Research on Transmission Characteristics of Hydromechanical Continuously Variable Transmission of Tractor

)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 characteristic of HMCVTare theoretically derived. On the basis of HMCVTpower flow direction, the Lrfkoes formula is used to calculate the transmission efficiency. )en, the image analysis method is used to study the influence of parameters on the transmission efficiency of HMCVT, and the main influencing factors are found. )e results of theoretical derivation demonstrate that, by coordinating control of the HST displacement ratio and the engagement conditions of shifting clutches, the stepless speed regulation of HMCVT at the tractor speed of 0–50 km/h can be realized. )e proposed HMCVT has the ability to continuously transmit and change torque over all working ranges. )e overall transmission efficiency of HMCVT is at a high level. To verify the theoretical derivation, Amesim simulation software is used for the modeling and simulation of HMCVT. )e simulation results are consistent with the theoretical analysis results. )erefore, the HMCVTproposed in this paper has the advantages of compact structure and high transmission efficiency, and it is suitable for matching tractors.


Introduction
In addition to transportation, the main function of tractors is to drive agricultural machinery needed for agricultural production, which needs the transmission of tractors to provide constantly changing speed and torque to adapt to frequent external load changes [1].
In this context, the use of a continuously variable transmission (CVT) for tractors has become a trend. e most common type of CVTs is the hydraulic static transmission (HST) [2]; it mainly includes hydraulic pumps, hydraulic motors, and control mechanism. HST has good performance at low speed and can easily switch between positive rotation and negative rotation. However, its transmission efficiency is much lower than that of gear transmission, which leads to the fact that it is rarely used in tractors.
In order to overcome the above shortcomings, the HST is connected in parallel with the mechanical components to form a hydromechanical continuously variable transmission (HMCVT) [3][4][5][6]. HMCVT transmits only part of the power through the HST; the remaining power is transmitted through the mechanical components, so the efficiency is much higher than HST. HMCVT can realize stepless speed regulation through the HST, while relying on mechanical components to achieve high efficiency transmission. In addition, the HMCVT can control the engine to operate on an optimal power curve or optimal economic curve, thus achieving optimum power performance or optimum fuel economy for tractors [7][8][9]. e research on power split continuously variable transmission technology began in the early 20th century in the world. However, limited by the level of hydraulic system manufacturing, this type of transmission was not used in armored and heavy construction vehicles until the late 1960s [10]. Literature [8] first proposes a transmission scheme for single-row planetary gears with two hydromechanical working ranges. In each of the two working ranges, the maximum speed of the tractor can reach 32 km/h and 50 km/ h, respectively. en, according to the market demand, only one hydromechanical working range transmission scheme is proposed, which can match the 75 kw engine. e forward driving speed is from 0 to 40 km/h, and the reverse driving speed is from 0 to 25 km/h [11]. e transmission developed by Steyr company has three planetary gear rows, controlled by 4 clutches and 2 brakes. is transmission can achieve speed adjustments from 0 to 50 km/h [12]. e authors of [13,14] study a HMCVT that has four planetary gear rows and can achieve speed adjustment from 0 to 40 km/h. e author of [15] proposes a transmission scheme in which a single-row planetary gear is connected in series with a mechanical stepped transmission. is transmission has 6 ranges in the forward direction and 3 ranges in the reverse direction.
In general, the more the transmission ranges, the greater the expansion of the hydraulic power [10], but it will lead to frequent switching during the ranges and reducing the transmission performance. e single-row planetary scheme helps to reduce the size of the transmission, but the hydraulic motor needs to be reversed quickly at the moment of switching, which needs high performance of the hydraulic motor.
is paper will propose a transmission scheme for a new 4-range HMCVT suitable for wheeled tractors and then conduct a detailed theoretical analysis on the transmission characteristics (speed characteristic, torque characteristic, and efficiency characteristic) of HMCVT and find out the important factors that affect the efficiency of HMCVT; finally, the Amesim simulation software is used to verify the transmission scheme.

Transmission Scheme of HMCVT
e structure of HMCVT proposed in this paper is shown in Figure 1. It includes the input shaft, pump-motor hydraulic system, planetary gear mechanism, gear pairs, shifting clutches, and directional clutches. e hydraulic path of HMCVT is the HST, and its mechanical path is the Ravigneaux planetary gear mechanism (PGM). e hydraulic path and the mechanical path are connected in parallel to form a closed loop and then are connected in series with the range switching mechanism. In the PGM, when the output component is the carrier c, it is an ordinary PGM; when the output component is the small sun gear s 2 , then it is a Ravigneaux PGM. In addition, part of the engine power is through the HST into the PGM, and the HST realizes the stepless transmission.
ere are 6 clutches in the HMCVT, which are 4 shifting clutches and 2 directional clutches. Different clutches can realize 4 working ranges in the front or rear directions. In order to ensure that engine power can be transferred from the input shaft to the output shaft, one shifting clutch and one directional clutch must be engaged. e engagement conditions of clutches in each range of HMCVT are shown in Table 1.

Theoretical Analysis on Transmission
Characteristics of HMCVT e power transmission routes of ranges HM 1 and HM 3 are basically the same, except for the difference between gear pairs i 7 and i 8 . Under these two conditions, the power is input by the large sun gear s 1 and the ring gear r and output by the carrier c.
If n e denotes the engine speed, the variable pump input shaft speed n p is Ignoring the effect of the volumetric efficiency of the volumetric speed control loop, the relationship between the motor output shaft speed n m , the variable pump input shaft speed n p , and the pump-motor displacement ratio ε is where ε � Q p /Q m −1 ≤ ε ≤ 1, Q p is the displacement of the variable pump, and Q m is the displacement of the quantitative motor. e input speed of the large sun gear in PGM n s 1 is e input speed of the ring gear n r equals n e . According to the PGM speed characteristic formula [16], the relationship between the input speed of large sun gear n s 1 , the input speed of ring gear n r , the output speed of carrier n c , and the PGM standing transmission ratio −k 1 is satisfied: where −k 1 � Z r /Z s 1 > 1, Z r is the number of teeth of the ring gear, and Z s 1 is the number of teeth of the large sun gear. erefore, the output speed of the carrier n c is e intermediate shaft speed n z is Finally, the output shaft speed of HMCVT n o is 2 Mathematical Problems in Engineering e transmission ratio of HMCVT i g is 3.1.2. Ranges HM2 and HM4. e power transmission routes of ranges HM 2 and HM 4 are the same, except for the difference between gear pairs i 6 and i 9 . Under these two conditions, the power is input by the large sun gear s 1 and the ring gear r and output by the small sun gear s 2 . e same as the previous case, the input speed of the large sun gear n s 1 is n s 1 � −εn e /(i 1 i 2 i 3 ); the input speed of the ring gear n r equals n e . According to the PGM speed characteristic formula (16), the relationship between the input speed of large sun gear n s 1 , the input speed of ring gear n r , the output speed of the small sun gear n s 2 , and the PGM standing transmission ratios k 2 and −k 3 is satisfied: erefore, the output speed of the small sun gear n s 2 is Finally, the output shaft speed of HMCVT n o is e transmission ratio of HMCVT i g is

Determination of Parameters of HMCVT.
e rated speed of the engine matching with the HMCVT is 2200 r/ min, the minimum speed of the tractor is 4 km/h, the maximum speed is 50 km/h, the main reducer ratio of the rear axle is 3.7, the reduction ratio of the wheel side is 5.6, the total transmission ratio of the rear axle is 20.72, and the drive wheel radius is 0.8579 m. e range of the transmission ratio in the drive train is determined by parameters such as the speed of the tractor, the rated speed of the engine, and the drive wheel radius. So the transmission ratio of each range of HMCVT is calculated as follows: where j � 1, 2, 3, 4; n e is the rated speed of the engine; r d is the drive wheel radius; v is the theoretical driving speed of the tractor. e HMCVT has 4 working ranges, so the common ratio of the transmission ratios is (15) e HMCVT adopts a proportional transmission, so, in each working range, the transmission ratio has the following relationship: When HMCVT is switching between adjacent ranges, the conditions must be met at the changing point: the transmission ratios are equal and the pump-motor displacement ratios are equal: e gear ratio of the connecting gears between the engine and the hydraulic variable pump needs to be met: According to the previous content, if i 10 , i 11 , and i 12 are set to 1, then the relationship between the various parameters of HMCVT can be obtained: ese parameters are all expressions about the PGM standing transmission ratio −k 1 . Once the value of −k 1 is determined, the values of these parameters will also be determined, which will be described later.

Ranges HM1 and HM3. Ignoring the power losses, if T o denotes the load torque of output shaft, the intermediate shaft torque T z is
and then the output torque of carrier in PGM T c is According to the PGM torque characteristic formula (16), the relationship between the input torque of large sun gear T s 1 , the input torque of ring gear T r , the output torque of carrier T c , and the PGM standing transmission ratio −k 1 is satisfied: where −k 1 � Z r /Z s 1 > 1, Z r is the number of teeth of the ring gear, and Z s 1 is the number of teeth of the large sun gear. erefore, the large sun gear torque T s 1 is en, the input torque of ring gear T r is us, the motor shaft load torque T m is Finally, the ratio of the output torque of HMCVT to the input torque of the engine μ is

Ranges HM2 and HM4.
e same as the previous case, T o denotes the load torque of output shaft, and the intermediate shaft torque T z is T z � T o /(i 11 i 12 ). en the output torque of small sun gear in PGM T s 2 is e external torque algebra sum of PGM is zero, so T s 1 + T r + T s 2 � 0. According to the PGM torque characteristic formula (16), the relationship between the input torque of large sun gear T s 1 , the output torque of small sun gear T s 2 , and the PGM standing transmission ratios k 2 and −k 3 is satisfied: Finally, the ratio of the output torque of HMCVT to the input torque of the engine μ is

Ranges HM1 and HM3
. e speed ratio of the power transmitting from the engine to the carrier i ce is e speed ratios of the power transmitting from the engine to the large sun gear and ring gear i re /i s 1 e are i re � 1, In the PGM, the speed ratios between the input elements (large sun gear s 1 and ring gear r) and the output element (carrier c) are Mathematical Problems in Engineering en, we can get Next two situations will be discussed as follows [17]: (1) When ε < 0, i s 1 ce > 0, i r ce > 0, and i s 1 ce i r ce > 0, the PGM has no circulating power P L � 0.
(2) When ε > 0, i s 1 ce > 0, i r ce < 0, and i s 1 ce i r ce < 0, the PGM has circulating power; substituting equation (19) for Because of P e > 0, P r � (i en, e circulating power P L is Assume that the closed power factor K L is where is a monotonically decreasing function. en K L (0) � 0, K L (1) � 0.44, and the maximum value of the circulating power is 0.44P e .

Ranges HM2 and HM4.
e speed ratio of the power transmitting from the engine to the small sun gear i s 2 e is e speed ratios of the power transmitting from the engine to the large sun gear and ring gear i re /i s 1 e are i re � 1, In the PGM, the speed ratios between the input elements (large sun gear s 1 and ring gear r) and the output element (small sun gear s 2 ) are en, we can get Next two situations will be discussed as follows [17]: (1) When ε < 0, i s 1 s 2 e > 0, i r s 2 e < 0, and i s 1 s 2 e i r s 2 e < 0, the PGM has circulating power; substituting equation (19) for Because of P e > 0, P r � i s 1 ce /i ce P e > 0, P s 1 � i r ce /i ce P e < 0, the output power P is P � P e � −P s 2 � P r + P s 1 � P r − P s 1 .
(47) en, e circulating power P L is Assume that the closed power factor K L is where ϕ(ε) � −ε/k 1 i 1 i 2 i 3 − ε and the derivative of

Hydraulic Split Power Calculation.
Hydraulic powersplit ratio ρ � P H /P e � P s 1 /P e ; hydraulic power is input to the PGM through the large sun gear s 1 .

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Mathematical Problems in Engineering

Ranges HM1 and HM3
and as can be obtained from the above, function ϕ(ε) is a monotonically decreasing function, so

Ranges HM2 and HM4
As can be obtained from the above, function ϕ(ε) is a monotonically increasing function, so According to the theoretical analysis on the circulating power flow and the hydraulic split power, the power flow direction of HMCVT in each working range is shown in Figure 2. is result lays the foundation for the analysis on the efficiency characteristic of HMCVT.

Efficiency Characteristic.
e Lrfkoes formula (17) is used to calculate the transmission efficiency of HMCVT: where μ is the ratio of output torque to input torque of HMCVT considering power losses and i g is the transmission ratio of HMCVT. e transmission efficiency calculation process of each working range of HMCVT is shown in Tables 2 and 3. Among them, the calculation processes of the first range and the third range are listed in Table 2, and the calculation processes of the second range and the fourth range are listed in Table 3. In Tables 2 and 3, i 1 − i 12 are the gear ratios of the gear pairs, η 1 − η 12 are the transmission efficiency of the gear pairs, η H is the hydraulic system transmission efficiency, and η s 1 r , η s 1 s 2 , and η rs 2 are the transmission efficiencies of the large sun gear to the ring gear, the large sun gear to the small sun gear, and the ring gear to the small sun gear in the PGM, respectively.

Analysis on the Influence of Parameters on the Transmission Efficiency of HMCVT
e results of the transmission efficiency in each working range of HMCVT are as follows:

Mathematical Problems in Engineering
When HMCVT is working in HM 1 or HM 3 range, if the transmission efficiencies of the gear pairs i 9 and i 8 are equal (η 7 � η 8 ), the transmission efficiencies of HMCVT in HM 1 and HM 3 ranges are equal.
When HMCVT is working in HM 2 or HM 4 range, if the transmission efficiencies of the gear pairs i 9 and i 6 are equal (η 9 � η 6 ), the transmission efficiencies of HMCVT in HM 2 and HM 4 ranges are also equal.
Due to the complexity of the above results, it is necessary to evaluate the influence of displacement ratio ε, standing transmission ratio of PGM −k 1 , gear ratio i 1 i 2 i 3 , and hydraulic system efficiency η H on the transmission efficiency of HMCVT by means of image analysis method. We assume that η 1 − η 12 are 0.98, η s 1 r is 0.98, η s 1 s 2 and η rs 2 are 0.97, and e evaluation results are shown in Figure 3.
According to the analysis on Figure 3, the following rules are obtained.
In each range of HMCVT, when the displacement ratio ε is 0, the transmission efficiency reaches the maximum value. Now the hydraulic path (HST) does not transmit power, and the power is completely transmitted by the mechanical path. When the displacement ratio ε is −1 or 1, the transmission efficiency reaches the minimum value, and the hydraulic      Mathematical Problems in Engineering path transmits more power at this time, resulting in a lower overall efficiency of HMCVT.
In each range of HMCVT, the transmission efficiency value in the power circulation phase is lower than the transmission efficiency value in the power split phase, which indicates that the power circulation will cause the transmission efficiency of HMCVT to decrease. e larger the value of the PGM standing transmission ratio −k 1 , the higher the transmission efficiency of HMCVT, but the influence on improving the transmission efficiency is not obvious. e PGM standing transmission ratio −k 1 ranges from 1.5 to 3. e gear ratio i 1 i 2 i 3 and the PGM standing transmission ratio −k 1 are in an inversely proportional relationship; the smaller the value of the gear ratio i 1 i 2 i 3 , the higher the transmission efficiency of HMCVT, and the influence on improving the transmission efficiency is also not obvious. e gear ratio i 1 i 2 i 3 ranges from 1.1 to 2.2. e greater the value of HST efficiency η H , the higher the overall transmission efficiency of HMCVT. e influence of η H on improving transmission efficiency of HMCVT is significant.

Analysis on the Transmission Characteristics Results of HMCVT
According to the previous content, the value of the PGM standing transmission ratio −k 1 is set to 2.1875, and the HST efficiency value η H is set to 0.81; then the other parameters of HMCVT are k 2 � 0.2714, −k 3 � 0.5938, i 1 � i 2 � 1, i 3 � 1.5, i 4 � i 5 � 1, i 6 � 1.66, i 7 � 1.16, i 8 � 4.10, i 9 � 5.84, and i 10 � i 11 � i 12 � 1. e relationship between the transmission ratio and displacement ratio is shown in Figure 4. e curve of the tractor speed is shown in Figure 5. In the first range, the displacement ratio changes from 1 to −1, the transmission ratio decreases from 8.62 to 4.59, and the tractor speed increases from 0 to 7.50 km/h. In the second range, the displacement ratio changes from −1 to 1, the transmission ratio decreases from 4.59 to 2.43, and the tractor speed increases from 7.50 km/h to 14.12 km/h. In the third range, the displacement ratio changes from 1 to −1, the transmission ratio decreases from 2.43 to 1.30, and the tractor speed increases from 14.12 km/h to 26.44 km/h. In the fourth range, the displacement ratio changes from −1 to 1, the transmission ratio decreases from 1.30 to 0.69, and the tractor speed increases from 26.44 km/h to 50 km/h. e results show that the stepless speed regulation of HMCVT in the tractor speed range of 0-50 km/h can be realized by the coordinated control of the HST displacement ratio and the clutches. e ratios of the output shaft torque to the input shaft torque in each range of HMCVT are shown in Figure 6(a). e continuous change of torque can be realized in the entire working ranges. e ratios of the motor shaft torque to the output shaft torque in each range of HMCVT are shown in Figure 6(b). e torque ratio remains constant in each range and varies between different ranges. e values of torque ratios in the four ranges are 5.14%, −9.72%, 18.49%, and −35.18%, respectively. e relationship between the tractor speed and the overall transmission efficiency of HMCVT is shown in Figure 7. Each of the four ranges has a transmission efficiency maximum value of 0.917, 0.902, 0.917, and 0.902, respectively. e efficiency value always increases first with the increase of the tractor speed and then decreases after reaching the maximum value. e maximum transmission efficiency in each range is obtained when the HST displacement ratio is 0. Now, the hydraulic path does not transmit power and all power is transmitted by the mechanical path. e transmission efficiency of HMCVT is higher than 83%, so the overall transmission efficiency is at a high level.

Verification of HMCVT Simulation Model
e modeling and simulation of HMCVT are accomplished by using Amesim simulation software. According to Figure 1, a simulation model of HMCVT is established, as shown in Figure 8.

Hydraulic System.
e hydraulic system is mainly composed of a variable displacement hydraulic pump, a fixed displacement hydraulic motor, two relief valves, two check valves, and a charge hydraulic pump with a relief valve. e aim of the charge pump is to compensate for the leakage loss in the pipeline and to establish the lowest pressure in the pump's suction pipeline. Two relief valves are used to limit the maximum pressure in the system piping, and its adjustment pressure should be above the normal working pressure of the system. Volumetric losses of the pump and motor are neglected, and their hydromechanical efficiencies are set at 0.9.

Mechanical Components.
Mechanical components are assumed to be constant efficiency components throughout the transmission. e power loss of each gear pair is 2%, so the efficiency is set to 0.98. In the Ravigneaux planetary gear mechanism, the efficiencies of ring gear-long planetary gear couple, long planetary gear-large sun gear couple, long planetary gear-short planetary gear couple, and short planetary gear-small sun gear couple are set at 0.99. e main reducer, differential, and wheel reducer are combined and modeled as a gear reducer with an efficiency of 0.965. To simplify the model, the friction losses of the shaft, gears, and bearings were ignored during the simulation modeling process.

Model Control and Simulation Results.
e engine speed is assumed to have a constant value of 2200 r/min; the load torque of the output shaft of HMCVT is set to 100 Nm; the model simulation time is 32 s, including range HM 1 : 0-8 s, range HM 2 : 8-16 s, range HM 3 : 16-24 s, and range HM 4 : 24-32 s. e simulation results are shown in Figure 9; through the variable pump displacement and coordinated control of the clutches [18,19], the transmission gradually changes from range HM 1 to range HM 4 , and the tractor achieves stepless speed change from 0 to 50 km/h. e simulation results are consistent with the previous theoretical analysis results, which show that the 4-range HMCVT transmission scheme proposed in this paper is suitable for matching wheeled tractors.

Comparison.
Xu LY's transmission scheme [15] has been applied to mass production tractors. is transmission scheme has 6 working ranges in the forward direction and 3 working ranges in the reverse direction, which is controlled by 8 clutches. e speed range in the forward direction is 0∼30 km/h, while the reverse direction is only 0∼10 km/h. e transmission efficiency of the hydraulic range is low (less than 80%); the transmission efficiency of the hydromechanical ranges is high (higher than 90%) but is only in the forward direction. e scheme proposed in this paper has 4 working ranges and a speed range of 0∼50 km/h in the forward and reverse directions, which is controlled by 6 clutches. e fewer the transmission ranges, the less switching during the working ranges, so the transmission performance is improved. e smaller number of clutches ensures that the system is easier to control. e transmission efficiencies in the forward and reverse directions are close, the overall forward transmission efficiency is higher than 83%, and the highest transmission efficiency is higher than 91%. erefore, the application of HMCVT proposed in this paper is wider; it can not only match conventional tractors but also match two-way operation tractors and other engineering vehicles.

Conclusions
is paper proposes a new type of transmission scheme of HMCVT. e Ravigneaux PGM and the HST are connected in parallel to form a closed loop and then connected in series with the range switching mechanism.
is new HMCVT features a double-row PGM with 4 hydromechanical working ranges. By coordinating control of the HST displacement ratio and the engagement conditions of shifting clutches, the stepless speed regulation of HMCVT at the tractor speed of 0-50 km/h is realized.
According to the theoretical analysis on the new HMCVT, the speed characteristic curve and the torque characteristic curve are obtained. Based on the power flow direction of HMCVT, the transmission efficiency curve is obtained by using the Lrfkoes formula.
Using image analysis method to study the influence of four parameters (displacement ratio ε, PGM standing transmission ratio −k 1 , gear ratio i 1 i 2 i 3 , and hydraulic system efficiency η H ) on the transmission efficiency of HMCVT, the results show that the standing transmission ratio −k 1 and gear ratio i 1 i 2 i 3 have little effect on the transmission efficiency, whereas the displacement ratio ε and hydraulic system efficiency η H have a greater impact on the transmission efficiency. e parameters of the new HMCVT are finally determined. e results of the transmission characteristics show that the new HMCVT proposed in this paper has the stepless speed regulation capability for tractors, which can also continuously transmit and change the torque. e transmission efficiency of HMCVT is at a high level.
In order to verify the theoretical derivation, Amesim simulation software is used for the modeling and simulation of HMCVT. e simulation results are consistent with the theoretical analysis results. erefore, we believe that the transmission scheme is reasonable and effective for tractors.

Data Availability
e Simcenter Amesim System File data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.