ADE682435 1 14

ADE682435 1 14 Research Article Advances in Mechanical Engineering 2016, Vol 8(12) 1–14 � The Author(s) 2016 DOI 10 1177/1687814016682435 aime sagepub com Development of a new sub shift schedule and c[.]

Research Article Development of a new sub-shift schedule and control algorithm for a hydro-mechanical transmission Advances in Mechanical Engineering 2016, Vol 8(12) 1–14 Ó The Author(s) 2016 DOI: 10.1177/1687814016682435 aime.sagepub.com Sunghyun Ahn1, Jingyu Choi1, Hanho Son1, Suchul Kim2, Jinwoong Lee2 and Hyunsoo Kim1 Abstract In this study, a new sub-shift schedule was proposed for a hydro-mechanical transmission To develop the sub-shift schedule, a network analysis was performed by considering the hydrostatic unit loss and mechanical component losses In the new sub-shift schedule, the sub-shift gear can be selected with respect to the demanded wheel torque and vehicle speed, which provides improved system efficiency for the given vehicle operating condition Since the sub-shift can only be carried out at a speed ratio where the off-going and on-coming clutch speeds are synchronized in the existing subshift control, a sub-shift control algorithm without the clutch speed synchronization was proposed to apply the new subshift schedule using the forward clutch pressure and hydrostatic unit stroke control The performance of the sub-shift control algorithm without the clutch speed synchronization was evaluated by the simulation and experiment It was found from the simulation and experimental results that the sub-shift can be achieved, showing an acceptable peak-topeak torque variation in the driveshaft Keywords Hydro-mechanical transmission, transmission efficiency, transmission component loss, sub-shift schedule, shift control Date received: 12 September 2016; accepted: 12 November 2016 Academic Editor: Francesco Massi Introduction To improve fuel efficiency and provide enhanced convenience for workers, many studies have been conducted on tractor transmissions Among them, hydro-mechanical transmission (HMT) is expected to be a viable solution since it can provide improved fuel efficiency by changing the engine operation point on the high-efficiency region using the continuously variable transmission (CVT) function, and it offers convenience in its working operation due to its automatic shifting feature The HMT transmits the engine power through the mechanical path and hydraulic path using the planetary gear and hydrostatic unit (HSU) The HMT transmits relatively higher power with higher efficiency than the hydrostatic transmission (HST) Because of these advantages, many tractor manufacturers have developed their own type of HMT.1,2 The HMT consists of an HSU, planetary gear sets, and sub-shift part The engine power is split at the planetary gear and is transmitted to the sub-shift gear The sub-shift gear provides an expanded gear ratio range The sub-shift gear is composed of multi-step School of Mechanical Engineering, Sungkyunkwan University, Suwon, Korea Machinery Technology Group, LS Mtron, Gunpo, Korea Corresponding author: Hyunsoo Kim, School of Mechanical Engineering, Sungkyunkwan University, 2066 Seobu-ro, Suwon 16419, Korea Email: hskim@me.skku.ac.kr Creative Commons CC-BY: This article is distributed under the terms of the Creative Commons Attribution 3.0 License (http://www.creativecommons.org/licenses/by/3.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/ open-access-at-sage) 2 gears, which are implemented by the clutch and brake operations In the target HMT, the sub-shift is performed at a specific speed ratio of the HMT, where the speeds of the oncoming gear (clutch) and off-going gear are synchronized using the on/off clutch.3–6 Otherwise, the shift shock occurs due to the speed difference between the on-coming and the off-going clutch In a conventional automotive transmission, the shift point is determined by considering the system efficiency with respect to the vehicle speed and acceleration pedal position However, in the target tractor, the shift point is determined according to the HMT speed ratio regardless of the efficiency For the control of the HMT with multi-speed subshift gears, various studies have been performed including the shift logic and tractor velocity control for the HMT with two-speed sub-shift gears,3 control architecture of the HMT using proportional–integral (PI) control and current compensation,4 sub-shift schedule to obtain the maximum driving capacity and to prevent the shift circulation due to the frequent shift,5 and HSU control for the sub-shift of the HMT with dog clutch.6 For smooth sub-shift, control strategy of the HMT was investigated based on the orthogonal test.7 In most of the aforementioned works, the sub-shift is only possible under the constraint that the sub-shift should be carried out when the clutch input and output speeds are synchronized The HMT efficiency varies depending on the speed ratio and input power.2,8 The power in the HMT is transmitted through the hydraulic path and mechanical path and the efficiency of each path depends on the hydraulic and mechanical component losses.8 Therefore, to perform a sub-shift-to-a gear ratio that provides better HMT efficiency, a new sub-shift schedule is required that considers the HMT efficiency, and a sub-shift control algorithm that can be applied using the new sub-shift schedule needs to be developed In this study, a sub-shift schedule that provides more efficient HMT operation and a sub-shift control algorithm without clutch speed synchronization were proposed To derive the sub-shift gear schedule, power losses of the hydraulic path and mechanical path were calculated including the transmission components The efficiency of each sub-shift gear was compared and a new sub-shift schedule was developed To apply the new sub-shift schedule, a sub-shift control algorithm was developed using the forward clutch pressure and HSU stroke control without the clutch speed synchronization Configuration of the HMT Figure shows the configuration of the target HMT and transmission components The target HMT Advances in Mechanical Engineering consists of two planetary gears (PG1 and PG2), an HSU, four sub-shift clutches (CL1–4), and forward (FWD)/reverse (REV) clutches The engine is connected to the sun gear of the first planetary gear and power take-off (PTO) shaft When the tractor operates with the attachment in the PTO workings such as the baler and rotary workings, the engine power is directly transmitted through the PTO shaft to the attachment, which is connected to the PTO shaft The hydraulic pump of the HSU is driven by the engine The hydraulic motor of the HSU is connected to the ring gear of the second planetary gear When the odd sub-shift clutch (first or third) is engaged, the power from the engine is transmitted through the carrier of the planetary gear set When the even sub-shift clutch (second or fourth) is engaged, the power flows through the second sun gear of the planetary gear For the dynamic model of the powertrain, the inertias of the engine, hydraulic pump, hydraulic motor, sub-shift shaft, and vehicle were considered (Figure 1) The dynamic equations of the target HMT are derived as follows Je v_ e = Te  TS1  iep Tp ð1Þ Jp v_ p = hmech Tp  Dp ist P ð2Þ P_ = b VHSU (hvol Dp ist vp  Dm vm ) Jm v_ m = Dm P  imR TR ð3Þ ð4Þ Jsub v_ sub = (iCL1  TCL1 + iCL2  TCL2 + iCL3  TCL3 + iCL4  TCL4 )  TDS iFWD  iRGs (Jwh + mvehicle r2wh )v_ wh = TDS  Tload ð5Þ ð6Þ where J is the inertia, i is the gear ratio, h is the efficiency, D is the HSU displacement, P is the pressure, V is the volume, m is the mass, and r is the radius The subscripts e, ep, p, mech, st, vol, m, and mR represent the engine, from the engine to the pump, the HSU pump, mechanical efficiency, HSU stroke, volumetric efficiency, HSU motor, and from the motor to the ring gear, respectively The subscripts sub, CL, wh, FWD, RGs, DS, and load indicate the sub-shift shaft, clutch, wheel, forward gear, reduction gears, driveshaft, and the load, respectively The numbers combined with CL represent the sub-shift gear number The HSU stroke (ist) is defined as the ratio of the HSU motor speed (vm) to the pump speed (vp) as ist = vm/vp The mechanical efficiency and the volumetric efficiency of the HSU were obtained from the experiments.8 From equations (1)–(6), the driveshaft torque TDS of the second sub-shift gear (even shaft) and third subshift gear (odd shaft) can be represented as follows Ahn et al Figure Configuration of the target HMT  iCL2 ieven (a + b) + b + (a  1)iep imR ist    dist Jm ve  Jsub v_ sub Te  J e v_ e  iep ist dt TDS = iRGs iFWD second sub  shift gear J0 e = Je + i2ep (Jp + i2st  Jm ) ZS2 ZP1 ZS2 b= ZS1 ZP2 ZR  iCL3 iodd (a + b) TDS = iRGs iFWD b + aiep imR ist    dist Jm ve  Jsub v_ sub Te  J e v_ e  iep ist dt a= third sub  shift gear ð7Þ represent the sun gear, the pinion gear, and the ring gear of the planetary gear, respectively The driveshaft torque of the first and fourth sub-shift gear can be derived in a similar way The HSU stroke ist in equations (2), (3), (7), and (8) is determined by the HSU swash plate angle The swash plate angle is controlled by the solenoid valve and linkage mechanism The dynamic model of the HSU stroke can be represented as a second-order system as follows ist ist ð8Þ where J0 e is the equivalent inertia, and Z is the number of gear teeth The subscripts S1, S2, P1, P2, and R cmd = KeTd s s2 + 2zvn s + v2n ð9Þ where K is the system gain, Td is the time delay, z is the damping ratio, and is the natural frequency The subscript cmd represents the command value The parameters of the transfer function (equation (9)) were determined from the time domain analysis in Figure 2(a) and Advances in Mechanical Engineering Figure Mechanical and volumetric efficiency of the HSU (DPHSU = 400 bar) Figure Comparison of the transfer function with experimental results ist: HSU stroke; ist_cmd: HSU stroke command verified in the frequency domain (Figure 2(b)) It is seen from Figure that the second-order transfer function shows a good agreement with the experiments Modeling of the HMT losses The efficiency of the HMT changes depending on the engine power, speed ratio, and the sub-shift gear In PTO workings, since a constant speed is required for the attachment, the engine and vehicle speeds need to be maintained at a constant value, in other words, at a constant speed ratio In this working condition, if the HMT is operated at a sub-shift gear that provides better system efficiency, improved fuel economy can be achieved In this study, to find a sub-shift schedule that provides better system efficiency for the given wheel load and speed, HMT losses were investigated, including the HSU loss and mechanical losses HSU loss The HSU in the target HMT is composed of a variable displacement type of hydraulic pump and a fixed displacement type of hydraulic motor The speed and torque of the HSU depend on the volumetric efficiency (hvol) and the mechanical efficiency (hmech) as follows vm = ist vp (hvol )sign(Tp ) ð10Þ Tm = Tp (ist )1 (hmech )sign(Tp ) ð11Þ where v is the rotational speed, T is the torque, and ist is the HSU stroke The subscripts m, p, vol, and mech indicate the motor, pump, volumetric, and mechanical, respectively Sign(Tp) represents the direction of the power in the HSU The mathematical models for the loss of the hydraulic pump and motor were proposed.9,10 However, in this study, an HSU efficiency map (Figure 3) was used, which was obtained from the experiment In the experiments, the HSU efficiency was measured for various HSU stroke and input speed at the oil temperature of 20°C Since the tractor can be used in variable environmental conditions, the HSU efficiency map needs to be constructed for various temperatures in the actual application Mechanical component loss in the HMT The target HMT consists of mechanical components such as gears, clutches, shafts, and bearings In the mechanical components, load-dependent loss and noload loss occur Both losses vary depending on the torque and speed, in other words, the power changes with the sub-shift gear In this study, to analyze the mechanical losses of the HMT, individual models of each loss Ahn et al were derived based on the mathematical governing equations and experiments Bearing loss There are two types of bearing loss: (1) noload loss which depends on the rotational speed of the bearing and (2) load-dependent loss which is proportional to the bearing load.11–13 The bearing torque loss is calculated as follows12,13 TBL0 = > > < 1:6  108  f0  dm > > : 1010  f0  (v  v)23  dm mm2 s  mm2 if (v  v)  2000 s  ð12Þ if (v  v)\2000 TBL1 = f1  F1  dm  103 ð13Þ TBL = TBL0 + TBL1 ð14Þ where f0 and f1 are the coefficients for the no-load loss and load-dependent loss, respectively; v is the kinematic viscosity of oil; dm is the bearing mean diameter; and F1 is the equivalent bearing load The subscripts BL0, BL1, and BL represent no-load loss, load-dependent loss, and total bearing loss, respectively Forty-six bearings are used in the target HMT The equivalent bearing load is obtained from the force equilibrium and moment equilibrium for each shaft Figure shows the power flow and free-body diagram of the odd shaft at the first sub-shift gear The third gear rotates freely because the third sub-shift clutch is not engaged At the first sub-shift gear, the torque from the carrier of the planetary gear comes into the odd shaft (Tin) and goes out to the sub-shift shaft through the first sub-shift gear (Tload) There are six bearings (B1–B6) in the odd shaft The equivalent load (F1) of each bearing can be calculated using Tin and Tload Since spur gears were used in the target HMT, only the radial direction was considered when calculating the equivalent bearing load Gear loss The gear loss is separated into friction loss and churning loss The rolling and sliding frictions cause the gear friction loss on the contact surface, and its magnitude is proportional to the torque transmitting through the gear.14 The target HMT uses a spur gear, which has an efficiency of 98%–98.5% in general.15 In this study, the gear friction loss was obtained assuming that the efficiency of the spur gear is 98% The gear churning loss occurs due to the viscous friction of the lubricant oil Its magnitude is determined from the rotational speed of the gear, the emersion depth in the lubricant oil, and so on There are many Figure Power flow and free-body diagram for the odd shaft at the first sub-shift gear studies to calculate the gear churning loss.16–18 In this study, the British Standards formulas are used18 PCHL = PCHL = 1:474  fg  v  v3  d 5:7 Ag  1026 for smooth sides of gears   Rf ffi 7:37  fg  v  v3  d 4:7  fw  pffiffiffiffiffiffiffi tan b ð15Þ Ag  1026 for tooth surfaces ð16Þ where PCHL is the power loss for gear churning, fg is the gear emersion factor, d is the diameter, fw is the face width of the gear, Ag is the arrangement constant, and b is the helix angle Rf is the roughness factor The gear churning loss is calculated as the sum of the loss for the smooth sides of the gears and the loss for the toothed surfaces using equations (15) and (16) An emersion factor fg = was used when the gear was fully submerged, and fg = 0.5 was used when the gear was submerged halfway Clutch drag torque The wet-type clutch was used for the sub-shift and FWD/REV clutches in the target HMT A lot of research discusses the mathematical model for the clutch drag torque.14,19,20 In this study, an alternative Advances in Mechanical Engineering shrinking model14 was used to calculate the clutch drag torque as follows Tdrag = N  p  m  Dv (rS  ri4 ) 2h ð17Þ where Tdrag is the drag torque, N is the number of clutch plates, m is the absolute viscosity, h is the clearance between clutch plates, rS is the equivalent effective radius, and ri is the inner radius of the clutch The equivalent effective radius depends on the centrifugal force of the clutch, the viscous force, and the surface tension forces When the relative velocity of the clutch is low, the oil film between the clutch plates is maintained at full immersion because the viscous and surface tension forces are larger than the centrifugal force For that reason, the equivalent effective radius is the same with the outer radius of the clutch at a low relative velocity However, when the relative velocity of the clutch is high, the equivalent effective radius becomes smaller because the oil film is decreased The equivalent effective radius is calculated as follows rS = ro for Q  Qre sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi Qre Q re r + ri2  for Q\Qre rS = Q o Q ð18Þ where Q is the input flow rate, and Qre is the required input flow rate.14 Oil pump loss The oil pump supplies the oil flow to the transmission components through the valve body The oil pump in the target HMT is installed directly to the engine shaft (Figure 1) The regulator valve maintains a constant line pressure at 20 bar The oil pump torque loss is determined as follows TP loss = hmech  DP  Pl 20p ð19Þ Figure Network model of the HMT (second gear engaged) where TP_loss is the oil pump torque loss, DP is the displacement, and Pl is the line pressure The mechanical efficiency of the oil pump (hmech) varies with the pump speed, pressure, and temperature In this study, experimental data from the manufacturing company were used Network analysis and sub-shift schedule The efficiency of the target HMT changes depending on the speed, torque, and the HSU stroke (ist) In addition, the direction and magnitude of the power flow are changed by ist To investigate the HMT efficiency considering the component losses, a network analysis was performed In the network analysis, the connection relationships between the transmission components were configured as a network, and the speed, torque, and power of the HMT system were analyzed.21–24 Figure shows the network model of the HMT when the second gear is engaged Numbers 1–26 are the torque nodes, which are the same as the shaft nodes, and numbers (1)–(14) are the speed nodes S3 and C3, between (13) and (14), denote the sun gear and carrier of the single-pinion planetary gear, respectively, which acts as the final reduction gear When the sub-shift second gear is engaged, the carrier is not connected Using the network model (Figure 5), the torque and the speed matrices are obtained as T =~ bT ẵMT   ~ 20ị ẵMv   ~ v =~ bv ð21Þ where [MT] and [Mv] denote the torque and the speed matrices, respectively; ~ T and ~ v denote the torque and bv denote speed at each node, respectively; and ~ bT and ~ the boundary vectors, that is, the input torque and speed, respectively These equations can be used to determine the torque and speed of each node, as well as the output torque and speed Ahn et al Using the boundary vectors (~ bT and ~ bv ), an initial network analysis is performed without the component losses From the initial network analysis, the torque and speed of each shaft are obtained, and then the hydraulic loss and mechanical losses are determined using the torque and speed of each shaft as follows n Tmech loss = X n TBL + P PnCHL n + Tdrag + TP vnshaft loss ð22Þ where the superscript n is the number of the shaft The torque and speed matrices, as well as the boundary vectors in equations (20)–(21), are modified by the amount of loss Using modified matrices and vectors, network analyses are conducted repeatedly until the torque and speed error between the previous and current stages converse within an allowable range Figure shows the network analysis results of the HMT efficiency in the presence or absence of mechanical losses for the sub-shift gear when the engine torque Te is 200 N m, and the engine speed ve is 1600 r/min When the mechanical losses are not included, the HMT efficiency of each sub-shift gear increases with the vehicle speed to 100% and then decreases The point where the mechanical efficiency shows 100% is called the ‘‘mechanical point’’ (MP) because the power from the engine is transmitted only through the mechanical path without flowing through the hydraulic path As shown in Figure 6, the HMT efficiency decreases rapidly when it departs from the MP This is because the HSU loss increases due to the relatively lower efficiency of the hydraulic path as the power ratio of the hydraulic path to the engine input power increases When the mechanical losses are included, the HMT efficiency drops by 14%–28% at the MP depending on the sub-shift gear It is also noted that the reduction in the HMT efficiency increases as the vehicle speed increases This is because the mechanical losses increase with the vehicle speed ‘‘A’’ (6.3 km/h), ‘‘B’’ (9 km/h), and ‘‘C’’ (16.3 km/h) in Figure are the vehicle speeds where the sub-shift can be performed, in other words, where the speeds of the sub-shift clutches are synchronized Each sub-shift can be performed at A (1 $ sub-shift), B (2 $3 subshift), and C (3 $ sub-shift) When the vehicle speed is km/h, the HMT efficiency of the third sub-shift gear is the highest However, only the second sub-shift gear can be used at v = km/h because the sub-shift to the third gear is impossible in the region A-B in the existing sub-shift control algorithm The HMT losses are compared for the second and third sub-shift gears at v = km/h in Figure It is seen that the total HMT loss of the third gear is less than that of the second gear, of which the third gear provides a higher efficiency compared to that of the second gear at v = km/h, as shown in Figure It is noted that the Figure Network analysis results of the HMT efficiency in the presence or absence of mechanical losses for sub-shift gear (Te = 200 N m, ve = 1600 r/min) Figure Comparison of HMT losses for the second and third sub-shift gears (Te = 200 N m, ve = 1600 r/min, n= km/h) HSU loss of the third gear and the gear losses are less than those of the second sub-shift gear This is because the power transmitting through the HSU in the third sub-shift gear is less than that of the second sub-shift gear The validity of the HMT loss models developed in this study was evaluated by experiment In the experiment, since it is only possible to measure the total HMT system efficiency instead of the loss of each component, the HMT efficiency was measured using the test bench in Figure and was then compared with the network analysis results, which consider the loss models The test was performed at an engine speed of 1600 r/min Advances in Mechanical Engineering Figure Test bench of the HMT Figure Comparison of the HMT efficiency between test and network analysis at the first sub-shift gear and a dynamo load torque of 150 N m for the first subshift gear Figure shows a comparison of the HMT efficiency between the test and network analysis It is seen that the efficiency of the network analysis agreed well with the test results for vehicle speed The mechanical point can be found at v = 3.5 km/h for the first sub-shift gear, where the HMT has the highest efficiency in the test and network analysis The gap between the observed efficiency and an efficiency of 100% at the mechanical point is due to the mechanical losses Using the network analysis, the HMT efficiency can be obtained for various engine speeds, engine torques, and vehicle speeds, and the sub-shift gear that provides the highest HMT efficiency for the given wheel torque and vehicle speed can be selected to construct the sub-shift schedule In Figure 10, the sub-shift schedule developed in this study is shown and compared with the existing subshift schedule when the engine speed is 1600 r/min As shown in Figure 10, in the existing sub-shift schedule, the sub-shift is performed according to the vehicle speed regardless of the wheel torque because the sub-shift can be carried out only when the speeds of the on-coming and off-going clutch are synchronized On the other hand, in the sub-shift schedule proposed in this study, the sub-shift gear is selected according to the vehicle speed and the demanded wheel torque It is also noted that for the same vehicle speed, that is, the same speed ratio, a different subshift gear is selected depending on the wheel torque For Ahn et al Driveshaft torque variation: the driveshaft torque TDS will experience torque variation due to the transient terms in equations (7) and (8) such as the engine torque variation, HSU stroke differential, and the engine speed differential In particular, the engine speed differential, which has a negative value during the upshift, causes the driveshaft torque to increase in the positive direction HSU stroke change: the HSU stroke should be changed to satisfy the given speed ratio Forward clutch pressure control Figure 10 Proposed sub-shift schedule at ve = 1600 r/min Figure 11 Schematic of the forward clutch pressure control example, at point P where the wheel torque is 14,000 N m and the vehicle speed v is km/h, the second sub-shift gear is used in the existing sub-shift control However, it is seen that the third sub-shift gear has better HMT efficiency in the proposed sub-shift schedule When the vehicle speed is low (v \ km/h) or high (v 20 km/h), only the first subshift gear or the fourth sub-shift gear can be used regardless of the HMT efficiency because of the sub-shift gear range limitation Sub-shift control algorithm without clutch speed synchronization As shown in Figure 10, to apply the new sub-shift schedule proposed in this study, the sub-shift needs to be carried out at a vehicle speed (speed ratio) where the clutch speeds are not synchronized To perform the sub-shift without the clutch speed synchronization, the following problems should be solved: As shown in Figure 1, the target HMT has the forward clutch, which is controlled by a proportional valve In the target HMT system, two types of hydraulic valves are used to control the clutch pressure: (1) an on/offtype valve for the sub-shift clutches and (2) a proportional valve for the forward and reverse clutch The on/ off-type valve has the advantage of a relatively low cost; however, precise pressure control is impossible In contrast, the proportional valve can control the pressure in a proportional manner In this study, the driveshaft torque control algorithm was presented by limiting the transfer torque of the forward clutch In the target tractor, the forward clutch pressure is maintained as a lock-up pressure, which is quite higher than the clutch pressure, to transmit the required torque Since the magnitude of the forward clutch torque TFWD is proportional to the clutch pressure, the forward clutch pressure is controlled to reduce the torque variation while maintaining the demanded driveshaft torque TDS_dmd during the sub-shift Since the PTO workings require a constant vehicle speed,25 the forward clutch pressure is controlled only to transmit the same torque during the sub-shift The forward clutch pressure to transmit TDS can be obtained as  PFWD dmd = TDS dmd iRGs N mReff + kpis xpis Apis  ð23Þ where TDS_dmd is the demanded driveshaft torque before the sub-shift, m is the friction coefficient, R is the radius, A is the area, k is the stiffness of the return spring, and x is the displacement Subscripts eff and pis indicate the effective radius of the clutch disk and the clutch piston, respectively The demanded driveshaft torque TDS_dmd for each sub-shift gear is determined from the HMT gear ratio and the engine torque Figure 11 shows a schematic of the forward clutch pressure control algorithm proposed in this study When the sub-shift signal is applied, PFWD_cmd decreases in a stepwise manner to PFWD_dmd and is 10 Advances in Mechanical Engineering Figure 12 Performance simulator of the HMT tractor maintained until the forward clutch speeds are synchronized At that point, PFWD_cmd increases up to the lock-up pressure Even if the forward clutch pressure PFWD decreases to PFWD_dmd, which can transmit the demanded driveshaft torque, the clutch slip occurs because the driveshaft torque fluctuates around the demanded torque due to the effect of the rotational inertias connected to the driveshaft SR = ZP2 ZS1  iimR ZP1 ZR ist ep ZP1 ZR + ZS1 ZP2 iodd iCL3 iFWD iRGs for the third sub  shift gear ð25Þ The relationship between the speed ratio SR and HSU stroke ist for the first and fourth sub-shift gears can be obtained in a similar way HSU stroke control Simulation and discussion To perform the sub-shift without the speed synchronization, the HSU stroke should be changed to satisfy the given engine and vehicle speeds, in other words, the given speed ratio, SR When the tractor works using the PTO, in general, the engine and vehicle speeds need to be maintained at a constant value.25 The HSU stroke ist, which maintains the demanded engine speed for the given vehicle speed, can be obtained The speed ratio SR for the second and third sub-shift gears is derived using the HSU stroke as follows The performance of the sub-shift control algorithm without the sub-shift clutch speed synchronization was evaluated To investigate the dynamic characteristics of the target tractor with HMT, a performance simulator was developed based on the dynamic model of the tractor using AMESim In Figure 12, the performance simulator developed in this study is shown The vehicle specifications used in the simulator are listed in Table The controller, which determines the command values of the engine and the HMT, was developed using MATLAB/Simulink The simulation to evaluate the sub-shift performance without the clutch speed synchronization was carried out at point P in Figure 10 from the second to third gear When the HMT is operating at P, in the existing SR = ZS1 ZP2 (ZR + ZS2 )  ZR (ZS2 ZP1  ZS1 ZP2 ) iimR ist ep ieven iCL2 iFWD iRGs ZS2 (ZP1 ZR + ZS1 ZP2 ) for the second sub  shift gear ð24Þ Ahn et al 11 Table Specifications used in the simulator Specification Engine HSU Sub-shift part Planetary gear set Hydraulic valves Vehicle Rated power: 70 kW Max speed: 2800 r/min Max torque: 320 N m Pump size: 245 to 45 cm3/rev Motor size: 45 cm3/rev Max pressure: 450 bar Number of clutch disk: EA Area of clutch disk: 70.8 cm2 Operating pressure: 20 bar Operating valve type: on/off type Gear efficiency: 98% Two planetary gears Single pinion planetary gear Gear efficiency: 96% HSU control valve - Two proportional solenoid valves with feedback lever Sub-shift part valve - On/off type, three-way, two-position FWD/REV part valve - Proportional valve Vehicle mass: 3200 kg Tire radius: 0.76 m Final reduction gear ratio: 5.58 HSU: hydrostatic unit; SPPG: single-pinion planetary gear; FWD/REV: forward/reverse sub-shift control, the second sub-shift gear should be used according to the shift schedule, which is determined only by the vehicle speed (speed ratio) However, in the new sub-shift schedule, the third sub-shift gear needs to be selected to achieve the improved HMT efficiency To perform the sub-shift from the second to third gear, the sub-shift control algorithm without the clutch speed synchronization was applied using the forward clutch pressure control and HSU stroke control proposed in this study The forward clutch pressure PFWD_dmd was determined from equation (23) using the driveshaft torque TDS_dmd, which can be calculated using the engine torque, the speed ratio, and the HMT efficiency When obtaining the driveshaft torque, the inertial torque of the rotational inertias can be neglected, since the vehicle speed is maintained at a nearly constant value The HSU stroke, which satisfies the HMT operation at point P of the third sub-shift gear, can be determined from equation (25) as follows   iep ZS1 ZP2 ZP1 ZR + ZS1 ZP2 ist cmd =  SR ð26Þ imR ZP1 ZR iodd iCL3 iFWD iRGs ZP1 ZR Figure 13 shows the simulation results for the second ! third sub-shift by the proposed sub-shift control algorithm without clutch speed synchronization Before the sub-shift, the engine speed was maintained at ve = 1600 r/min, and the vehicle speed v was km/h When the sub-shift signal was applied at t = 0.2 s, the HSU stroke command (ist_cmd) obtained from equation (26) was applied in a stepwise manner (a) The actual HSU stroke followed the command stroke, exhibiting the second-order dynamic characteristics in equation (9) When the sub-shift signal was applied, PFWD_cmd decreased in a stepwise manner to PFWD_dmd and was maintained until the forward clutch speeds were synchronized At that point, PFWD_cmd increased to the lock-up pressure Even if the forward clutch pressure PFWD decreased to PFWD_dmd, which can transmit the demanded driveshaft torque, the clutch slip occurred because the driveshaft torque fluctuated around the demanded torque due to the inertial torque of the rotational inertias connected to the driveshaft As shown in Figure 13(e), the vehicle speed was almost maintained at the demanded speed The peak-to-peak of the driveshaft torque was obtained as 445 N m Unlike conventional transmissions in automobiles, it is difficult to evaluate the shift quality of the tractor since the shift quality is affected greatly by the type of workings, the type of transmission, and the class of tractor.26 It was reported that the maximum peakto-peak of the driveshaft torque is 10,000 N m in a 62.5-kW class tractor equipped with power shift transmission when driving on a flat road.27 Compared with the peak-to-peak torque of the power shift transmission, the 445 N m of the target HMT tractor by the proposed sub-shift control is considered to be in the acceptable range Experimental results and discussion To investigate the performance of the proposed subshift control algorithm, the experiments were performed using the test bench in Figure In the test bench, since the dynamo motor was connected to the HMT output shaft, the HMT output torque was measured instead of the driveshaft torque In Figure 14, the experimental results are shown In the experiment, the second ! third sub-shift was carried out by the proposed sub-shift control algorithm without clutch speed synchronization When the sub-shift signal was applied at t = 0.1 s, the forward clutch pressure PFWD (d) decreased in a stepwise manner, maintained 2.1 bar for 0.6 s, and increased again The HSU stroke ist increased from ist = 20.55 to ist = 0.2 to meet the same speed ratio before and after the sub-shift Since the speed sensor for the sub-shift shaft could not be installed in the test bench, the time when the speeds of the input and output clutches become equal could not be measured Therefore, in the experiment, the duration time that was obtained from the simulation was used 12 Figure 13 Simulation results for second ! third sub-shift ist: HSU stroke; ist_cmd: HSU stroke command; vFWD_out: forward clutch output speed; vFWD_in: forward clutch input speed; PFWD: forward clutch pressure; PFWD_cmd: forward clutch pressure command; V: vehicle speed; Vdmd: demanded vehicle speed The peak-to-peak torque in the experimental results (Figure 14) was 57 N m Unlike the simulation, this peak-to-peak torque was measured at the HMT output shaft, not at the driveshaft because the driveshaft was not installed in the proto-type HMT used in the experiment Considering the final reduction gear ratio between the output shaft and the driveshaft, the HMT output shaft torque turned out to be 1550 N m of the driveshaft torque variation Even if this value is higher than the simulation result of 445 N m, it is considered to be acceptable compared with the experimental results for a flat road, 10,000 N m.27 The reason why the experimental result of the driveshaft is higher than Advances in Mechanical Engineering Figure 14 Experimental results for second ! third sub-shift without clutch speed synchronization ist: HSU stroke that of the simulation result is that the driveshaft and tire were not installed in the proto-type HMT used in the experiment If the stiffness effect of the driveshaft and tire was included, the peak-to-peak torque variation is expected to be reduced It was also seen that the engine speed (a) and vehicle speed (e) were maintained at almost constant values, which is required for PTO workings It was found from the experimental results that the sub-shift control algorithm without the clutch speed synchronization proposed in this study can provide a satisfactory sub-shift, while maintaining constant engine and vehicle speeds Ahn et al 13 Conclusion Declaration of conflicting interests A new sub-shift schedule that provides improved system efficiency was proposed for an HMT, which consists of an HSU, planetary gear set, and four sub-shift gears In addition, to apply the sub-shift schedule, a sub-shift control algorithm without clutch speed synchronization was developed To construct the sub-shift schedule, a network analysis was performed by considering the HSU loss and mechanical component losses For the HSU loss, an HSU efficiency map was used from the experiment To obtain the mechanical component losses, individual models were constructed for the bearing loss, gear churning loss, clutch drag loss, and oil pump loss based on the mathematical governing equations and experiments Using the network analysis results, a new sub-shift schedule was proposed for the demanded wheel torque and vehicle speed Unlike the existing sub-shift schedule, which selects the gear ratio based only on the vehicle speed, the new sub-shift schedule selects the sub-shift gear according to the demanded wheel torque and vehicle speed Since the sub-shift can only be performed at a speed ratio where the clutch speeds are synchronized in the existing sub-shift control, a sub-shift control algorithm without clutch speed synchronization was proposed to apply the new sub-shift schedule First, an HSU stroke control was developed to maintain the demanded engine speed for the given vehicle speed The HSU stroke at the target sub-shift gear ratio was determined from the lever relationship for each sub-shift gear Next, the forward clutch pressure control was proposed, which reduces the torque variation during the sub-shift while transmitting the demanded driveshaft torque The forward clutch pressure was decreased in a stepwise manner to a pressure enough to transmit the demanded driveshaft torque and was maintained until the forward clutch speeds were synchronized This pressure drop caused a slip between the clutch plates, which prevented excessive torque fluctuation After the forward clutch speed synchronization, the clutch pressure was increased to the lock-up pressure The performance of the sub-shift control algorithm without the clutch speed synchronization was evaluated by the simulation and experiment It was found from the simulation and experimental results that the subshift can achieve acceptable peak-to-peak torque variation in the driveshaft It is expected that the sub-shift schedule and the sub-shift control algorithm without the clutch speed synchronization can provide improved fuel economy while maintaining the demanded speed ratio The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article Funding The author(s) received no financial support for the research, authorship, and/or publication of this article References Renius K and Resch R Continuously variable tractor transmissions In: 2005 Agricultural equipment technology conference, 2005, ASAE distinguished lecture no 29, pp.1–37, https://elibrary.asabe.org/data/pdf/6/cvtt2005/le ctureseries29rev.pdf Macor A and Rossetti A Optimization of hydromechanical power split transmissions Mech Mach Theory 2011; 46: 1901–1919 Yang S, Bao Y, Tang X, et al Integrated control of hydromechanical variable transmissions Math Probl Eng 2015; 2015: 1–11 Savaresi S, 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Korea Society of Agricultural Machinery  ... follows12 ,13 TBL0 = > > < 1: 6  10 8  f0  dm > > : 10 10  f0  (v  v)23  dm mm2 s  mm2 if (v  v)  2000 s  ? ?12 Þ if (v  v)\2000 TBL1 = f1  F1  dm  10 3 ? ?13 Þ TBL = TBL0 + TBL1 ? ?14 Þ where... Goriv Maz 2009; 48: 4 41? ??480 12 SKF general catalogue 6000 EN, November 2005 13 ISO/TR 14 179-2:20 01 Gears—thermal capacity—part 2: thermal load-carrying capacity (1st ed) 14 Zhou X, Walker P, Zhang... www.inacomm2 013 .ammindia.org/Papers/046-inacomm2 013 _submission_46.pdf 10 McCandlish D and Dorey R The mathematical modelling of hydrostatic pumps and motor Proc IMechE, Part B: J Engineering Manufacture 19 84; 19 8: 16 5? ?17 4 11 Hohn B,