between different ranges )e values of torqueratios in the four ranges are 514 minus972 1849 andminus3518 respectively

)e relationship between the tractor speed and theoverall transmission efficiency of HMCVT is shown inFigure 7 Each of the four ranges has a transmission effi-ciency maximum value of 0917 0902 0917 and 0902respectively )e efficiency value always increases first withthe increase of the tractor speed and then decreases afterreaching the maximum value )e maximum transmissionefficiency in each range is obtained when the HST dis-placement ratio is 0 Now the hydraulic path does nottransmit power and all power is transmitted by the me-chanical path )e transmission efficiency of HMCVT ishigher than 83 so the overall transmission efficiency is at ahigh level

ndash2

ndash4

ndash6

ndash8

ndash10

0

0 5 10 15 20 25 30 35 40 45 50

Torq

ue ra

tio (T

oTe

)

Tractor speed (kmh)

HM1HM2

HM3HM4

(a)

05

04

03

02

01

0

ndash01

ndash02

ndash03

ndash04

ndash05

Torq

ue ra

tio (T

mT

o)

0 5 10 15 20 25 30 35 40 45 50Tractor speed (kmh)

HM1HM2

HM3HM4

(b)

Figure 6)e curve of torque characteristic (a) Ratios of output shaft torque to input shaft torque (b) Ratios of motor shaft torque to outputshaft torque

095

09

085

08

075

07

Effici

ency

0 5 10 15 20 25 30 35 40 45 50Tractor speed (kmh)

HM1HM2

HM3HM4

Figure 7 Overall transmission efficiency of HMCVT

Mathematical Problems in Engineering 11

6 Verification of HMCVT Simulation Model

)e modeling and simulation of HMCVT are accom-plished by using Amesim simulation softwareAccording to Figure 1 a simulation model of HMCVTis established as shown in Figure 8

61 Hydraulic System )e hydraulic system is mainlycomposed of a variable displacement hydraulic pump afixed displacement hydraulic motor two relief valves twocheck valves and a charge hydraulic pump with a reliefvalve )e aim of the charge pump is to compensate for theleakage loss in the pipeline and to establish the lowestpressure in the pumprsquos suction pipeline Two relief valves areused to limit the maximum pressure in the system pipingand its adjustment pressure should be above the normalworking pressure of the system Volumetric losses of the

pump and motor are neglected and their hydromechanicalefficiencies are set at 09

62 Mechanical Components Mechanical components areassumed to be constant efficiency components throughoutthe transmission )e power loss of each gear pair is 2 sothe efficiency is set to 098 In the Ravigneaux planetary gearmechanism the efficiencies of ring gear-long planetary gearcouple long planetary gear-large sun gear couple longplanetary gear-short planetary gear couple and shortplanetary gear-small sun gear couple are set at 099 )emain reducer differential and wheel reducer are combinedand modeled as a gear reducer with an efficiency of 0965 Tosimplify the model the friction losses of the shaft gears andbearings were ignored during the simulation modelingprocess

Efficiency calculation unit

Speed calculation unit

Small sun gear

Short planet gear

Long planet gear

Large sun gear

Ring gear

i1

i2

Pump Motor

i3

i5

i4

i7 i6

i9i8

Carrier

C1

C2

C3

C4

ω ωT

TL

ωωT

T

L

PP

MOT

f (x y)

y

x

X

k

Figure 8 HMCVT simulation model

(km

h)

50

40

30

20

10

00 5 10 15 20 25 30 35

X Time (s)

Tractor linear velocity (kmh)

(a)

0 5 10 15 20 25 30 35X Time (s)

10

08

06

04

02

00

(nul

l)

Transmission efficiency of HMCVT (null)

(b)

Figure 9 Simulation results (a) Simulation result of the tractor speed (b) Simulation result of the transmission efficiency of HMCVT

12 Mathematical Problems in Engineering

63Model Control and Simulation Results )e engine speedis assumed to have a constant value of 2200 rmin the loadtorque of the output shaft of HMCVT is set to 100Nm themodel simulation time is 32 s including range HM1 0ndash8 srange HM2 8ndash16 s range HM3 16ndash24 s and range HM424ndash32 s

)e simulation results are shown in Figure 9 through thevariable pump displacement and coordinated control of theclutches [18 19] the transmission gradually changes fromrange HM1 to range HM4 and the tractor achieves steplessspeed change from 0 to 50 kmh )e simulation results areconsistent with the previous theoretical analysis results whichshow that the 4-range HMCVT transmission scheme pro-posed in this paper is suitable for matching wheeled tractors

64 Comparison Xu LYrsquos transmission scheme [15] hasbeen applied to mass production tractors )is transmissionscheme has 6 working ranges in the forward direction and 3working ranges in the reverse direction which is controlledby 8 clutches )e speed range in the forward direction is0sim30 kmh while the reverse direction is only 0sim10 kmh)e transmission efficiency of the hydraulic range is low (lessthan 80) the transmission efficiency of the hydrome-chanical ranges is high (higher than 90) but is only in theforward direction

)e scheme proposed in this paper has 4 working rangesand a speed range of 0sim50 kmh in the forward and reversedirections which is controlled by 6 clutches )e fewer thetransmission ranges the less switching during the workingranges so the transmission performance is improved )esmaller number of clutches ensures that the system is easierto control )e transmission efficiencies in the forward andreverse directions are close the overall forward transmissionefficiency is higher than 83 and the highest transmissionefficiency is higher than 91

)erefore the application of HMCVT proposed in thispaper is wider it can not only match conventional tractorsbut also match two-way operation tractors and other en-gineering vehicles

7 Conclusions

)is paper proposes a new type of transmission scheme ofHMCVT )e Ravigneaux PGM and the HSTare connectedin parallel to form a closed loop and then connected in serieswith the range switching mechanism )is new HMCVTfeatures a double-row PGM with 4 hydromechanicalworking ranges By coordinating control of the HST dis-placement ratio and the engagement conditions of shiftingclutches the stepless speed regulation of HMCVT at thetractor speed of 0ndash50 kmh is realized

According to the theoretical analysis on the newHMCVT the speed characteristic curve and the torquecharacteristic curve are obtained Based on the power flowdirection of HMCVT the transmission efficiency curve isobtained by using the Lrfkoes formula

Using image analysis method to study the influence offour parameters (displacement ratio ε PGM standing

transmission ratio minusk1 gear ratio i1i2i3 and hydraulic systemefficiency ηH) on the transmission efficiency of HMCVT theresults show that the standing transmission ratio minusk1 andgear ratio i1i2i3 have little effect on the transmission effi-ciency whereas the displacement ratio ε and hydraulicsystem efficiency ηH have a greater impact on the trans-mission efficiency

)e parameters of the new HMCVT are finally deter-mined )e results of the transmission characteristics showthat the new HMCVTproposed in this paper has the steplessspeed regulation capability for tractors which can alsocontinuously transmit and change the torque )e trans-mission efficiency of HMCVT is at a high level

In order to verify the theoretical derivation Amesimsimulation software is used for the modeling and simulationof HMCVT )e simulation results are consistent with thetheoretical analysis results )erefore we believe that thetransmission scheme is reasonable and effective for tractors

Data Availability

)e Simcenter Amesim System File data used to support thefindings of this study are available from the correspondingauthor upon request

Conflicts of Interest

)e authors declare that they have no conflicts of interest

Acknowledgments

)is research was financially supported by the National KeyResearch and Development Plan Project of China(2016YFD0700402) and Changzhou Sci amp Tech Program(no CE20185037)

References

[1] L Y Xu Z L Zhou M Z Zhang et al ldquoResearch and designof hydro-mechanical continuously variable transmission fortractorsrdquo Journal of Northeast Agricultural University vol 13no 2 pp 182ndash186 2006

[2] R Paoluzzi and L G Zarotti ldquo)e minimum size of hy-drostatic transmissions for locomotionrdquo Journal of Terra-mechanics vol 50 no 3 pp 153ndash164 2013

[3] J Kim J Kang Y Kim B Min and H Kim ldquoDesign of powersplit transmission design of dual mode power split trans-missionrdquo International Journal of Automotive Technologyvol 11 no 4 pp 565ndash571 2010

[4] A Kim and A Rossetti ldquoFuel consumption reduction inurban buses by using power split transmissionsrdquo EnergyConversion and Management vol 71 pp 159ndash171 2013

[5] A Rossetti and A Macor ldquoMulti-objective optimization ofhydro-mechanical power split transmissionsrdquo Mechanismand Machine lteory vol 62 pp 112ndash128 2013

[6] A Macor and A Rossetti ldquoOptimization of hydro-mechanicalpower split transmissionsrdquo Mechanism and Machine lteoryvol 46 no 12 pp 1901ndash1919 2011

[7] S Shamshirband D Petkovic A Amini et al ldquoSupport vectorregression methodology for wind turbine reaction torqueprediction with power-split hydrostatic continuous variabletransmissionrdquo Energy vol 67 pp 623ndash630 2014

Mathematical Problems in Engineering 13

[8] K T Anuar ldquoResch R Continuously variable tractor trans-missionsrdquo ASAE Distinguished Lecture No 29 pp 1ndash37 StJoseph American Society of Agricultural Engineers St JosephMI USA 2005

[9] S H Yuan and J B Hu ldquo)e efficiency of multi-range hydro-mechanical stepless transmissionrdquo Journal of Beijing Instituteof Technology vol 7 no 2 pp 129ndash133 1998

[10] X J LiuAnalysis of Vehicle Transmission System pp 258-259National Defence Industry Press Beijing China 1998

[11] M M Brenninger ldquoFendt vario CVT in agricultural tractorsrdquoSAE International Warrendale PA USA SAE TechnicalPaper 2007-01-4205 2007

[12] H Aitzetmuller ldquoSteyr S-Matic-the future CVT systemrdquo inProceedings of the Seoul 2000 FISITA world automotive con-gress pp 1ndash6 Seoul South Korea June 2000

[13] J Pohlenz and K Grad ldquoCVTfor input power above 300 kWrdquoVDI-berichte vol 2007 pp 59ndash65 2001

[14] J Pohlenz and K Grad ldquoCVT-System fur denGroszligflacheneinsatzrdquoVDI-berichte vol 2004 pp 23ndash33 1855

[15] L Y Xu Study on Characteristics of Hydro-MechanicalContinuously Variable Transmission of Tractor Xirsquoan Uni-versity of Technology Xirsquoan China 2007

[16] J Looman Zahnradgetriebe-Grundlagen KonstruktionenAnwendungen in Fahrzeugen Springer Berlin Germany2009

[17] Z Rao Planetary Gear Transmission Design Chemical In-dustry Press Beijing China 2003

[18] X Xu Y Liang M Jordan T Peter and D Peng ldquoOptimizedcontrol of engine start assisted by the disconnect clutch in a P2hybrid automatic transmissionrdquo Mechanical Systems andSignal Processing vol 124 2019

[19] S Wang Y Liu Z Wang et al ldquoAdaptive fuzzy iterativecontrol strategy for the wet-clutch filling of automatictransmissionrdquo Mechanical Systems and Signal Processingvol 130 pp 164ndash182 2019

14 Mathematical Problems in Engineering

no

minusk1i1i2i3 minus ε1 minus k1( 1113857i1i2i3i4i8i11i12

ne HM1( 1113857

minusk1i1i2i3 minus ε1 minus k1( 1113857i1i2i3i4i7i11i12

ne HM3( 1113857

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(7)

)e transmission ratio of HMCVT ig is

ig ne

no

1 minus k1( 1113857i1i2i3i4i8i11i12

minusk1i1i2i3 minus ε HM1( 1113857

1 minus k1( 1113857i1i2i3i4i7i11i12

minusk1i1i2i3 minus εHM3( 1113857

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(8)

312 Ranges HM2 and HM4 )e power transmissionroutes of ranges HM2 and HM4 are the same except for the

difference between gear pairs i6 and i9 Under these twoconditions the power is input by the large sun gear s1 andthe ring gear r and output by the small sun gear s2

)e same as the previous case the input speed of thelarge sun gear ns1

is ns1 minusεne(i1i2i3) the input speed of the

ring gear nr equals ne According to the PGM speed char-acteristic formula (16) the relationship between the inputspeed of large sun gear ns1

the input speed of ring gear nr theoutput speed of the small sun gear ns2

and the PGM standingtransmission ratios k2 and minusk3 is satisfied

1 minus k3

k2 minus k3nr minus

1 minus k2

k2 minus k3ns1

minus ns2 0 (9)

)erefore the output speed of the small sun gear ns2is

ns2

1 minus k3( 1113857i1i2i3 + 1 minus k2( 1113857εk2 minus k3( 1113857i1i2i3

ne (10)

Table 1 Engagement conditions of clutches of HMCVT

Range Directional clutch Shifting clutchCF CR C1 C2 C3 C4

Forward

First + +Second + +)ird + +Fourth + +

Reverse

First + +Second + +)ird + +Fourth + +

Engine

Tei1

i2

p1

r

p2

S1

S2c

i5

i4

i6

i7

i9

i8

i3

i12

i11

i10

ne

Tp

npTm

nm

Tz

nz

To

no

C1C3

C2C4

Pump Motor

CR CF

Figure 1 )e structure of HMCVT

Mathematical Problems in Engineering 3

where k2 Zs2Zr gt 0 minusk3 Zs2

Zs1gt 0 and Zr Zs1

Zs2are

the numbers of teeth of ring gear large sun gear and smallsun gear respectively

)e intermediate shaft speed nz is

nz

1 minus k3( 1113857i1i2i3 + 1 minus k2( 1113857εk2 minus k3( 1113857i1i2i3i5i9

ne HM2( 1113857

1 minus k3( 1113857i1i2i3 + 1 minus k2( 1113857εk2 minus k3( 1113857i1i2i3i5i6

ne HM4( 1113857

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(11)

Finally the output shaft speed of HMCVT no is

no

1 minus k3( 1113857i1i2i3 + 1 minus k2( 1113857εk2 minus k3( 1113857i1i2i3i5i9i11i12

ne HM2( 1113857

1 minus k3( 1113857i1i2i3 + 1 minus k2( 1113857εk2 minus k3( 1113857i1i2i3i5i6i11i12

ne HM4( 1113857

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(12)

)e transmission ratio of HMCVT ig is

ig ne

no

k2 minus k3( 1113857i1i2i3i5i9i11i121 minus k3( 1113857i1i2i3 + 1 minus k2( 1113857ε

HM2( 1113857

k2 minus k3( 1113857i1i2i3i5i6i11i12

1 minus k3( 1113857i1i2i3 + 1 minus k2( 1113857ε HM4( 1113857

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(13)

32 Determination of Parameters of HMCVT )e ratedspeed of the engine matching with the HMCVT is 2200 rmin the minimum speed of the tractor is 4 kmh themaximum speed is 50 kmh the main reducer ratio of therear axle is 37 the reduction ratio of the wheel side is 56 thetotal transmission ratio of the rear axle is 2072 and the drivewheel radius is 08579m

)e range of the transmission ratio in the drive train isdetermined by parameters such as the speed of the tractorthe rated speed of the engine and the drive wheel radius Sothe transmission ratio of each range of HMCVT is cal-culated as follows

igj 0377nerd

v (14)

where j 1 2 3 4 ne is the rated speed of the engine rd is thedrive wheel radius v is the theoretical driving speed of thetractor

)e HMCVT has 4 working ranges so the common ratioof the transmission ratios is

φ

igmax

igmin

4

1113971

(15)

)e HMCVT adopts a proportional transmission so ineach working range the transmission ratio has the followingrelationship

igjmax

igjmin φ (16)

When HMCVT is switching between adjacent rangesthe conditions must be met at the changing point thetransmission ratios are equal and the pump-motor dis-placement ratios are equal

ig1min(minus1) ig2min(minus1)

ig2min(1) ig3min(1)

ig3min(minus1) ig4min(minus1)

ig4min(1) imin

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(17)

)e gear ratio of the connecting gears between theengine and the hydraulic variable pump needs to be met

i1i2 genemax

npmax (18)

According to the previous content if i10 i11 and i12 areset to 1 then the relationship between the various param-eters of HMCVT can be obtained

k1i1i2i3 minus32727

1 minus k3 k1 k2 minus 1( 1113857

i5i6 08969k1

1 + k1( 1113857

i4i7 minus16865k1

1 minus k1( 1113857

i4i8 minus59615k1

1 minus k1( 1113857

i5i9 31702k1

1 + k1( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(19)

)ese parameters are all expressions about the PGMstanding transmission ratio minusk1 Once the value of minusk1 isdetermined the values of these parameters will also bedetermined which will be described later

33 Torque Characteristic

331 Ranges HM1 and HM3 Ignoring the power losses ifTo denotes the load torque of output shaft the intermediateshaft torque Tz is

Tz To

i11i12 (20)

and then the output torque of carrier in PGM Tc is

Tc

Tz

i4i8

To

i4i8i11i12 HM1( 1113857

Tz

i4i7

To

i4i7i11i12 HM3( 1113857

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(21)

4 Mathematical Problems in Engineering

According to the PGM torque characteristic formula(16) the relationship between the input torque of large sungear Ts1

the input torque of ring gear Tr the output torqueof carrierTc and the PGM standing transmission ratio minusk1 issatisfied

Ts1

Tr

minusk1 minus

Tc

1 minus k1 (22)

where minusk1 ZrZs1gt 1 Zr is the number of teeth of the ring

gear and Zs1is the number of teeth of the large sun gear

)erefore the large sun gear torque Ts1is

Ts1 minus

Tc

1 minus k1

minusTo

i4i8i11i12 1 minus k1( 1113857 HM1( 1113857

minusTo

i4i7i11i12 1 minus k1( 1113857 HM3( 1113857

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(23)

)en the input torque of ring gear Tr is

Tr minusk1Ts1

k1To

i4i8i11i12 1 minus k1( 1113857 HM1( 1113857

k1To

i4i7i11i12 1 minus k1( 1113857 HM3( 1113857

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(24)

)us the motor shaft load torque Tm is

Tm Ts1

minusi3

To

i3i4i8i11i12 1 minus k1( 1113857 HM1( 1113857

To

i3i4i7i11i12 1 minus k1( 1113857 HM3( 1113857

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(25)

)e pump shaft torque Tp is

Tp εTm

εTo

i3i4i8i11i12 1 minus k1( 1113857 HM1( 1113857

εTo

i3i4i7i11i12 1 minus k1( 1113857 HM3( 1113857

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(26)

)e engine output torque Te is

Te Tr +Tp

i1i2

k1i1i2i3 + ε( 1113857To

i1i2i3i4i8i11i12 1 minus k1( 1113857 HM1( 1113857

k1i1i2i3 + ε( 1113857To

i1i2i3i4i7i11i12 1 minus k1( 1113857 HM3( 1113857

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(27)

Finally the ratio of the output torque of HMCVT to theinput torque of the engine μ is

μ To

Te

1 minus k1( i1i2i3i4i8i11i12

ε + k1i1i2i3 HM1( 1113857

1 minus k1( 1113857i1i2i3i4i7i11i12ε + k1i1i2i3

HM3( 1113857

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(28)

332 Ranges HM2 and HM4 )e same as the previous caseTo denotes the load torque of output shaft and the inter-mediate shaft torque Tz is Tz To(i11i12) )en the outputtorque of small sun gear in PGM Ts2

is

Ts2

Tz

i5i9

To

i5i9i11i12 HM2( 1113857

Tz

i5i6

To

i5i6i11i12 HM4( 1113857

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(29)

)e external torque algebra sum of PGM is zero soTs1

+ Tr + Ts2 0 According to the PGM torque character-

istic formula (16) the relationship between the input torque oflarge sun gear Ts1

the output torque of small sun gear Ts2 and

the PGM standing transmission ratios k2 and minusk3 is satisfied

Ts1

k2 minus 1k3 minus k2

Ts2

k2 minus 1( 1113857To

k3 minus k2( 1113857i5i9i11i12 HM2( 1113857

k2 minus 1( 1113857To

k3 minus k2( 1113857i5i6i11i12 HM4( 1113857

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(30)

)e engine output torque Te is

Te

1 minus k2( 1113857ε + 1 minus k3( 1113857i1i2i3

k3 minus k2( 1113857i1i2i3i5i9i11i12To HM2( 1113857

1 minus k2( 1113857ε + 1 minus k3( 1113857i1i2i3

k3 minus k2( 1113857i1i2i3i5i6i11i12To HM4( 1113857

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(31)

Finally the ratio of the output torque of HMCVT to theinput torque of the engine μ is

μ To

Te

k3 minus k2( 1113857i1i2i3i5i9i11i12

1 minus k2( 1113857ε + 1 minus k3( 1113857i1i2i3 HM2( 1113857

k3 minus k2( 1113857i1i2i3i5i6i11i12

1 minus k2( 1113857ε + 1 minus k3( 1113857i1i2i3 HM4( 1113857

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(32)

34 Circulating Power Flow Calculation

341 Ranges HM1 and HM3 )e speed ratio of the powertransmitting from the engine to the carrier ice is

ice nc

ne

minusk1i1i2i3 minus ε1 minus k1( 1113857i1i2i3

irce + i

s1ce (33)

)e speed ratios of the power transmitting from theengine to the large sun gear and ring gear ireis1e are

ire 1

is1e minusε

i1i2i3

(34)

In the PGM the speed ratios between the input elements(large sun gear s1 and ring gear r) and the output element(carrier c) are

Mathematical Problems in Engineering 5

ircs1

nc

ns1

1

1 minus k1

is1cr

nc

nr

minusk1

1 minus k1

(35)

)en we can get

irce i

rcs1

is1e minusε

i1i2i3 1 minus k1( 1113857

is1ce i

s1crire minus

k1

1 minus k1

(36)

Next two situations will be discussed as follows [17]

(1) When εlt 0 is1ce gt 0 irce gt 0 and i

s1cei

rce gt 0 the PGM has

no circulating power PL 0(2) When εgt 0 i

s1ce gt 0 irce lt 0 and i

s1cei

rce lt 0 the PGM has

circulating power substituting equation (19) fork1i1i2i3 in equation (33) ice is

ice 32727 minus ε

32727 + i1i2i3gt 0 (37)

Because ofPe gt 0 Pr (is1ceice)Pe gt 0 Ps1

(irceice)Pe lt 0the output power P is

P Pe minusPc Pr + Ps1 Pr minus Ps1

11138681113868111386811138681113868

11138681113868111386811138681113868 (38)

)en

Pr P + Ps1

11138681113868111386811138681113868

11138681113868111386811138681113868 P + PL

11138681113868111386811138681113868111386811138681113868 (39)

)e circulating power PL is

PL Ps1

irce

ice

Pe (40)

Assume that the closed power factor KL is

KL PL

11138681113868111386811138681113868111386811138681113868

Pe

irce

ice

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

εk1i1i2i3 + ε

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 |ϕ(ε)| (41)

where ϕ(ε) εk1i1i2i3 + ε and the derivative of ϕ(ε) isdϕ(ε)dε k1i1i2i3(k1i1i2i3 + ε)2 lt 0 so function ϕ(ε) is amonotonically decreasing function )en KL(0) 0KL(1) 044 and the maximum value of the circulatingpower is 044Pe

342 Ranges HM2 and HM4 )e speed ratio of the powertransmitting from the engine to the small sun gear is2e is

is2e ns2

ne

1 minus k3( 1113857i1i2i3 + 1 minus k2( 1113857ε

k2 minus k3( 1113857i1i2i3 i

rs2e + i

s1s2e (42)

)e speed ratios of the power transmitting from theengine to the large sun gear and ring gear ireis1e are

ire 1

is1e minusε

i1i2i3

(43)

In the PGM the speed ratios between the input elements(large sun gear s1 and ring gear r) and the output element(small sun gear s2) are

irs2s1

k2 minus 1k2 minus k3

is1s2r

1 minus k3

k2 minus k3

(44)

)en we can get

irs2e i

rs2s1

is1e minusk2 minus 1( 1113857ε

i1i2i3 k2 minus k3( 1113857

is1s2e i

s1s2rire

1 minus k3

k2 minus k3

(45)

Next two situations will be discussed as follows [17]

(1) When εlt 0 is1s2e gt 0 irs2e lt 0 and i

s1s2ei

rs2e lt 0 the PGM

has circulating power substituting equation (19) fork2 k3 i1i2i3 in equation (42) is2e is

is2e 32727 + ε

32727 minus 32727 minusk1( 1113857gt 0 (46)

Because of Pe gt 0 Pr is1ceicePe gt 0 Ps1

irceicePe lt 0the output power P is

P Pe minusPs2 Pr + Ps1

Pr minus Ps1

11138681113868111386811138681113868

11138681113868111386811138681113868 (47)

)en

Pr P + Ps1

11138681113868111386811138681113868

11138681113868111386811138681113868 P + PL

11138681113868111386811138681113868111386811138681113868 (48)

)e circulating power PL is

PL Ps1 irs2e

is2e

Pe (49)

Assume that the closed power factor KL is

KL PL

11138681113868111386811138681113868111386811138681113868

Pe

irs2e

is2e

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

minusεk1i1i2i3 minus ε

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 |ϕ(ε)| (50)

where ϕ(ε) minusεk1i1i2i3 minus ε and the derivative ofϕ(ε) is dϕ(ε)dε minusk1i1i2i3(k1i1i2i3 minus ε)2 gt 0 sofunction ϕ(ε) is a monotonically increasing function)en KL(0) 0 KL(1) 044 and the maximumvalue of the circulating power is 044Pe

(2) When εgt 0 is1s2e gt 0 irs2e gt 0 and i

s1s2ei

rs2e gt 0 the PGM

has no circulating power PL 0

35 Hydraulic Split Power Calculation Hydraulic power-split ratio ρ PHPe Ps1

Pe hydraulic power is input tothe PGM through the large sun gear s1

6 Mathematical Problems in Engineering

351 Ranges HM1 and HM3

ρ Ps1

Pe

irce

ice

ε

k1i1i2i3 + ε ϕ(ε) (51)

and as can be obtained from the above function ϕ(ε) is amonotonically decreasing function so

ρ(minus1) 0234gt 0

ρ(0) 0

ρ(1) minus044lt 0

⎧⎪⎪⎨

⎪⎪⎩(52)

352 Ranges HM2 and HM4

ρ Ps1

Pe

irs2e

is2e

minusε

k1i1i2i3 minus ε ϕ(ε) (53)

As can be obtained from the above function ϕ(ε) is amonotonically increasing function so

ρ(minus1) minus044lt 0

ρ(0) 0

ρ(1) 0234gt 0

⎧⎪⎪⎨

⎪⎪⎩(54)

According to the theoretical analysis on the circulatingpower flow and the hydraulic split power the power flowdirection of HMCVT in each working range is shown inFigure 2 )is result lays the foundation for the analysis onthe efficiency characteristic of HMCVT

36 Efficiency Characteristic )e Lrfkoes formula (17) isused to calculate the transmission efficiency of HMCVT

η minus1113957μig

minusToTe

neno

(55)

where 1113957μ is the ratio of output torque to input torque ofHMCVTconsidering power losses and ig is the transmissionratio of HMCVT

)e transmission efficiency calculation process of eachworking range of HMCVT is shown in Tables 2 and 3Among them the calculation processes of the first range andthe third range are listed in Table 2 and the calculationprocesses of the second range and the fourth range are listedin Table 3 In Tables 2 and 3 i1 minus i12 are the gear ratios of thegear pairs η1 minus η12 are the transmission efficiency of the gearpairs ηH is the hydraulic system transmission efficiency andηs1r ηs1s2

and ηrs2are the transmission efficiencies of the

large sun gear to the ring gear the large sun gear to the smallsun gear and the ring gear to the small sun gear in the PGMrespectively

4 Analysis on the Influence of Parameters onthe Transmission Efficiency of HMCVT

)e results of the transmission efficiency in each workingrange of HMCVT are as follows

ηHM1

ηs1r minus k11113872 1113873η4η8η11η12 ε + k1i1i2i3( 1113857

1 minus k1( 1113857 η1η2η3ηHηs1rε + k1i1i2i31113872 1113873 (εgt 0)

ηs1r minus k11113872 1113873η1η2η3ηHη4η8η11η12 ε + k1i1i2i3( 1113857

1 minus k1( 1113857 i1i2i3η1η2η3ηHk1 + εηs1r1113872 1113873 (εle 0)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

ηHM2

k3ηs1s2minus k2ηrs2

1113872 1113873η5η9η11η12 1 minus k3( 1113857i1i2i3 + 1 minus k2( 1113857ε1113858 1113859

k3 minus k2( 1113857 1 minus k3ηs1s21113872 1113873i1i2i3 minus η1η2η3ηHε k2ηrs2

minus 11113872 11138731113960 1113961 (εlt 0)

k3ηs1s2minus k2ηrs2

1113872 1113873η1η2η3η5η9η11η12ηH 1 minus k3( 1113857i1i2i3 + 1 minus k2( 1113857ε1113858 1113859

k3 minus k2( 1113857 1 minus k3ηs1s21113872 1113873i1i2i3η1η2η3ηH minus ε k2ηrs2

minus 11113872 11138731113960 1113961 (εge 0)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

ηHM3

ηs1r minus k11113872 1113873η4η7η11η12 ε + k1i1i2i3( 1113857

1 minus k1( 1113857 η1η2η3ηHηs1rε + k1i1i2i31113872 1113873 (εgt 0)

ηs1r minus k11113872 η1η2η3ηHη4η7η11η12 ε + k1i1i2i3( 1113857

1 minus k1( 1113857 i1i2i3η1η2η3ηHk1 + εηs1r1113872 1113873 (εle 0)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

ηHM4

k3ηs1s2minus k2ηrs2

1113872 1113873η5η6η11η12 1 minus k3( 1113857i1i2i3 + 1 minus k2( 1113857ε1113858 1113859

k3 minus k2( 1113857 1 minus k3ηs1s21113872 1113873i1i2i3 minus η1η2η3ηHε k2ηrs2

minus 11113872 11138731113960 1113961 (εlt 0)

k3ηs1s2minus k2ηrs2

1113872 1113873η1η2η3η5η6η11η12ηH 1 minus k3( 1113857i1i2i3 + 1 minus k2( 1113857ε1113858 1113859

k3 minus k2( 1113857 1 minus k3ηs1s21113872 1113873i1i2i3η1η2η3ηH minus ε k2ηrs2

minus 11113872 11138731113960 1113961 (εge 0)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(56)

Mathematical Problems in Engineering 7

When HMCVT is working in HM1 or HM3 range if thetransmission efficiencies of the gear pairs i9 and i8 are equal(η7 η8) the transmission efficiencies of HMCVT in HM1and HM3 ranges are equal

When HMCVT is working in HM2 or HM4 range if thetransmission efficiencies of the gear pairs i9 and i6 are equal(η9 η6) the transmission efficiencies of HMCVT in HM2and HM4 ranges are also equal

Due to the complexity of the above results it is necessaryto evaluate the influence of displacement ratio ε standingtransmission ratio of PGM minusk1 gear ratio i1i2i3 and hy-draulic system efficiency ηH on the transmission efficiency of

HMCVT by means of image analysis method We assumethat η1 minus η12 are 098 ηs1r is 098 ηs1s2

and ηrs2are 097 and

k1i1i2i3 is minus32727 )e evaluation results are shown inFigure 3

According to the analysis on Figure 3 the following rulesare obtained

In each range of HMCVT when the displacement ratio εis 0 the transmission efficiency reaches the maximum valueNow the hydraulic path (HST) does not transmit power andthe power is completely transmitted by the mechanical pathWhen the displacement ratio ε is minus1 or 1 the transmissionefficiency reaches the minimum value and the hydraulic

Table 2 Equation of efficiency calculation for ranges HM1 and HM3

ElementsTorque equation considering power losses

εgt 0 εlt 0HM1 HM3 HM1 HM3

Engine Te Tr + η1η2Tpi1i2 Te Tr + Tpi1i2η1η2Pump Tp εηHTm Tp εTmηH

Motor Tm η3Ts1(minusi3) Tm Ts1

(minusη3i3)PGM Ts1

+ Tr + Tc 0PGM Tr minusk1Ts1

ηs1r

Intermediate shaft Tc Tzi4i8η4η8 Tc Tzi4i7η4η7 Tc Tzi4i8η4η8 Tc Tzi4i7η4η7Output shaft Tz To(i11i12η11η12)Torque ratio 1113957μ ToTe

Efficiency η minus1113957μig

144 max

100 44 max

44 max

Engine

r

S1

S2c

100

(a)

766 min

100

100

234 max

234 max

Engine

r

S1

S2c

(b)

766 min

100 234 max

234 max

Engine

r

S1

S2c

100

(c)

144 max

100

100

44 max

44 max

Engine

r

S1

S2c

(d)

Figure 2 Power flow direction of the PGM of HMCVT (a) Range HM1HM3 εgt 0 power-circulation (b) Range HM1HM3 εlt 0 power-split (c) Range HM2HM4 εgt 0 power-split (d) Range HM2HM4 εlt 0 power-circulation

8 Mathematical Problems in Engineering

Table 3 Equation of efficiency calculation for ranges HM2 and HM4

ElementsTorque equation considering power losses

εgt 0 εlt 0HM2 HM4 HM2 HM4

Engine Te Tr + Tpi1i2η1η2 Te Tr + η1η2Tpi1i2Pump Tp εTmηH Tp εηHTm

Motor Tm Ts1(minusη3i3) Tm η3Ts1

(minusi3)

PGM Ts1+ Tr + Ts2

0PGM Tr 1 minus k3ηs1s2

k3ηs1s2minus k2ηrs2

Ts2Intermediate shaft Ts2

Tzi5i9η5η9 Ts2 Tzi5i6η5η6 Ts2

Tzi5i9η5η9 Ts2 Tzi5i6η5η6

Output shaft Tz To(i11i12η11η12)Torque ratio 1113957μ ToTe

Efficiency η minus1113957μig

ndashk1 = 25172ndashk1 = 29130

ndashk1 = 17805ndashk1 = 21875

095

09

085

08

075

07ndash1 ndash08 ndash06 ndash04 ndash02 0 02 04 06 08 1

Displacement ratio

ηH = 081

Effici

ency

(a)

ndashk1 = 25172ndashk1 = 29130

ndashk1 = 17805ndashk1 = 21875

ηH = 081

095

09

085

08

075

07ndash1 ndash08 ndash06 ndash04 ndash02 0 02 04 06 08 1

Displacement ratio

Effici

ency

(b)

ηH = 081ηH = 090

ηH = 064ηH = 072

095

09

085

08

075

07ndash1 ndash08 ndash06 ndash04 ndash02 0 02 04 06 08 1

Displacement ratio

Effici

ency

ndashk1 = 21875

(c)

ηH = 081ηH = 090

ηH = 064ηH = 072

095

09

085

08

075

07ndash1 ndash08 ndash06 ndash04 ndash02 0 02 04 06 08 1

Displacement ratio

Effici

ency

ndashk1 = 21875

(d)

Figure 3 Continued

Mathematical Problems in Engineering 9

path transmits more power at this time resulting in a loweroverall efficiency of HMCVT

In each range of HMCVT the transmission efficiencyvalue in the power circulation phase is lower than thetransmission efficiency value in the power split phase whichindicates that the power circulation will cause the trans-mission efficiency of HMCVT to decrease

)e larger the value of the PGM standing transmissionratio minusk1 the higher the transmission efficiency of HMCVTbut the influence on improving the transmission efficiency is

not obvious )e PGM standing transmission ratio minusk1ranges from 15 to 3 )e gear ratio i1i2i3 and the PGMstanding transmission ratio minusk1 are in an inversely pro-portional relationship the smaller the value of the gear ratioi1i2i3 the higher the transmission efficiency of HMCVT andthe influence on improving the transmission efficiency isalso not obvious )e gear ratio i1i2i3 ranges from 11 to 22

)e greater the value of HSTefficiency ηH the higher theoverall transmission efficiency of HMCVT )e influence ofηH on improving transmission efficiency of HMCVT issignificant

i1i2i3 = 13001i1i2i3 = 11234

i1i2i3 = 18381i1i2i3 = 14961

ηH = 081

095

09

085

08

075

07ndash1 ndash08 ndash06 ndash04 ndash02 0 02 04 06 08 1

Displacement ratio

Effici

ency

(e)

i1i2i3 = 13001i1i2i3 = 11234

i1i2i3 = 18381i1i2i3 = 14961

ηH = 081

095

09

085

08

075

07ndash1 ndash08 ndash06 ndash04 ndash02 0 02 04 06 08 1

Displacement ratio

Effici

ency

(f )

Figure 3)e influence of parameters on the transmission efficiency of HMCVT (a))e influence of minusk1 on HM1 andHM3 ranges (b))einfluence of minusk1 on HM2 and HM4 ranges (c) )e influence of ηH on HM1 and HM3 ranges (d) )e influence of ηH on HM2 and HM4ranges (e) )e influence of i1i2i3 on HM1 and HM3 ranges (f ) )e influence of i1i2i3 on HM2 and HM4 ranges

9

8

7

6

5

4

3

2

1

0

Tran

smiss

ion

ratio

ndash15 ndash1 ndash05 0 05 1 15Displacement ratio

HM1HM2

HM3HM4

Figure 4 Relationship between transmission ratio and displace-ment ratio

60

50

40

30

20

10

0

Trac

tor s

peed

(km

h)

ndash15 ndash1 ndash05 0 05 1 15Displacement ratio

HM1HM2

HM3HM4

Figure 5 )e curve of the tractor speed

10 Mathematical Problems in Engineering

5 Analysis on the Transmission CharacteristicsResults of HMCVT

According to the previous content the value of the PGMstanding transmission ratio minusk1 is set to 21875 and the HSTefficiency value ηH is set to 081 then the other parameters ofHMCVT are k2 02714 minusk3 05938 i1 i2 1 i3 15i4 i5 1 i6 166 i7 116 i8 410 i9 584 andi10 i11 i12 1

)e relationship between the transmission ratio anddisplacement ratio is shown in Figure 4 )e curve of thetractor speed is shown in Figure 5 In the first range thedisplacement ratio changes from 1 to minus1 the transmissionratio decreases from 862 to 459 and the tractor speedincreases from 0 to 750 kmh In the second range thedisplacement ratio changes from minus1 to 1 the transmissionratio decreases from 459 to 243 and the tractor speedincreases from 750 kmh to 1412 kmh In the third rangethe displacement ratio changes from 1 to minus1 the trans-mission ratio decreases from 243 to 130 and the tractorspeed increases from 1412 kmh to 2644 kmh In thefourth range the displacement ratio changes from minus1 to 1the transmission ratio decreases from 130 to 069 and thetractor speed increases from 2644 kmh to 50 kmh )eresults show that the stepless speed regulation of HMCVTin the tractor speed range of 0ndash50 kmh can be realized bythe coordinated control of the HST displacement ratio andthe clutches

)e ratios of the output shaft torque to the input shafttorque in each range of HMCVT are shown in Figure 6(a))e continuous change of torque can be realized in the entireworking ranges )e ratios of the motor shaft torque to theoutput shaft torque in each range of HMCVT are shown inFigure 6(b) )e torque ratio remains constant in each range

and varies between different ranges )e values of torqueratios in the four ranges are 514 minus972 1849 andminus3518 respectively

)e relationship between the tractor speed and theoverall transmission efficiency of HMCVT is shown inFigure 7 Each of the four ranges has a transmission effi-ciency maximum value of 0917 0902 0917 and 0902respectively )e efficiency value always increases first withthe increase of the tractor speed and then decreases afterreaching the maximum value )e maximum transmissionefficiency in each range is obtained when the HST dis-placement ratio is 0 Now the hydraulic path does nottransmit power and all power is transmitted by the me-chanical path )e transmission efficiency of HMCVT ishigher than 83 so the overall transmission efficiency is at ahigh level

ndash2

ndash4

ndash6

ndash8

ndash10

0

0 5 10 15 20 25 30 35 40 45 50

Torq

ue ra

tio (T

oTe

)

Tractor speed (kmh)

HM1HM2

HM3HM4

(a)

05

04

03

02

01

0

ndash01

ndash02

ndash03

ndash04

ndash05

Torq

ue ra

tio (T

mT

o)

0 5 10 15 20 25 30 35 40 45 50Tractor speed (kmh)

HM1HM2

HM3HM4

(b)

Figure 6)e curve of torque characteristic (a) Ratios of output shaft torque to input shaft torque (b) Ratios of motor shaft torque to outputshaft torque

095

09

085

08

075

07

Effici

ency

0 5 10 15 20 25 30 35 40 45 50Tractor speed (kmh)

HM1HM2

HM3HM4

Figure 7 Overall transmission efficiency of HMCVT

Mathematical Problems in Engineering 11

6 Verification of HMCVT Simulation Model

)e modeling and simulation of HMCVT are accom-plished by using Amesim simulation softwareAccording to Figure 1 a simulation model of HMCVTis established as shown in Figure 8

61 Hydraulic System )e hydraulic system is mainlycomposed of a variable displacement hydraulic pump afixed displacement hydraulic motor two relief valves twocheck valves and a charge hydraulic pump with a reliefvalve )e aim of the charge pump is to compensate for theleakage loss in the pipeline and to establish the lowestpressure in the pumprsquos suction pipeline Two relief valves areused to limit the maximum pressure in the system pipingand its adjustment pressure should be above the normalworking pressure of the system Volumetric losses of the

pump and motor are neglected and their hydromechanicalefficiencies are set at 09

62 Mechanical Components Mechanical components areassumed to be constant efficiency components throughoutthe transmission )e power loss of each gear pair is 2 sothe efficiency is set to 098 In the Ravigneaux planetary gearmechanism the efficiencies of ring gear-long planetary gearcouple long planetary gear-large sun gear couple longplanetary gear-short planetary gear couple and shortplanetary gear-small sun gear couple are set at 099 )emain reducer differential and wheel reducer are combinedand modeled as a gear reducer with an efficiency of 0965 Tosimplify the model the friction losses of the shaft gears andbearings were ignored during the simulation modelingprocess

Efficiency calculation unit

Speed calculation unit

Small sun gear

Short planet gear

Long planet gear

Large sun gear

Ring gear

i1

i2

Pump Motor

i3

i5

i4

i7 i6

i9i8

Carrier

C1

C2

C3

C4

ω ωT

TL

ωωT

T

L

PP

MOT

f (x y)

y

x

X

k

Figure 8 HMCVT simulation model

(km

h)

50

40

30

20

10

00 5 10 15 20 25 30 35

X Time (s)

Tractor linear velocity (kmh)

(a)

0 5 10 15 20 25 30 35X Time (s)

10

08

06

04

02

00

(nul

l)

Transmission efficiency of HMCVT (null)

(b)

Figure 9 Simulation results (a) Simulation result of the tractor speed (b) Simulation result of the transmission efficiency of HMCVT

12 Mathematical Problems in Engineering

63Model Control and Simulation Results )e engine speedis assumed to have a constant value of 2200 rmin the loadtorque of the output shaft of HMCVT is set to 100Nm themodel simulation time is 32 s including range HM1 0ndash8 srange HM2 8ndash16 s range HM3 16ndash24 s and range HM424ndash32 s

)e simulation results are shown in Figure 9 through thevariable pump displacement and coordinated control of theclutches [18 19] the transmission gradually changes fromrange HM1 to range HM4 and the tractor achieves steplessspeed change from 0 to 50 kmh )e simulation results areconsistent with the previous theoretical analysis results whichshow that the 4-range HMCVT transmission scheme pro-posed in this paper is suitable for matching wheeled tractors

64 Comparison Xu LYrsquos transmission scheme [15] hasbeen applied to mass production tractors )is transmissionscheme has 6 working ranges in the forward direction and 3working ranges in the reverse direction which is controlledby 8 clutches )e speed range in the forward direction is0sim30 kmh while the reverse direction is only 0sim10 kmh)e transmission efficiency of the hydraulic range is low (lessthan 80) the transmission efficiency of the hydrome-chanical ranges is high (higher than 90) but is only in theforward direction

)e scheme proposed in this paper has 4 working rangesand a speed range of 0sim50 kmh in the forward and reversedirections which is controlled by 6 clutches )e fewer thetransmission ranges the less switching during the workingranges so the transmission performance is improved )esmaller number of clutches ensures that the system is easierto control )e transmission efficiencies in the forward andreverse directions are close the overall forward transmissionefficiency is higher than 83 and the highest transmissionefficiency is higher than 91

)erefore the application of HMCVT proposed in thispaper is wider it can not only match conventional tractorsbut also match two-way operation tractors and other en-gineering vehicles

7 Conclusions

)is paper proposes a new type of transmission scheme ofHMCVT )e Ravigneaux PGM and the HSTare connectedin parallel to form a closed loop and then connected in serieswith the range switching mechanism )is new HMCVTfeatures a double-row PGM with 4 hydromechanicalworking ranges By coordinating control of the HST dis-placement ratio and the engagement conditions of shiftingclutches the stepless speed regulation of HMCVT at thetractor speed of 0ndash50 kmh is realized

According to the theoretical analysis on the newHMCVT the speed characteristic curve and the torquecharacteristic curve are obtained Based on the power flowdirection of HMCVT the transmission efficiency curve isobtained by using the Lrfkoes formula

Using image analysis method to study the influence offour parameters (displacement ratio ε PGM standing

transmission ratio minusk1 gear ratio i1i2i3 and hydraulic systemefficiency ηH) on the transmission efficiency of HMCVT theresults show that the standing transmission ratio minusk1 andgear ratio i1i2i3 have little effect on the transmission effi-ciency whereas the displacement ratio ε and hydraulicsystem efficiency ηH have a greater impact on the trans-mission efficiency

)e parameters of the new HMCVT are finally deter-mined )e results of the transmission characteristics showthat the new HMCVTproposed in this paper has the steplessspeed regulation capability for tractors which can alsocontinuously transmit and change the torque )e trans-mission efficiency of HMCVT is at a high level

In order to verify the theoretical derivation Amesimsimulation software is used for the modeling and simulationof HMCVT )e simulation results are consistent with thetheoretical analysis results )erefore we believe that thetransmission scheme is reasonable and effective for tractors

Data Availability

)e Simcenter Amesim System File data used to support thefindings of this study are available from the correspondingauthor upon request

Conflicts of Interest

)e authors declare that they have no conflicts of interest

Acknowledgments

)is research was financially supported by the National KeyResearch and Development Plan Project of China(2016YFD0700402) and Changzhou Sci amp Tech Program(no CE20185037)

References

[1] L Y Xu Z L Zhou M Z Zhang et al ldquoResearch and designof hydro-mechanical continuously variable transmission fortractorsrdquo Journal of Northeast Agricultural University vol 13no 2 pp 182ndash186 2006

[2] R Paoluzzi and L G Zarotti ldquo)e minimum size of hy-drostatic transmissions for locomotionrdquo Journal of Terra-mechanics vol 50 no 3 pp 153ndash164 2013

[3] J Kim J Kang Y Kim B Min and H Kim ldquoDesign of powersplit transmission design of dual mode power split trans-missionrdquo International Journal of Automotive Technologyvol 11 no 4 pp 565ndash571 2010

[4] A Kim and A Rossetti ldquoFuel consumption reduction inurban buses by using power split transmissionsrdquo EnergyConversion and Management vol 71 pp 159ndash171 2013

[5] A Rossetti and A Macor ldquoMulti-objective optimization ofhydro-mechanical power split transmissionsrdquo Mechanismand Machine lteory vol 62 pp 112ndash128 2013

[6] A Macor and A Rossetti ldquoOptimization of hydro-mechanicalpower split transmissionsrdquo Mechanism and Machine lteoryvol 46 no 12 pp 1901ndash1919 2011

[7] S Shamshirband D Petkovic A Amini et al ldquoSupport vectorregression methodology for wind turbine reaction torqueprediction with power-split hydrostatic continuous variabletransmissionrdquo Energy vol 67 pp 623ndash630 2014

Mathematical Problems in Engineering 13

[8] K T Anuar ldquoResch R Continuously variable tractor trans-missionsrdquo ASAE Distinguished Lecture No 29 pp 1ndash37 StJoseph American Society of Agricultural Engineers St JosephMI USA 2005

[9] S H Yuan and J B Hu ldquo)e efficiency of multi-range hydro-mechanical stepless transmissionrdquo Journal of Beijing Instituteof Technology vol 7 no 2 pp 129ndash133 1998

[10] X J LiuAnalysis of Vehicle Transmission System pp 258-259National Defence Industry Press Beijing China 1998

[11] M M Brenninger ldquoFendt vario CVT in agricultural tractorsrdquoSAE International Warrendale PA USA SAE TechnicalPaper 2007-01-4205 2007

[12] H Aitzetmuller ldquoSteyr S-Matic-the future CVT systemrdquo inProceedings of the Seoul 2000 FISITA world automotive con-gress pp 1ndash6 Seoul South Korea June 2000

[13] J Pohlenz and K Grad ldquoCVTfor input power above 300 kWrdquoVDI-berichte vol 2007 pp 59ndash65 2001

[14] J Pohlenz and K Grad ldquoCVT-System fur denGroszligflacheneinsatzrdquoVDI-berichte vol 2004 pp 23ndash33 1855

[15] L Y Xu Study on Characteristics of Hydro-MechanicalContinuously Variable Transmission of Tractor Xirsquoan Uni-versity of Technology Xirsquoan China 2007

[16] J Looman Zahnradgetriebe-Grundlagen KonstruktionenAnwendungen in Fahrzeugen Springer Berlin Germany2009

[17] Z Rao Planetary Gear Transmission Design Chemical In-dustry Press Beijing China 2003

[18] X Xu Y Liang M Jordan T Peter and D Peng ldquoOptimizedcontrol of engine start assisted by the disconnect clutch in a P2hybrid automatic transmissionrdquo Mechanical Systems andSignal Processing vol 124 2019

[19] S Wang Y Liu Z Wang et al ldquoAdaptive fuzzy iterativecontrol strategy for the wet-clutch filling of automatictransmissionrdquo Mechanical Systems and Signal Processingvol 130 pp 164ndash182 2019

14 Mathematical Problems in Engineering

where k2 Zs2Zr gt 0 minusk3 Zs2

Zs1gt 0 and Zr Zs1

Zs2are

the numbers of teeth of ring gear large sun gear and smallsun gear respectively

)e intermediate shaft speed nz is

nz

1 minus k3( 1113857i1i2i3 + 1 minus k2( 1113857εk2 minus k3( 1113857i1i2i3i5i9

ne HM2( 1113857

1 minus k3( 1113857i1i2i3 + 1 minus k2( 1113857εk2 minus k3( 1113857i1i2i3i5i6

ne HM4( 1113857

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(11)

Finally the output shaft speed of HMCVT no is

no

1 minus k3( 1113857i1i2i3 + 1 minus k2( 1113857εk2 minus k3( 1113857i1i2i3i5i9i11i12

ne HM2( 1113857

1 minus k3( 1113857i1i2i3 + 1 minus k2( 1113857εk2 minus k3( 1113857i1i2i3i5i6i11i12

ne HM4( 1113857

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(12)

)e transmission ratio of HMCVT ig is

ig ne

no

k2 minus k3( 1113857i1i2i3i5i9i11i121 minus k3( 1113857i1i2i3 + 1 minus k2( 1113857ε

HM2( 1113857

k2 minus k3( 1113857i1i2i3i5i6i11i12

1 minus k3( 1113857i1i2i3 + 1 minus k2( 1113857ε HM4( 1113857

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(13)

32 Determination of Parameters of HMCVT )e ratedspeed of the engine matching with the HMCVT is 2200 rmin the minimum speed of the tractor is 4 kmh themaximum speed is 50 kmh the main reducer ratio of therear axle is 37 the reduction ratio of the wheel side is 56 thetotal transmission ratio of the rear axle is 2072 and the drivewheel radius is 08579m

)e range of the transmission ratio in the drive train isdetermined by parameters such as the speed of the tractorthe rated speed of the engine and the drive wheel radius Sothe transmission ratio of each range of HMCVT is cal-culated as follows

igj 0377nerd

v (14)

where j 1 2 3 4 ne is the rated speed of the engine rd is thedrive wheel radius v is the theoretical driving speed of thetractor

)e HMCVT has 4 working ranges so the common ratioof the transmission ratios is

φ

igmax

igmin

4

1113971

(15)

)e HMCVT adopts a proportional transmission so ineach working range the transmission ratio has the followingrelationship

igjmax

igjmin φ (16)

When HMCVT is switching between adjacent rangesthe conditions must be met at the changing point thetransmission ratios are equal and the pump-motor dis-placement ratios are equal

ig1min(minus1) ig2min(minus1)

ig2min(1) ig3min(1)

ig3min(minus1) ig4min(minus1)

ig4min(1) imin

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(17)

)e gear ratio of the connecting gears between theengine and the hydraulic variable pump needs to be met

i1i2 genemax

npmax (18)

According to the previous content if i10 i11 and i12 areset to 1 then the relationship between the various param-eters of HMCVT can be obtained

k1i1i2i3 minus32727

1 minus k3 k1 k2 minus 1( 1113857

i5i6 08969k1

1 + k1( 1113857

i4i7 minus16865k1

1 minus k1( 1113857

i4i8 minus59615k1

1 minus k1( 1113857

i5i9 31702k1

1 + k1( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(19)

)ese parameters are all expressions about the PGMstanding transmission ratio minusk1 Once the value of minusk1 isdetermined the values of these parameters will also bedetermined which will be described later

33 Torque Characteristic

331 Ranges HM1 and HM3 Ignoring the power losses ifTo denotes the load torque of output shaft the intermediateshaft torque Tz is

Tz To

i11i12 (20)

and then the output torque of carrier in PGM Tc is

Tc

Tz

i4i8

To

i4i8i11i12 HM1( 1113857

Tz

i4i7

To

i4i7i11i12 HM3( 1113857

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(21)

4 Mathematical Problems in Engineering

According to the PGM torque characteristic formula(16) the relationship between the input torque of large sungear Ts1

the input torque of ring gear Tr the output torqueof carrierTc and the PGM standing transmission ratio minusk1 issatisfied

Ts1

Tr

minusk1 minus

Tc

1 minus k1 (22)

where minusk1 ZrZs1gt 1 Zr is the number of teeth of the ring

gear and Zs1is the number of teeth of the large sun gear

)erefore the large sun gear torque Ts1is

Ts1 minus

Tc

1 minus k1

minusTo

i4i8i11i12 1 minus k1( 1113857 HM1( 1113857

minusTo

i4i7i11i12 1 minus k1( 1113857 HM3( 1113857

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(23)

)en the input torque of ring gear Tr is

Tr minusk1Ts1

k1To

i4i8i11i12 1 minus k1( 1113857 HM1( 1113857

k1To

i4i7i11i12 1 minus k1( 1113857 HM3( 1113857

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(24)

)us the motor shaft load torque Tm is

Tm Ts1

minusi3

To

i3i4i8i11i12 1 minus k1( 1113857 HM1( 1113857

To

i3i4i7i11i12 1 minus k1( 1113857 HM3( 1113857

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(25)

)e pump shaft torque Tp is

Tp εTm

εTo

i3i4i8i11i12 1 minus k1( 1113857 HM1( 1113857

εTo

i3i4i7i11i12 1 minus k1( 1113857 HM3( 1113857

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(26)

)e engine output torque Te is

Te Tr +Tp

i1i2

k1i1i2i3 + ε( 1113857To

i1i2i3i4i8i11i12 1 minus k1( 1113857 HM1( 1113857

k1i1i2i3 + ε( 1113857To

i1i2i3i4i7i11i12 1 minus k1( 1113857 HM3( 1113857

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(27)

Finally the ratio of the output torque of HMCVT to theinput torque of the engine μ is

μ To

Te

1 minus k1( i1i2i3i4i8i11i12

ε + k1i1i2i3 HM1( 1113857

1 minus k1( 1113857i1i2i3i4i7i11i12ε + k1i1i2i3

HM3( 1113857

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(28)

332 Ranges HM2 and HM4 )e same as the previous caseTo denotes the load torque of output shaft and the inter-mediate shaft torque Tz is Tz To(i11i12) )en the outputtorque of small sun gear in PGM Ts2

is

Ts2

Tz

i5i9

To

i5i9i11i12 HM2( 1113857

Tz

i5i6

To

i5i6i11i12 HM4( 1113857

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(29)

)e external torque algebra sum of PGM is zero soTs1

+ Tr + Ts2 0 According to the PGM torque character-

istic formula (16) the relationship between the input torque oflarge sun gear Ts1

the output torque of small sun gear Ts2 and

the PGM standing transmission ratios k2 and minusk3 is satisfied

Ts1

k2 minus 1k3 minus k2

Ts2

k2 minus 1( 1113857To

k3 minus k2( 1113857i5i9i11i12 HM2( 1113857

k2 minus 1( 1113857To

k3 minus k2( 1113857i5i6i11i12 HM4( 1113857

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(30)

)e engine output torque Te is

Te

1 minus k2( 1113857ε + 1 minus k3( 1113857i1i2i3

k3 minus k2( 1113857i1i2i3i5i9i11i12To HM2( 1113857

1 minus k2( 1113857ε + 1 minus k3( 1113857i1i2i3

k3 minus k2( 1113857i1i2i3i5i6i11i12To HM4( 1113857

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(31)

Finally the ratio of the output torque of HMCVT to theinput torque of the engine μ is

μ To

Te

k3 minus k2( 1113857i1i2i3i5i9i11i12

1 minus k2( 1113857ε + 1 minus k3( 1113857i1i2i3 HM2( 1113857

k3 minus k2( 1113857i1i2i3i5i6i11i12

1 minus k2( 1113857ε + 1 minus k3( 1113857i1i2i3 HM4( 1113857

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(32)

34 Circulating Power Flow Calculation

341 Ranges HM1 and HM3 )e speed ratio of the powertransmitting from the engine to the carrier ice is

ice nc

ne

minusk1i1i2i3 minus ε1 minus k1( 1113857i1i2i3

irce + i

s1ce (33)

)e speed ratios of the power transmitting from theengine to the large sun gear and ring gear ireis1e are

ire 1

is1e minusε

i1i2i3

(34)

In the PGM the speed ratios between the input elements(large sun gear s1 and ring gear r) and the output element(carrier c) are

Mathematical Problems in Engineering 5

ircs1

nc

ns1

1

1 minus k1

is1cr

nc

nr

minusk1

1 minus k1

(35)

)en we can get

irce i

rcs1

is1e minusε

i1i2i3 1 minus k1( 1113857

is1ce i

s1crire minus

k1

1 minus k1

(36)

Next two situations will be discussed as follows [17]

(1) When εlt 0 is1ce gt 0 irce gt 0 and i

s1cei

rce gt 0 the PGM has

no circulating power PL 0(2) When εgt 0 i

s1ce gt 0 irce lt 0 and i

s1cei

rce lt 0 the PGM has

circulating power substituting equation (19) fork1i1i2i3 in equation (33) ice is

ice 32727 minus ε

32727 + i1i2i3gt 0 (37)

Because ofPe gt 0 Pr (is1ceice)Pe gt 0 Ps1

(irceice)Pe lt 0the output power P is

P Pe minusPc Pr + Ps1 Pr minus Ps1

11138681113868111386811138681113868

11138681113868111386811138681113868 (38)

)en

Pr P + Ps1

11138681113868111386811138681113868

11138681113868111386811138681113868 P + PL

11138681113868111386811138681113868111386811138681113868 (39)

)e circulating power PL is

PL Ps1

irce

ice

Pe (40)

Assume that the closed power factor KL is

KL PL

11138681113868111386811138681113868111386811138681113868

Pe

irce

ice

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

εk1i1i2i3 + ε

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 |ϕ(ε)| (41)

where ϕ(ε) εk1i1i2i3 + ε and the derivative of ϕ(ε) isdϕ(ε)dε k1i1i2i3(k1i1i2i3 + ε)2 lt 0 so function ϕ(ε) is amonotonically decreasing function )en KL(0) 0KL(1) 044 and the maximum value of the circulatingpower is 044Pe

342 Ranges HM2 and HM4 )e speed ratio of the powertransmitting from the engine to the small sun gear is2e is

is2e ns2

ne

1 minus k3( 1113857i1i2i3 + 1 minus k2( 1113857ε

k2 minus k3( 1113857i1i2i3 i

rs2e + i

s1s2e (42)

)e speed ratios of the power transmitting from theengine to the large sun gear and ring gear ireis1e are

ire 1

is1e minusε

i1i2i3

(43)

In the PGM the speed ratios between the input elements(large sun gear s1 and ring gear r) and the output element(small sun gear s2) are

irs2s1

k2 minus 1k2 minus k3

is1s2r

1 minus k3

k2 minus k3

(44)

)en we can get

irs2e i

rs2s1

is1e minusk2 minus 1( 1113857ε

i1i2i3 k2 minus k3( 1113857

is1s2e i

s1s2rire

1 minus k3

k2 minus k3

(45)

Next two situations will be discussed as follows [17]

(1) When εlt 0 is1s2e gt 0 irs2e lt 0 and i

s1s2ei

rs2e lt 0 the PGM

has circulating power substituting equation (19) fork2 k3 i1i2i3 in equation (42) is2e is

is2e 32727 + ε

32727 minus 32727 minusk1( 1113857gt 0 (46)

Because of Pe gt 0 Pr is1ceicePe gt 0 Ps1

irceicePe lt 0the output power P is

P Pe minusPs2 Pr + Ps1

Pr minus Ps1

11138681113868111386811138681113868

11138681113868111386811138681113868 (47)

)en

Pr P + Ps1

11138681113868111386811138681113868

11138681113868111386811138681113868 P + PL

11138681113868111386811138681113868111386811138681113868 (48)

)e circulating power PL is

PL Ps1 irs2e

is2e

Pe (49)

Assume that the closed power factor KL is

KL PL

11138681113868111386811138681113868111386811138681113868

Pe

irs2e

is2e

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

minusεk1i1i2i3 minus ε

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 |ϕ(ε)| (50)

where ϕ(ε) minusεk1i1i2i3 minus ε and the derivative ofϕ(ε) is dϕ(ε)dε minusk1i1i2i3(k1i1i2i3 minus ε)2 gt 0 sofunction ϕ(ε) is a monotonically increasing function)en KL(0) 0 KL(1) 044 and the maximumvalue of the circulating power is 044Pe

(2) When εgt 0 is1s2e gt 0 irs2e gt 0 and i

s1s2ei

rs2e gt 0 the PGM

has no circulating power PL 0

35 Hydraulic Split Power Calculation Hydraulic power-split ratio ρ PHPe Ps1

Pe hydraulic power is input tothe PGM through the large sun gear s1

6 Mathematical Problems in Engineering

351 Ranges HM1 and HM3

ρ Ps1

Pe

irce

ice

ε

k1i1i2i3 + ε ϕ(ε) (51)

and as can be obtained from the above function ϕ(ε) is amonotonically decreasing function so

ρ(minus1) 0234gt 0

ρ(0) 0

ρ(1) minus044lt 0

⎧⎪⎪⎨

⎪⎪⎩(52)

352 Ranges HM2 and HM4

ρ Ps1

Pe

irs2e

is2e

minusε

k1i1i2i3 minus ε ϕ(ε) (53)

As can be obtained from the above function ϕ(ε) is amonotonically increasing function so

ρ(minus1) minus044lt 0

ρ(0) 0

ρ(1) 0234gt 0

⎧⎪⎪⎨

⎪⎪⎩(54)

According to the theoretical analysis on the circulatingpower flow and the hydraulic split power the power flowdirection of HMCVT in each working range is shown inFigure 2 )is result lays the foundation for the analysis onthe efficiency characteristic of HMCVT

36 Efficiency Characteristic )e Lrfkoes formula (17) isused to calculate the transmission efficiency of HMCVT

η minus1113957μig

minusToTe

neno

(55)

where 1113957μ is the ratio of output torque to input torque ofHMCVTconsidering power losses and ig is the transmissionratio of HMCVT

)e transmission efficiency calculation process of eachworking range of HMCVT is shown in Tables 2 and 3Among them the calculation processes of the first range andthe third range are listed in Table 2 and the calculationprocesses of the second range and the fourth range are listedin Table 3 In Tables 2 and 3 i1 minus i12 are the gear ratios of thegear pairs η1 minus η12 are the transmission efficiency of the gearpairs ηH is the hydraulic system transmission efficiency andηs1r ηs1s2

and ηrs2are the transmission efficiencies of the

large sun gear to the ring gear the large sun gear to the smallsun gear and the ring gear to the small sun gear in the PGMrespectively

4 Analysis on the Influence of Parameters onthe Transmission Efficiency of HMCVT

)e results of the transmission efficiency in each workingrange of HMCVT are as follows

ηHM1

ηs1r minus k11113872 1113873η4η8η11η12 ε + k1i1i2i3( 1113857

1 minus k1( 1113857 η1η2η3ηHηs1rε + k1i1i2i31113872 1113873 (εgt 0)

ηs1r minus k11113872 1113873η1η2η3ηHη4η8η11η12 ε + k1i1i2i3( 1113857

1 minus k1( 1113857 i1i2i3η1η2η3ηHk1 + εηs1r1113872 1113873 (εle 0)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

ηHM2

k3ηs1s2minus k2ηrs2

1113872 1113873η5η9η11η12 1 minus k3( 1113857i1i2i3 + 1 minus k2( 1113857ε1113858 1113859

k3 minus k2( 1113857 1 minus k3ηs1s21113872 1113873i1i2i3 minus η1η2η3ηHε k2ηrs2

minus 11113872 11138731113960 1113961 (εlt 0)

k3ηs1s2minus k2ηrs2

1113872 1113873η1η2η3η5η9η11η12ηH 1 minus k3( 1113857i1i2i3 + 1 minus k2( 1113857ε1113858 1113859

k3 minus k2( 1113857 1 minus k3ηs1s21113872 1113873i1i2i3η1η2η3ηH minus ε k2ηrs2

minus 11113872 11138731113960 1113961 (εge 0)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

ηHM3

ηs1r minus k11113872 1113873η4η7η11η12 ε + k1i1i2i3( 1113857

1 minus k1( 1113857 η1η2η3ηHηs1rε + k1i1i2i31113872 1113873 (εgt 0)

ηs1r minus k11113872 η1η2η3ηHη4η7η11η12 ε + k1i1i2i3( 1113857

1 minus k1( 1113857 i1i2i3η1η2η3ηHk1 + εηs1r1113872 1113873 (εle 0)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

ηHM4

k3ηs1s2minus k2ηrs2

1113872 1113873η5η6η11η12 1 minus k3( 1113857i1i2i3 + 1 minus k2( 1113857ε1113858 1113859

k3 minus k2( 1113857 1 minus k3ηs1s21113872 1113873i1i2i3 minus η1η2η3ηHε k2ηrs2

minus 11113872 11138731113960 1113961 (εlt 0)

k3ηs1s2minus k2ηrs2

1113872 1113873η1η2η3η5η6η11η12ηH 1 minus k3( 1113857i1i2i3 + 1 minus k2( 1113857ε1113858 1113859

k3 minus k2( 1113857 1 minus k3ηs1s21113872 1113873i1i2i3η1η2η3ηH minus ε k2ηrs2

minus 11113872 11138731113960 1113961 (εge 0)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(56)

Mathematical Problems in Engineering 7

When HMCVT is working in HM1 or HM3 range if thetransmission efficiencies of the gear pairs i9 and i8 are equal(η7 η8) the transmission efficiencies of HMCVT in HM1and HM3 ranges are equal

When HMCVT is working in HM2 or HM4 range if thetransmission efficiencies of the gear pairs i9 and i6 are equal(η9 η6) the transmission efficiencies of HMCVT in HM2and HM4 ranges are also equal

Due to the complexity of the above results it is necessaryto evaluate the influence of displacement ratio ε standingtransmission ratio of PGM minusk1 gear ratio i1i2i3 and hy-draulic system efficiency ηH on the transmission efficiency of

HMCVT by means of image analysis method We assumethat η1 minus η12 are 098 ηs1r is 098 ηs1s2

and ηrs2are 097 and

k1i1i2i3 is minus32727 )e evaluation results are shown inFigure 3

According to the analysis on Figure 3 the following rulesare obtained

In each range of HMCVT when the displacement ratio εis 0 the transmission efficiency reaches the maximum valueNow the hydraulic path (HST) does not transmit power andthe power is completely transmitted by the mechanical pathWhen the displacement ratio ε is minus1 or 1 the transmissionefficiency reaches the minimum value and the hydraulic

Table 2 Equation of efficiency calculation for ranges HM1 and HM3

ElementsTorque equation considering power losses

εgt 0 εlt 0HM1 HM3 HM1 HM3

Engine Te Tr + η1η2Tpi1i2 Te Tr + Tpi1i2η1η2Pump Tp εηHTm Tp εTmηH

Motor Tm η3Ts1(minusi3) Tm Ts1

(minusη3i3)PGM Ts1

+ Tr + Tc 0PGM Tr minusk1Ts1

ηs1r

Intermediate shaft Tc Tzi4i8η4η8 Tc Tzi4i7η4η7 Tc Tzi4i8η4η8 Tc Tzi4i7η4η7Output shaft Tz To(i11i12η11η12)Torque ratio 1113957μ ToTe

Efficiency η minus1113957μig

144 max

100 44 max

44 max

Engine

r

S1

S2c

100

(a)

766 min

100

100

234 max

234 max

Engine

r

S1

S2c

(b)

766 min

100 234 max

234 max

Engine

r

S1

S2c

100

(c)

144 max

100

100

44 max

44 max

Engine

r

S1

S2c

(d)

Figure 2 Power flow direction of the PGM of HMCVT (a) Range HM1HM3 εgt 0 power-circulation (b) Range HM1HM3 εlt 0 power-split (c) Range HM2HM4 εgt 0 power-split (d) Range HM2HM4 εlt 0 power-circulation

8 Mathematical Problems in Engineering

Table 3 Equation of efficiency calculation for ranges HM2 and HM4

ElementsTorque equation considering power losses

εgt 0 εlt 0HM2 HM4 HM2 HM4

Engine Te Tr + Tpi1i2η1η2 Te Tr + η1η2Tpi1i2Pump Tp εTmηH Tp εηHTm

Motor Tm Ts1(minusη3i3) Tm η3Ts1

(minusi3)

PGM Ts1+ Tr + Ts2

0PGM Tr 1 minus k3ηs1s2

k3ηs1s2minus k2ηrs2

Ts2Intermediate shaft Ts2

Tzi5i9η5η9 Ts2 Tzi5i6η5η6 Ts2

Tzi5i9η5η9 Ts2 Tzi5i6η5η6

Output shaft Tz To(i11i12η11η12)Torque ratio 1113957μ ToTe

Efficiency η minus1113957μig

ndashk1 = 25172ndashk1 = 29130

ndashk1 = 17805ndashk1 = 21875

095

09

085

08

075

07ndash1 ndash08 ndash06 ndash04 ndash02 0 02 04 06 08 1

Displacement ratio

ηH = 081

Effici

ency

(a)

ndashk1 = 25172ndashk1 = 29130

ndashk1 = 17805ndashk1 = 21875

ηH = 081

095

09

085

08

075

07ndash1 ndash08 ndash06 ndash04 ndash02 0 02 04 06 08 1

Displacement ratio

Effici

ency

(b)

ηH = 081ηH = 090

ηH = 064ηH = 072

095

09

085

08

075

07ndash1 ndash08 ndash06 ndash04 ndash02 0 02 04 06 08 1

Displacement ratio

Effici

ency

ndashk1 = 21875

(c)

ηH = 081ηH = 090

ηH = 064ηH = 072

095

09

085

08

075

07ndash1 ndash08 ndash06 ndash04 ndash02 0 02 04 06 08 1

Displacement ratio

Effici

ency

ndashk1 = 21875

(d)

Figure 3 Continued

Mathematical Problems in Engineering 9

path transmits more power at this time resulting in a loweroverall efficiency of HMCVT

In each range of HMCVT the transmission efficiencyvalue in the power circulation phase is lower than thetransmission efficiency value in the power split phase whichindicates that the power circulation will cause the trans-mission efficiency of HMCVT to decrease

)e larger the value of the PGM standing transmissionratio minusk1 the higher the transmission efficiency of HMCVTbut the influence on improving the transmission efficiency is

not obvious )e PGM standing transmission ratio minusk1ranges from 15 to 3 )e gear ratio i1i2i3 and the PGMstanding transmission ratio minusk1 are in an inversely pro-portional relationship the smaller the value of the gear ratioi1i2i3 the higher the transmission efficiency of HMCVT andthe influence on improving the transmission efficiency isalso not obvious )e gear ratio i1i2i3 ranges from 11 to 22

)e greater the value of HSTefficiency ηH the higher theoverall transmission efficiency of HMCVT )e influence ofηH on improving transmission efficiency of HMCVT issignificant

i1i2i3 = 13001i1i2i3 = 11234

i1i2i3 = 18381i1i2i3 = 14961

ηH = 081

095

09

085

08

075

07ndash1 ndash08 ndash06 ndash04 ndash02 0 02 04 06 08 1

Displacement ratio

Effici

ency

(e)

i1i2i3 = 13001i1i2i3 = 11234

i1i2i3 = 18381i1i2i3 = 14961

ηH = 081

095

09

085

08

075

07ndash1 ndash08 ndash06 ndash04 ndash02 0 02 04 06 08 1

Displacement ratio

Effici

ency

(f )

Figure 3)e influence of parameters on the transmission efficiency of HMCVT (a))e influence of minusk1 on HM1 andHM3 ranges (b))einfluence of minusk1 on HM2 and HM4 ranges (c) )e influence of ηH on HM1 and HM3 ranges (d) )e influence of ηH on HM2 and HM4ranges (e) )e influence of i1i2i3 on HM1 and HM3 ranges (f ) )e influence of i1i2i3 on HM2 and HM4 ranges

9

8

7

6

5

4

3

2

1

0

Tran

smiss

ion

ratio

ndash15 ndash1 ndash05 0 05 1 15Displacement ratio

HM1HM2

HM3HM4

Figure 4 Relationship between transmission ratio and displace-ment ratio

60

50

40

30

20

10

0

Trac

tor s

peed

(km

h)

ndash15 ndash1 ndash05 0 05 1 15Displacement ratio

HM1HM2

HM3HM4

Figure 5 )e curve of the tractor speed

10 Mathematical Problems in Engineering

5 Analysis on the Transmission CharacteristicsResults of HMCVT

According to the previous content the value of the PGMstanding transmission ratio minusk1 is set to 21875 and the HSTefficiency value ηH is set to 081 then the other parameters ofHMCVT are k2 02714 minusk3 05938 i1 i2 1 i3 15i4 i5 1 i6 166 i7 116 i8 410 i9 584 andi10 i11 i12 1

)e relationship between the transmission ratio anddisplacement ratio is shown in Figure 4 )e curve of thetractor speed is shown in Figure 5 In the first range thedisplacement ratio changes from 1 to minus1 the transmissionratio decreases from 862 to 459 and the tractor speedincreases from 0 to 750 kmh In the second range thedisplacement ratio changes from minus1 to 1 the transmissionratio decreases from 459 to 243 and the tractor speedincreases from 750 kmh to 1412 kmh In the third rangethe displacement ratio changes from 1 to minus1 the trans-mission ratio decreases from 243 to 130 and the tractorspeed increases from 1412 kmh to 2644 kmh In thefourth range the displacement ratio changes from minus1 to 1the transmission ratio decreases from 130 to 069 and thetractor speed increases from 2644 kmh to 50 kmh )eresults show that the stepless speed regulation of HMCVTin the tractor speed range of 0ndash50 kmh can be realized bythe coordinated control of the HST displacement ratio andthe clutches

)e ratios of the output shaft torque to the input shafttorque in each range of HMCVT are shown in Figure 6(a))e continuous change of torque can be realized in the entireworking ranges )e ratios of the motor shaft torque to theoutput shaft torque in each range of HMCVT are shown inFigure 6(b) )e torque ratio remains constant in each range

and varies between different ranges )e values of torqueratios in the four ranges are 514 minus972 1849 andminus3518 respectively

)e relationship between the tractor speed and theoverall transmission efficiency of HMCVT is shown inFigure 7 Each of the four ranges has a transmission effi-ciency maximum value of 0917 0902 0917 and 0902respectively )e efficiency value always increases first withthe increase of the tractor speed and then decreases afterreaching the maximum value )e maximum transmissionefficiency in each range is obtained when the HST dis-placement ratio is 0 Now the hydraulic path does nottransmit power and all power is transmitted by the me-chanical path )e transmission efficiency of HMCVT ishigher than 83 so the overall transmission efficiency is at ahigh level

ndash2

ndash4

ndash6

ndash8

ndash10

0

0 5 10 15 20 25 30 35 40 45 50

Torq

ue ra

tio (T

oTe

)

Tractor speed (kmh)

HM1HM2

HM3HM4

(a)

05

04

03

02

01

0

ndash01

ndash02

ndash03

ndash04

ndash05

Torq

ue ra

tio (T

mT

o)

0 5 10 15 20 25 30 35 40 45 50Tractor speed (kmh)

HM1HM2

HM3HM4

(b)

Figure 6)e curve of torque characteristic (a) Ratios of output shaft torque to input shaft torque (b) Ratios of motor shaft torque to outputshaft torque

095

09

085

08

075

07

Effici

ency

0 5 10 15 20 25 30 35 40 45 50Tractor speed (kmh)

HM1HM2

HM3HM4

Figure 7 Overall transmission efficiency of HMCVT

Mathematical Problems in Engineering 11

6 Verification of HMCVT Simulation Model

)e modeling and simulation of HMCVT are accom-plished by using Amesim simulation softwareAccording to Figure 1 a simulation model of HMCVTis established as shown in Figure 8

61 Hydraulic System )e hydraulic system is mainlycomposed of a variable displacement hydraulic pump afixed displacement hydraulic motor two relief valves twocheck valves and a charge hydraulic pump with a reliefvalve )e aim of the charge pump is to compensate for theleakage loss in the pipeline and to establish the lowestpressure in the pumprsquos suction pipeline Two relief valves areused to limit the maximum pressure in the system pipingand its adjustment pressure should be above the normalworking pressure of the system Volumetric losses of the

pump and motor are neglected and their hydromechanicalefficiencies are set at 09

62 Mechanical Components Mechanical components areassumed to be constant efficiency components throughoutthe transmission )e power loss of each gear pair is 2 sothe efficiency is set to 098 In the Ravigneaux planetary gearmechanism the efficiencies of ring gear-long planetary gearcouple long planetary gear-large sun gear couple longplanetary gear-short planetary gear couple and shortplanetary gear-small sun gear couple are set at 099 )emain reducer differential and wheel reducer are combinedand modeled as a gear reducer with an efficiency of 0965 Tosimplify the model the friction losses of the shaft gears andbearings were ignored during the simulation modelingprocess

Efficiency calculation unit

Speed calculation unit

Small sun gear

Short planet gear

Long planet gear

Large sun gear

Ring gear

i1

i2

Pump Motor

i3

i5

i4

i7 i6

i9i8

Carrier

C1

C2

C3

C4

ω ωT

TL

ωωT

T

L

PP

MOT

f (x y)

y

x

X

k

Figure 8 HMCVT simulation model

(km

h)

50

40

30

20

10

00 5 10 15 20 25 30 35

X Time (s)

Tractor linear velocity (kmh)

(a)

0 5 10 15 20 25 30 35X Time (s)

10

08

06

04

02

00

(nul

l)

Transmission efficiency of HMCVT (null)

(b)

Figure 9 Simulation results (a) Simulation result of the tractor speed (b) Simulation result of the transmission efficiency of HMCVT

12 Mathematical Problems in Engineering

63Model Control and Simulation Results )e engine speedis assumed to have a constant value of 2200 rmin the loadtorque of the output shaft of HMCVT is set to 100Nm themodel simulation time is 32 s including range HM1 0ndash8 srange HM2 8ndash16 s range HM3 16ndash24 s and range HM424ndash32 s

)e simulation results are shown in Figure 9 through thevariable pump displacement and coordinated control of theclutches [18 19] the transmission gradually changes fromrange HM1 to range HM4 and the tractor achieves steplessspeed change from 0 to 50 kmh )e simulation results areconsistent with the previous theoretical analysis results whichshow that the 4-range HMCVT transmission scheme pro-posed in this paper is suitable for matching wheeled tractors

64 Comparison Xu LYrsquos transmission scheme [15] hasbeen applied to mass production tractors )is transmissionscheme has 6 working ranges in the forward direction and 3working ranges in the reverse direction which is controlledby 8 clutches )e speed range in the forward direction is0sim30 kmh while the reverse direction is only 0sim10 kmh)e transmission efficiency of the hydraulic range is low (lessthan 80) the transmission efficiency of the hydrome-chanical ranges is high (higher than 90) but is only in theforward direction

)e scheme proposed in this paper has 4 working rangesand a speed range of 0sim50 kmh in the forward and reversedirections which is controlled by 6 clutches )e fewer thetransmission ranges the less switching during the workingranges so the transmission performance is improved )esmaller number of clutches ensures that the system is easierto control )e transmission efficiencies in the forward andreverse directions are close the overall forward transmissionefficiency is higher than 83 and the highest transmissionefficiency is higher than 91

)erefore the application of HMCVT proposed in thispaper is wider it can not only match conventional tractorsbut also match two-way operation tractors and other en-gineering vehicles

7 Conclusions

)is paper proposes a new type of transmission scheme ofHMCVT )e Ravigneaux PGM and the HSTare connectedin parallel to form a closed loop and then connected in serieswith the range switching mechanism )is new HMCVTfeatures a double-row PGM with 4 hydromechanicalworking ranges By coordinating control of the HST dis-placement ratio and the engagement conditions of shiftingclutches the stepless speed regulation of HMCVT at thetractor speed of 0ndash50 kmh is realized

According to the theoretical analysis on the newHMCVT the speed characteristic curve and the torquecharacteristic curve are obtained Based on the power flowdirection of HMCVT the transmission efficiency curve isobtained by using the Lrfkoes formula

Using image analysis method to study the influence offour parameters (displacement ratio ε PGM standing

transmission ratio minusk1 gear ratio i1i2i3 and hydraulic systemefficiency ηH) on the transmission efficiency of HMCVT theresults show that the standing transmission ratio minusk1 andgear ratio i1i2i3 have little effect on the transmission effi-ciency whereas the displacement ratio ε and hydraulicsystem efficiency ηH have a greater impact on the trans-mission efficiency

)e parameters of the new HMCVT are finally deter-mined )e results of the transmission characteristics showthat the new HMCVTproposed in this paper has the steplessspeed regulation capability for tractors which can alsocontinuously transmit and change the torque )e trans-mission efficiency of HMCVT is at a high level

In order to verify the theoretical derivation Amesimsimulation software is used for the modeling and simulationof HMCVT )e simulation results are consistent with thetheoretical analysis results )erefore we believe that thetransmission scheme is reasonable and effective for tractors

Data Availability

)e Simcenter Amesim System File data used to support thefindings of this study are available from the correspondingauthor upon request

Conflicts of Interest

)e authors declare that they have no conflicts of interest

Acknowledgments

)is research was financially supported by the National KeyResearch and Development Plan Project of China(2016YFD0700402) and Changzhou Sci amp Tech Program(no CE20185037)

References

[1] L Y Xu Z L Zhou M Z Zhang et al ldquoResearch and designof hydro-mechanical continuously variable transmission fortractorsrdquo Journal of Northeast Agricultural University vol 13no 2 pp 182ndash186 2006

[2] R Paoluzzi and L G Zarotti ldquo)e minimum size of hy-drostatic transmissions for locomotionrdquo Journal of Terra-mechanics vol 50 no 3 pp 153ndash164 2013

[3] J Kim J Kang Y Kim B Min and H Kim ldquoDesign of powersplit transmission design of dual mode power split trans-missionrdquo International Journal of Automotive Technologyvol 11 no 4 pp 565ndash571 2010

[4] A Kim and A Rossetti ldquoFuel consumption reduction inurban buses by using power split transmissionsrdquo EnergyConversion and Management vol 71 pp 159ndash171 2013

[5] A Rossetti and A Macor ldquoMulti-objective optimization ofhydro-mechanical power split transmissionsrdquo Mechanismand Machine lteory vol 62 pp 112ndash128 2013

[6] A Macor and A Rossetti ldquoOptimization of hydro-mechanicalpower split transmissionsrdquo Mechanism and Machine lteoryvol 46 no 12 pp 1901ndash1919 2011

[7] S Shamshirband D Petkovic A Amini et al ldquoSupport vectorregression methodology for wind turbine reaction torqueprediction with power-split hydrostatic continuous variabletransmissionrdquo Energy vol 67 pp 623ndash630 2014

Mathematical Problems in Engineering 13

[8] K T Anuar ldquoResch R Continuously variable tractor trans-missionsrdquo ASAE Distinguished Lecture No 29 pp 1ndash37 StJoseph American Society of Agricultural Engineers St JosephMI USA 2005

[9] S H Yuan and J B Hu ldquo)e efficiency of multi-range hydro-mechanical stepless transmissionrdquo Journal of Beijing Instituteof Technology vol 7 no 2 pp 129ndash133 1998

[10] X J LiuAnalysis of Vehicle Transmission System pp 258-259National Defence Industry Press Beijing China 1998

[11] M M Brenninger ldquoFendt vario CVT in agricultural tractorsrdquoSAE International Warrendale PA USA SAE TechnicalPaper 2007-01-4205 2007

[12] H Aitzetmuller ldquoSteyr S-Matic-the future CVT systemrdquo inProceedings of the Seoul 2000 FISITA world automotive con-gress pp 1ndash6 Seoul South Korea June 2000

[13] J Pohlenz and K Grad ldquoCVTfor input power above 300 kWrdquoVDI-berichte vol 2007 pp 59ndash65 2001

[14] J Pohlenz and K Grad ldquoCVT-System fur denGroszligflacheneinsatzrdquoVDI-berichte vol 2004 pp 23ndash33 1855

[15] L Y Xu Study on Characteristics of Hydro-MechanicalContinuously Variable Transmission of Tractor Xirsquoan Uni-versity of Technology Xirsquoan China 2007

[16] J Looman Zahnradgetriebe-Grundlagen KonstruktionenAnwendungen in Fahrzeugen Springer Berlin Germany2009

[17] Z Rao Planetary Gear Transmission Design Chemical In-dustry Press Beijing China 2003

[18] X Xu Y Liang M Jordan T Peter and D Peng ldquoOptimizedcontrol of engine start assisted by the disconnect clutch in a P2hybrid automatic transmissionrdquo Mechanical Systems andSignal Processing vol 124 2019

[19] S Wang Y Liu Z Wang et al ldquoAdaptive fuzzy iterativecontrol strategy for the wet-clutch filling of automatictransmissionrdquo Mechanical Systems and Signal Processingvol 130 pp 164ndash182 2019

14 Mathematical Problems in Engineering

According to the PGM torque characteristic formula(16) the relationship between the input torque of large sungear Ts1

the input torque of ring gear Tr the output torqueof carrierTc and the PGM standing transmission ratio minusk1 issatisfied

Ts1

Tr

minusk1 minus

Tc

1 minus k1 (22)

where minusk1 ZrZs1gt 1 Zr is the number of teeth of the ring

gear and Zs1is the number of teeth of the large sun gear

)erefore the large sun gear torque Ts1is

Ts1 minus

Tc

1 minus k1

minusTo

i4i8i11i12 1 minus k1( 1113857 HM1( 1113857

minusTo

i4i7i11i12 1 minus k1( 1113857 HM3( 1113857

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(23)

)en the input torque of ring gear Tr is

Tr minusk1Ts1

k1To

i4i8i11i12 1 minus k1( 1113857 HM1( 1113857

k1To

i4i7i11i12 1 minus k1( 1113857 HM3( 1113857

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(24)

)us the motor shaft load torque Tm is

Tm Ts1

minusi3

To

i3i4i8i11i12 1 minus k1( 1113857 HM1( 1113857

To

i3i4i7i11i12 1 minus k1( 1113857 HM3( 1113857

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(25)

)e pump shaft torque Tp is

Tp εTm

εTo

i3i4i8i11i12 1 minus k1( 1113857 HM1( 1113857

εTo

i3i4i7i11i12 1 minus k1( 1113857 HM3( 1113857

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(26)

)e engine output torque Te is

Te Tr +Tp

i1i2

k1i1i2i3 + ε( 1113857To

i1i2i3i4i8i11i12 1 minus k1( 1113857 HM1( 1113857

k1i1i2i3 + ε( 1113857To

i1i2i3i4i7i11i12 1 minus k1( 1113857 HM3( 1113857

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(27)

Finally the ratio of the output torque of HMCVT to theinput torque of the engine μ is

μ To

Te

1 minus k1( i1i2i3i4i8i11i12

ε + k1i1i2i3 HM1( 1113857

1 minus k1( 1113857i1i2i3i4i7i11i12ε + k1i1i2i3

HM3( 1113857

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(28)

332 Ranges HM2 and HM4 )e same as the previous caseTo denotes the load torque of output shaft and the inter-mediate shaft torque Tz is Tz To(i11i12) )en the outputtorque of small sun gear in PGM Ts2

is

Ts2

Tz

i5i9

To

i5i9i11i12 HM2( 1113857

Tz

i5i6

To

i5i6i11i12 HM4( 1113857

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(29)

)e external torque algebra sum of PGM is zero soTs1

+ Tr + Ts2 0 According to the PGM torque character-

istic formula (16) the relationship between the input torque oflarge sun gear Ts1

the output torque of small sun gear Ts2 and

the PGM standing transmission ratios k2 and minusk3 is satisfied

Ts1

k2 minus 1k3 minus k2

Ts2

k2 minus 1( 1113857To

k3 minus k2( 1113857i5i9i11i12 HM2( 1113857

k2 minus 1( 1113857To

k3 minus k2( 1113857i5i6i11i12 HM4( 1113857

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(30)

)e engine output torque Te is

Te

1 minus k2( 1113857ε + 1 minus k3( 1113857i1i2i3

k3 minus k2( 1113857i1i2i3i5i9i11i12To HM2( 1113857

1 minus k2( 1113857ε + 1 minus k3( 1113857i1i2i3

k3 minus k2( 1113857i1i2i3i5i6i11i12To HM4( 1113857

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(31)

Finally the ratio of the output torque of HMCVT to theinput torque of the engine μ is

μ To

Te

k3 minus k2( 1113857i1i2i3i5i9i11i12

1 minus k2( 1113857ε + 1 minus k3( 1113857i1i2i3 HM2( 1113857

k3 minus k2( 1113857i1i2i3i5i6i11i12

1 minus k2( 1113857ε + 1 minus k3( 1113857i1i2i3 HM4( 1113857

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(32)

34 Circulating Power Flow Calculation

341 Ranges HM1 and HM3 )e speed ratio of the powertransmitting from the engine to the carrier ice is

ice nc

ne

minusk1i1i2i3 minus ε1 minus k1( 1113857i1i2i3

irce + i

s1ce (33)

)e speed ratios of the power transmitting from theengine to the large sun gear and ring gear ireis1e are

ire 1

is1e minusε

i1i2i3

(34)

In the PGM the speed ratios between the input elements(large sun gear s1 and ring gear r) and the output element(carrier c) are

Mathematical Problems in Engineering 5

ircs1

nc

ns1

1

1 minus k1

is1cr

nc

nr

minusk1

1 minus k1

(35)

)en we can get

irce i

rcs1

is1e minusε

i1i2i3 1 minus k1( 1113857

is1ce i

s1crire minus

k1

1 minus k1

(36)

Next two situations will be discussed as follows [17]

(1) When εlt 0 is1ce gt 0 irce gt 0 and i

s1cei

rce gt 0 the PGM has

no circulating power PL 0(2) When εgt 0 i

s1ce gt 0 irce lt 0 and i

s1cei

rce lt 0 the PGM has

circulating power substituting equation (19) fork1i1i2i3 in equation (33) ice is

ice 32727 minus ε

32727 + i1i2i3gt 0 (37)

Because ofPe gt 0 Pr (is1ceice)Pe gt 0 Ps1

(irceice)Pe lt 0the output power P is

P Pe minusPc Pr + Ps1 Pr minus Ps1

11138681113868111386811138681113868

11138681113868111386811138681113868 (38)

)en

Pr P + Ps1

11138681113868111386811138681113868

11138681113868111386811138681113868 P + PL

11138681113868111386811138681113868111386811138681113868 (39)

)e circulating power PL is

PL Ps1

irce

ice

Pe (40)

Assume that the closed power factor KL is

KL PL

11138681113868111386811138681113868111386811138681113868

Pe

irce

ice

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

εk1i1i2i3 + ε

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 |ϕ(ε)| (41)

where ϕ(ε) εk1i1i2i3 + ε and the derivative of ϕ(ε) isdϕ(ε)dε k1i1i2i3(k1i1i2i3 + ε)2 lt 0 so function ϕ(ε) is amonotonically decreasing function )en KL(0) 0KL(1) 044 and the maximum value of the circulatingpower is 044Pe

342 Ranges HM2 and HM4 )e speed ratio of the powertransmitting from the engine to the small sun gear is2e is

is2e ns2

ne

1 minus k3( 1113857i1i2i3 + 1 minus k2( 1113857ε

k2 minus k3( 1113857i1i2i3 i

rs2e + i

s1s2e (42)

)e speed ratios of the power transmitting from theengine to the large sun gear and ring gear ireis1e are

ire 1

is1e minusε

i1i2i3

(43)

In the PGM the speed ratios between the input elements(large sun gear s1 and ring gear r) and the output element(small sun gear s2) are

irs2s1

k2 minus 1k2 minus k3

is1s2r

1 minus k3

k2 minus k3

(44)

)en we can get

irs2e i

rs2s1

is1e minusk2 minus 1( 1113857ε

i1i2i3 k2 minus k3( 1113857

is1s2e i

s1s2rire

1 minus k3

k2 minus k3

(45)

Next two situations will be discussed as follows [17]

(1) When εlt 0 is1s2e gt 0 irs2e lt 0 and i

s1s2ei

rs2e lt 0 the PGM

has circulating power substituting equation (19) fork2 k3 i1i2i3 in equation (42) is2e is

is2e 32727 + ε

32727 minus 32727 minusk1( 1113857gt 0 (46)

Because of Pe gt 0 Pr is1ceicePe gt 0 Ps1

irceicePe lt 0the output power P is

P Pe minusPs2 Pr + Ps1

Pr minus Ps1

11138681113868111386811138681113868

11138681113868111386811138681113868 (47)

)en

Pr P + Ps1

11138681113868111386811138681113868

11138681113868111386811138681113868 P + PL

11138681113868111386811138681113868111386811138681113868 (48)

)e circulating power PL is

PL Ps1 irs2e

is2e

Pe (49)

Assume that the closed power factor KL is

KL PL

11138681113868111386811138681113868111386811138681113868

Pe

irs2e

is2e

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

minusεk1i1i2i3 minus ε

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 |ϕ(ε)| (50)

where ϕ(ε) minusεk1i1i2i3 minus ε and the derivative ofϕ(ε) is dϕ(ε)dε minusk1i1i2i3(k1i1i2i3 minus ε)2 gt 0 sofunction ϕ(ε) is a monotonically increasing function)en KL(0) 0 KL(1) 044 and the maximumvalue of the circulating power is 044Pe

(2) When εgt 0 is1s2e gt 0 irs2e gt 0 and i

s1s2ei

rs2e gt 0 the PGM

has no circulating power PL 0

35 Hydraulic Split Power Calculation Hydraulic power-split ratio ρ PHPe Ps1

Pe hydraulic power is input tothe PGM through the large sun gear s1

6 Mathematical Problems in Engineering

351 Ranges HM1 and HM3

ρ Ps1

Pe

irce

ice

ε

k1i1i2i3 + ε ϕ(ε) (51)

and as can be obtained from the above function ϕ(ε) is amonotonically decreasing function so

ρ(minus1) 0234gt 0

ρ(0) 0

ρ(1) minus044lt 0

⎧⎪⎪⎨

⎪⎪⎩(52)

352 Ranges HM2 and HM4

ρ Ps1

Pe

irs2e

is2e

minusε

k1i1i2i3 minus ε ϕ(ε) (53)

As can be obtained from the above function ϕ(ε) is amonotonically increasing function so

ρ(minus1) minus044lt 0

ρ(0) 0

ρ(1) 0234gt 0

⎧⎪⎪⎨

⎪⎪⎩(54)

According to the theoretical analysis on the circulatingpower flow and the hydraulic split power the power flowdirection of HMCVT in each working range is shown inFigure 2 )is result lays the foundation for the analysis onthe efficiency characteristic of HMCVT

36 Efficiency Characteristic )e Lrfkoes formula (17) isused to calculate the transmission efficiency of HMCVT

η minus1113957μig

minusToTe

neno

(55)

where 1113957μ is the ratio of output torque to input torque ofHMCVTconsidering power losses and ig is the transmissionratio of HMCVT

)e transmission efficiency calculation process of eachworking range of HMCVT is shown in Tables 2 and 3Among them the calculation processes of the first range andthe third range are listed in Table 2 and the calculationprocesses of the second range and the fourth range are listedin Table 3 In Tables 2 and 3 i1 minus i12 are the gear ratios of thegear pairs η1 minus η12 are the transmission efficiency of the gearpairs ηH is the hydraulic system transmission efficiency andηs1r ηs1s2

and ηrs2are the transmission efficiencies of the

large sun gear to the ring gear the large sun gear to the smallsun gear and the ring gear to the small sun gear in the PGMrespectively

4 Analysis on the Influence of Parameters onthe Transmission Efficiency of HMCVT

)e results of the transmission efficiency in each workingrange of HMCVT are as follows

ηHM1

ηs1r minus k11113872 1113873η4η8η11η12 ε + k1i1i2i3( 1113857

1 minus k1( 1113857 η1η2η3ηHηs1rε + k1i1i2i31113872 1113873 (εgt 0)

ηs1r minus k11113872 1113873η1η2η3ηHη4η8η11η12 ε + k1i1i2i3( 1113857

1 minus k1( 1113857 i1i2i3η1η2η3ηHk1 + εηs1r1113872 1113873 (εle 0)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

ηHM2

k3ηs1s2minus k2ηrs2

1113872 1113873η5η9η11η12 1 minus k3( 1113857i1i2i3 + 1 minus k2( 1113857ε1113858 1113859

k3 minus k2( 1113857 1 minus k3ηs1s21113872 1113873i1i2i3 minus η1η2η3ηHε k2ηrs2

minus 11113872 11138731113960 1113961 (εlt 0)

k3ηs1s2minus k2ηrs2

1113872 1113873η1η2η3η5η9η11η12ηH 1 minus k3( 1113857i1i2i3 + 1 minus k2( 1113857ε1113858 1113859

k3 minus k2( 1113857 1 minus k3ηs1s21113872 1113873i1i2i3η1η2η3ηH minus ε k2ηrs2

minus 11113872 11138731113960 1113961 (εge 0)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

ηHM3

ηs1r minus k11113872 1113873η4η7η11η12 ε + k1i1i2i3( 1113857

1 minus k1( 1113857 η1η2η3ηHηs1rε + k1i1i2i31113872 1113873 (εgt 0)

ηs1r minus k11113872 η1η2η3ηHη4η7η11η12 ε + k1i1i2i3( 1113857

1 minus k1( 1113857 i1i2i3η1η2η3ηHk1 + εηs1r1113872 1113873 (εle 0)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

ηHM4

k3ηs1s2minus k2ηrs2

1113872 1113873η5η6η11η12 1 minus k3( 1113857i1i2i3 + 1 minus k2( 1113857ε1113858 1113859

k3 minus k2( 1113857 1 minus k3ηs1s21113872 1113873i1i2i3 minus η1η2η3ηHε k2ηrs2

minus 11113872 11138731113960 1113961 (εlt 0)

k3ηs1s2minus k2ηrs2

1113872 1113873η1η2η3η5η6η11η12ηH 1 minus k3( 1113857i1i2i3 + 1 minus k2( 1113857ε1113858 1113859

k3 minus k2( 1113857 1 minus k3ηs1s21113872 1113873i1i2i3η1η2η3ηH minus ε k2ηrs2

minus 11113872 11138731113960 1113961 (εge 0)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(56)

Mathematical Problems in Engineering 7

When HMCVT is working in HM1 or HM3 range if thetransmission efficiencies of the gear pairs i9 and i8 are equal(η7 η8) the transmission efficiencies of HMCVT in HM1and HM3 ranges are equal

When HMCVT is working in HM2 or HM4 range if thetransmission efficiencies of the gear pairs i9 and i6 are equal(η9 η6) the transmission efficiencies of HMCVT in HM2and HM4 ranges are also equal

Due to the complexity of the above results it is necessaryto evaluate the influence of displacement ratio ε standingtransmission ratio of PGM minusk1 gear ratio i1i2i3 and hy-draulic system efficiency ηH on the transmission efficiency of

HMCVT by means of image analysis method We assumethat η1 minus η12 are 098 ηs1r is 098 ηs1s2

and ηrs2are 097 and

k1i1i2i3 is minus32727 )e evaluation results are shown inFigure 3

According to the analysis on Figure 3 the following rulesare obtained

In each range of HMCVT when the displacement ratio εis 0 the transmission efficiency reaches the maximum valueNow the hydraulic path (HST) does not transmit power andthe power is completely transmitted by the mechanical pathWhen the displacement ratio ε is minus1 or 1 the transmissionefficiency reaches the minimum value and the hydraulic

Table 2 Equation of efficiency calculation for ranges HM1 and HM3

ElementsTorque equation considering power losses

εgt 0 εlt 0HM1 HM3 HM1 HM3

Engine Te Tr + η1η2Tpi1i2 Te Tr + Tpi1i2η1η2Pump Tp εηHTm Tp εTmηH

Motor Tm η3Ts1(minusi3) Tm Ts1

(minusη3i3)PGM Ts1

+ Tr + Tc 0PGM Tr minusk1Ts1

ηs1r

Intermediate shaft Tc Tzi4i8η4η8 Tc Tzi4i7η4η7 Tc Tzi4i8η4η8 Tc Tzi4i7η4η7Output shaft Tz To(i11i12η11η12)Torque ratio 1113957μ ToTe

Efficiency η minus1113957μig

144 max

100 44 max

44 max

Engine

r

S1

S2c

100

(a)

766 min

100

100

234 max

234 max

Engine

r

S1

S2c

(b)

766 min

100 234 max

234 max

Engine

r

S1

S2c

100

(c)

144 max

100

100

44 max

44 max

Engine

r

S1

S2c

(d)

Figure 2 Power flow direction of the PGM of HMCVT (a) Range HM1HM3 εgt 0 power-circulation (b) Range HM1HM3 εlt 0 power-split (c) Range HM2HM4 εgt 0 power-split (d) Range HM2HM4 εlt 0 power-circulation

8 Mathematical Problems in Engineering

Table 3 Equation of efficiency calculation for ranges HM2 and HM4

ElementsTorque equation considering power losses

εgt 0 εlt 0HM2 HM4 HM2 HM4

Engine Te Tr + Tpi1i2η1η2 Te Tr + η1η2Tpi1i2Pump Tp εTmηH Tp εηHTm

Motor Tm Ts1(minusη3i3) Tm η3Ts1

(minusi3)

PGM Ts1+ Tr + Ts2

0PGM Tr 1 minus k3ηs1s2

k3ηs1s2minus k2ηrs2

Ts2Intermediate shaft Ts2

Tzi5i9η5η9 Ts2 Tzi5i6η5η6 Ts2

Tzi5i9η5η9 Ts2 Tzi5i6η5η6

Output shaft Tz To(i11i12η11η12)Torque ratio 1113957μ ToTe

Efficiency η minus1113957μig

ndashk1 = 25172ndashk1 = 29130

ndashk1 = 17805ndashk1 = 21875

095

09

085

08

075

07ndash1 ndash08 ndash06 ndash04 ndash02 0 02 04 06 08 1

Displacement ratio

ηH = 081

Effici

ency

(a)

ndashk1 = 25172ndashk1 = 29130

ndashk1 = 17805ndashk1 = 21875

ηH = 081

095

09

085

08

075

07ndash1 ndash08 ndash06 ndash04 ndash02 0 02 04 06 08 1

Displacement ratio

Effici

ency

(b)

ηH = 081ηH = 090

ηH = 064ηH = 072

095

09

085

08

075

07ndash1 ndash08 ndash06 ndash04 ndash02 0 02 04 06 08 1

Displacement ratio

Effici

ency

ndashk1 = 21875

(c)

ηH = 081ηH = 090

ηH = 064ηH = 072

095

09

085

08

075

07ndash1 ndash08 ndash06 ndash04 ndash02 0 02 04 06 08 1

Displacement ratio

Effici

ency

ndashk1 = 21875

(d)

Figure 3 Continued

Mathematical Problems in Engineering 9

path transmits more power at this time resulting in a loweroverall efficiency of HMCVT

In each range of HMCVT the transmission efficiencyvalue in the power circulation phase is lower than thetransmission efficiency value in the power split phase whichindicates that the power circulation will cause the trans-mission efficiency of HMCVT to decrease

)e larger the value of the PGM standing transmissionratio minusk1 the higher the transmission efficiency of HMCVTbut the influence on improving the transmission efficiency is

not obvious )e PGM standing transmission ratio minusk1ranges from 15 to 3 )e gear ratio i1i2i3 and the PGMstanding transmission ratio minusk1 are in an inversely pro-portional relationship the smaller the value of the gear ratioi1i2i3 the higher the transmission efficiency of HMCVT andthe influence on improving the transmission efficiency isalso not obvious )e gear ratio i1i2i3 ranges from 11 to 22

)e greater the value of HSTefficiency ηH the higher theoverall transmission efficiency of HMCVT )e influence ofηH on improving transmission efficiency of HMCVT issignificant

i1i2i3 = 13001i1i2i3 = 11234

i1i2i3 = 18381i1i2i3 = 14961

ηH = 081

095

09

085

08

075

07ndash1 ndash08 ndash06 ndash04 ndash02 0 02 04 06 08 1

Displacement ratio

Effici

ency

(e)

i1i2i3 = 13001i1i2i3 = 11234

i1i2i3 = 18381i1i2i3 = 14961

ηH = 081

095

09

085

08

075

07ndash1 ndash08 ndash06 ndash04 ndash02 0 02 04 06 08 1

Displacement ratio

Effici

ency

(f )

Figure 3)e influence of parameters on the transmission efficiency of HMCVT (a))e influence of minusk1 on HM1 andHM3 ranges (b))einfluence of minusk1 on HM2 and HM4 ranges (c) )e influence of ηH on HM1 and HM3 ranges (d) )e influence of ηH on HM2 and HM4ranges (e) )e influence of i1i2i3 on HM1 and HM3 ranges (f ) )e influence of i1i2i3 on HM2 and HM4 ranges

9

8

7

6

5

4

3

2

1

0

Tran

smiss

ion

ratio

ndash15 ndash1 ndash05 0 05 1 15Displacement ratio

HM1HM2

HM3HM4

Figure 4 Relationship between transmission ratio and displace-ment ratio

60

50

40

30

20

10

0

Trac

tor s

peed

(km

h)

ndash15 ndash1 ndash05 0 05 1 15Displacement ratio

HM1HM2

HM3HM4

Figure 5 )e curve of the tractor speed

10 Mathematical Problems in Engineering

5 Analysis on the Transmission CharacteristicsResults of HMCVT

According to the previous content the value of the PGMstanding transmission ratio minusk1 is set to 21875 and the HSTefficiency value ηH is set to 081 then the other parameters ofHMCVT are k2 02714 minusk3 05938 i1 i2 1 i3 15i4 i5 1 i6 166 i7 116 i8 410 i9 584 andi10 i11 i12 1

)e relationship between the transmission ratio anddisplacement ratio is shown in Figure 4 )e curve of thetractor speed is shown in Figure 5 In the first range thedisplacement ratio changes from 1 to minus1 the transmissionratio decreases from 862 to 459 and the tractor speedincreases from 0 to 750 kmh In the second range thedisplacement ratio changes from minus1 to 1 the transmissionratio decreases from 459 to 243 and the tractor speedincreases from 750 kmh to 1412 kmh In the third rangethe displacement ratio changes from 1 to minus1 the trans-mission ratio decreases from 243 to 130 and the tractorspeed increases from 1412 kmh to 2644 kmh In thefourth range the displacement ratio changes from minus1 to 1the transmission ratio decreases from 130 to 069 and thetractor speed increases from 2644 kmh to 50 kmh )eresults show that the stepless speed regulation of HMCVTin the tractor speed range of 0ndash50 kmh can be realized bythe coordinated control of the HST displacement ratio andthe clutches

)e ratios of the output shaft torque to the input shafttorque in each range of HMCVT are shown in Figure 6(a))e continuous change of torque can be realized in the entireworking ranges )e ratios of the motor shaft torque to theoutput shaft torque in each range of HMCVT are shown inFigure 6(b) )e torque ratio remains constant in each range

and varies between different ranges )e values of torqueratios in the four ranges are 514 minus972 1849 andminus3518 respectively

)e relationship between the tractor speed and theoverall transmission efficiency of HMCVT is shown inFigure 7 Each of the four ranges has a transmission effi-ciency maximum value of 0917 0902 0917 and 0902respectively )e efficiency value always increases first withthe increase of the tractor speed and then decreases afterreaching the maximum value )e maximum transmissionefficiency in each range is obtained when the HST dis-placement ratio is 0 Now the hydraulic path does nottransmit power and all power is transmitted by the me-chanical path )e transmission efficiency of HMCVT ishigher than 83 so the overall transmission efficiency is at ahigh level

ndash2

ndash4

ndash6

ndash8

ndash10

0

0 5 10 15 20 25 30 35 40 45 50

Torq

ue ra

tio (T

oTe

)

Tractor speed (kmh)

HM1HM2

HM3HM4

(a)

05

04

03

02

01

0

ndash01

ndash02

ndash03

ndash04

ndash05

Torq

ue ra

tio (T

mT

o)

0 5 10 15 20 25 30 35 40 45 50Tractor speed (kmh)

HM1HM2

HM3HM4

(b)

Figure 6)e curve of torque characteristic (a) Ratios of output shaft torque to input shaft torque (b) Ratios of motor shaft torque to outputshaft torque

095

09

085

08

075

07

Effici

ency

0 5 10 15 20 25 30 35 40 45 50Tractor speed (kmh)

HM1HM2

HM3HM4

Figure 7 Overall transmission efficiency of HMCVT

Mathematical Problems in Engineering 11

6 Verification of HMCVT Simulation Model

)e modeling and simulation of HMCVT are accom-plished by using Amesim simulation softwareAccording to Figure 1 a simulation model of HMCVTis established as shown in Figure 8

61 Hydraulic System )e hydraulic system is mainlycomposed of a variable displacement hydraulic pump afixed displacement hydraulic motor two relief valves twocheck valves and a charge hydraulic pump with a reliefvalve )e aim of the charge pump is to compensate for theleakage loss in the pipeline and to establish the lowestpressure in the pumprsquos suction pipeline Two relief valves areused to limit the maximum pressure in the system pipingand its adjustment pressure should be above the normalworking pressure of the system Volumetric losses of the

pump and motor are neglected and their hydromechanicalefficiencies are set at 09

62 Mechanical Components Mechanical components areassumed to be constant efficiency components throughoutthe transmission )e power loss of each gear pair is 2 sothe efficiency is set to 098 In the Ravigneaux planetary gearmechanism the efficiencies of ring gear-long planetary gearcouple long planetary gear-large sun gear couple longplanetary gear-short planetary gear couple and shortplanetary gear-small sun gear couple are set at 099 )emain reducer differential and wheel reducer are combinedand modeled as a gear reducer with an efficiency of 0965 Tosimplify the model the friction losses of the shaft gears andbearings were ignored during the simulation modelingprocess

Efficiency calculation unit

Speed calculation unit

Small sun gear

Short planet gear

Long planet gear

Large sun gear

Ring gear

i1

i2

Pump Motor

i3

i5

i4

i7 i6

i9i8

Carrier

C1

C2

C3

C4

ω ωT

TL

ωωT

T

L

PP

MOT

f (x y)

y

x

X

k

Figure 8 HMCVT simulation model

(km

h)

50

40

30

20

10

00 5 10 15 20 25 30 35

X Time (s)

Tractor linear velocity (kmh)

(a)

0 5 10 15 20 25 30 35X Time (s)

10

08

06

04

02

00

(nul

l)

Transmission efficiency of HMCVT (null)

(b)

Figure 9 Simulation results (a) Simulation result of the tractor speed (b) Simulation result of the transmission efficiency of HMCVT

12 Mathematical Problems in Engineering

63Model Control and Simulation Results )e engine speedis assumed to have a constant value of 2200 rmin the loadtorque of the output shaft of HMCVT is set to 100Nm themodel simulation time is 32 s including range HM1 0ndash8 srange HM2 8ndash16 s range HM3 16ndash24 s and range HM424ndash32 s

)e simulation results are shown in Figure 9 through thevariable pump displacement and coordinated control of theclutches [18 19] the transmission gradually changes fromrange HM1 to range HM4 and the tractor achieves steplessspeed change from 0 to 50 kmh )e simulation results areconsistent with the previous theoretical analysis results whichshow that the 4-range HMCVT transmission scheme pro-posed in this paper is suitable for matching wheeled tractors

64 Comparison Xu LYrsquos transmission scheme [15] hasbeen applied to mass production tractors )is transmissionscheme has 6 working ranges in the forward direction and 3working ranges in the reverse direction which is controlledby 8 clutches )e speed range in the forward direction is0sim30 kmh while the reverse direction is only 0sim10 kmh)e transmission efficiency of the hydraulic range is low (lessthan 80) the transmission efficiency of the hydrome-chanical ranges is high (higher than 90) but is only in theforward direction

)e scheme proposed in this paper has 4 working rangesand a speed range of 0sim50 kmh in the forward and reversedirections which is controlled by 6 clutches )e fewer thetransmission ranges the less switching during the workingranges so the transmission performance is improved )esmaller number of clutches ensures that the system is easierto control )e transmission efficiencies in the forward andreverse directions are close the overall forward transmissionefficiency is higher than 83 and the highest transmissionefficiency is higher than 91

)erefore the application of HMCVT proposed in thispaper is wider it can not only match conventional tractorsbut also match two-way operation tractors and other en-gineering vehicles

7 Conclusions

)is paper proposes a new type of transmission scheme ofHMCVT )e Ravigneaux PGM and the HSTare connectedin parallel to form a closed loop and then connected in serieswith the range switching mechanism )is new HMCVTfeatures a double-row PGM with 4 hydromechanicalworking ranges By coordinating control of the HST dis-placement ratio and the engagement conditions of shiftingclutches the stepless speed regulation of HMCVT at thetractor speed of 0ndash50 kmh is realized

According to the theoretical analysis on the newHMCVT the speed characteristic curve and the torquecharacteristic curve are obtained Based on the power flowdirection of HMCVT the transmission efficiency curve isobtained by using the Lrfkoes formula

Using image analysis method to study the influence offour parameters (displacement ratio ε PGM standing

transmission ratio minusk1 gear ratio i1i2i3 and hydraulic systemefficiency ηH) on the transmission efficiency of HMCVT theresults show that the standing transmission ratio minusk1 andgear ratio i1i2i3 have little effect on the transmission effi-ciency whereas the displacement ratio ε and hydraulicsystem efficiency ηH have a greater impact on the trans-mission efficiency

)e parameters of the new HMCVT are finally deter-mined )e results of the transmission characteristics showthat the new HMCVTproposed in this paper has the steplessspeed regulation capability for tractors which can alsocontinuously transmit and change the torque )e trans-mission efficiency of HMCVT is at a high level

In order to verify the theoretical derivation Amesimsimulation software is used for the modeling and simulationof HMCVT )e simulation results are consistent with thetheoretical analysis results )erefore we believe that thetransmission scheme is reasonable and effective for tractors

Data Availability

)e Simcenter Amesim System File data used to support thefindings of this study are available from the correspondingauthor upon request

Conflicts of Interest

)e authors declare that they have no conflicts of interest

Acknowledgments

)is research was financially supported by the National KeyResearch and Development Plan Project of China(2016YFD0700402) and Changzhou Sci amp Tech Program(no CE20185037)

References

[1] L Y Xu Z L Zhou M Z Zhang et al ldquoResearch and designof hydro-mechanical continuously variable transmission fortractorsrdquo Journal of Northeast Agricultural University vol 13no 2 pp 182ndash186 2006

[2] R Paoluzzi and L G Zarotti ldquo)e minimum size of hy-drostatic transmissions for locomotionrdquo Journal of Terra-mechanics vol 50 no 3 pp 153ndash164 2013

[3] J Kim J Kang Y Kim B Min and H Kim ldquoDesign of powersplit transmission design of dual mode power split trans-missionrdquo International Journal of Automotive Technologyvol 11 no 4 pp 565ndash571 2010

[4] A Kim and A Rossetti ldquoFuel consumption reduction inurban buses by using power split transmissionsrdquo EnergyConversion and Management vol 71 pp 159ndash171 2013

[5] A Rossetti and A Macor ldquoMulti-objective optimization ofhydro-mechanical power split transmissionsrdquo Mechanismand Machine lteory vol 62 pp 112ndash128 2013

[6] A Macor and A Rossetti ldquoOptimization of hydro-mechanicalpower split transmissionsrdquo Mechanism and Machine lteoryvol 46 no 12 pp 1901ndash1919 2011

[7] S Shamshirband D Petkovic A Amini et al ldquoSupport vectorregression methodology for wind turbine reaction torqueprediction with power-split hydrostatic continuous variabletransmissionrdquo Energy vol 67 pp 623ndash630 2014

Mathematical Problems in Engineering 13

[8] K T Anuar ldquoResch R Continuously variable tractor trans-missionsrdquo ASAE Distinguished Lecture No 29 pp 1ndash37 StJoseph American Society of Agricultural Engineers St JosephMI USA 2005

[9] S H Yuan and J B Hu ldquo)e efficiency of multi-range hydro-mechanical stepless transmissionrdquo Journal of Beijing Instituteof Technology vol 7 no 2 pp 129ndash133 1998

[10] X J LiuAnalysis of Vehicle Transmission System pp 258-259National Defence Industry Press Beijing China 1998

[11] M M Brenninger ldquoFendt vario CVT in agricultural tractorsrdquoSAE International Warrendale PA USA SAE TechnicalPaper 2007-01-4205 2007

[12] H Aitzetmuller ldquoSteyr S-Matic-the future CVT systemrdquo inProceedings of the Seoul 2000 FISITA world automotive con-gress pp 1ndash6 Seoul South Korea June 2000

[13] J Pohlenz and K Grad ldquoCVTfor input power above 300 kWrdquoVDI-berichte vol 2007 pp 59ndash65 2001

[14] J Pohlenz and K Grad ldquoCVT-System fur denGroszligflacheneinsatzrdquoVDI-berichte vol 2004 pp 23ndash33 1855

[15] L Y Xu Study on Characteristics of Hydro-MechanicalContinuously Variable Transmission of Tractor Xirsquoan Uni-versity of Technology Xirsquoan China 2007

[16] J Looman Zahnradgetriebe-Grundlagen KonstruktionenAnwendungen in Fahrzeugen Springer Berlin Germany2009

[17] Z Rao Planetary Gear Transmission Design Chemical In-dustry Press Beijing China 2003

[18] X Xu Y Liang M Jordan T Peter and D Peng ldquoOptimizedcontrol of engine start assisted by the disconnect clutch in a P2hybrid automatic transmissionrdquo Mechanical Systems andSignal Processing vol 124 2019

[19] S Wang Y Liu Z Wang et al ldquoAdaptive fuzzy iterativecontrol strategy for the wet-clutch filling of automatictransmissionrdquo Mechanical Systems and Signal Processingvol 130 pp 164ndash182 2019

14 Mathematical Problems in Engineering

ircs1

nc

ns1

1

1 minus k1

is1cr

nc

nr

minusk1

1 minus k1

(35)

)en we can get

irce i

rcs1

is1e minusε

i1i2i3 1 minus k1( 1113857

is1ce i

s1crire minus

k1

1 minus k1

(36)

Next two situations will be discussed as follows [17]

(1) When εlt 0 is1ce gt 0 irce gt 0 and i

s1cei

rce gt 0 the PGM has

no circulating power PL 0(2) When εgt 0 i

s1ce gt 0 irce lt 0 and i

s1cei

rce lt 0 the PGM has

circulating power substituting equation (19) fork1i1i2i3 in equation (33) ice is

ice 32727 minus ε

32727 + i1i2i3gt 0 (37)

Because ofPe gt 0 Pr (is1ceice)Pe gt 0 Ps1

(irceice)Pe lt 0the output power P is

P Pe minusPc Pr + Ps1 Pr minus Ps1

11138681113868111386811138681113868

11138681113868111386811138681113868 (38)

)en

Pr P + Ps1

11138681113868111386811138681113868

11138681113868111386811138681113868 P + PL

11138681113868111386811138681113868111386811138681113868 (39)

)e circulating power PL is

PL Ps1

irce

ice

Pe (40)

Assume that the closed power factor KL is

KL PL

11138681113868111386811138681113868111386811138681113868

Pe

irce

ice

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

εk1i1i2i3 + ε

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 |ϕ(ε)| (41)

where ϕ(ε) εk1i1i2i3 + ε and the derivative of ϕ(ε) isdϕ(ε)dε k1i1i2i3(k1i1i2i3 + ε)2 lt 0 so function ϕ(ε) is amonotonically decreasing function )en KL(0) 0KL(1) 044 and the maximum value of the circulatingpower is 044Pe

342 Ranges HM2 and HM4 )e speed ratio of the powertransmitting from the engine to the small sun gear is2e is

is2e ns2

ne

1 minus k3( 1113857i1i2i3 + 1 minus k2( 1113857ε

k2 minus k3( 1113857i1i2i3 i

rs2e + i

s1s2e (42)

)e speed ratios of the power transmitting from theengine to the large sun gear and ring gear ireis1e are

ire 1

is1e minusε

i1i2i3

(43)

In the PGM the speed ratios between the input elements(large sun gear s1 and ring gear r) and the output element(small sun gear s2) are

irs2s1

k2 minus 1k2 minus k3

is1s2r

1 minus k3

k2 minus k3

(44)

)en we can get

irs2e i

rs2s1

is1e minusk2 minus 1( 1113857ε

i1i2i3 k2 minus k3( 1113857

is1s2e i

s1s2rire

1 minus k3

k2 minus k3

(45)

Next two situations will be discussed as follows [17]

(1) When εlt 0 is1s2e gt 0 irs2e lt 0 and i

s1s2ei

rs2e lt 0 the PGM

has circulating power substituting equation (19) fork2 k3 i1i2i3 in equation (42) is2e is

is2e 32727 + ε

32727 minus 32727 minusk1( 1113857gt 0 (46)

Because of Pe gt 0 Pr is1ceicePe gt 0 Ps1

irceicePe lt 0the output power P is

P Pe minusPs2 Pr + Ps1

Pr minus Ps1

11138681113868111386811138681113868

11138681113868111386811138681113868 (47)

)en

Pr P + Ps1

11138681113868111386811138681113868

11138681113868111386811138681113868 P + PL

11138681113868111386811138681113868111386811138681113868 (48)

)e circulating power PL is

PL Ps1 irs2e

is2e

Pe (49)

Assume that the closed power factor KL is

KL PL

11138681113868111386811138681113868111386811138681113868

Pe

irs2e

is2e

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

minusεk1i1i2i3 minus ε

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 |ϕ(ε)| (50)

where ϕ(ε) minusεk1i1i2i3 minus ε and the derivative ofϕ(ε) is dϕ(ε)dε minusk1i1i2i3(k1i1i2i3 minus ε)2 gt 0 sofunction ϕ(ε) is a monotonically increasing function)en KL(0) 0 KL(1) 044 and the maximumvalue of the circulating power is 044Pe

(2) When εgt 0 is1s2e gt 0 irs2e gt 0 and i

s1s2ei

rs2e gt 0 the PGM

has no circulating power PL 0

35 Hydraulic Split Power Calculation Hydraulic power-split ratio ρ PHPe Ps1

Pe hydraulic power is input tothe PGM through the large sun gear s1

6 Mathematical Problems in Engineering

351 Ranges HM1 and HM3

ρ Ps1

Pe

irce

ice

ε

k1i1i2i3 + ε ϕ(ε) (51)

and as can be obtained from the above function ϕ(ε) is amonotonically decreasing function so

ρ(minus1) 0234gt 0

ρ(0) 0

ρ(1) minus044lt 0

⎧⎪⎪⎨

⎪⎪⎩(52)

352 Ranges HM2 and HM4

ρ Ps1

Pe

irs2e

is2e

minusε

k1i1i2i3 minus ε ϕ(ε) (53)

As can be obtained from the above function ϕ(ε) is amonotonically increasing function so

ρ(minus1) minus044lt 0

ρ(0) 0

ρ(1) 0234gt 0

⎧⎪⎪⎨

⎪⎪⎩(54)

According to the theoretical analysis on the circulatingpower flow and the hydraulic split power the power flowdirection of HMCVT in each working range is shown inFigure 2 )is result lays the foundation for the analysis onthe efficiency characteristic of HMCVT

36 Efficiency Characteristic )e Lrfkoes formula (17) isused to calculate the transmission efficiency of HMCVT

η minus1113957μig

minusToTe

neno

(55)

where 1113957μ is the ratio of output torque to input torque ofHMCVTconsidering power losses and ig is the transmissionratio of HMCVT

)e transmission efficiency calculation process of eachworking range of HMCVT is shown in Tables 2 and 3Among them the calculation processes of the first range andthe third range are listed in Table 2 and the calculationprocesses of the second range and the fourth range are listedin Table 3 In Tables 2 and 3 i1 minus i12 are the gear ratios of thegear pairs η1 minus η12 are the transmission efficiency of the gearpairs ηH is the hydraulic system transmission efficiency andηs1r ηs1s2

and ηrs2are the transmission efficiencies of the

large sun gear to the ring gear the large sun gear to the smallsun gear and the ring gear to the small sun gear in the PGMrespectively

4 Analysis on the Influence of Parameters onthe Transmission Efficiency of HMCVT

)e results of the transmission efficiency in each workingrange of HMCVT are as follows

ηHM1

ηs1r minus k11113872 1113873η4η8η11η12 ε + k1i1i2i3( 1113857

1 minus k1( 1113857 η1η2η3ηHηs1rε + k1i1i2i31113872 1113873 (εgt 0)

ηs1r minus k11113872 1113873η1η2η3ηHη4η8η11η12 ε + k1i1i2i3( 1113857

1 minus k1( 1113857 i1i2i3η1η2η3ηHk1 + εηs1r1113872 1113873 (εle 0)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

ηHM2

k3ηs1s2minus k2ηrs2

1113872 1113873η5η9η11η12 1 minus k3( 1113857i1i2i3 + 1 minus k2( 1113857ε1113858 1113859

k3 minus k2( 1113857 1 minus k3ηs1s21113872 1113873i1i2i3 minus η1η2η3ηHε k2ηrs2

minus 11113872 11138731113960 1113961 (εlt 0)

k3ηs1s2minus k2ηrs2

1113872 1113873η1η2η3η5η9η11η12ηH 1 minus k3( 1113857i1i2i3 + 1 minus k2( 1113857ε1113858 1113859

k3 minus k2( 1113857 1 minus k3ηs1s21113872 1113873i1i2i3η1η2η3ηH minus ε k2ηrs2

minus 11113872 11138731113960 1113961 (εge 0)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

ηHM3

ηs1r minus k11113872 1113873η4η7η11η12 ε + k1i1i2i3( 1113857

1 minus k1( 1113857 η1η2η3ηHηs1rε + k1i1i2i31113872 1113873 (εgt 0)

ηs1r minus k11113872 η1η2η3ηHη4η7η11η12 ε + k1i1i2i3( 1113857

1 minus k1( 1113857 i1i2i3η1η2η3ηHk1 + εηs1r1113872 1113873 (εle 0)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

ηHM4

k3ηs1s2minus k2ηrs2

1113872 1113873η5η6η11η12 1 minus k3( 1113857i1i2i3 + 1 minus k2( 1113857ε1113858 1113859

k3 minus k2( 1113857 1 minus k3ηs1s21113872 1113873i1i2i3 minus η1η2η3ηHε k2ηrs2

minus 11113872 11138731113960 1113961 (εlt 0)

k3ηs1s2minus k2ηrs2

1113872 1113873η1η2η3η5η6η11η12ηH 1 minus k3( 1113857i1i2i3 + 1 minus k2( 1113857ε1113858 1113859

k3 minus k2( 1113857 1 minus k3ηs1s21113872 1113873i1i2i3η1η2η3ηH minus ε k2ηrs2

minus 11113872 11138731113960 1113961 (εge 0)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(56)

Mathematical Problems in Engineering 7

When HMCVT is working in HM1 or HM3 range if thetransmission efficiencies of the gear pairs i9 and i8 are equal(η7 η8) the transmission efficiencies of HMCVT in HM1and HM3 ranges are equal

When HMCVT is working in HM2 or HM4 range if thetransmission efficiencies of the gear pairs i9 and i6 are equal(η9 η6) the transmission efficiencies of HMCVT in HM2and HM4 ranges are also equal

Due to the complexity of the above results it is necessaryto evaluate the influence of displacement ratio ε standingtransmission ratio of PGM minusk1 gear ratio i1i2i3 and hy-draulic system efficiency ηH on the transmission efficiency of

HMCVT by means of image analysis method We assumethat η1 minus η12 are 098 ηs1r is 098 ηs1s2

and ηrs2are 097 and

k1i1i2i3 is minus32727 )e evaluation results are shown inFigure 3

According to the analysis on Figure 3 the following rulesare obtained

In each range of HMCVT when the displacement ratio εis 0 the transmission efficiency reaches the maximum valueNow the hydraulic path (HST) does not transmit power andthe power is completely transmitted by the mechanical pathWhen the displacement ratio ε is minus1 or 1 the transmissionefficiency reaches the minimum value and the hydraulic

Table 2 Equation of efficiency calculation for ranges HM1 and HM3

ElementsTorque equation considering power losses

εgt 0 εlt 0HM1 HM3 HM1 HM3

Engine Te Tr + η1η2Tpi1i2 Te Tr + Tpi1i2η1η2Pump Tp εηHTm Tp εTmηH

Motor Tm η3Ts1(minusi3) Tm Ts1

(minusη3i3)PGM Ts1

+ Tr + Tc 0PGM Tr minusk1Ts1

ηs1r

Intermediate shaft Tc Tzi4i8η4η8 Tc Tzi4i7η4η7 Tc Tzi4i8η4η8 Tc Tzi4i7η4η7Output shaft Tz To(i11i12η11η12)Torque ratio 1113957μ ToTe

Efficiency η minus1113957μig

144 max

100 44 max

44 max

Engine

r

S1

S2c

100

(a)

766 min

100

100

234 max

234 max

Engine

r

S1

S2c

(b)

766 min

100 234 max

234 max

Engine

r

S1

S2c

100

(c)

144 max

100

100

44 max

44 max

Engine

r

S1

S2c

(d)

Figure 2 Power flow direction of the PGM of HMCVT (a) Range HM1HM3 εgt 0 power-circulation (b) Range HM1HM3 εlt 0 power-split (c) Range HM2HM4 εgt 0 power-split (d) Range HM2HM4 εlt 0 power-circulation

8 Mathematical Problems in Engineering

Table 3 Equation of efficiency calculation for ranges HM2 and HM4

ElementsTorque equation considering power losses

εgt 0 εlt 0HM2 HM4 HM2 HM4

Engine Te Tr + Tpi1i2η1η2 Te Tr + η1η2Tpi1i2Pump Tp εTmηH Tp εηHTm

Motor Tm Ts1(minusη3i3) Tm η3Ts1

(minusi3)

PGM Ts1+ Tr + Ts2

0PGM Tr 1 minus k3ηs1s2

k3ηs1s2minus k2ηrs2

Ts2Intermediate shaft Ts2

Tzi5i9η5η9 Ts2 Tzi5i6η5η6 Ts2

Tzi5i9η5η9 Ts2 Tzi5i6η5η6

Output shaft Tz To(i11i12η11η12)Torque ratio 1113957μ ToTe

Efficiency η minus1113957μig

ndashk1 = 25172ndashk1 = 29130

ndashk1 = 17805ndashk1 = 21875

095

09

085

08

075

07ndash1 ndash08 ndash06 ndash04 ndash02 0 02 04 06 08 1

Displacement ratio

ηH = 081

Effici

ency

(a)

ndashk1 = 25172ndashk1 = 29130

ndashk1 = 17805ndashk1 = 21875

ηH = 081

095

09

085

08

075

07ndash1 ndash08 ndash06 ndash04 ndash02 0 02 04 06 08 1

Displacement ratio

Effici

ency

(b)

ηH = 081ηH = 090

ηH = 064ηH = 072

095

09

085

08

075

07ndash1 ndash08 ndash06 ndash04 ndash02 0 02 04 06 08 1

Displacement ratio

Effici

ency

ndashk1 = 21875

(c)

ηH = 081ηH = 090

ηH = 064ηH = 072

095

09

085

08

075

07ndash1 ndash08 ndash06 ndash04 ndash02 0 02 04 06 08 1

Displacement ratio

Effici

ency

ndashk1 = 21875

(d)

Figure 3 Continued

Mathematical Problems in Engineering 9

path transmits more power at this time resulting in a loweroverall efficiency of HMCVT

In each range of HMCVT the transmission efficiencyvalue in the power circulation phase is lower than thetransmission efficiency value in the power split phase whichindicates that the power circulation will cause the trans-mission efficiency of HMCVT to decrease

)e larger the value of the PGM standing transmissionratio minusk1 the higher the transmission efficiency of HMCVTbut the influence on improving the transmission efficiency is

not obvious )e PGM standing transmission ratio minusk1ranges from 15 to 3 )e gear ratio i1i2i3 and the PGMstanding transmission ratio minusk1 are in an inversely pro-portional relationship the smaller the value of the gear ratioi1i2i3 the higher the transmission efficiency of HMCVT andthe influence on improving the transmission efficiency isalso not obvious )e gear ratio i1i2i3 ranges from 11 to 22

)e greater the value of HSTefficiency ηH the higher theoverall transmission efficiency of HMCVT )e influence ofηH on improving transmission efficiency of HMCVT issignificant

i1i2i3 = 13001i1i2i3 = 11234

i1i2i3 = 18381i1i2i3 = 14961

ηH = 081

095

09

085

08

075

07ndash1 ndash08 ndash06 ndash04 ndash02 0 02 04 06 08 1

Displacement ratio

Effici

ency

(e)

i1i2i3 = 13001i1i2i3 = 11234

i1i2i3 = 18381i1i2i3 = 14961

ηH = 081

095

09

085

08

075

07ndash1 ndash08 ndash06 ndash04 ndash02 0 02 04 06 08 1

Displacement ratio

Effici

ency

(f )

Figure 3)e influence of parameters on the transmission efficiency of HMCVT (a))e influence of minusk1 on HM1 andHM3 ranges (b))einfluence of minusk1 on HM2 and HM4 ranges (c) )e influence of ηH on HM1 and HM3 ranges (d) )e influence of ηH on HM2 and HM4ranges (e) )e influence of i1i2i3 on HM1 and HM3 ranges (f ) )e influence of i1i2i3 on HM2 and HM4 ranges

9

8

7

6

5

4

3

2

1

0

Tran

smiss

ion

ratio

ndash15 ndash1 ndash05 0 05 1 15Displacement ratio

HM1HM2

HM3HM4

Figure 4 Relationship between transmission ratio and displace-ment ratio

60

50

40

30

20

10

0

Trac

tor s

peed

(km

h)

ndash15 ndash1 ndash05 0 05 1 15Displacement ratio

HM1HM2

HM3HM4

Figure 5 )e curve of the tractor speed

10 Mathematical Problems in Engineering

5 Analysis on the Transmission CharacteristicsResults of HMCVT

According to the previous content the value of the PGMstanding transmission ratio minusk1 is set to 21875 and the HSTefficiency value ηH is set to 081 then the other parameters ofHMCVT are k2 02714 minusk3 05938 i1 i2 1 i3 15i4 i5 1 i6 166 i7 116 i8 410 i9 584 andi10 i11 i12 1

)e relationship between the transmission ratio anddisplacement ratio is shown in Figure 4 )e curve of thetractor speed is shown in Figure 5 In the first range thedisplacement ratio changes from 1 to minus1 the transmissionratio decreases from 862 to 459 and the tractor speedincreases from 0 to 750 kmh In the second range thedisplacement ratio changes from minus1 to 1 the transmissionratio decreases from 459 to 243 and the tractor speedincreases from 750 kmh to 1412 kmh In the third rangethe displacement ratio changes from 1 to minus1 the trans-mission ratio decreases from 243 to 130 and the tractorspeed increases from 1412 kmh to 2644 kmh In thefourth range the displacement ratio changes from minus1 to 1the transmission ratio decreases from 130 to 069 and thetractor speed increases from 2644 kmh to 50 kmh )eresults show that the stepless speed regulation of HMCVTin the tractor speed range of 0ndash50 kmh can be realized bythe coordinated control of the HST displacement ratio andthe clutches

)e ratios of the output shaft torque to the input shafttorque in each range of HMCVT are shown in Figure 6(a))e continuous change of torque can be realized in the entireworking ranges )e ratios of the motor shaft torque to theoutput shaft torque in each range of HMCVT are shown inFigure 6(b) )e torque ratio remains constant in each range

and varies between different ranges )e values of torqueratios in the four ranges are 514 minus972 1849 andminus3518 respectively

)e relationship between the tractor speed and theoverall transmission efficiency of HMCVT is shown inFigure 7 Each of the four ranges has a transmission effi-ciency maximum value of 0917 0902 0917 and 0902respectively )e efficiency value always increases first withthe increase of the tractor speed and then decreases afterreaching the maximum value )e maximum transmissionefficiency in each range is obtained when the HST dis-placement ratio is 0 Now the hydraulic path does nottransmit power and all power is transmitted by the me-chanical path )e transmission efficiency of HMCVT ishigher than 83 so the overall transmission efficiency is at ahigh level

ndash2

ndash4

ndash6

ndash8

ndash10

0

0 5 10 15 20 25 30 35 40 45 50

Torq

ue ra

tio (T

oTe

)

Tractor speed (kmh)

HM1HM2

HM3HM4

(a)

05

04

03

02

01

0

ndash01

ndash02

ndash03

ndash04

ndash05

Torq

ue ra

tio (T

mT

o)

0 5 10 15 20 25 30 35 40 45 50Tractor speed (kmh)

HM1HM2

HM3HM4

(b)

Figure 6)e curve of torque characteristic (a) Ratios of output shaft torque to input shaft torque (b) Ratios of motor shaft torque to outputshaft torque

095

09

085

08

075

07

Effici

ency

0 5 10 15 20 25 30 35 40 45 50Tractor speed (kmh)

HM1HM2

HM3HM4

Figure 7 Overall transmission efficiency of HMCVT

Mathematical Problems in Engineering 11

6 Verification of HMCVT Simulation Model

)e modeling and simulation of HMCVT are accom-plished by using Amesim simulation softwareAccording to Figure 1 a simulation model of HMCVTis established as shown in Figure 8

61 Hydraulic System )e hydraulic system is mainlycomposed of a variable displacement hydraulic pump afixed displacement hydraulic motor two relief valves twocheck valves and a charge hydraulic pump with a reliefvalve )e aim of the charge pump is to compensate for theleakage loss in the pipeline and to establish the lowestpressure in the pumprsquos suction pipeline Two relief valves areused to limit the maximum pressure in the system pipingand its adjustment pressure should be above the normalworking pressure of the system Volumetric losses of the

pump and motor are neglected and their hydromechanicalefficiencies are set at 09

62 Mechanical Components Mechanical components areassumed to be constant efficiency components throughoutthe transmission )e power loss of each gear pair is 2 sothe efficiency is set to 098 In the Ravigneaux planetary gearmechanism the efficiencies of ring gear-long planetary gearcouple long planetary gear-large sun gear couple longplanetary gear-short planetary gear couple and shortplanetary gear-small sun gear couple are set at 099 )emain reducer differential and wheel reducer are combinedand modeled as a gear reducer with an efficiency of 0965 Tosimplify the model the friction losses of the shaft gears andbearings were ignored during the simulation modelingprocess

Efficiency calculation unit

Speed calculation unit

Small sun gear

Short planet gear

Long planet gear

Large sun gear

Ring gear

i1

i2

Pump Motor

i3

i5

i4

i7 i6

i9i8

Carrier

C1

C2

C3

C4

ω ωT

TL

ωωT

T

L

PP

MOT

f (x y)

y

x

X

k

Figure 8 HMCVT simulation model

(km

h)

50

40

30

20

10

00 5 10 15 20 25 30 35

X Time (s)

Tractor linear velocity (kmh)

(a)

0 5 10 15 20 25 30 35X Time (s)

10

08

06

04

02

00

(nul

l)

Transmission efficiency of HMCVT (null)

(b)

Figure 9 Simulation results (a) Simulation result of the tractor speed (b) Simulation result of the transmission efficiency of HMCVT

12 Mathematical Problems in Engineering

63Model Control and Simulation Results )e engine speedis assumed to have a constant value of 2200 rmin the loadtorque of the output shaft of HMCVT is set to 100Nm themodel simulation time is 32 s including range HM1 0ndash8 srange HM2 8ndash16 s range HM3 16ndash24 s and range HM424ndash32 s

)e simulation results are shown in Figure 9 through thevariable pump displacement and coordinated control of theclutches [18 19] the transmission gradually changes fromrange HM1 to range HM4 and the tractor achieves steplessspeed change from 0 to 50 kmh )e simulation results areconsistent with the previous theoretical analysis results whichshow that the 4-range HMCVT transmission scheme pro-posed in this paper is suitable for matching wheeled tractors

64 Comparison Xu LYrsquos transmission scheme [15] hasbeen applied to mass production tractors )is transmissionscheme has 6 working ranges in the forward direction and 3working ranges in the reverse direction which is controlledby 8 clutches )e speed range in the forward direction is0sim30 kmh while the reverse direction is only 0sim10 kmh)e transmission efficiency of the hydraulic range is low (lessthan 80) the transmission efficiency of the hydrome-chanical ranges is high (higher than 90) but is only in theforward direction

)e scheme proposed in this paper has 4 working rangesand a speed range of 0sim50 kmh in the forward and reversedirections which is controlled by 6 clutches )e fewer thetransmission ranges the less switching during the workingranges so the transmission performance is improved )esmaller number of clutches ensures that the system is easierto control )e transmission efficiencies in the forward andreverse directions are close the overall forward transmissionefficiency is higher than 83 and the highest transmissionefficiency is higher than 91

)erefore the application of HMCVT proposed in thispaper is wider it can not only match conventional tractorsbut also match two-way operation tractors and other en-gineering vehicles

7 Conclusions

)is paper proposes a new type of transmission scheme ofHMCVT )e Ravigneaux PGM and the HSTare connectedin parallel to form a closed loop and then connected in serieswith the range switching mechanism )is new HMCVTfeatures a double-row PGM with 4 hydromechanicalworking ranges By coordinating control of the HST dis-placement ratio and the engagement conditions of shiftingclutches the stepless speed regulation of HMCVT at thetractor speed of 0ndash50 kmh is realized

According to the theoretical analysis on the newHMCVT the speed characteristic curve and the torquecharacteristic curve are obtained Based on the power flowdirection of HMCVT the transmission efficiency curve isobtained by using the Lrfkoes formula

Using image analysis method to study the influence offour parameters (displacement ratio ε PGM standing

transmission ratio minusk1 gear ratio i1i2i3 and hydraulic systemefficiency ηH) on the transmission efficiency of HMCVT theresults show that the standing transmission ratio minusk1 andgear ratio i1i2i3 have little effect on the transmission effi-ciency whereas the displacement ratio ε and hydraulicsystem efficiency ηH have a greater impact on the trans-mission efficiency

)e parameters of the new HMCVT are finally deter-mined )e results of the transmission characteristics showthat the new HMCVTproposed in this paper has the steplessspeed regulation capability for tractors which can alsocontinuously transmit and change the torque )e trans-mission efficiency of HMCVT is at a high level

In order to verify the theoretical derivation Amesimsimulation software is used for the modeling and simulationof HMCVT )e simulation results are consistent with thetheoretical analysis results )erefore we believe that thetransmission scheme is reasonable and effective for tractors

Data Availability

)e Simcenter Amesim System File data used to support thefindings of this study are available from the correspondingauthor upon request

Conflicts of Interest

)e authors declare that they have no conflicts of interest

Acknowledgments

)is research was financially supported by the National KeyResearch and Development Plan Project of China(2016YFD0700402) and Changzhou Sci amp Tech Program(no CE20185037)

References

[1] L Y Xu Z L Zhou M Z Zhang et al ldquoResearch and designof hydro-mechanical continuously variable transmission fortractorsrdquo Journal of Northeast Agricultural University vol 13no 2 pp 182ndash186 2006

[2] R Paoluzzi and L G Zarotti ldquo)e minimum size of hy-drostatic transmissions for locomotionrdquo Journal of Terra-mechanics vol 50 no 3 pp 153ndash164 2013

[3] J Kim J Kang Y Kim B Min and H Kim ldquoDesign of powersplit transmission design of dual mode power split trans-missionrdquo International Journal of Automotive Technologyvol 11 no 4 pp 565ndash571 2010

[4] A Kim and A Rossetti ldquoFuel consumption reduction inurban buses by using power split transmissionsrdquo EnergyConversion and Management vol 71 pp 159ndash171 2013

[5] A Rossetti and A Macor ldquoMulti-objective optimization ofhydro-mechanical power split transmissionsrdquo Mechanismand Machine lteory vol 62 pp 112ndash128 2013

[6] A Macor and A Rossetti ldquoOptimization of hydro-mechanicalpower split transmissionsrdquo Mechanism and Machine lteoryvol 46 no 12 pp 1901ndash1919 2011

[7] S Shamshirband D Petkovic A Amini et al ldquoSupport vectorregression methodology for wind turbine reaction torqueprediction with power-split hydrostatic continuous variabletransmissionrdquo Energy vol 67 pp 623ndash630 2014

Mathematical Problems in Engineering 13

[8] K T Anuar ldquoResch R Continuously variable tractor trans-missionsrdquo ASAE Distinguished Lecture No 29 pp 1ndash37 StJoseph American Society of Agricultural Engineers St JosephMI USA 2005

[9] S H Yuan and J B Hu ldquo)e efficiency of multi-range hydro-mechanical stepless transmissionrdquo Journal of Beijing Instituteof Technology vol 7 no 2 pp 129ndash133 1998

[10] X J LiuAnalysis of Vehicle Transmission System pp 258-259National Defence Industry Press Beijing China 1998

[11] M M Brenninger ldquoFendt vario CVT in agricultural tractorsrdquoSAE International Warrendale PA USA SAE TechnicalPaper 2007-01-4205 2007

[12] H Aitzetmuller ldquoSteyr S-Matic-the future CVT systemrdquo inProceedings of the Seoul 2000 FISITA world automotive con-gress pp 1ndash6 Seoul South Korea June 2000

[13] J Pohlenz and K Grad ldquoCVTfor input power above 300 kWrdquoVDI-berichte vol 2007 pp 59ndash65 2001

[14] J Pohlenz and K Grad ldquoCVT-System fur denGroszligflacheneinsatzrdquoVDI-berichte vol 2004 pp 23ndash33 1855

[15] L Y Xu Study on Characteristics of Hydro-MechanicalContinuously Variable Transmission of Tractor Xirsquoan Uni-versity of Technology Xirsquoan China 2007

[16] J Looman Zahnradgetriebe-Grundlagen KonstruktionenAnwendungen in Fahrzeugen Springer Berlin Germany2009

[17] Z Rao Planetary Gear Transmission Design Chemical In-dustry Press Beijing China 2003

[18] X Xu Y Liang M Jordan T Peter and D Peng ldquoOptimizedcontrol of engine start assisted by the disconnect clutch in a P2hybrid automatic transmissionrdquo Mechanical Systems andSignal Processing vol 124 2019

[19] S Wang Y Liu Z Wang et al ldquoAdaptive fuzzy iterativecontrol strategy for the wet-clutch filling of automatictransmissionrdquo Mechanical Systems and Signal Processingvol 130 pp 164ndash182 2019

14 Mathematical Problems in Engineering

351 Ranges HM1 and HM3

ρ Ps1

Pe

irce

ice

ε

k1i1i2i3 + ε ϕ(ε) (51)

and as can be obtained from the above function ϕ(ε) is amonotonically decreasing function so

ρ(minus1) 0234gt 0

ρ(0) 0

ρ(1) minus044lt 0

⎧⎪⎪⎨

⎪⎪⎩(52)

352 Ranges HM2 and HM4

ρ Ps1

Pe

irs2e

is2e

minusε

k1i1i2i3 minus ε ϕ(ε) (53)

As can be obtained from the above function ϕ(ε) is amonotonically increasing function so

ρ(minus1) minus044lt 0

ρ(0) 0

ρ(1) 0234gt 0

⎧⎪⎪⎨

⎪⎪⎩(54)

According to the theoretical analysis on the circulatingpower flow and the hydraulic split power the power flowdirection of HMCVT in each working range is shown inFigure 2 )is result lays the foundation for the analysis onthe efficiency characteristic of HMCVT

36 Efficiency Characteristic )e Lrfkoes formula (17) isused to calculate the transmission efficiency of HMCVT

η minus1113957μig

minusToTe

neno

(55)

where 1113957μ is the ratio of output torque to input torque ofHMCVTconsidering power losses and ig is the transmissionratio of HMCVT

)e transmission efficiency calculation process of eachworking range of HMCVT is shown in Tables 2 and 3Among them the calculation processes of the first range andthe third range are listed in Table 2 and the calculationprocesses of the second range and the fourth range are listedin Table 3 In Tables 2 and 3 i1 minus i12 are the gear ratios of thegear pairs η1 minus η12 are the transmission efficiency of the gearpairs ηH is the hydraulic system transmission efficiency andηs1r ηs1s2

and ηrs2are the transmission efficiencies of the

large sun gear to the ring gear the large sun gear to the smallsun gear and the ring gear to the small sun gear in the PGMrespectively

4 Analysis on the Influence of Parameters onthe Transmission Efficiency of HMCVT

)e results of the transmission efficiency in each workingrange of HMCVT are as follows

ηHM1

ηs1r minus k11113872 1113873η4η8η11η12 ε + k1i1i2i3( 1113857

1 minus k1( 1113857 η1η2η3ηHηs1rε + k1i1i2i31113872 1113873 (εgt 0)

ηs1r minus k11113872 1113873η1η2η3ηHη4η8η11η12 ε + k1i1i2i3( 1113857

1 minus k1( 1113857 i1i2i3η1η2η3ηHk1 + εηs1r1113872 1113873 (εle 0)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

ηHM2

k3ηs1s2minus k2ηrs2

1113872 1113873η5η9η11η12 1 minus k3( 1113857i1i2i3 + 1 minus k2( 1113857ε1113858 1113859

k3 minus k2( 1113857 1 minus k3ηs1s21113872 1113873i1i2i3 minus η1η2η3ηHε k2ηrs2

minus 11113872 11138731113960 1113961 (εlt 0)

k3ηs1s2minus k2ηrs2

1113872 1113873η1η2η3η5η9η11η12ηH 1 minus k3( 1113857i1i2i3 + 1 minus k2( 1113857ε1113858 1113859

k3 minus k2( 1113857 1 minus k3ηs1s21113872 1113873i1i2i3η1η2η3ηH minus ε k2ηrs2

minus 11113872 11138731113960 1113961 (εge 0)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

ηHM3

ηs1r minus k11113872 1113873η4η7η11η12 ε + k1i1i2i3( 1113857

1 minus k1( 1113857 η1η2η3ηHηs1rε + k1i1i2i31113872 1113873 (εgt 0)

ηs1r minus k11113872 η1η2η3ηHη4η7η11η12 ε + k1i1i2i3( 1113857

1 minus k1( 1113857 i1i2i3η1η2η3ηHk1 + εηs1r1113872 1113873 (εle 0)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

ηHM4

k3ηs1s2minus k2ηrs2

1113872 1113873η5η6η11η12 1 minus k3( 1113857i1i2i3 + 1 minus k2( 1113857ε1113858 1113859

k3 minus k2( 1113857 1 minus k3ηs1s21113872 1113873i1i2i3 minus η1η2η3ηHε k2ηrs2

minus 11113872 11138731113960 1113961 (εlt 0)

k3ηs1s2minus k2ηrs2

1113872 1113873η1η2η3η5η6η11η12ηH 1 minus k3( 1113857i1i2i3 + 1 minus k2( 1113857ε1113858 1113859

k3 minus k2( 1113857 1 minus k3ηs1s21113872 1113873i1i2i3η1η2η3ηH minus ε k2ηrs2

minus 11113872 11138731113960 1113961 (εge 0)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(56)

Mathematical Problems in Engineering 7

When HMCVT is working in HM1 or HM3 range if thetransmission efficiencies of the gear pairs i9 and i8 are equal(η7 η8) the transmission efficiencies of HMCVT in HM1and HM3 ranges are equal

When HMCVT is working in HM2 or HM4 range if thetransmission efficiencies of the gear pairs i9 and i6 are equal(η9 η6) the transmission efficiencies of HMCVT in HM2and HM4 ranges are also equal

Due to the complexity of the above results it is necessaryto evaluate the influence of displacement ratio ε standingtransmission ratio of PGM minusk1 gear ratio i1i2i3 and hy-draulic system efficiency ηH on the transmission efficiency of

HMCVT by means of image analysis method We assumethat η1 minus η12 are 098 ηs1r is 098 ηs1s2

and ηrs2are 097 and

k1i1i2i3 is minus32727 )e evaluation results are shown inFigure 3

According to the analysis on Figure 3 the following rulesare obtained

In each range of HMCVT when the displacement ratio εis 0 the transmission efficiency reaches the maximum valueNow the hydraulic path (HST) does not transmit power andthe power is completely transmitted by the mechanical pathWhen the displacement ratio ε is minus1 or 1 the transmissionefficiency reaches the minimum value and the hydraulic

Table 2 Equation of efficiency calculation for ranges HM1 and HM3

ElementsTorque equation considering power losses

εgt 0 εlt 0HM1 HM3 HM1 HM3

Engine Te Tr + η1η2Tpi1i2 Te Tr + Tpi1i2η1η2Pump Tp εηHTm Tp εTmηH

Motor Tm η3Ts1(minusi3) Tm Ts1

(minusη3i3)PGM Ts1

+ Tr + Tc 0PGM Tr minusk1Ts1

ηs1r

Intermediate shaft Tc Tzi4i8η4η8 Tc Tzi4i7η4η7 Tc Tzi4i8η4η8 Tc Tzi4i7η4η7Output shaft Tz To(i11i12η11η12)Torque ratio 1113957μ ToTe

Efficiency η minus1113957μig

144 max

100 44 max

44 max

Engine

r

S1

S2c

100

(a)

766 min

100

100

234 max

234 max

Engine

r

S1

S2c

(b)

766 min

100 234 max

234 max

Engine

r

S1

S2c

100

(c)

144 max

100

100

44 max

44 max

Engine

r

S1

S2c

(d)

Figure 2 Power flow direction of the PGM of HMCVT (a) Range HM1HM3 εgt 0 power-circulation (b) Range HM1HM3 εlt 0 power-split (c) Range HM2HM4 εgt 0 power-split (d) Range HM2HM4 εlt 0 power-circulation

8 Mathematical Problems in Engineering

Table 3 Equation of efficiency calculation for ranges HM2 and HM4

ElementsTorque equation considering power losses

εgt 0 εlt 0HM2 HM4 HM2 HM4

Engine Te Tr + Tpi1i2η1η2 Te Tr + η1η2Tpi1i2Pump Tp εTmηH Tp εηHTm

Motor Tm Ts1(minusη3i3) Tm η3Ts1

(minusi3)

PGM Ts1+ Tr + Ts2

0PGM Tr 1 minus k3ηs1s2

k3ηs1s2minus k2ηrs2

Ts2Intermediate shaft Ts2

Tzi5i9η5η9 Ts2 Tzi5i6η5η6 Ts2

Tzi5i9η5η9 Ts2 Tzi5i6η5η6

Output shaft Tz To(i11i12η11η12)Torque ratio 1113957μ ToTe

Efficiency η minus1113957μig

ndashk1 = 25172ndashk1 = 29130

ndashk1 = 17805ndashk1 = 21875

095

09

085

08

075

07ndash1 ndash08 ndash06 ndash04 ndash02 0 02 04 06 08 1

Displacement ratio

ηH = 081

Effici

ency

(a)

ndashk1 = 25172ndashk1 = 29130

ndashk1 = 17805ndashk1 = 21875

ηH = 081

095

09

085

08

075

07ndash1 ndash08 ndash06 ndash04 ndash02 0 02 04 06 08 1

Displacement ratio

Effici

ency

(b)

ηH = 081ηH = 090

ηH = 064ηH = 072

095

09

085

08

075

07ndash1 ndash08 ndash06 ndash04 ndash02 0 02 04 06 08 1

Displacement ratio

Effici

ency

ndashk1 = 21875

(c)

ηH = 081ηH = 090

ηH = 064ηH = 072

095

09

085

08

075

07ndash1 ndash08 ndash06 ndash04 ndash02 0 02 04 06 08 1

Displacement ratio

Effici

ency

ndashk1 = 21875

(d)

Figure 3 Continued

Mathematical Problems in Engineering 9

path transmits more power at this time resulting in a loweroverall efficiency of HMCVT

In each range of HMCVT the transmission efficiencyvalue in the power circulation phase is lower than thetransmission efficiency value in the power split phase whichindicates that the power circulation will cause the trans-mission efficiency of HMCVT to decrease

)e larger the value of the PGM standing transmissionratio minusk1 the higher the transmission efficiency of HMCVTbut the influence on improving the transmission efficiency is

not obvious )e PGM standing transmission ratio minusk1ranges from 15 to 3 )e gear ratio i1i2i3 and the PGMstanding transmission ratio minusk1 are in an inversely pro-portional relationship the smaller the value of the gear ratioi1i2i3 the higher the transmission efficiency of HMCVT andthe influence on improving the transmission efficiency isalso not obvious )e gear ratio i1i2i3 ranges from 11 to 22

)e greater the value of HSTefficiency ηH the higher theoverall transmission efficiency of HMCVT )e influence ofηH on improving transmission efficiency of HMCVT issignificant

i1i2i3 = 13001i1i2i3 = 11234

i1i2i3 = 18381i1i2i3 = 14961

ηH = 081

095

09

085

08

075

07ndash1 ndash08 ndash06 ndash04 ndash02 0 02 04 06 08 1

Displacement ratio

Effici

ency

(e)

i1i2i3 = 13001i1i2i3 = 11234

i1i2i3 = 18381i1i2i3 = 14961

ηH = 081

095

09

085

08

075

07ndash1 ndash08 ndash06 ndash04 ndash02 0 02 04 06 08 1

Displacement ratio

Effici

ency

(f )

Figure 3)e influence of parameters on the transmission efficiency of HMCVT (a))e influence of minusk1 on HM1 andHM3 ranges (b))einfluence of minusk1 on HM2 and HM4 ranges (c) )e influence of ηH on HM1 and HM3 ranges (d) )e influence of ηH on HM2 and HM4ranges (e) )e influence of i1i2i3 on HM1 and HM3 ranges (f ) )e influence of i1i2i3 on HM2 and HM4 ranges

9

8

7

6

5

4

3

2

1

0

Tran

smiss

ion

ratio

ndash15 ndash1 ndash05 0 05 1 15Displacement ratio

HM1HM2

HM3HM4

Figure 4 Relationship between transmission ratio and displace-ment ratio

60

50

40

30

20

10

0

Trac

tor s

peed

(km

h)

ndash15 ndash1 ndash05 0 05 1 15Displacement ratio

HM1HM2

HM3HM4

Figure 5 )e curve of the tractor speed

10 Mathematical Problems in Engineering

5 Analysis on the Transmission CharacteristicsResults of HMCVT

According to the previous content the value of the PGMstanding transmission ratio minusk1 is set to 21875 and the HSTefficiency value ηH is set to 081 then the other parameters ofHMCVT are k2 02714 minusk3 05938 i1 i2 1 i3 15i4 i5 1 i6 166 i7 116 i8 410 i9 584 andi10 i11 i12 1

)e relationship between the transmission ratio anddisplacement ratio is shown in Figure 4 )e curve of thetractor speed is shown in Figure 5 In the first range thedisplacement ratio changes from 1 to minus1 the transmissionratio decreases from 862 to 459 and the tractor speedincreases from 0 to 750 kmh In the second range thedisplacement ratio changes from minus1 to 1 the transmissionratio decreases from 459 to 243 and the tractor speedincreases from 750 kmh to 1412 kmh In the third rangethe displacement ratio changes from 1 to minus1 the trans-mission ratio decreases from 243 to 130 and the tractorspeed increases from 1412 kmh to 2644 kmh In thefourth range the displacement ratio changes from minus1 to 1the transmission ratio decreases from 130 to 069 and thetractor speed increases from 2644 kmh to 50 kmh )eresults show that the stepless speed regulation of HMCVTin the tractor speed range of 0ndash50 kmh can be realized bythe coordinated control of the HST displacement ratio andthe clutches

)e ratios of the output shaft torque to the input shafttorque in each range of HMCVT are shown in Figure 6(a))e continuous change of torque can be realized in the entireworking ranges )e ratios of the motor shaft torque to theoutput shaft torque in each range of HMCVT are shown inFigure 6(b) )e torque ratio remains constant in each range

and varies between different ranges )e values of torqueratios in the four ranges are 514 minus972 1849 andminus3518 respectively

)e relationship between the tractor speed and theoverall transmission efficiency of HMCVT is shown inFigure 7 Each of the four ranges has a transmission effi-ciency maximum value of 0917 0902 0917 and 0902respectively )e efficiency value always increases first withthe increase of the tractor speed and then decreases afterreaching the maximum value )e maximum transmissionefficiency in each range is obtained when the HST dis-placement ratio is 0 Now the hydraulic path does nottransmit power and all power is transmitted by the me-chanical path )e transmission efficiency of HMCVT ishigher than 83 so the overall transmission efficiency is at ahigh level

ndash2

ndash4

ndash6

ndash8

ndash10

0

0 5 10 15 20 25 30 35 40 45 50

Torq

ue ra

tio (T

oTe

)

Tractor speed (kmh)

HM1HM2

HM3HM4

(a)

05

04

03

02

01

0

ndash01

ndash02

ndash03

ndash04

ndash05

Torq

ue ra

tio (T

mT

o)

0 5 10 15 20 25 30 35 40 45 50Tractor speed (kmh)

HM1HM2

HM3HM4

(b)

Figure 6)e curve of torque characteristic (a) Ratios of output shaft torque to input shaft torque (b) Ratios of motor shaft torque to outputshaft torque

095

09

085

08

075

07

Effici

ency

0 5 10 15 20 25 30 35 40 45 50Tractor speed (kmh)

HM1HM2

HM3HM4

Figure 7 Overall transmission efficiency of HMCVT

Mathematical Problems in Engineering 11

6 Verification of HMCVT Simulation Model

)e modeling and simulation of HMCVT are accom-plished by using Amesim simulation softwareAccording to Figure 1 a simulation model of HMCVTis established as shown in Figure 8

61 Hydraulic System )e hydraulic system is mainlycomposed of a variable displacement hydraulic pump afixed displacement hydraulic motor two relief valves twocheck valves and a charge hydraulic pump with a reliefvalve )e aim of the charge pump is to compensate for theleakage loss in the pipeline and to establish the lowestpressure in the pumprsquos suction pipeline Two relief valves areused to limit the maximum pressure in the system pipingand its adjustment pressure should be above the normalworking pressure of the system Volumetric losses of the

pump and motor are neglected and their hydromechanicalefficiencies are set at 09

62 Mechanical Components Mechanical components areassumed to be constant efficiency components throughoutthe transmission )e power loss of each gear pair is 2 sothe efficiency is set to 098 In the Ravigneaux planetary gearmechanism the efficiencies of ring gear-long planetary gearcouple long planetary gear-large sun gear couple longplanetary gear-short planetary gear couple and shortplanetary gear-small sun gear couple are set at 099 )emain reducer differential and wheel reducer are combinedand modeled as a gear reducer with an efficiency of 0965 Tosimplify the model the friction losses of the shaft gears andbearings were ignored during the simulation modelingprocess

Efficiency calculation unit

Speed calculation unit

Small sun gear

Short planet gear

Long planet gear

Large sun gear

Ring gear

i1

i2

Pump Motor

i3

i5

i4

i7 i6

i9i8

Carrier

C1

C2

C3

C4

ω ωT

TL

ωωT

T

L

PP

MOT

f (x y)

y

x

X

k

Figure 8 HMCVT simulation model

(km

h)

50

40

30

20

10

00 5 10 15 20 25 30 35

X Time (s)

Tractor linear velocity (kmh)

(a)

0 5 10 15 20 25 30 35X Time (s)

10

08

06

04

02

00

(nul

l)

Transmission efficiency of HMCVT (null)

(b)

Figure 9 Simulation results (a) Simulation result of the tractor speed (b) Simulation result of the transmission efficiency of HMCVT

12 Mathematical Problems in Engineering

63Model Control and Simulation Results )e engine speedis assumed to have a constant value of 2200 rmin the loadtorque of the output shaft of HMCVT is set to 100Nm themodel simulation time is 32 s including range HM1 0ndash8 srange HM2 8ndash16 s range HM3 16ndash24 s and range HM424ndash32 s

)e simulation results are shown in Figure 9 through thevariable pump displacement and coordinated control of theclutches [18 19] the transmission gradually changes fromrange HM1 to range HM4 and the tractor achieves steplessspeed change from 0 to 50 kmh )e simulation results areconsistent with the previous theoretical analysis results whichshow that the 4-range HMCVT transmission scheme pro-posed in this paper is suitable for matching wheeled tractors

64 Comparison Xu LYrsquos transmission scheme [15] hasbeen applied to mass production tractors )is transmissionscheme has 6 working ranges in the forward direction and 3working ranges in the reverse direction which is controlledby 8 clutches )e speed range in the forward direction is0sim30 kmh while the reverse direction is only 0sim10 kmh)e transmission efficiency of the hydraulic range is low (lessthan 80) the transmission efficiency of the hydrome-chanical ranges is high (higher than 90) but is only in theforward direction

)e scheme proposed in this paper has 4 working rangesand a speed range of 0sim50 kmh in the forward and reversedirections which is controlled by 6 clutches )e fewer thetransmission ranges the less switching during the workingranges so the transmission performance is improved )esmaller number of clutches ensures that the system is easierto control )e transmission efficiencies in the forward andreverse directions are close the overall forward transmissionefficiency is higher than 83 and the highest transmissionefficiency is higher than 91

)erefore the application of HMCVT proposed in thispaper is wider it can not only match conventional tractorsbut also match two-way operation tractors and other en-gineering vehicles

7 Conclusions

)is paper proposes a new type of transmission scheme ofHMCVT )e Ravigneaux PGM and the HSTare connectedin parallel to form a closed loop and then connected in serieswith the range switching mechanism )is new HMCVTfeatures a double-row PGM with 4 hydromechanicalworking ranges By coordinating control of the HST dis-placement ratio and the engagement conditions of shiftingclutches the stepless speed regulation of HMCVT at thetractor speed of 0ndash50 kmh is realized

According to the theoretical analysis on the newHMCVT the speed characteristic curve and the torquecharacteristic curve are obtained Based on the power flowdirection of HMCVT the transmission efficiency curve isobtained by using the Lrfkoes formula

Using image analysis method to study the influence offour parameters (displacement ratio ε PGM standing

transmission ratio minusk1 gear ratio i1i2i3 and hydraulic systemefficiency ηH) on the transmission efficiency of HMCVT theresults show that the standing transmission ratio minusk1 andgear ratio i1i2i3 have little effect on the transmission effi-ciency whereas the displacement ratio ε and hydraulicsystem efficiency ηH have a greater impact on the trans-mission efficiency

)e parameters of the new HMCVT are finally deter-mined )e results of the transmission characteristics showthat the new HMCVTproposed in this paper has the steplessspeed regulation capability for tractors which can alsocontinuously transmit and change the torque )e trans-mission efficiency of HMCVT is at a high level

In order to verify the theoretical derivation Amesimsimulation software is used for the modeling and simulationof HMCVT )e simulation results are consistent with thetheoretical analysis results )erefore we believe that thetransmission scheme is reasonable and effective for tractors

Data Availability

)e Simcenter Amesim System File data used to support thefindings of this study are available from the correspondingauthor upon request

Conflicts of Interest

)e authors declare that they have no conflicts of interest

Acknowledgments

)is research was financially supported by the National KeyResearch and Development Plan Project of China(2016YFD0700402) and Changzhou Sci amp Tech Program(no CE20185037)

References

[1] L Y Xu Z L Zhou M Z Zhang et al ldquoResearch and designof hydro-mechanical continuously variable transmission fortractorsrdquo Journal of Northeast Agricultural University vol 13no 2 pp 182ndash186 2006

[2] R Paoluzzi and L G Zarotti ldquo)e minimum size of hy-drostatic transmissions for locomotionrdquo Journal of Terra-mechanics vol 50 no 3 pp 153ndash164 2013

[3] J Kim J Kang Y Kim B Min and H Kim ldquoDesign of powersplit transmission design of dual mode power split trans-missionrdquo International Journal of Automotive Technologyvol 11 no 4 pp 565ndash571 2010

[4] A Kim and A Rossetti ldquoFuel consumption reduction inurban buses by using power split transmissionsrdquo EnergyConversion and Management vol 71 pp 159ndash171 2013

[5] A Rossetti and A Macor ldquoMulti-objective optimization ofhydro-mechanical power split transmissionsrdquo Mechanismand Machine lteory vol 62 pp 112ndash128 2013

[6] A Macor and A Rossetti ldquoOptimization of hydro-mechanicalpower split transmissionsrdquo Mechanism and Machine lteoryvol 46 no 12 pp 1901ndash1919 2011

[7] S Shamshirband D Petkovic A Amini et al ldquoSupport vectorregression methodology for wind turbine reaction torqueprediction with power-split hydrostatic continuous variabletransmissionrdquo Energy vol 67 pp 623ndash630 2014

Mathematical Problems in Engineering 13

[8] K T Anuar ldquoResch R Continuously variable tractor trans-missionsrdquo ASAE Distinguished Lecture No 29 pp 1ndash37 StJoseph American Society of Agricultural Engineers St JosephMI USA 2005

[9] S H Yuan and J B Hu ldquo)e efficiency of multi-range hydro-mechanical stepless transmissionrdquo Journal of Beijing Instituteof Technology vol 7 no 2 pp 129ndash133 1998

[10] X J LiuAnalysis of Vehicle Transmission System pp 258-259National Defence Industry Press Beijing China 1998

[11] M M Brenninger ldquoFendt vario CVT in agricultural tractorsrdquoSAE International Warrendale PA USA SAE TechnicalPaper 2007-01-4205 2007

[12] H Aitzetmuller ldquoSteyr S-Matic-the future CVT systemrdquo inProceedings of the Seoul 2000 FISITA world automotive con-gress pp 1ndash6 Seoul South Korea June 2000

[13] J Pohlenz and K Grad ldquoCVTfor input power above 300 kWrdquoVDI-berichte vol 2007 pp 59ndash65 2001

[14] J Pohlenz and K Grad ldquoCVT-System fur denGroszligflacheneinsatzrdquoVDI-berichte vol 2004 pp 23ndash33 1855

[15] L Y Xu Study on Characteristics of Hydro-MechanicalContinuously Variable Transmission of Tractor Xirsquoan Uni-versity of Technology Xirsquoan China 2007

[16] J Looman Zahnradgetriebe-Grundlagen KonstruktionenAnwendungen in Fahrzeugen Springer Berlin Germany2009

[17] Z Rao Planetary Gear Transmission Design Chemical In-dustry Press Beijing China 2003

[18] X Xu Y Liang M Jordan T Peter and D Peng ldquoOptimizedcontrol of engine start assisted by the disconnect clutch in a P2hybrid automatic transmissionrdquo Mechanical Systems andSignal Processing vol 124 2019

[19] S Wang Y Liu Z Wang et al ldquoAdaptive fuzzy iterativecontrol strategy for the wet-clutch filling of automatictransmissionrdquo Mechanical Systems and Signal Processingvol 130 pp 164ndash182 2019

14 Mathematical Problems in Engineering

When HMCVT is working in HM1 or HM3 range if thetransmission efficiencies of the gear pairs i9 and i8 are equal(η7 η8) the transmission efficiencies of HMCVT in HM1and HM3 ranges are equal

When HMCVT is working in HM2 or HM4 range if thetransmission efficiencies of the gear pairs i9 and i6 are equal(η9 η6) the transmission efficiencies of HMCVT in HM2and HM4 ranges are also equal

Due to the complexity of the above results it is necessaryto evaluate the influence of displacement ratio ε standingtransmission ratio of PGM minusk1 gear ratio i1i2i3 and hy-draulic system efficiency ηH on the transmission efficiency of

HMCVT by means of image analysis method We assumethat η1 minus η12 are 098 ηs1r is 098 ηs1s2

and ηrs2are 097 and

k1i1i2i3 is minus32727 )e evaluation results are shown inFigure 3

According to the analysis on Figure 3 the following rulesare obtained

In each range of HMCVT when the displacement ratio εis 0 the transmission efficiency reaches the maximum valueNow the hydraulic path (HST) does not transmit power andthe power is completely transmitted by the mechanical pathWhen the displacement ratio ε is minus1 or 1 the transmissionefficiency reaches the minimum value and the hydraulic

Table 2 Equation of efficiency calculation for ranges HM1 and HM3

ElementsTorque equation considering power losses

εgt 0 εlt 0HM1 HM3 HM1 HM3

Engine Te Tr + η1η2Tpi1i2 Te Tr + Tpi1i2η1η2Pump Tp εηHTm Tp εTmηH

Motor Tm η3Ts1(minusi3) Tm Ts1

(minusη3i3)PGM Ts1

+ Tr + Tc 0PGM Tr minusk1Ts1

ηs1r

Intermediate shaft Tc Tzi4i8η4η8 Tc Tzi4i7η4η7 Tc Tzi4i8η4η8 Tc Tzi4i7η4η7Output shaft Tz To(i11i12η11η12)Torque ratio 1113957μ ToTe

Efficiency η minus1113957μig

144 max

100 44 max

44 max

Engine

r

S1

S2c

100

(a)

766 min

100

100

234 max

234 max

Engine

r

S1

S2c

(b)

766 min

100 234 max

234 max

Engine

r

S1

S2c

100

(c)

144 max

100

100

44 max

44 max

Engine

r

S1

S2c

(d)

Figure 2 Power flow direction of the PGM of HMCVT (a) Range HM1HM3 εgt 0 power-circulation (b) Range HM1HM3 εlt 0 power-split (c) Range HM2HM4 εgt 0 power-split (d) Range HM2HM4 εlt 0 power-circulation

8 Mathematical Problems in Engineering

Table 3 Equation of efficiency calculation for ranges HM2 and HM4

ElementsTorque equation considering power losses

εgt 0 εlt 0HM2 HM4 HM2 HM4

Engine Te Tr + Tpi1i2η1η2 Te Tr + η1η2Tpi1i2Pump Tp εTmηH Tp εηHTm

Motor Tm Ts1(minusη3i3) Tm η3Ts1

(minusi3)

PGM Ts1+ Tr + Ts2

0PGM Tr 1 minus k3ηs1s2

k3ηs1s2minus k2ηrs2

Ts2Intermediate shaft Ts2

Tzi5i9η5η9 Ts2 Tzi5i6η5η6 Ts2

Tzi5i9η5η9 Ts2 Tzi5i6η5η6

Output shaft Tz To(i11i12η11η12)Torque ratio 1113957μ ToTe

Efficiency η minus1113957μig

ndashk1 = 25172ndashk1 = 29130

ndashk1 = 17805ndashk1 = 21875

095

09

085

08

075

07ndash1 ndash08 ndash06 ndash04 ndash02 0 02 04 06 08 1

Displacement ratio

ηH = 081

Effici

ency

(a)

ndashk1 = 25172ndashk1 = 29130

ndashk1 = 17805ndashk1 = 21875

ηH = 081

095

09

085

08

075

07ndash1 ndash08 ndash06 ndash04 ndash02 0 02 04 06 08 1

Displacement ratio

Effici

ency

(b)

ηH = 081ηH = 090

ηH = 064ηH = 072

095

09

085

08

075

07ndash1 ndash08 ndash06 ndash04 ndash02 0 02 04 06 08 1

Displacement ratio

Effici

ency

ndashk1 = 21875

(c)

ηH = 081ηH = 090

ηH = 064ηH = 072

095

09

085

08

075

07ndash1 ndash08 ndash06 ndash04 ndash02 0 02 04 06 08 1

Displacement ratio

Effici

ency

ndashk1 = 21875

(d)

Figure 3 Continued

Mathematical Problems in Engineering 9

path transmits more power at this time resulting in a loweroverall efficiency of HMCVT

In each range of HMCVT the transmission efficiencyvalue in the power circulation phase is lower than thetransmission efficiency value in the power split phase whichindicates that the power circulation will cause the trans-mission efficiency of HMCVT to decrease

)e larger the value of the PGM standing transmissionratio minusk1 the higher the transmission efficiency of HMCVTbut the influence on improving the transmission efficiency is

not obvious )e PGM standing transmission ratio minusk1ranges from 15 to 3 )e gear ratio i1i2i3 and the PGMstanding transmission ratio minusk1 are in an inversely pro-portional relationship the smaller the value of the gear ratioi1i2i3 the higher the transmission efficiency of HMCVT andthe influence on improving the transmission efficiency isalso not obvious )e gear ratio i1i2i3 ranges from 11 to 22

)e greater the value of HSTefficiency ηH the higher theoverall transmission efficiency of HMCVT )e influence ofηH on improving transmission efficiency of HMCVT issignificant

i1i2i3 = 13001i1i2i3 = 11234

i1i2i3 = 18381i1i2i3 = 14961

ηH = 081

095

09

085

08

075

07ndash1 ndash08 ndash06 ndash04 ndash02 0 02 04 06 08 1

Displacement ratio

Effici

ency

(e)

i1i2i3 = 13001i1i2i3 = 11234

i1i2i3 = 18381i1i2i3 = 14961

ηH = 081

095

09

085

08

075

07ndash1 ndash08 ndash06 ndash04 ndash02 0 02 04 06 08 1

Displacement ratio

Effici

ency

(f )

Figure 3)e influence of parameters on the transmission efficiency of HMCVT (a))e influence of minusk1 on HM1 andHM3 ranges (b))einfluence of minusk1 on HM2 and HM4 ranges (c) )e influence of ηH on HM1 and HM3 ranges (d) )e influence of ηH on HM2 and HM4ranges (e) )e influence of i1i2i3 on HM1 and HM3 ranges (f ) )e influence of i1i2i3 on HM2 and HM4 ranges

9

8

7

6

5

4

3

2

1

0

Tran

smiss

ion

ratio

ndash15 ndash1 ndash05 0 05 1 15Displacement ratio

HM1HM2

HM3HM4

Figure 4 Relationship between transmission ratio and displace-ment ratio

60

50

40

30

20

10

0

Trac

tor s

peed

(km

h)

ndash15 ndash1 ndash05 0 05 1 15Displacement ratio

HM1HM2

HM3HM4

Figure 5 )e curve of the tractor speed

10 Mathematical Problems in Engineering

5 Analysis on the Transmission CharacteristicsResults of HMCVT

According to the previous content the value of the PGMstanding transmission ratio minusk1 is set to 21875 and the HSTefficiency value ηH is set to 081 then the other parameters ofHMCVT are k2 02714 minusk3 05938 i1 i2 1 i3 15i4 i5 1 i6 166 i7 116 i8 410 i9 584 andi10 i11 i12 1

)e relationship between the transmission ratio anddisplacement ratio is shown in Figure 4 )e curve of thetractor speed is shown in Figure 5 In the first range thedisplacement ratio changes from 1 to minus1 the transmissionratio decreases from 862 to 459 and the tractor speedincreases from 0 to 750 kmh In the second range thedisplacement ratio changes from minus1 to 1 the transmissionratio decreases from 459 to 243 and the tractor speedincreases from 750 kmh to 1412 kmh In the third rangethe displacement ratio changes from 1 to minus1 the trans-mission ratio decreases from 243 to 130 and the tractorspeed increases from 1412 kmh to 2644 kmh In thefourth range the displacement ratio changes from minus1 to 1the transmission ratio decreases from 130 to 069 and thetractor speed increases from 2644 kmh to 50 kmh )eresults show that the stepless speed regulation of HMCVTin the tractor speed range of 0ndash50 kmh can be realized bythe coordinated control of the HST displacement ratio andthe clutches

)e ratios of the output shaft torque to the input shafttorque in each range of HMCVT are shown in Figure 6(a))e continuous change of torque can be realized in the entireworking ranges )e ratios of the motor shaft torque to theoutput shaft torque in each range of HMCVT are shown inFigure 6(b) )e torque ratio remains constant in each range

and varies between different ranges )e values of torqueratios in the four ranges are 514 minus972 1849 andminus3518 respectively

)e relationship between the tractor speed and theoverall transmission efficiency of HMCVT is shown inFigure 7 Each of the four ranges has a transmission effi-ciency maximum value of 0917 0902 0917 and 0902respectively )e efficiency value always increases first withthe increase of the tractor speed and then decreases afterreaching the maximum value )e maximum transmissionefficiency in each range is obtained when the HST dis-placement ratio is 0 Now the hydraulic path does nottransmit power and all power is transmitted by the me-chanical path )e transmission efficiency of HMCVT ishigher than 83 so the overall transmission efficiency is at ahigh level

ndash2

ndash4

ndash6

ndash8

ndash10

0

0 5 10 15 20 25 30 35 40 45 50

Torq

ue ra

tio (T

oTe

)

Tractor speed (kmh)

HM1HM2

HM3HM4

(a)

05

04

03

02

01

0

ndash01

ndash02

ndash03

ndash04

ndash05

Torq

ue ra

tio (T

mT

o)

0 5 10 15 20 25 30 35 40 45 50Tractor speed (kmh)

HM1HM2

HM3HM4

(b)

Figure 6)e curve of torque characteristic (a) Ratios of output shaft torque to input shaft torque (b) Ratios of motor shaft torque to outputshaft torque

095

09

085

08

075

07

Effici

ency

0 5 10 15 20 25 30 35 40 45 50Tractor speed (kmh)

HM1HM2

HM3HM4

Figure 7 Overall transmission efficiency of HMCVT

Mathematical Problems in Engineering 11

6 Verification of HMCVT Simulation Model

)e modeling and simulation of HMCVT are accom-plished by using Amesim simulation softwareAccording to Figure 1 a simulation model of HMCVTis established as shown in Figure 8

61 Hydraulic System )e hydraulic system is mainlycomposed of a variable displacement hydraulic pump afixed displacement hydraulic motor two relief valves twocheck valves and a charge hydraulic pump with a reliefvalve )e aim of the charge pump is to compensate for theleakage loss in the pipeline and to establish the lowestpressure in the pumprsquos suction pipeline Two relief valves areused to limit the maximum pressure in the system pipingand its adjustment pressure should be above the normalworking pressure of the system Volumetric losses of the

pump and motor are neglected and their hydromechanicalefficiencies are set at 09

62 Mechanical Components Mechanical components areassumed to be constant efficiency components throughoutthe transmission )e power loss of each gear pair is 2 sothe efficiency is set to 098 In the Ravigneaux planetary gearmechanism the efficiencies of ring gear-long planetary gearcouple long planetary gear-large sun gear couple longplanetary gear-short planetary gear couple and shortplanetary gear-small sun gear couple are set at 099 )emain reducer differential and wheel reducer are combinedand modeled as a gear reducer with an efficiency of 0965 Tosimplify the model the friction losses of the shaft gears andbearings were ignored during the simulation modelingprocess

Efficiency calculation unit

Speed calculation unit

Small sun gear

Short planet gear

Long planet gear

Large sun gear

Ring gear

i1

i2

Pump Motor

i3

i5

i4

i7 i6

i9i8

Carrier

C1

C2

C3

C4

ω ωT

TL

ωωT

T

L

PP

MOT

f (x y)

y

x

X

k

Figure 8 HMCVT simulation model

(km

h)

50

40

30

20

10

00 5 10 15 20 25 30 35

X Time (s)

Tractor linear velocity (kmh)

(a)

0 5 10 15 20 25 30 35X Time (s)

10

08

06

04

02

00

(nul

l)

Transmission efficiency of HMCVT (null)

(b)

Figure 9 Simulation results (a) Simulation result of the tractor speed (b) Simulation result of the transmission efficiency of HMCVT

12 Mathematical Problems in Engineering

63Model Control and Simulation Results )e engine speedis assumed to have a constant value of 2200 rmin the loadtorque of the output shaft of HMCVT is set to 100Nm themodel simulation time is 32 s including range HM1 0ndash8 srange HM2 8ndash16 s range HM3 16ndash24 s and range HM424ndash32 s

)e simulation results are shown in Figure 9 through thevariable pump displacement and coordinated control of theclutches [18 19] the transmission gradually changes fromrange HM1 to range HM4 and the tractor achieves steplessspeed change from 0 to 50 kmh )e simulation results areconsistent with the previous theoretical analysis results whichshow that the 4-range HMCVT transmission scheme pro-posed in this paper is suitable for matching wheeled tractors

64 Comparison Xu LYrsquos transmission scheme [15] hasbeen applied to mass production tractors )is transmissionscheme has 6 working ranges in the forward direction and 3working ranges in the reverse direction which is controlledby 8 clutches )e speed range in the forward direction is0sim30 kmh while the reverse direction is only 0sim10 kmh)e transmission efficiency of the hydraulic range is low (lessthan 80) the transmission efficiency of the hydrome-chanical ranges is high (higher than 90) but is only in theforward direction

)e scheme proposed in this paper has 4 working rangesand a speed range of 0sim50 kmh in the forward and reversedirections which is controlled by 6 clutches )e fewer thetransmission ranges the less switching during the workingranges so the transmission performance is improved )esmaller number of clutches ensures that the system is easierto control )e transmission efficiencies in the forward andreverse directions are close the overall forward transmissionefficiency is higher than 83 and the highest transmissionefficiency is higher than 91

)erefore the application of HMCVT proposed in thispaper is wider it can not only match conventional tractorsbut also match two-way operation tractors and other en-gineering vehicles

7 Conclusions

)is paper proposes a new type of transmission scheme ofHMCVT )e Ravigneaux PGM and the HSTare connectedin parallel to form a closed loop and then connected in serieswith the range switching mechanism )is new HMCVTfeatures a double-row PGM with 4 hydromechanicalworking ranges By coordinating control of the HST dis-placement ratio and the engagement conditions of shiftingclutches the stepless speed regulation of HMCVT at thetractor speed of 0ndash50 kmh is realized

According to the theoretical analysis on the newHMCVT the speed characteristic curve and the torquecharacteristic curve are obtained Based on the power flowdirection of HMCVT the transmission efficiency curve isobtained by using the Lrfkoes formula

Using image analysis method to study the influence offour parameters (displacement ratio ε PGM standing

transmission ratio minusk1 gear ratio i1i2i3 and hydraulic systemefficiency ηH) on the transmission efficiency of HMCVT theresults show that the standing transmission ratio minusk1 andgear ratio i1i2i3 have little effect on the transmission effi-ciency whereas the displacement ratio ε and hydraulicsystem efficiency ηH have a greater impact on the trans-mission efficiency

)e parameters of the new HMCVT are finally deter-mined )e results of the transmission characteristics showthat the new HMCVTproposed in this paper has the steplessspeed regulation capability for tractors which can alsocontinuously transmit and change the torque )e trans-mission efficiency of HMCVT is at a high level

In order to verify the theoretical derivation Amesimsimulation software is used for the modeling and simulationof HMCVT )e simulation results are consistent with thetheoretical analysis results )erefore we believe that thetransmission scheme is reasonable and effective for tractors

Data Availability

)e Simcenter Amesim System File data used to support thefindings of this study are available from the correspondingauthor upon request

Conflicts of Interest

)e authors declare that they have no conflicts of interest

Acknowledgments

)is research was financially supported by the National KeyResearch and Development Plan Project of China(2016YFD0700402) and Changzhou Sci amp Tech Program(no CE20185037)

References

[1] L Y Xu Z L Zhou M Z Zhang et al ldquoResearch and designof hydro-mechanical continuously variable transmission fortractorsrdquo Journal of Northeast Agricultural University vol 13no 2 pp 182ndash186 2006

[2] R Paoluzzi and L G Zarotti ldquo)e minimum size of hy-drostatic transmissions for locomotionrdquo Journal of Terra-mechanics vol 50 no 3 pp 153ndash164 2013

[3] J Kim J Kang Y Kim B Min and H Kim ldquoDesign of powersplit transmission design of dual mode power split trans-missionrdquo International Journal of Automotive Technologyvol 11 no 4 pp 565ndash571 2010

[4] A Kim and A Rossetti ldquoFuel consumption reduction inurban buses by using power split transmissionsrdquo EnergyConversion and Management vol 71 pp 159ndash171 2013

[5] A Rossetti and A Macor ldquoMulti-objective optimization ofhydro-mechanical power split transmissionsrdquo Mechanismand Machine lteory vol 62 pp 112ndash128 2013

[6] A Macor and A Rossetti ldquoOptimization of hydro-mechanicalpower split transmissionsrdquo Mechanism and Machine lteoryvol 46 no 12 pp 1901ndash1919 2011

[7] S Shamshirband D Petkovic A Amini et al ldquoSupport vectorregression methodology for wind turbine reaction torqueprediction with power-split hydrostatic continuous variabletransmissionrdquo Energy vol 67 pp 623ndash630 2014

Mathematical Problems in Engineering 13

[8] K T Anuar ldquoResch R Continuously variable tractor trans-missionsrdquo ASAE Distinguished Lecture No 29 pp 1ndash37 StJoseph American Society of Agricultural Engineers St JosephMI USA 2005

[9] S H Yuan and J B Hu ldquo)e efficiency of multi-range hydro-mechanical stepless transmissionrdquo Journal of Beijing Instituteof Technology vol 7 no 2 pp 129ndash133 1998

[10] X J LiuAnalysis of Vehicle Transmission System pp 258-259National Defence Industry Press Beijing China 1998

[11] M M Brenninger ldquoFendt vario CVT in agricultural tractorsrdquoSAE International Warrendale PA USA SAE TechnicalPaper 2007-01-4205 2007

[12] H Aitzetmuller ldquoSteyr S-Matic-the future CVT systemrdquo inProceedings of the Seoul 2000 FISITA world automotive con-gress pp 1ndash6 Seoul South Korea June 2000

[13] J Pohlenz and K Grad ldquoCVTfor input power above 300 kWrdquoVDI-berichte vol 2007 pp 59ndash65 2001

[14] J Pohlenz and K Grad ldquoCVT-System fur denGroszligflacheneinsatzrdquoVDI-berichte vol 2004 pp 23ndash33 1855

[15] L Y Xu Study on Characteristics of Hydro-MechanicalContinuously Variable Transmission of Tractor Xirsquoan Uni-versity of Technology Xirsquoan China 2007

[16] J Looman Zahnradgetriebe-Grundlagen KonstruktionenAnwendungen in Fahrzeugen Springer Berlin Germany2009

[17] Z Rao Planetary Gear Transmission Design Chemical In-dustry Press Beijing China 2003

[18] X Xu Y Liang M Jordan T Peter and D Peng ldquoOptimizedcontrol of engine start assisted by the disconnect clutch in a P2hybrid automatic transmissionrdquo Mechanical Systems andSignal Processing vol 124 2019

[19] S Wang Y Liu Z Wang et al ldquoAdaptive fuzzy iterativecontrol strategy for the wet-clutch filling of automatictransmissionrdquo Mechanical Systems and Signal Processingvol 130 pp 164ndash182 2019

14 Mathematical Problems in Engineering

Table 3 Equation of efficiency calculation for ranges HM2 and HM4

ElementsTorque equation considering power losses

εgt 0 εlt 0HM2 HM4 HM2 HM4

Engine Te Tr + Tpi1i2η1η2 Te Tr + η1η2Tpi1i2Pump Tp εTmηH Tp εηHTm

Motor Tm Ts1(minusη3i3) Tm η3Ts1

(minusi3)

PGM Ts1+ Tr + Ts2

0PGM Tr 1 minus k3ηs1s2

k3ηs1s2minus k2ηrs2

Ts2Intermediate shaft Ts2

Tzi5i9η5η9 Ts2 Tzi5i6η5η6 Ts2

Tzi5i9η5η9 Ts2 Tzi5i6η5η6

Output shaft Tz To(i11i12η11η12)Torque ratio 1113957μ ToTe

Efficiency η minus1113957μig

ndashk1 = 25172ndashk1 = 29130

ndashk1 = 17805ndashk1 = 21875

095

09

085

08

075

07ndash1 ndash08 ndash06 ndash04 ndash02 0 02 04 06 08 1

Displacement ratio

ηH = 081

Effici

ency

(a)

ndashk1 = 25172ndashk1 = 29130

ndashk1 = 17805ndashk1 = 21875

ηH = 081

095

09

085

08

075

07ndash1 ndash08 ndash06 ndash04 ndash02 0 02 04 06 08 1

Displacement ratio

Effici

ency

(b)

ηH = 081ηH = 090

ηH = 064ηH = 072

095

09

085

08

075

07ndash1 ndash08 ndash06 ndash04 ndash02 0 02 04 06 08 1

Displacement ratio

Effici

ency

ndashk1 = 21875

(c)

ηH = 081ηH = 090

ηH = 064ηH = 072

095

09

085

08

075

07ndash1 ndash08 ndash06 ndash04 ndash02 0 02 04 06 08 1

Displacement ratio

Effici

ency

ndashk1 = 21875

(d)

Figure 3 Continued

Mathematical Problems in Engineering 9

path transmits more power at this time resulting in a loweroverall efficiency of HMCVT

In each range of HMCVT the transmission efficiencyvalue in the power circulation phase is lower than thetransmission efficiency value in the power split phase whichindicates that the power circulation will cause the trans-mission efficiency of HMCVT to decrease

)e larger the value of the PGM standing transmissionratio minusk1 the higher the transmission efficiency of HMCVTbut the influence on improving the transmission efficiency is

not obvious )e PGM standing transmission ratio minusk1ranges from 15 to 3 )e gear ratio i1i2i3 and the PGMstanding transmission ratio minusk1 are in an inversely pro-portional relationship the smaller the value of the gear ratioi1i2i3 the higher the transmission efficiency of HMCVT andthe influence on improving the transmission efficiency isalso not obvious )e gear ratio i1i2i3 ranges from 11 to 22

)e greater the value of HSTefficiency ηH the higher theoverall transmission efficiency of HMCVT )e influence ofηH on improving transmission efficiency of HMCVT issignificant

i1i2i3 = 13001i1i2i3 = 11234

i1i2i3 = 18381i1i2i3 = 14961

ηH = 081

095

09

085

08

075

07ndash1 ndash08 ndash06 ndash04 ndash02 0 02 04 06 08 1

Displacement ratio

Effici

ency

(e)

i1i2i3 = 13001i1i2i3 = 11234

i1i2i3 = 18381i1i2i3 = 14961

ηH = 081

095

09

085

08

075

07ndash1 ndash08 ndash06 ndash04 ndash02 0 02 04 06 08 1

Displacement ratio

Effici

ency

(f )

Figure 3)e influence of parameters on the transmission efficiency of HMCVT (a))e influence of minusk1 on HM1 andHM3 ranges (b))einfluence of minusk1 on HM2 and HM4 ranges (c) )e influence of ηH on HM1 and HM3 ranges (d) )e influence of ηH on HM2 and HM4ranges (e) )e influence of i1i2i3 on HM1 and HM3 ranges (f ) )e influence of i1i2i3 on HM2 and HM4 ranges

9

8

7

6

5

4

3

2

1

0

Tran

smiss

ion

ratio

ndash15 ndash1 ndash05 0 05 1 15Displacement ratio

HM1HM2

HM3HM4

Figure 4 Relationship between transmission ratio and displace-ment ratio

60

50

40

30

20

10

0

Trac

tor s

peed

(km

h)

ndash15 ndash1 ndash05 0 05 1 15Displacement ratio

HM1HM2

HM3HM4

Figure 5 )e curve of the tractor speed

10 Mathematical Problems in Engineering

5 Analysis on the Transmission CharacteristicsResults of HMCVT

According to the previous content the value of the PGMstanding transmission ratio minusk1 is set to 21875 and the HSTefficiency value ηH is set to 081 then the other parameters ofHMCVT are k2 02714 minusk3 05938 i1 i2 1 i3 15i4 i5 1 i6 166 i7 116 i8 410 i9 584 andi10 i11 i12 1

)e relationship between the transmission ratio anddisplacement ratio is shown in Figure 4 )e curve of thetractor speed is shown in Figure 5 In the first range thedisplacement ratio changes from 1 to minus1 the transmissionratio decreases from 862 to 459 and the tractor speedincreases from 0 to 750 kmh In the second range thedisplacement ratio changes from minus1 to 1 the transmissionratio decreases from 459 to 243 and the tractor speedincreases from 750 kmh to 1412 kmh In the third rangethe displacement ratio changes from 1 to minus1 the trans-mission ratio decreases from 243 to 130 and the tractorspeed increases from 1412 kmh to 2644 kmh In thefourth range the displacement ratio changes from minus1 to 1the transmission ratio decreases from 130 to 069 and thetractor speed increases from 2644 kmh to 50 kmh )eresults show that the stepless speed regulation of HMCVTin the tractor speed range of 0ndash50 kmh can be realized bythe coordinated control of the HST displacement ratio andthe clutches

)e ratios of the output shaft torque to the input shafttorque in each range of HMCVT are shown in Figure 6(a))e continuous change of torque can be realized in the entireworking ranges )e ratios of the motor shaft torque to theoutput shaft torque in each range of HMCVT are shown inFigure 6(b) )e torque ratio remains constant in each range

and varies between different ranges )e values of torqueratios in the four ranges are 514 minus972 1849 andminus3518 respectively

)e relationship between the tractor speed and theoverall transmission efficiency of HMCVT is shown inFigure 7 Each of the four ranges has a transmission effi-ciency maximum value of 0917 0902 0917 and 0902respectively )e efficiency value always increases first withthe increase of the tractor speed and then decreases afterreaching the maximum value )e maximum transmissionefficiency in each range is obtained when the HST dis-placement ratio is 0 Now the hydraulic path does nottransmit power and all power is transmitted by the me-chanical path )e transmission efficiency of HMCVT ishigher than 83 so the overall transmission efficiency is at ahigh level

ndash2

ndash4

ndash6

ndash8

ndash10

0

0 5 10 15 20 25 30 35 40 45 50

Torq

ue ra

tio (T

oTe

)

Tractor speed (kmh)

HM1HM2

HM3HM4

(a)

05

04

03

02

01

0

ndash01

ndash02

ndash03

ndash04

ndash05

Torq

ue ra

tio (T

mT

o)

0 5 10 15 20 25 30 35 40 45 50Tractor speed (kmh)

HM1HM2

HM3HM4

(b)

Figure 6)e curve of torque characteristic (a) Ratios of output shaft torque to input shaft torque (b) Ratios of motor shaft torque to outputshaft torque

095

09

085

08

075

07

Effici

ency

0 5 10 15 20 25 30 35 40 45 50Tractor speed (kmh)

HM1HM2

HM3HM4

Figure 7 Overall transmission efficiency of HMCVT

Mathematical Problems in Engineering 11

6 Verification of HMCVT Simulation Model

)e modeling and simulation of HMCVT are accom-plished by using Amesim simulation softwareAccording to Figure 1 a simulation model of HMCVTis established as shown in Figure 8

61 Hydraulic System )e hydraulic system is mainlycomposed of a variable displacement hydraulic pump afixed displacement hydraulic motor two relief valves twocheck valves and a charge hydraulic pump with a reliefvalve )e aim of the charge pump is to compensate for theleakage loss in the pipeline and to establish the lowestpressure in the pumprsquos suction pipeline Two relief valves areused to limit the maximum pressure in the system pipingand its adjustment pressure should be above the normalworking pressure of the system Volumetric losses of the

pump and motor are neglected and their hydromechanicalefficiencies are set at 09

62 Mechanical Components Mechanical components areassumed to be constant efficiency components throughoutthe transmission )e power loss of each gear pair is 2 sothe efficiency is set to 098 In the Ravigneaux planetary gearmechanism the efficiencies of ring gear-long planetary gearcouple long planetary gear-large sun gear couple longplanetary gear-short planetary gear couple and shortplanetary gear-small sun gear couple are set at 099 )emain reducer differential and wheel reducer are combinedand modeled as a gear reducer with an efficiency of 0965 Tosimplify the model the friction losses of the shaft gears andbearings were ignored during the simulation modelingprocess

Efficiency calculation unit

Speed calculation unit

Small sun gear

Short planet gear

Long planet gear

Large sun gear

Ring gear

i1

i2

Pump Motor

i3

i5

i4

i7 i6

i9i8

Carrier

C1

C2

C3

C4

ω ωT

TL

ωωT

T

L

PP

MOT

f (x y)

y

x

X

k

Figure 8 HMCVT simulation model

(km

h)

50

40

30

20

10

00 5 10 15 20 25 30 35

X Time (s)

Tractor linear velocity (kmh)

(a)

0 5 10 15 20 25 30 35X Time (s)

10

08

06

04

02

00

(nul

l)

Transmission efficiency of HMCVT (null)

(b)

Figure 9 Simulation results (a) Simulation result of the tractor speed (b) Simulation result of the transmission efficiency of HMCVT

12 Mathematical Problems in Engineering

63Model Control and Simulation Results )e engine speedis assumed to have a constant value of 2200 rmin the loadtorque of the output shaft of HMCVT is set to 100Nm themodel simulation time is 32 s including range HM1 0ndash8 srange HM2 8ndash16 s range HM3 16ndash24 s and range HM424ndash32 s

)e simulation results are shown in Figure 9 through thevariable pump displacement and coordinated control of theclutches [18 19] the transmission gradually changes fromrange HM1 to range HM4 and the tractor achieves steplessspeed change from 0 to 50 kmh )e simulation results areconsistent with the previous theoretical analysis results whichshow that the 4-range HMCVT transmission scheme pro-posed in this paper is suitable for matching wheeled tractors

64 Comparison Xu LYrsquos transmission scheme [15] hasbeen applied to mass production tractors )is transmissionscheme has 6 working ranges in the forward direction and 3working ranges in the reverse direction which is controlledby 8 clutches )e speed range in the forward direction is0sim30 kmh while the reverse direction is only 0sim10 kmh)e transmission efficiency of the hydraulic range is low (lessthan 80) the transmission efficiency of the hydrome-chanical ranges is high (higher than 90) but is only in theforward direction

)e scheme proposed in this paper has 4 working rangesand a speed range of 0sim50 kmh in the forward and reversedirections which is controlled by 6 clutches )e fewer thetransmission ranges the less switching during the workingranges so the transmission performance is improved )esmaller number of clutches ensures that the system is easierto control )e transmission efficiencies in the forward andreverse directions are close the overall forward transmissionefficiency is higher than 83 and the highest transmissionefficiency is higher than 91

)erefore the application of HMCVT proposed in thispaper is wider it can not only match conventional tractorsbut also match two-way operation tractors and other en-gineering vehicles

7 Conclusions

)is paper proposes a new type of transmission scheme ofHMCVT )e Ravigneaux PGM and the HSTare connectedin parallel to form a closed loop and then connected in serieswith the range switching mechanism )is new HMCVTfeatures a double-row PGM with 4 hydromechanicalworking ranges By coordinating control of the HST dis-placement ratio and the engagement conditions of shiftingclutches the stepless speed regulation of HMCVT at thetractor speed of 0ndash50 kmh is realized

According to the theoretical analysis on the newHMCVT the speed characteristic curve and the torquecharacteristic curve are obtained Based on the power flowdirection of HMCVT the transmission efficiency curve isobtained by using the Lrfkoes formula

Using image analysis method to study the influence offour parameters (displacement ratio ε PGM standing

transmission ratio minusk1 gear ratio i1i2i3 and hydraulic systemefficiency ηH) on the transmission efficiency of HMCVT theresults show that the standing transmission ratio minusk1 andgear ratio i1i2i3 have little effect on the transmission effi-ciency whereas the displacement ratio ε and hydraulicsystem efficiency ηH have a greater impact on the trans-mission efficiency

)e parameters of the new HMCVT are finally deter-mined )e results of the transmission characteristics showthat the new HMCVTproposed in this paper has the steplessspeed regulation capability for tractors which can alsocontinuously transmit and change the torque )e trans-mission efficiency of HMCVT is at a high level

In order to verify the theoretical derivation Amesimsimulation software is used for the modeling and simulationof HMCVT )e simulation results are consistent with thetheoretical analysis results )erefore we believe that thetransmission scheme is reasonable and effective for tractors

Data Availability

)e Simcenter Amesim System File data used to support thefindings of this study are available from the correspondingauthor upon request

Conflicts of Interest

)e authors declare that they have no conflicts of interest

Acknowledgments

)is research was financially supported by the National KeyResearch and Development Plan Project of China(2016YFD0700402) and Changzhou Sci amp Tech Program(no CE20185037)

References

[1] L Y Xu Z L Zhou M Z Zhang et al ldquoResearch and designof hydro-mechanical continuously variable transmission fortractorsrdquo Journal of Northeast Agricultural University vol 13no 2 pp 182ndash186 2006

[2] R Paoluzzi and L G Zarotti ldquo)e minimum size of hy-drostatic transmissions for locomotionrdquo Journal of Terra-mechanics vol 50 no 3 pp 153ndash164 2013

[3] J Kim J Kang Y Kim B Min and H Kim ldquoDesign of powersplit transmission design of dual mode power split trans-missionrdquo International Journal of Automotive Technologyvol 11 no 4 pp 565ndash571 2010

[4] A Kim and A Rossetti ldquoFuel consumption reduction inurban buses by using power split transmissionsrdquo EnergyConversion and Management vol 71 pp 159ndash171 2013

[5] A Rossetti and A Macor ldquoMulti-objective optimization ofhydro-mechanical power split transmissionsrdquo Mechanismand Machine lteory vol 62 pp 112ndash128 2013

[6] A Macor and A Rossetti ldquoOptimization of hydro-mechanicalpower split transmissionsrdquo Mechanism and Machine lteoryvol 46 no 12 pp 1901ndash1919 2011

[7] S Shamshirband D Petkovic A Amini et al ldquoSupport vectorregression methodology for wind turbine reaction torqueprediction with power-split hydrostatic continuous variabletransmissionrdquo Energy vol 67 pp 623ndash630 2014

Mathematical Problems in Engineering 13

[8] K T Anuar ldquoResch R Continuously variable tractor trans-missionsrdquo ASAE Distinguished Lecture No 29 pp 1ndash37 StJoseph American Society of Agricultural Engineers St JosephMI USA 2005

[9] S H Yuan and J B Hu ldquo)e efficiency of multi-range hydro-mechanical stepless transmissionrdquo Journal of Beijing Instituteof Technology vol 7 no 2 pp 129ndash133 1998

[10] X J LiuAnalysis of Vehicle Transmission System pp 258-259National Defence Industry Press Beijing China 1998

[11] M M Brenninger ldquoFendt vario CVT in agricultural tractorsrdquoSAE International Warrendale PA USA SAE TechnicalPaper 2007-01-4205 2007

[12] H Aitzetmuller ldquoSteyr S-Matic-the future CVT systemrdquo inProceedings of the Seoul 2000 FISITA world automotive con-gress pp 1ndash6 Seoul South Korea June 2000

[13] J Pohlenz and K Grad ldquoCVTfor input power above 300 kWrdquoVDI-berichte vol 2007 pp 59ndash65 2001

[14] J Pohlenz and K Grad ldquoCVT-System fur denGroszligflacheneinsatzrdquoVDI-berichte vol 2004 pp 23ndash33 1855

[15] L Y Xu Study on Characteristics of Hydro-MechanicalContinuously Variable Transmission of Tractor Xirsquoan Uni-versity of Technology Xirsquoan China 2007

[16] J Looman Zahnradgetriebe-Grundlagen KonstruktionenAnwendungen in Fahrzeugen Springer Berlin Germany2009

[17] Z Rao Planetary Gear Transmission Design Chemical In-dustry Press Beijing China 2003

[18] X Xu Y Liang M Jordan T Peter and D Peng ldquoOptimizedcontrol of engine start assisted by the disconnect clutch in a P2hybrid automatic transmissionrdquo Mechanical Systems andSignal Processing vol 124 2019

[19] S Wang Y Liu Z Wang et al ldquoAdaptive fuzzy iterativecontrol strategy for the wet-clutch filling of automatictransmissionrdquo Mechanical Systems and Signal Processingvol 130 pp 164ndash182 2019

14 Mathematical Problems in Engineering

path transmits more power at this time resulting in a loweroverall efficiency of HMCVT

In each range of HMCVT the transmission efficiencyvalue in the power circulation phase is lower than thetransmission efficiency value in the power split phase whichindicates that the power circulation will cause the trans-mission efficiency of HMCVT to decrease

)e larger the value of the PGM standing transmissionratio minusk1 the higher the transmission efficiency of HMCVTbut the influence on improving the transmission efficiency is

not obvious )e PGM standing transmission ratio minusk1ranges from 15 to 3 )e gear ratio i1i2i3 and the PGMstanding transmission ratio minusk1 are in an inversely pro-portional relationship the smaller the value of the gear ratioi1i2i3 the higher the transmission efficiency of HMCVT andthe influence on improving the transmission efficiency isalso not obvious )e gear ratio i1i2i3 ranges from 11 to 22

)e greater the value of HSTefficiency ηH the higher theoverall transmission efficiency of HMCVT )e influence ofηH on improving transmission efficiency of HMCVT issignificant

i1i2i3 = 13001i1i2i3 = 11234

i1i2i3 = 18381i1i2i3 = 14961

ηH = 081

095

09

085

08

075

07ndash1 ndash08 ndash06 ndash04 ndash02 0 02 04 06 08 1

Displacement ratio

Effici

ency

(e)

i1i2i3 = 13001i1i2i3 = 11234

i1i2i3 = 18381i1i2i3 = 14961

ηH = 081

095

09

085

08

075

07ndash1 ndash08 ndash06 ndash04 ndash02 0 02 04 06 08 1

Displacement ratio

Effici

ency

(f )

Figure 3)e influence of parameters on the transmission efficiency of HMCVT (a))e influence of minusk1 on HM1 andHM3 ranges (b))einfluence of minusk1 on HM2 and HM4 ranges (c) )e influence of ηH on HM1 and HM3 ranges (d) )e influence of ηH on HM2 and HM4ranges (e) )e influence of i1i2i3 on HM1 and HM3 ranges (f ) )e influence of i1i2i3 on HM2 and HM4 ranges

9

8

7

6

5

4

3

2

1

0

Tran

smiss

ion

ratio

ndash15 ndash1 ndash05 0 05 1 15Displacement ratio

HM1HM2

HM3HM4

Figure 4 Relationship between transmission ratio and displace-ment ratio

60

50

40

30

20

10

0

Trac

tor s

peed

(km

h)

ndash15 ndash1 ndash05 0 05 1 15Displacement ratio

HM1HM2

HM3HM4

Figure 5 )e curve of the tractor speed

10 Mathematical Problems in Engineering

5 Analysis on the Transmission CharacteristicsResults of HMCVT

According to the previous content the value of the PGMstanding transmission ratio minusk1 is set to 21875 and the HSTefficiency value ηH is set to 081 then the other parameters ofHMCVT are k2 02714 minusk3 05938 i1 i2 1 i3 15i4 i5 1 i6 166 i7 116 i8 410 i9 584 andi10 i11 i12 1

)e relationship between the transmission ratio anddisplacement ratio is shown in Figure 4 )e curve of thetractor speed is shown in Figure 5 In the first range thedisplacement ratio changes from 1 to minus1 the transmissionratio decreases from 862 to 459 and the tractor speedincreases from 0 to 750 kmh In the second range thedisplacement ratio changes from minus1 to 1 the transmissionratio decreases from 459 to 243 and the tractor speedincreases from 750 kmh to 1412 kmh In the third rangethe displacement ratio changes from 1 to minus1 the trans-mission ratio decreases from 243 to 130 and the tractorspeed increases from 1412 kmh to 2644 kmh In thefourth range the displacement ratio changes from minus1 to 1the transmission ratio decreases from 130 to 069 and thetractor speed increases from 2644 kmh to 50 kmh )eresults show that the stepless speed regulation of HMCVTin the tractor speed range of 0ndash50 kmh can be realized bythe coordinated control of the HST displacement ratio andthe clutches

)e ratios of the output shaft torque to the input shafttorque in each range of HMCVT are shown in Figure 6(a))e continuous change of torque can be realized in the entireworking ranges )e ratios of the motor shaft torque to theoutput shaft torque in each range of HMCVT are shown inFigure 6(b) )e torque ratio remains constant in each range

and varies between different ranges )e values of torqueratios in the four ranges are 514 minus972 1849 andminus3518 respectively

)e relationship between the tractor speed and theoverall transmission efficiency of HMCVT is shown inFigure 7 Each of the four ranges has a transmission effi-ciency maximum value of 0917 0902 0917 and 0902respectively )e efficiency value always increases first withthe increase of the tractor speed and then decreases afterreaching the maximum value )e maximum transmissionefficiency in each range is obtained when the HST dis-placement ratio is 0 Now the hydraulic path does nottransmit power and all power is transmitted by the me-chanical path )e transmission efficiency of HMCVT ishigher than 83 so the overall transmission efficiency is at ahigh level

ndash2

ndash4

ndash6

ndash8

ndash10

0

0 5 10 15 20 25 30 35 40 45 50

Torq

ue ra

tio (T

oTe

)

Tractor speed (kmh)

HM1HM2

HM3HM4

(a)

05

04

03

02

01

0

ndash01

ndash02

ndash03

ndash04

ndash05

Torq

ue ra

tio (T

mT

o)

0 5 10 15 20 25 30 35 40 45 50Tractor speed (kmh)

HM1HM2

HM3HM4

(b)

Figure 6)e curve of torque characteristic (a) Ratios of output shaft torque to input shaft torque (b) Ratios of motor shaft torque to outputshaft torque

095

09

085

08

075

07

Effici

ency

0 5 10 15 20 25 30 35 40 45 50Tractor speed (kmh)

HM1HM2

HM3HM4

Figure 7 Overall transmission efficiency of HMCVT

Mathematical Problems in Engineering 11

6 Verification of HMCVT Simulation Model

)e modeling and simulation of HMCVT are accom-plished by using Amesim simulation softwareAccording to Figure 1 a simulation model of HMCVTis established as shown in Figure 8

61 Hydraulic System )e hydraulic system is mainlycomposed of a variable displacement hydraulic pump afixed displacement hydraulic motor two relief valves twocheck valves and a charge hydraulic pump with a reliefvalve )e aim of the charge pump is to compensate for theleakage loss in the pipeline and to establish the lowestpressure in the pumprsquos suction pipeline Two relief valves areused to limit the maximum pressure in the system pipingand its adjustment pressure should be above the normalworking pressure of the system Volumetric losses of the

pump and motor are neglected and their hydromechanicalefficiencies are set at 09

62 Mechanical Components Mechanical components areassumed to be constant efficiency components throughoutthe transmission )e power loss of each gear pair is 2 sothe efficiency is set to 098 In the Ravigneaux planetary gearmechanism the efficiencies of ring gear-long planetary gearcouple long planetary gear-large sun gear couple longplanetary gear-short planetary gear couple and shortplanetary gear-small sun gear couple are set at 099 )emain reducer differential and wheel reducer are combinedand modeled as a gear reducer with an efficiency of 0965 Tosimplify the model the friction losses of the shaft gears andbearings were ignored during the simulation modelingprocess

Efficiency calculation unit

Speed calculation unit

Small sun gear

Short planet gear

Long planet gear

Large sun gear

Ring gear

i1

i2

Pump Motor

i3

i5

i4

i7 i6

i9i8

Carrier

C1

C2

C3

C4

ω ωT

TL

ωωT

T

L

PP

MOT

f (x y)

y

x

X

k

Figure 8 HMCVT simulation model

(km

h)

50

40

30

20

10

00 5 10 15 20 25 30 35

X Time (s)

Tractor linear velocity (kmh)

(a)

0 5 10 15 20 25 30 35X Time (s)

10

08

06

04

02

00

(nul

l)

Transmission efficiency of HMCVT (null)

(b)

Figure 9 Simulation results (a) Simulation result of the tractor speed (b) Simulation result of the transmission efficiency of HMCVT

12 Mathematical Problems in Engineering

63Model Control and Simulation Results )e engine speedis assumed to have a constant value of 2200 rmin the loadtorque of the output shaft of HMCVT is set to 100Nm themodel simulation time is 32 s including range HM1 0ndash8 srange HM2 8ndash16 s range HM3 16ndash24 s and range HM424ndash32 s

)e simulation results are shown in Figure 9 through thevariable pump displacement and coordinated control of theclutches [18 19] the transmission gradually changes fromrange HM1 to range HM4 and the tractor achieves steplessspeed change from 0 to 50 kmh )e simulation results areconsistent with the previous theoretical analysis results whichshow that the 4-range HMCVT transmission scheme pro-posed in this paper is suitable for matching wheeled tractors

64 Comparison Xu LYrsquos transmission scheme [15] hasbeen applied to mass production tractors )is transmissionscheme has 6 working ranges in the forward direction and 3working ranges in the reverse direction which is controlledby 8 clutches )e speed range in the forward direction is0sim30 kmh while the reverse direction is only 0sim10 kmh)e transmission efficiency of the hydraulic range is low (lessthan 80) the transmission efficiency of the hydrome-chanical ranges is high (higher than 90) but is only in theforward direction

)e scheme proposed in this paper has 4 working rangesand a speed range of 0sim50 kmh in the forward and reversedirections which is controlled by 6 clutches )e fewer thetransmission ranges the less switching during the workingranges so the transmission performance is improved )esmaller number of clutches ensures that the system is easierto control )e transmission efficiencies in the forward andreverse directions are close the overall forward transmissionefficiency is higher than 83 and the highest transmissionefficiency is higher than 91

)erefore the application of HMCVT proposed in thispaper is wider it can not only match conventional tractorsbut also match two-way operation tractors and other en-gineering vehicles

7 Conclusions

)is paper proposes a new type of transmission scheme ofHMCVT )e Ravigneaux PGM and the HSTare connectedin parallel to form a closed loop and then connected in serieswith the range switching mechanism )is new HMCVTfeatures a double-row PGM with 4 hydromechanicalworking ranges By coordinating control of the HST dis-placement ratio and the engagement conditions of shiftingclutches the stepless speed regulation of HMCVT at thetractor speed of 0ndash50 kmh is realized

According to the theoretical analysis on the newHMCVT the speed characteristic curve and the torquecharacteristic curve are obtained Based on the power flowdirection of HMCVT the transmission efficiency curve isobtained by using the Lrfkoes formula

Using image analysis method to study the influence offour parameters (displacement ratio ε PGM standing

transmission ratio minusk1 gear ratio i1i2i3 and hydraulic systemefficiency ηH) on the transmission efficiency of HMCVT theresults show that the standing transmission ratio minusk1 andgear ratio i1i2i3 have little effect on the transmission effi-ciency whereas the displacement ratio ε and hydraulicsystem efficiency ηH have a greater impact on the trans-mission efficiency

)e parameters of the new HMCVT are finally deter-mined )e results of the transmission characteristics showthat the new HMCVTproposed in this paper has the steplessspeed regulation capability for tractors which can alsocontinuously transmit and change the torque )e trans-mission efficiency of HMCVT is at a high level

In order to verify the theoretical derivation Amesimsimulation software is used for the modeling and simulationof HMCVT )e simulation results are consistent with thetheoretical analysis results )erefore we believe that thetransmission scheme is reasonable and effective for tractors

Data Availability

)e Simcenter Amesim System File data used to support thefindings of this study are available from the correspondingauthor upon request

Conflicts of Interest

)e authors declare that they have no conflicts of interest

Acknowledgments

)is research was financially supported by the National KeyResearch and Development Plan Project of China(2016YFD0700402) and Changzhou Sci amp Tech Program(no CE20185037)

References

[1] L Y Xu Z L Zhou M Z Zhang et al ldquoResearch and designof hydro-mechanical continuously variable transmission fortractorsrdquo Journal of Northeast Agricultural University vol 13no 2 pp 182ndash186 2006

[2] R Paoluzzi and L G Zarotti ldquo)e minimum size of hy-drostatic transmissions for locomotionrdquo Journal of Terra-mechanics vol 50 no 3 pp 153ndash164 2013

[3] J Kim J Kang Y Kim B Min and H Kim ldquoDesign of powersplit transmission design of dual mode power split trans-missionrdquo International Journal of Automotive Technologyvol 11 no 4 pp 565ndash571 2010

[4] A Kim and A Rossetti ldquoFuel consumption reduction inurban buses by using power split transmissionsrdquo EnergyConversion and Management vol 71 pp 159ndash171 2013

[5] A Rossetti and A Macor ldquoMulti-objective optimization ofhydro-mechanical power split transmissionsrdquo Mechanismand Machine lteory vol 62 pp 112ndash128 2013

[6] A Macor and A Rossetti ldquoOptimization of hydro-mechanicalpower split transmissionsrdquo Mechanism and Machine lteoryvol 46 no 12 pp 1901ndash1919 2011

[7] S Shamshirband D Petkovic A Amini et al ldquoSupport vectorregression methodology for wind turbine reaction torqueprediction with power-split hydrostatic continuous variabletransmissionrdquo Energy vol 67 pp 623ndash630 2014

Mathematical Problems in Engineering 13

[8] K T Anuar ldquoResch R Continuously variable tractor trans-missionsrdquo ASAE Distinguished Lecture No 29 pp 1ndash37 StJoseph American Society of Agricultural Engineers St JosephMI USA 2005

[9] S H Yuan and J B Hu ldquo)e efficiency of multi-range hydro-mechanical stepless transmissionrdquo Journal of Beijing Instituteof Technology vol 7 no 2 pp 129ndash133 1998

[10] X J LiuAnalysis of Vehicle Transmission System pp 258-259National Defence Industry Press Beijing China 1998

[11] M M Brenninger ldquoFendt vario CVT in agricultural tractorsrdquoSAE International Warrendale PA USA SAE TechnicalPaper 2007-01-4205 2007

[12] H Aitzetmuller ldquoSteyr S-Matic-the future CVT systemrdquo inProceedings of the Seoul 2000 FISITA world automotive con-gress pp 1ndash6 Seoul South Korea June 2000

[13] J Pohlenz and K Grad ldquoCVTfor input power above 300 kWrdquoVDI-berichte vol 2007 pp 59ndash65 2001

[14] J Pohlenz and K Grad ldquoCVT-System fur denGroszligflacheneinsatzrdquoVDI-berichte vol 2004 pp 23ndash33 1855

[15] L Y Xu Study on Characteristics of Hydro-MechanicalContinuously Variable Transmission of Tractor Xirsquoan Uni-versity of Technology Xirsquoan China 2007

[16] J Looman Zahnradgetriebe-Grundlagen KonstruktionenAnwendungen in Fahrzeugen Springer Berlin Germany2009

[17] Z Rao Planetary Gear Transmission Design Chemical In-dustry Press Beijing China 2003

[18] X Xu Y Liang M Jordan T Peter and D Peng ldquoOptimizedcontrol of engine start assisted by the disconnect clutch in a P2hybrid automatic transmissionrdquo Mechanical Systems andSignal Processing vol 124 2019

[19] S Wang Y Liu Z Wang et al ldquoAdaptive fuzzy iterativecontrol strategy for the wet-clutch filling of automatictransmissionrdquo Mechanical Systems and Signal Processingvol 130 pp 164ndash182 2019

14 Mathematical Problems in Engineering

5 Analysis on the Transmission CharacteristicsResults of HMCVT

According to the previous content the value of the PGMstanding transmission ratio minusk1 is set to 21875 and the HSTefficiency value ηH is set to 081 then the other parameters ofHMCVT are k2 02714 minusk3 05938 i1 i2 1 i3 15i4 i5 1 i6 166 i7 116 i8 410 i9 584 andi10 i11 i12 1

)e relationship between the transmission ratio anddisplacement ratio is shown in Figure 4 )e curve of thetractor speed is shown in Figure 5 In the first range thedisplacement ratio changes from 1 to minus1 the transmissionratio decreases from 862 to 459 and the tractor speedincreases from 0 to 750 kmh In the second range thedisplacement ratio changes from minus1 to 1 the transmissionratio decreases from 459 to 243 and the tractor speedincreases from 750 kmh to 1412 kmh In the third rangethe displacement ratio changes from 1 to minus1 the trans-mission ratio decreases from 243 to 130 and the tractorspeed increases from 1412 kmh to 2644 kmh In thefourth range the displacement ratio changes from minus1 to 1the transmission ratio decreases from 130 to 069 and thetractor speed increases from 2644 kmh to 50 kmh )eresults show that the stepless speed regulation of HMCVTin the tractor speed range of 0ndash50 kmh can be realized bythe coordinated control of the HST displacement ratio andthe clutches

)e ratios of the output shaft torque to the input shafttorque in each range of HMCVT are shown in Figure 6(a))e continuous change of torque can be realized in the entireworking ranges )e ratios of the motor shaft torque to theoutput shaft torque in each range of HMCVT are shown inFigure 6(b) )e torque ratio remains constant in each range

and varies between different ranges )e values of torqueratios in the four ranges are 514 minus972 1849 andminus3518 respectively

)e relationship between the tractor speed and theoverall transmission efficiency of HMCVT is shown inFigure 7 Each of the four ranges has a transmission effi-ciency maximum value of 0917 0902 0917 and 0902respectively )e efficiency value always increases first withthe increase of the tractor speed and then decreases afterreaching the maximum value )e maximum transmissionefficiency in each range is obtained when the HST dis-placement ratio is 0 Now the hydraulic path does nottransmit power and all power is transmitted by the me-chanical path )e transmission efficiency of HMCVT ishigher than 83 so the overall transmission efficiency is at ahigh level

ndash2

ndash4

ndash6

ndash8

ndash10

0

0 5 10 15 20 25 30 35 40 45 50

Torq

ue ra

tio (T

oTe

)

Tractor speed (kmh)

HM1HM2

HM3HM4

(a)

05

04

03

02

01

0

ndash01

ndash02

ndash03

ndash04

ndash05

Torq

ue ra

tio (T

mT

o)

0 5 10 15 20 25 30 35 40 45 50Tractor speed (kmh)

HM1HM2

HM3HM4

(b)

Figure 6)e curve of torque characteristic (a) Ratios of output shaft torque to input shaft torque (b) Ratios of motor shaft torque to outputshaft torque

095

09

085

08

075

07

Effici

ency

0 5 10 15 20 25 30 35 40 45 50Tractor speed (kmh)

HM1HM2

HM3HM4

Figure 7 Overall transmission efficiency of HMCVT

Mathematical Problems in Engineering 11

6 Verification of HMCVT Simulation Model

)e modeling and simulation of HMCVT are accom-plished by using Amesim simulation softwareAccording to Figure 1 a simulation model of HMCVTis established as shown in Figure 8

61 Hydraulic System )e hydraulic system is mainlycomposed of a variable displacement hydraulic pump afixed displacement hydraulic motor two relief valves twocheck valves and a charge hydraulic pump with a reliefvalve )e aim of the charge pump is to compensate for theleakage loss in the pipeline and to establish the lowestpressure in the pumprsquos suction pipeline Two relief valves areused to limit the maximum pressure in the system pipingand its adjustment pressure should be above the normalworking pressure of the system Volumetric losses of the

pump and motor are neglected and their hydromechanicalefficiencies are set at 09

62 Mechanical Components Mechanical components areassumed to be constant efficiency components throughoutthe transmission )e power loss of each gear pair is 2 sothe efficiency is set to 098 In the Ravigneaux planetary gearmechanism the efficiencies of ring gear-long planetary gearcouple long planetary gear-large sun gear couple longplanetary gear-short planetary gear couple and shortplanetary gear-small sun gear couple are set at 099 )emain reducer differential and wheel reducer are combinedand modeled as a gear reducer with an efficiency of 0965 Tosimplify the model the friction losses of the shaft gears andbearings were ignored during the simulation modelingprocess

Efficiency calculation unit

Speed calculation unit

Small sun gear

Short planet gear

Long planet gear

Large sun gear

Ring gear

i1

i2

Pump Motor

i3

i5

i4

i7 i6

i9i8

Carrier

C1

C2

C3

C4

ω ωT

TL

ωωT

T

L

PP

MOT

f (x y)

y

x

X

k

Figure 8 HMCVT simulation model

(km

h)

50

40

30

20

10

00 5 10 15 20 25 30 35

X Time (s)

Tractor linear velocity (kmh)

(a)

0 5 10 15 20 25 30 35X Time (s)

10

08

06

04

02

00

(nul

l)

Transmission efficiency of HMCVT (null)

(b)

Figure 9 Simulation results (a) Simulation result of the tractor speed (b) Simulation result of the transmission efficiency of HMCVT

12 Mathematical Problems in Engineering

63Model Control and Simulation Results )e engine speedis assumed to have a constant value of 2200 rmin the loadtorque of the output shaft of HMCVT is set to 100Nm themodel simulation time is 32 s including range HM1 0ndash8 srange HM2 8ndash16 s range HM3 16ndash24 s and range HM424ndash32 s

)e simulation results are shown in Figure 9 through thevariable pump displacement and coordinated control of theclutches [18 19] the transmission gradually changes fromrange HM1 to range HM4 and the tractor achieves steplessspeed change from 0 to 50 kmh )e simulation results areconsistent with the previous theoretical analysis results whichshow that the 4-range HMCVT transmission scheme pro-posed in this paper is suitable for matching wheeled tractors

64 Comparison Xu LYrsquos transmission scheme [15] hasbeen applied to mass production tractors )is transmissionscheme has 6 working ranges in the forward direction and 3working ranges in the reverse direction which is controlledby 8 clutches )e speed range in the forward direction is0sim30 kmh while the reverse direction is only 0sim10 kmh)e transmission efficiency of the hydraulic range is low (lessthan 80) the transmission efficiency of the hydrome-chanical ranges is high (higher than 90) but is only in theforward direction

)e scheme proposed in this paper has 4 working rangesand a speed range of 0sim50 kmh in the forward and reversedirections which is controlled by 6 clutches )e fewer thetransmission ranges the less switching during the workingranges so the transmission performance is improved )esmaller number of clutches ensures that the system is easierto control )e transmission efficiencies in the forward andreverse directions are close the overall forward transmissionefficiency is higher than 83 and the highest transmissionefficiency is higher than 91

)erefore the application of HMCVT proposed in thispaper is wider it can not only match conventional tractorsbut also match two-way operation tractors and other en-gineering vehicles

7 Conclusions

)is paper proposes a new type of transmission scheme ofHMCVT )e Ravigneaux PGM and the HSTare connectedin parallel to form a closed loop and then connected in serieswith the range switching mechanism )is new HMCVTfeatures a double-row PGM with 4 hydromechanicalworking ranges By coordinating control of the HST dis-placement ratio and the engagement conditions of shiftingclutches the stepless speed regulation of HMCVT at thetractor speed of 0ndash50 kmh is realized

According to the theoretical analysis on the newHMCVT the speed characteristic curve and the torquecharacteristic curve are obtained Based on the power flowdirection of HMCVT the transmission efficiency curve isobtained by using the Lrfkoes formula

Using image analysis method to study the influence offour parameters (displacement ratio ε PGM standing

transmission ratio minusk1 gear ratio i1i2i3 and hydraulic systemefficiency ηH) on the transmission efficiency of HMCVT theresults show that the standing transmission ratio minusk1 andgear ratio i1i2i3 have little effect on the transmission effi-ciency whereas the displacement ratio ε and hydraulicsystem efficiency ηH have a greater impact on the trans-mission efficiency

)e parameters of the new HMCVT are finally deter-mined )e results of the transmission characteristics showthat the new HMCVTproposed in this paper has the steplessspeed regulation capability for tractors which can alsocontinuously transmit and change the torque )e trans-mission efficiency of HMCVT is at a high level

In order to verify the theoretical derivation Amesimsimulation software is used for the modeling and simulationof HMCVT )e simulation results are consistent with thetheoretical analysis results )erefore we believe that thetransmission scheme is reasonable and effective for tractors

Data Availability

)e Simcenter Amesim System File data used to support thefindings of this study are available from the correspondingauthor upon request

Conflicts of Interest

)e authors declare that they have no conflicts of interest

Acknowledgments

)is research was financially supported by the National KeyResearch and Development Plan Project of China(2016YFD0700402) and Changzhou Sci amp Tech Program(no CE20185037)

References

[1] L Y Xu Z L Zhou M Z Zhang et al ldquoResearch and designof hydro-mechanical continuously variable transmission fortractorsrdquo Journal of Northeast Agricultural University vol 13no 2 pp 182ndash186 2006

[2] R Paoluzzi and L G Zarotti ldquo)e minimum size of hy-drostatic transmissions for locomotionrdquo Journal of Terra-mechanics vol 50 no 3 pp 153ndash164 2013

[3] J Kim J Kang Y Kim B Min and H Kim ldquoDesign of powersplit transmission design of dual mode power split trans-missionrdquo International Journal of Automotive Technologyvol 11 no 4 pp 565ndash571 2010

[4] A Kim and A Rossetti ldquoFuel consumption reduction inurban buses by using power split transmissionsrdquo EnergyConversion and Management vol 71 pp 159ndash171 2013

[5] A Rossetti and A Macor ldquoMulti-objective optimization ofhydro-mechanical power split transmissionsrdquo Mechanismand Machine lteory vol 62 pp 112ndash128 2013

[6] A Macor and A Rossetti ldquoOptimization of hydro-mechanicalpower split transmissionsrdquo Mechanism and Machine lteoryvol 46 no 12 pp 1901ndash1919 2011

[7] S Shamshirband D Petkovic A Amini et al ldquoSupport vectorregression methodology for wind turbine reaction torqueprediction with power-split hydrostatic continuous variabletransmissionrdquo Energy vol 67 pp 623ndash630 2014

Mathematical Problems in Engineering 13

[8] K T Anuar ldquoResch R Continuously variable tractor trans-missionsrdquo ASAE Distinguished Lecture No 29 pp 1ndash37 StJoseph American Society of Agricultural Engineers St JosephMI USA 2005

[9] S H Yuan and J B Hu ldquo)e efficiency of multi-range hydro-mechanical stepless transmissionrdquo Journal of Beijing Instituteof Technology vol 7 no 2 pp 129ndash133 1998

[10] X J LiuAnalysis of Vehicle Transmission System pp 258-259National Defence Industry Press Beijing China 1998

[11] M M Brenninger ldquoFendt vario CVT in agricultural tractorsrdquoSAE International Warrendale PA USA SAE TechnicalPaper 2007-01-4205 2007

[12] H Aitzetmuller ldquoSteyr S-Matic-the future CVT systemrdquo inProceedings of the Seoul 2000 FISITA world automotive con-gress pp 1ndash6 Seoul South Korea June 2000

[13] J Pohlenz and K Grad ldquoCVTfor input power above 300 kWrdquoVDI-berichte vol 2007 pp 59ndash65 2001

[14] J Pohlenz and K Grad ldquoCVT-System fur denGroszligflacheneinsatzrdquoVDI-berichte vol 2004 pp 23ndash33 1855

[15] L Y Xu Study on Characteristics of Hydro-MechanicalContinuously Variable Transmission of Tractor Xirsquoan Uni-versity of Technology Xirsquoan China 2007

[16] J Looman Zahnradgetriebe-Grundlagen KonstruktionenAnwendungen in Fahrzeugen Springer Berlin Germany2009

[17] Z Rao Planetary Gear Transmission Design Chemical In-dustry Press Beijing China 2003

[18] X Xu Y Liang M Jordan T Peter and D Peng ldquoOptimizedcontrol of engine start assisted by the disconnect clutch in a P2hybrid automatic transmissionrdquo Mechanical Systems andSignal Processing vol 124 2019

[19] S Wang Y Liu Z Wang et al ldquoAdaptive fuzzy iterativecontrol strategy for the wet-clutch filling of automatictransmissionrdquo Mechanical Systems and Signal Processingvol 130 pp 164ndash182 2019

14 Mathematical Problems in Engineering

6 Verification of HMCVT Simulation Model

)e modeling and simulation of HMCVT are accom-plished by using Amesim simulation softwareAccording to Figure 1 a simulation model of HMCVTis established as shown in Figure 8

61 Hydraulic System )e hydraulic system is mainlycomposed of a variable displacement hydraulic pump afixed displacement hydraulic motor two relief valves twocheck valves and a charge hydraulic pump with a reliefvalve )e aim of the charge pump is to compensate for theleakage loss in the pipeline and to establish the lowestpressure in the pumprsquos suction pipeline Two relief valves areused to limit the maximum pressure in the system pipingand its adjustment pressure should be above the normalworking pressure of the system Volumetric losses of the

pump and motor are neglected and their hydromechanicalefficiencies are set at 09

62 Mechanical Components Mechanical components areassumed to be constant efficiency components throughoutthe transmission )e power loss of each gear pair is 2 sothe efficiency is set to 098 In the Ravigneaux planetary gearmechanism the efficiencies of ring gear-long planetary gearcouple long planetary gear-large sun gear couple longplanetary gear-short planetary gear couple and shortplanetary gear-small sun gear couple are set at 099 )emain reducer differential and wheel reducer are combinedand modeled as a gear reducer with an efficiency of 0965 Tosimplify the model the friction losses of the shaft gears andbearings were ignored during the simulation modelingprocess

Efficiency calculation unit

Speed calculation unit

Small sun gear

Short planet gear

Long planet gear

Large sun gear

Ring gear

i1

i2

Pump Motor

i3

i5

i4

i7 i6

i9i8

Carrier

C1

C2

C3

C4

ω ωT

TL

ωωT

T

L

PP

MOT

f (x y)

y

x

X

k

Figure 8 HMCVT simulation model

(km

h)

50

40

30

20

10

00 5 10 15 20 25 30 35

X Time (s)

Tractor linear velocity (kmh)

(a)

0 5 10 15 20 25 30 35X Time (s)

10

08

06

04

02

00

(nul

l)

Transmission efficiency of HMCVT (null)

(b)

Figure 9 Simulation results (a) Simulation result of the tractor speed (b) Simulation result of the transmission efficiency of HMCVT

12 Mathematical Problems in Engineering

63Model Control and Simulation Results )e engine speedis assumed to have a constant value of 2200 rmin the loadtorque of the output shaft of HMCVT is set to 100Nm themodel simulation time is 32 s including range HM1 0ndash8 srange HM2 8ndash16 s range HM3 16ndash24 s and range HM424ndash32 s

)e simulation results are shown in Figure 9 through thevariable pump displacement and coordinated control of theclutches [18 19] the transmission gradually changes fromrange HM1 to range HM4 and the tractor achieves steplessspeed change from 0 to 50 kmh )e simulation results areconsistent with the previous theoretical analysis results whichshow that the 4-range HMCVT transmission scheme pro-posed in this paper is suitable for matching wheeled tractors

64 Comparison Xu LYrsquos transmission scheme [15] hasbeen applied to mass production tractors )is transmissionscheme has 6 working ranges in the forward direction and 3working ranges in the reverse direction which is controlledby 8 clutches )e speed range in the forward direction is0sim30 kmh while the reverse direction is only 0sim10 kmh)e transmission efficiency of the hydraulic range is low (lessthan 80) the transmission efficiency of the hydrome-chanical ranges is high (higher than 90) but is only in theforward direction

)e scheme proposed in this paper has 4 working rangesand a speed range of 0sim50 kmh in the forward and reversedirections which is controlled by 6 clutches )e fewer thetransmission ranges the less switching during the workingranges so the transmission performance is improved )esmaller number of clutches ensures that the system is easierto control )e transmission efficiencies in the forward andreverse directions are close the overall forward transmissionefficiency is higher than 83 and the highest transmissionefficiency is higher than 91

)erefore the application of HMCVT proposed in thispaper is wider it can not only match conventional tractorsbut also match two-way operation tractors and other en-gineering vehicles

7 Conclusions

)is paper proposes a new type of transmission scheme ofHMCVT )e Ravigneaux PGM and the HSTare connectedin parallel to form a closed loop and then connected in serieswith the range switching mechanism )is new HMCVTfeatures a double-row PGM with 4 hydromechanicalworking ranges By coordinating control of the HST dis-placement ratio and the engagement conditions of shiftingclutches the stepless speed regulation of HMCVT at thetractor speed of 0ndash50 kmh is realized

According to the theoretical analysis on the newHMCVT the speed characteristic curve and the torquecharacteristic curve are obtained Based on the power flowdirection of HMCVT the transmission efficiency curve isobtained by using the Lrfkoes formula

Using image analysis method to study the influence offour parameters (displacement ratio ε PGM standing

transmission ratio minusk1 gear ratio i1i2i3 and hydraulic systemefficiency ηH) on the transmission efficiency of HMCVT theresults show that the standing transmission ratio minusk1 andgear ratio i1i2i3 have little effect on the transmission effi-ciency whereas the displacement ratio ε and hydraulicsystem efficiency ηH have a greater impact on the trans-mission efficiency

)e parameters of the new HMCVT are finally deter-mined )e results of the transmission characteristics showthat the new HMCVTproposed in this paper has the steplessspeed regulation capability for tractors which can alsocontinuously transmit and change the torque )e trans-mission efficiency of HMCVT is at a high level

In order to verify the theoretical derivation Amesimsimulation software is used for the modeling and simulationof HMCVT )e simulation results are consistent with thetheoretical analysis results )erefore we believe that thetransmission scheme is reasonable and effective for tractors

Data Availability

)e Simcenter Amesim System File data used to support thefindings of this study are available from the correspondingauthor upon request

Conflicts of Interest

)e authors declare that they have no conflicts of interest

Acknowledgments

)is research was financially supported by the National KeyResearch and Development Plan Project of China(2016YFD0700402) and Changzhou Sci amp Tech Program(no CE20185037)

References

[1] L Y Xu Z L Zhou M Z Zhang et al ldquoResearch and designof hydro-mechanical continuously variable transmission fortractorsrdquo Journal of Northeast Agricultural University vol 13no 2 pp 182ndash186 2006

[2] R Paoluzzi and L G Zarotti ldquo)e minimum size of hy-drostatic transmissions for locomotionrdquo Journal of Terra-mechanics vol 50 no 3 pp 153ndash164 2013

[3] J Kim J Kang Y Kim B Min and H Kim ldquoDesign of powersplit transmission design of dual mode power split trans-missionrdquo International Journal of Automotive Technologyvol 11 no 4 pp 565ndash571 2010

[4] A Kim and A Rossetti ldquoFuel consumption reduction inurban buses by using power split transmissionsrdquo EnergyConversion and Management vol 71 pp 159ndash171 2013

[5] A Rossetti and A Macor ldquoMulti-objective optimization ofhydro-mechanical power split transmissionsrdquo Mechanismand Machine lteory vol 62 pp 112ndash128 2013

[6] A Macor and A Rossetti ldquoOptimization of hydro-mechanicalpower split transmissionsrdquo Mechanism and Machine lteoryvol 46 no 12 pp 1901ndash1919 2011

[7] S Shamshirband D Petkovic A Amini et al ldquoSupport vectorregression methodology for wind turbine reaction torqueprediction with power-split hydrostatic continuous variabletransmissionrdquo Energy vol 67 pp 623ndash630 2014

Mathematical Problems in Engineering 13

[8] K T Anuar ldquoResch R Continuously variable tractor trans-missionsrdquo ASAE Distinguished Lecture No 29 pp 1ndash37 StJoseph American Society of Agricultural Engineers St JosephMI USA 2005

[9] S H Yuan and J B Hu ldquo)e efficiency of multi-range hydro-mechanical stepless transmissionrdquo Journal of Beijing Instituteof Technology vol 7 no 2 pp 129ndash133 1998

[10] X J LiuAnalysis of Vehicle Transmission System pp 258-259National Defence Industry Press Beijing China 1998

[11] M M Brenninger ldquoFendt vario CVT in agricultural tractorsrdquoSAE International Warrendale PA USA SAE TechnicalPaper 2007-01-4205 2007

[12] H Aitzetmuller ldquoSteyr S-Matic-the future CVT systemrdquo inProceedings of the Seoul 2000 FISITA world automotive con-gress pp 1ndash6 Seoul South Korea June 2000

[13] J Pohlenz and K Grad ldquoCVTfor input power above 300 kWrdquoVDI-berichte vol 2007 pp 59ndash65 2001

[14] J Pohlenz and K Grad ldquoCVT-System fur denGroszligflacheneinsatzrdquoVDI-berichte vol 2004 pp 23ndash33 1855

[15] L Y Xu Study on Characteristics of Hydro-MechanicalContinuously Variable Transmission of Tractor Xirsquoan Uni-versity of Technology Xirsquoan China 2007

[16] J Looman Zahnradgetriebe-Grundlagen KonstruktionenAnwendungen in Fahrzeugen Springer Berlin Germany2009

[17] Z Rao Planetary Gear Transmission Design Chemical In-dustry Press Beijing China 2003

[18] X Xu Y Liang M Jordan T Peter and D Peng ldquoOptimizedcontrol of engine start assisted by the disconnect clutch in a P2hybrid automatic transmissionrdquo Mechanical Systems andSignal Processing vol 124 2019

[19] S Wang Y Liu Z Wang et al ldquoAdaptive fuzzy iterativecontrol strategy for the wet-clutch filling of automatictransmissionrdquo Mechanical Systems and Signal Processingvol 130 pp 164ndash182 2019

14 Mathematical Problems in Engineering

63Model Control and Simulation Results )e engine speedis assumed to have a constant value of 2200 rmin the loadtorque of the output shaft of HMCVT is set to 100Nm themodel simulation time is 32 s including range HM1 0ndash8 srange HM2 8ndash16 s range HM3 16ndash24 s and range HM424ndash32 s

)e simulation results are shown in Figure 9 through thevariable pump displacement and coordinated control of theclutches [18 19] the transmission gradually changes fromrange HM1 to range HM4 and the tractor achieves steplessspeed change from 0 to 50 kmh )e simulation results areconsistent with the previous theoretical analysis results whichshow that the 4-range HMCVT transmission scheme pro-posed in this paper is suitable for matching wheeled tractors

64 Comparison Xu LYrsquos transmission scheme [15] hasbeen applied to mass production tractors )is transmissionscheme has 6 working ranges in the forward direction and 3working ranges in the reverse direction which is controlledby 8 clutches )e speed range in the forward direction is0sim30 kmh while the reverse direction is only 0sim10 kmh)e transmission efficiency of the hydraulic range is low (lessthan 80) the transmission efficiency of the hydrome-chanical ranges is high (higher than 90) but is only in theforward direction

)e scheme proposed in this paper has 4 working rangesand a speed range of 0sim50 kmh in the forward and reversedirections which is controlled by 6 clutches )e fewer thetransmission ranges the less switching during the workingranges so the transmission performance is improved )esmaller number of clutches ensures that the system is easierto control )e transmission efficiencies in the forward andreverse directions are close the overall forward transmissionefficiency is higher than 83 and the highest transmissionefficiency is higher than 91

)erefore the application of HMCVT proposed in thispaper is wider it can not only match conventional tractorsbut also match two-way operation tractors and other en-gineering vehicles

7 Conclusions

)is paper proposes a new type of transmission scheme ofHMCVT )e Ravigneaux PGM and the HSTare connectedin parallel to form a closed loop and then connected in serieswith the range switching mechanism )is new HMCVTfeatures a double-row PGM with 4 hydromechanicalworking ranges By coordinating control of the HST dis-placement ratio and the engagement conditions of shiftingclutches the stepless speed regulation of HMCVT at thetractor speed of 0ndash50 kmh is realized

According to the theoretical analysis on the newHMCVT the speed characteristic curve and the torquecharacteristic curve are obtained Based on the power flowdirection of HMCVT the transmission efficiency curve isobtained by using the Lrfkoes formula

Using image analysis method to study the influence offour parameters (displacement ratio ε PGM standing

transmission ratio minusk1 gear ratio i1i2i3 and hydraulic systemefficiency ηH) on the transmission efficiency of HMCVT theresults show that the standing transmission ratio minusk1 andgear ratio i1i2i3 have little effect on the transmission effi-ciency whereas the displacement ratio ε and hydraulicsystem efficiency ηH have a greater impact on the trans-mission efficiency

)e parameters of the new HMCVT are finally deter-mined )e results of the transmission characteristics showthat the new HMCVTproposed in this paper has the steplessspeed regulation capability for tractors which can alsocontinuously transmit and change the torque )e trans-mission efficiency of HMCVT is at a high level

In order to verify the theoretical derivation Amesimsimulation software is used for the modeling and simulationof HMCVT )e simulation results are consistent with thetheoretical analysis results )erefore we believe that thetransmission scheme is reasonable and effective for tractors

Data Availability

)e Simcenter Amesim System File data used to support thefindings of this study are available from the correspondingauthor upon request

Conflicts of Interest

)e authors declare that they have no conflicts of interest

Acknowledgments

)is research was financially supported by the National KeyResearch and Development Plan Project of China(2016YFD0700402) and Changzhou Sci amp Tech Program(no CE20185037)

References

[1] L Y Xu Z L Zhou M Z Zhang et al ldquoResearch and designof hydro-mechanical continuously variable transmission fortractorsrdquo Journal of Northeast Agricultural University vol 13no 2 pp 182ndash186 2006

[2] R Paoluzzi and L G Zarotti ldquo)e minimum size of hy-drostatic transmissions for locomotionrdquo Journal of Terra-mechanics vol 50 no 3 pp 153ndash164 2013

[3] J Kim J Kang Y Kim B Min and H Kim ldquoDesign of powersplit transmission design of dual mode power split trans-missionrdquo International Journal of Automotive Technologyvol 11 no 4 pp 565ndash571 2010

[4] A Kim and A Rossetti ldquoFuel consumption reduction inurban buses by using power split transmissionsrdquo EnergyConversion and Management vol 71 pp 159ndash171 2013

[5] A Rossetti and A Macor ldquoMulti-objective optimization ofhydro-mechanical power split transmissionsrdquo Mechanismand Machine lteory vol 62 pp 112ndash128 2013

[6] A Macor and A Rossetti ldquoOptimization of hydro-mechanicalpower split transmissionsrdquo Mechanism and Machine lteoryvol 46 no 12 pp 1901ndash1919 2011

[7] S Shamshirband D Petkovic A Amini et al ldquoSupport vectorregression methodology for wind turbine reaction torqueprediction with power-split hydrostatic continuous variabletransmissionrdquo Energy vol 67 pp 623ndash630 2014

Mathematical Problems in Engineering 13

[8] K T Anuar ldquoResch R Continuously variable tractor trans-missionsrdquo ASAE Distinguished Lecture No 29 pp 1ndash37 StJoseph American Society of Agricultural Engineers St JosephMI USA 2005

[9] S H Yuan and J B Hu ldquo)e efficiency of multi-range hydro-mechanical stepless transmissionrdquo Journal of Beijing Instituteof Technology vol 7 no 2 pp 129ndash133 1998

[10] X J LiuAnalysis of Vehicle Transmission System pp 258-259National Defence Industry Press Beijing China 1998

[11] M M Brenninger ldquoFendt vario CVT in agricultural tractorsrdquoSAE International Warrendale PA USA SAE TechnicalPaper 2007-01-4205 2007

[12] H Aitzetmuller ldquoSteyr S-Matic-the future CVT systemrdquo inProceedings of the Seoul 2000 FISITA world automotive con-gress pp 1ndash6 Seoul South Korea June 2000

[13] J Pohlenz and K Grad ldquoCVTfor input power above 300 kWrdquoVDI-berichte vol 2007 pp 59ndash65 2001

[14] J Pohlenz and K Grad ldquoCVT-System fur denGroszligflacheneinsatzrdquoVDI-berichte vol 2004 pp 23ndash33 1855

[15] L Y Xu Study on Characteristics of Hydro-MechanicalContinuously Variable Transmission of Tractor Xirsquoan Uni-versity of Technology Xirsquoan China 2007

[16] J Looman Zahnradgetriebe-Grundlagen KonstruktionenAnwendungen in Fahrzeugen Springer Berlin Germany2009

[17] Z Rao Planetary Gear Transmission Design Chemical In-dustry Press Beijing China 2003

[18] X Xu Y Liang M Jordan T Peter and D Peng ldquoOptimizedcontrol of engine start assisted by the disconnect clutch in a P2hybrid automatic transmissionrdquo Mechanical Systems andSignal Processing vol 124 2019

[19] S Wang Y Liu Z Wang et al ldquoAdaptive fuzzy iterativecontrol strategy for the wet-clutch filling of automatictransmissionrdquo Mechanical Systems and Signal Processingvol 130 pp 164ndash182 2019

14 Mathematical Problems in Engineering

[8] K T Anuar ldquoResch R Continuously variable tractor trans-missionsrdquo ASAE Distinguished Lecture No 29 pp 1ndash37 StJoseph American Society of Agricultural Engineers St JosephMI USA 2005

[9] S H Yuan and J B Hu ldquo)e efficiency of multi-range hydro-mechanical stepless transmissionrdquo Journal of Beijing Instituteof Technology vol 7 no 2 pp 129ndash133 1998

[10] X J LiuAnalysis of Vehicle Transmission System pp 258-259National Defence Industry Press Beijing China 1998

[11] M M Brenninger ldquoFendt vario CVT in agricultural tractorsrdquoSAE International Warrendale PA USA SAE TechnicalPaper 2007-01-4205 2007

[12] H Aitzetmuller ldquoSteyr S-Matic-the future CVT systemrdquo inProceedings of the Seoul 2000 FISITA world automotive con-gress pp 1ndash6 Seoul South Korea June 2000

[13] J Pohlenz and K Grad ldquoCVTfor input power above 300 kWrdquoVDI-berichte vol 2007 pp 59ndash65 2001

[14] J Pohlenz and K Grad ldquoCVT-System fur denGroszligflacheneinsatzrdquoVDI-berichte vol 2004 pp 23ndash33 1855

[15] L Y Xu Study on Characteristics of Hydro-MechanicalContinuously Variable Transmission of Tractor Xirsquoan Uni-versity of Technology Xirsquoan China 2007

[16] J Looman Zahnradgetriebe-Grundlagen KonstruktionenAnwendungen in Fahrzeugen Springer Berlin Germany2009

[17] Z Rao Planetary Gear Transmission Design Chemical In-dustry Press Beijing China 2003

[18] X Xu Y Liang M Jordan T Peter and D Peng ldquoOptimizedcontrol of engine start assisted by the disconnect clutch in a P2hybrid automatic transmissionrdquo Mechanical Systems andSignal Processing vol 124 2019

[19] S Wang Y Liu Z Wang et al ldquoAdaptive fuzzy iterativecontrol strategy for the wet-clutch filling of automatictransmissionrdquo Mechanical Systems and Signal Processingvol 130 pp 164ndash182 2019

14 Mathematical Problems in Engineering