Research on the Influence of Tractor Parameters on Shift Quality, Based on Uniform Design

Abstract

This paper mainly studies the influence of tractor parameters on shift quality. The kinetic theoretical model of the powertrain system was deduced, and then the simulation model was established by using AMESim simulation software. The acceptable jerk and shift time were selected as the evaluation indexes of shift quality. The influence of engine speed, the tractive resistance of the tractor, the clutch oil pressure and the orifice diameter on the above evaluation indexes were quantitatively studied. The simulation test, under the influence of a single parameter, was carried out using the parameter analysis method. After this, the uniform design experimentation method was used to carry out the simulation test, under the influence of combined parameters. The results showed that in order to improve the shift quality, it is necessary to focus on the selection of clutch oil pressure and orifice diameter.

1. Introduction

In recent years, hydro-mechanical continuously variable transmission (HMCVT) has been widely studied [1,2,3]. It not only uses hydraulic static transmission to realize stepless speed regulation, but also relies on mechanical transmission parts to improve transmission efficiency [4,5,6]. Although a tractor equipped with a front-end loader can achieve stepless speed change by matching the HMCVT described above, it is inevitable that the driver needs to operate and change the working direction frequently. This can make the driver uncomfortable and shorten the service life of the clutch in the transmission [7]. To overcome these drawbacks, in this study the HMCVT was combined with two multi-plate wet clutch assemblies to achieve fast commutation movements of the tractor powertrain [8]. The system methodology was used to model and simulate the power reversing system of an agricultural tractor [9]. For a long time, the research of fast power reversing transmission systems or HMCVT focused on the structure design and performance analysis [10,11,12,13]. A continuously variable transmission scheme with fast power reversing was proposed, and the transmission characteristics of the transmission scheme were studied theoretically and verified by a simulation [14]. With the increasing requirements for tractor smoothness, the study of dynamic characteristics of shift quality has become a key research focus.

In previous research on this topic [15], the effects of the design parameters of a power shuttle tractor on the shift performance have been investigated using simulations. The shift performance was evaluated in terms of the peak torques of the input shaft and the tractor axles, the power transmitted per unit area of the clutch, and in terms of the time required to transmit the power. This research only investigated the effect of a single design parameter on the evaluation index. Furthermore, previous research [16] has also studied the shift quality of a tractor in the acceleration phase. The shift quality was evaluated by the peak acceleration of the tractor and the frictional work of the clutch. Moreover, [17] other research has also reported that the shift quality was improved by controlling the clutch current profile, and that the relative amplitude of jerk was used to decide the regular shift performance. However, the indexes chosen by the above-mentioned studies to assess shift quality were all transient indicators and did not take into account the time course of the jerk. In addition, these studies [15,16] did not consider the effect of combined design parameters on the shift quality, which is far from the actual engineering situation. The orthogonal test method was widely used to study the influence of combined parameters on the shift quality [18,19]. However, this method is less suitable for applications with a large range of parameter values and many factor levels (greater than five).

Accordingly, this paper has selected acceptable jerk and shift time as the evaluation indexes of tractor shift quality. These indexes comprehensively considered the influence of the peak jerk and its continuous impact in the shift process. The uniform design experimentation method was used to carry out the simulation test, under the influence of combined parameters, and the interaction between these parameters was also taken into account. Moreover, the kinetic model was established according to the structure of the powertrain, and then the simulation model was established based on AMESim software. A research method based on simulation could simplify the research and development process of HMCVT. It could also provide the basis for follow-up research and development and help to avoid a variety of possible problems.

2. Powertrain Model

2.1. HMCVT Mechanical System

The HMCVT is shown as a stick diagram in Figure 1. As can be seen, the engine and the HMCVT were connected by an input shaft. The structure of the HMCVT was mainly composed of the following four parts: a variable displacement pump-fixed displacement motor system (pump-motor), a planetary gear mechanism (PGM), shifting clutches (C1, C2, C3, and C4), and directional clutches (CF and CR).

Table 1. Clutches’ engagement schedule for the HMCVT.

Range		Directional Clutch		Shifting Clutch				Gearbox Gear Ratio (INPUT to Output Shaft)	Rear Axle Gear Ratio
		CF	CR	C1	C2	C3	C4
Forward	First	+		+				8.62∼4.59	20.72
	Second	+			+			4.59∼2.43
	Third	+				+		2.43∼1.30
	Fourth	+					+	1.30∼0.69

shows the clutches’ engagement schedule and the gearbox and rear axle gear ratios. Please note, that only forward driving conditions were considered in this paper, and reverse driving conditions were not considered.

2.1.1. Rotational Dynamics of the HMCVT

To capture the low frequency dynamic effects that result from events such as gear shifting and clutch selection, an HMCVT model with lumped inertias was considered. A schematic of the aforementioned HMCVT is shown in Figure 2.

By applying Newton’s second law for rotation at each of the inertias, the following set of equations, describing the rotational dynamics of the HMCVT mechanical system was obtained:

$$\mathrm{Shaft}1:J_1\frac{d{\omega }_1}{dt}=T_1-T_r-\frac{T_P}{i_1{i}_2}$$

(1)

$$\mathrm{Shaft}2:J_2\frac{d{\omega }_2}{dt}=i_4{T}_c-T_{C1}-T_{C3}$$

(2)

$$\mathrm{Shaft}3:J_3\frac{d{\omega }_3}{dt}=i_5{T}_{s2}-T_{C2}-T_{C4}$$

(3)

$$\mathrm{Shaft}4:J_4\frac{d{\omega }_4}{dt}=i_8{T}_{C1}+i_9{T}_{C2}+i_7{T}_{C3}+i_6{T}_{C4}-T_{CF}$$

(4)

$$\mathrm{Shaft}5:J_5\frac{d{\omega }_5}{dt}=i_{11}{i}_{12}{T}_{CF}-T_5$$

(5)

$$\mathrm{Large}\mathrm{sun}\mathrm{gear}:J_{s1}\frac{d{\omega }_{s1}}{dt}=i_3{T}_M-T_{s1}$$

(6)

$$\mathrm{Small}\mathrm{sun}\mathrm{gear}:J_{s2}\frac{d{\omega }_{s2}}{dt}=T_{s2}-\frac{T_{C2}}{i_5}-\frac{T_{C4}}{i_5}$$

(7)

$$\mathrm{Ring}\mathrm{gear}:J_r\frac{d{\omega }_r}{dt}=T_1-\frac{T_P}{i_1{i}_2}-T_r$$

(8)

$$\mathrm{Carrier}:J_c\frac{d{\omega }_c}{dt}=T_c-\frac{T_{C1}}{i_4}-\frac{T_{C3}}{i_4}$$

(9)

where $J$ is lumped inertia, kg·m²; $\omega$ is angular velocity, rad/s; $T$ is torque, Nm;$i_1~i_{12}$ are gear ratios; the subscripts C1, C2, C3, C4, and CF denote the clutches; and the subscripts P and M denote the pump and motor, respectively.

2.1.2. Variable Displacement Pump-Fixed Displacement Motor System

Ignoring the influence of internal and external leakage and the internal resistance of the pump motor system, the simplified kinetic model is as follows:

$$\frac{V_1}{\beta }\frac{d_{p_1}}{dt}=D_P\epsilon {\omega }_P-D_M{\omega }_M$$

(10)

$$J_P\frac{d{\omega }_P}{dt}=T_P-D_P\epsilon \left(p_1-p_2\right)$$

(11)

$$J_M\frac{d{\omega }_M}{dt}=D_M\left(p_1-p_2\right)-T_M$$

(12)

where $p_1$ and $p_2$ denote the high and low side pressure values of the system, respectively, Pa; $p_2$ is equal to constant makeup pressure; $V_1$ is the volume of fluid on the high pressure side, ${\mathrm{m}}^3$, $V_1$ is equal to the sum of the volume of pump and motor oil cavity and the volume of the pipeline; $\beta$ is the oil bulk modulus, Pa; and $\epsilon$ is the swash value, $-1\le \epsilon \le 1$.

2.1.3. Dynamic Clutch Friction Model

In this paper, we used the reset-integrator friction model, considering the Stribeck effect [20]. The effect of viscous friction was ignored, so the friction torque evolved smoothly, according to the equation below:

$$T_{dry}=T_{dyn}+\left(T_{stat}-T_{dyn}\right)·e^{\left[\frac{-3\left|{\omega }_{rel}\right|}{astrib}\right]}$$

(13)

where $astrib$ is the Stribeck constant (positive value), r/min; $T_{dyn}$ is the maximum transmittable torque during sliding, Nm; $T_{stat}$ is the maximum transmittable torque during stiction, Nm; and ${\omega }_{rel}$ is the relative velocity, rad/s.

2.2. Longitudinal Tractor Dynamics

By ignoring tire slip, the tractor was simplified as a rigid body, with a mass of $m_t$. The resulting tractor dynamics were described by thr following:

$$m_t{R}_{wr}{}^2\frac{d{\omega }_{wr}}{dt}=T_{wr}-T_{tr}-4T_{brake}$$

(14)

where $R_{wr}$ is the radius of the driving wheel, m; ${\omega }_{wr}$ is the speed of the driving wheel, rad/s; $T_{wr}$ is the driving wheel torque, Nm; $T_{tr}$ is the road load torque, Nm; and $T_{brake}$ is the wheel braking torque, Nm.

The parameters used in the simulation of the HMCVT mechanical system and the tractor are summarized in

Table 2. Parameters used in simulation of the HMCVT mechanical system and the tractor.

Parameter	Value	Parameter	Value
$J_1$(kg·m2)	0.1283	$J_2$ (kg·m2)	0.2940
$J_3$ (kg·m2)	0.0940	$J_4$ (kg·m2)	0.2500
$J_5$ (kg·m2)	4.2333	$J_{s1}$ (kg·m2)	0.0022
$J_{s2}$ (kg·m2)	0.0040	$J_r$ (kg·m2)	0.0619
$J_c$ (kg·m2)	0.1893	$i_1{i}_2$	1
$i_3$	−1.5	$i_4$, $i_5$	−2
$i_6$	−0.825	$i_7$	−0.580
$i_8$	−2.046	$i_9$	−2.920
$i_{11}{i}_{12}$	1	$\beta$ (Pa)	1.7 × 109
$V_1$ (${\mathrm{m}}^3$)	6 × 10−4	$D_M$ (cm3/r)	54.8
$D_P$ (cm3/r)	54.8	$J_M$ (kg·m2)	0.0520
$J_P$ (kg·m2)	0.0060	$m_t$ (kg)	12,000
$astrib$(r/min)	0.1	$R_{wr}$ (m)	0.858

.

2.3. Hydraulic Component Actuation

The hydraulic actuation system significantly affected the operation of the HMCVT, a mathematical model of the hydraulic actuation system was necessary [21]. The simplified hydraulic actuation system in this paper included a pressure regulation system, a clutch actuation system, and a clutch cooling and lubrication system, as shown in Figure 3.

2.3.1. Pressure Regulation System

The pump was powered by the engine, and supplies flowed to the hydraulic system at the volumetric flow rate, $Q_{pump}$, as follows:

$$Q_{pump}=D_{pump}{\omega }_e$$

(15)

where $D_{pump}$ is the pump displacement, m³/rad; ${\omega }_e$ is the engine speed, rad/s. The volumetric flow rate through the pressure relief valve (PRV) was modeled as follows:

$$Q_{PRV}=\left\{\begin{array}{c}K{V}_{PRV}\left(P_{line}-P_{crack,PRV}\right)\\ 0\end{array}\right.\begin{array}{c}when P_{line}>P_{crack,PRV}\\ when P_{line}\le P_{crack,PRV}\end{array}$$

(16)

where $K{V}_{PRV}$is the pressure relief valve gain, m³/s/Pa; $P_{line}$ is the line pressure, Pa; and $P_{crack,PRV}$ is the cracking pressure, Pa.

2.3.2. Clutch Actuation System

The clutch actuation system consisted of six subsystems. Each system included a pilot-operated proportional pressure reducing valve (NC1, NC2, NC3, NC4, NCF or NCR) and a clutch piston (C1, C2, C3, C4, CF or CR). Thus, the C1 system shown in Figure 3 will be described in this section. The structure of the pilot-operated proportional pressure reducing valve, NC1, is shown in Figure 4.

Pilot-Operated Proportional Pressure Reducing Valve (NC1)

The mathematical model of NC1 consists of three subsystems. They are the spool mechanical system, the electromagnetic circuit, and the fluid flow system.

(1)

Spool mechanical systems

(a)

Pilot valve

The pilot valve spool motion depended on the difference in pressure forces acting on the spool, the magnetic force, the inertia force, the viscous damping force, and the steady-state flow forces due to flow through the exhaust and inlet ports. The static friction force and the transient flow force were assumed to be negligible. Thus, the mechanical dynamics of the pilot valve were described by the following:

$$m_1·\frac{d{x}_1{}^2}{d{t}^2}+B_1\frac{d{x}_1}{dt}=F_{mag,NC1}+F_{1,flow,NC1}-P_{NC1,X}{A}_{NC1,X}$$

(17)

where $m_1$ is the mass of the plunger, kg; $B_1$ is the viscous damping coefficient, N/(m/s); $F_{mag,NC1}$ is the magnetic force acting on the spool, N; $P_{NC1,X}$ is the pressure in chamber X, Pa; $A_{NC1,X}$ is the land cross-sectional areas at chamber X, m²; $x_1$ is the spool displacement, m; and $F_{1,flow,NC1}$ is the net steady flow force acting on the spool, N.

(b)

Main valve

The main valve spool motion depended on the difference in pressure forces acting on the spool, the net spring force, the inertia force, the viscous damping force, and the steady-state flow forces due to flow through the exhaust and inlet ports. The static friction force and the transient flow force were assumed to be negligible. Thus, the mechanical dynamics of the main valve were described by the following:

$$m_2\frac{d{x}_2{}^2}{d{t}^2}+B_2\frac{d{x}_2}{dt}+K_{NC1}{x}_2=P_{NC1,C}{A}_{NC1,C}-P_{NC1,A}{A}_{NC1,A}-F_{2,flow,NC1}$$

(18)

where $m_2$ is the mass of the plunger, kg; $B_2$ is the viscous damping coefficient, Nm/rad/s; $K_{NC1}$ is the spring constant, N/mm; $P_{NC1,A}$,$P_{NC1,C}$ are the pressures in chambers A and C, Pa; $A_{NC1,A}$,$A_{NC1,C}$ are the land cross-sectional areas at chambers A and C, m²; $x_2$ is the spool displacement, m; and $F_{2,flow,NC1}$ is the net steady flow force acting on the spool, N.

(2)

Electromagnetic circuit

In the range of the proportional electromagnetic force displacement characteristics, the output force equation of the proportional electromagnet was as follows:

$$F_{mag,NC1}=K_{i}i+K_{xe}{x}_e$$

(19)

The given voltage current equation of the proportional electromagnet was as follows:

$$U_g=\frac{1}{K_e}\left(\left(R_L+K_e{K}_{fi}\right)i+L_h\frac{di}{dt}\right)$$

(20)

where $K_i$ is the current-force gain coefficient of the proportional electromagnet, N/A; $K_{xe}$ is the displacement-force gain coefficient of the proportional electromagnet, N/m. The value was very small: about 0. $i$ is the current in the coil, A; $x_e$ is armature displacement, m; $R_L$ is the coil resistance and the internal resistance of the amplifier, $\mathsf{\Omega }$; $L_h$ is the coil inductance, H; $K_e$ is the voltage amplification factor of the amplifier; and $K_{fi}$ is the current feedback gain coefficient of the proportional electromagnet, V/A.

(3)

Fluid flow system

For the simplified hydraulic system, the net volumetric flow rate to the NC1 actuator came from the flow rate exiting the safety valve, $Q_{Line}$. By applying the continuity equation at the safety valve exit, the flow rate at the inlet of the NC1, $Q_{NC1,X}$ and $Q_{NC1,in}$ were described by the following:

$$Q_{NC1,X}+Q_{NC1,in}=Q_{Line}-Q_{NC2}-Q_{NC3}-Q_{NC4}-Q_{NCF}-Q_{NCR}$$

(21)

where $Q_{NC2},Q_{NC3},Q_{NC4},Q_{NCF}$, and $Q_{NCR}$ are the flow rates to the proportional pressure reducing valves—$NC2,NC3,NC4,NCF$ and $NCR$, respectively.

(a)

Pilot valve

The flow through chamber X of the NC1 was defined by applying the following continuity equation to the chamber:

$$Q_{NC1,X,in}-Q_{NC1,X,ex}=\frac{V_{NC1,X}}{\beta }\frac{d{P}_{NC1,X}}{dt}$$

(22)

$$Q_{NC1,X\&C}=Q_{NC1,X,in}+Q_{NC1,C}$$

(23)

$$\left\{\begin{array}{c}{Q}_{NC1,X\&C}=C_d\left(\lambda \right){A^{\prime }}_{O11}\sqrt{\frac{2}{\rho }\left|\mathsf{\Delta }P\right|}\mathrm{sgn}\left(\mathsf{\Delta }P\right)\\ \Delta P=P_{NC1,in}-P_{NC1,X}\end{array}\right.$$

(24)

where$Q_{NC1,X\&C}$ is the total flow rate into chambers X and C, m³/s; $C_d\left(\lambda \right)$ is the flow coefficient; ${A^{\prime }}_{O11}$ is the constant area of sharp-edged orifice 11, m²;$\rho$ is the density of the transmission fluid, kg/m³; $\beta$ is the transmission fluid bulk modulus, Pa; $Q_{NC1,X,ex}$ is the flow rate at the outlet of chamber X, m³/s; and $V_{NC1,X}$ is the volume of chamber X, m³.

(b)

Main valve

The flow through chamber, B, of the NC1 was defined by applying the following continuity equation to the chamber, while the clutch was either filling or exhausting:

$$Q_{NC1,in}-Q_{NC1,control}=\frac{V_{NC1,B}}{\beta }\frac{d{P}_{NC1,B}}{dt}$$

(25)

$$Q_{NC1,control}-Q_{NC1,ex}=\frac{V_{NC1,B}}{\beta }\frac{d{P}_{NC1,B}}{dt}$$

(26)

where $V_{NC1,B}$ is the volume of chamber B, m³; $P_{NC1,B}$ is the pressure in chamber B, Pa. $Q_{NC1,control}$ and $Q_{NC1,ex}$, are the metered flow rates at the output and the exhaust of $NC1$, respectively.

The flow into and out of chambers A and C was described by applying the continuity equation at those chambers. The resulting equations were given by the following:

$$Q_{NC1,A}+A_{NC1,A}·{\dot{x}}_2=\frac{V_{NC1,A}}{\beta }\frac{d{P}_{NC1,A}}{dt}$$

(27)

$$Q_{NC1,C}-A_{NC1,C}·{\dot{x}}_2=\frac{V_{NC1,C}}{\beta }\frac{d{P}_{NC1,C}}{dt}$$

(28)

$$\left\{\begin{array}{c}{Q}_{NC1,A}=C_d\left(\lambda \right){A^{\prime }}_{O12}\sqrt{\frac{2}{\rho }\left|\mathsf{\Delta }P\right|}\mathrm{sgn}\left(\mathsf{\Delta }P\right)\\ \Delta P=P_{NC1,control}-P_{NC1,A}\end{array}\right.$$

(29)

where$Q_{NC1,A}$ and $Q_{NC1,C}$ are the flow rates into or out of chambers A and C, m³/s; ${\dot{x}}_2$ is the spool speed, m/s; and $V_{NC1,A}$ and $V_{NC1,C}$ are the volumes in chambers A and C, m³.

Clutch Piston C1

The mathematical model of the clutch piston consists of two subsystems. They are the clutch piston mechanical dynamics and the fluid flow system.

(1)

Clutch piston mechanical dynamics

The C1 piston motion depended on the pressure force acting on the piston, they were as follows: the spring force, the inertia force, and the viscous damping force. The static friction force and the flow force were assumed to be negligible. Thus, the mechanical dynamics of the C1 piston were described by the following:

$$m_{C1}{\ddot{x}}_{C1}+B_{C1}{\dot{x}}_{C1}+K_{C1}{x}_{C1}=A_{C1}{P}_{C1}-F_{initial,C1}$$

(30)

where $m_{C1}$ is the mass of the clutch piston, kg; $B_{C1}$ is the viscous damping coefficient, N/(m/s); $K_{C1}$ is the spring constant, N/mm; $A_{C1}$ is the pressure acting area of the clutch piston, m²; $P_{C1}$ is the pressure applied to the clutch piston, Pa; $F_{initial,C1}$ is the spring preload, and N; $x_{C1}$, is defined as piston displacement and measured from the unloaded static equilibrium, m.

(2)

Fluid flow system

The net volumetric flow rate to or from the clutch piston cavity,$Q_{C1}$, was determined by applying the continuity equation at the outlet of the clutch pressure regulation valve. Furthermore, $Q_{C1}$ was also equal to the flow rate across accumulator #1. The resulting expression was given by the following equations:

$$Q_{C1}=Q_{NC1,control}-Q_{NC1,A}$$

(31)

$$\left\{\begin{array}{c}{Q}_{C1}=C_d\left(\lambda \right){A^{\prime }}_{O13}\sqrt{\frac{2}{\rho }\left|\mathsf{\Delta }P\right|}\mathrm{sgn}\left(\mathsf{\Delta }P\right)\\ \Delta P=P_{NC1,control}-P_{C1} \end{array}\right.$$

(32)

The parameters used in the simulation of the hydraulic actuation system are summarized in

Table 3. Parameters used in simulation of the hydraulic actuation system.

Parameter	Value	Parameter	Value
$D_{pump}$ (cm3/r)	30	$K{V}_{PRV}$ (m3/s/Pa)	8.33 × 10−7
$P_{crack,PRV}$ (Pa)	2 × 106	$m_1$ (kg)	0.01
$B_1$(N/(m/s))	10	$m_2$ (kg)	0.04
$B_2$(N/(m/s))	100	$K_{NC1}$ (N/mm)	10
$C_d\left(\lambda \right)$	0.7	$\rho$(kg/m3)	850
$\beta$ (Pa)	1.7 × 109	$m_{\mathrm{C}1}$ (kg)	0.5
$B_{\mathrm{C}1}$(N/(m/s))	200	$K_{C1}$ (N/mm)	438
$F_{initial,C1}$ (N)	500	$A_{C1}$ (m2)	0.0217

.

3. Simulation Analysis of the Shift Process

3.1. Evaluation Indexes of Shift Quality

Acceptable jerk and shift time were selected as the evaluation indexes of shift quality.

3.1.1. Acceptable Jerk

The traditional calculation formula of jerk, that is, the derivative of acceleration, is as follows [18,19]:

$$J\left(t\right)=\frac{da}{dt}=\frac{d^{2}v}{d{t}^2}$$

(33)

The traditional calculation of jerk does not consider the influence of frequency and action time, so, this paper presented an evaluation index (acceptable jerk), which considered the effect of the two factors, peak jerk and root mean square jerk, as follows:

$$J_{ajv}=0.004J_{max}{}^2+J_{rms}$$

(34)

where $J_{max}$ is the peak jerk,$J_{max}=\max \left(J\left(t\right)\right)$, $\mathrm{m}/{\mathrm{s}}^3$; $J_{rms}$ is the root mean square jerk, $J_{rms}=\sqrt{\frac{{{\displaystyle \int }}_0^{t_e}{J}^2\left(t\right)dt}{t_e}},\mathrm{m}/{\mathrm{s}}^3;t_e$ is the shift time, s.

3.1.2. Shift Time

Shift time is the time required after the steady state of the previous range changes to a new range. It is a comprehensive index, reflecting tractor shift quality. Good shift quality requires less shift time on the basis of a smooth shift. This paper held that the shift process was finished when the total torque capacity of the oncoming clutch reached 99% of the stable torque capacity.

3.2. Simulation Model of Powertrain

According to the design scheme of the HMCVT in Figure 1, the simulation model of the powertrain, based on AMESim, is shown in Figure 5. The simulation model mainly included the following four modules: the transmission module of the HMCVT, the hydraulic component actuation, the calculation module for evaluation indexes, and the rear axle and longitudinal tractor model. The transmission module of the HMCVT can be subdivided into the following three parts: the engine speed and tractive resistance of the tractor, the variable displacement pump-fixed displacement motor system, and the HMCVT mechanical system. All the clutches were controlled by the pilot-operated proportional pressure reducing valves. The calculation module for evaluation indexes was used to calculate the jerk of the HMCVT output shaft.

3.3. Continuous Shift Process of HMCVT

The simulation condition can be expressed as follows: $N_e$= 2000 r/min, $F_{r }$= 5 kN, $P_c$ = 2 MPa, $D_{o }$= 3.5 mm. The whole continuous shift process included the upshift from range 1 to range 4 in the first 40 s, and the downshift from range 4 back to range 1 in the next 40 s. The simulation time of each range of the HMCVT was 10 s, the engagement states of the clutches were changed in the 0th second of each range (0 s, 10 s, 20 s, 30 s, 50 s, 60 s, and 70 s). The swash value of the variable displacement pump-fixed displacement motor system was changed in the last 5 s of each range (5–10 s, 15–20 s, 25–30 s, 35–40 s, 45–50 s, 55–60 s, 65–70 s, and 75–80 s). The simulation curves of the continuous shift process are shown in Figure 6.

The simulation results showed that when the tractor’s range changed, there was a slight fluctuation in the tractor’s speed, and the jerk of the HMCVT output shaft was large. We can infer that the range switching from range 1 to range 4 are the typical operating conditions for improving the shift quality. The working speed of a tractor plowing is generally 5–10 km/h, while that of a tractor sowing is 6–15 km/h. In the whole working life cycle of tractors, the speed of 4–15 km/h accounts for about 75% [22]. As such, this paper took the range 1 to range 2 upshift to study the effect of tractor parameters on the shift quality.

3.4. Analysis on the Influence of Tractor Parameters on the Shift Quality (Range 1 to Range 2)

According to practical experience, the following factors may affect the shift quality: engine speed, the tractive resistance of the tractor, the clutch oil pressure, and the orifice diameter, etc. In this study, the engine speed was set at 1000–2000 r/min, and the range of the tractive resistance of the tractor was 5–20 kN. The range of clutch oil pressure was 1–2 MPa and the orifice diameter range was 2.5–3.75 mm. In the following tests, each parameter was further divided into six levels.

3.4.1. The Shift Process under the Influence of a Single Parameter

1.

Engine speed

The simulation condition can be expressed as follows: $F_{r }$ = 5 kN, $P_c$ = 2 MPa, $D_o$ = 3.5 mm. Six tests were conducted at engine speeds of 1000 r/min, 1200 r/min, 1400 r/min, 1600 r/min, 1800 r/min, and 2000 r/min. The test results are shown in Figure 7a. In general, the acceptable jerk value increased with the engine speed during the shift process. The shift time decreased with the increase in engine speed, and did not change after 0.24 s. Therefore, the engine speed had a significant effect on the acceptable jerk and shift time.

2.

Tractive resistance of the tractor

The simulation condition can be expressed as follows:$N_e$ = 2000 r/min,$P_c$ = 2 MPa, $D_o$ = 3.5 mm. Six tests were conducted at the tractive resistances of the tractor of 5 kN, 8 kN, 11 kN, 14 kN, 17 kN, and 20 kN. The test results are shown in Figure 7b. The acceptable jerk value increased with the increase of the tractive resistances of the tractor during the shift process; the shift times were all 0.24 s. Therefore, the tractive resistance of the tractor had no effect on the shift time, but it had a significant effect on the acceptable jerk.

3.

Clutch oil pressure

The simulation condition can be expressed as follows:$N_e$ = 2000 r/min, $F_r$ = 5 kN, $D_{o }$ = 3.5 mm. Six tests were conducted at the clutch oil pressure of 2 MPa, 1.8 MPa, 1.6 MPa, 1.4 MPa, 1.2 MPa, and 1 MPa. The test results are shown in Figure 7c. The acceptable jerk value decreased with the decrease of clutch oil pressure during the shift process. The shift time varied with the change of the clutch oil pressure during the shift process, but it did not reflect the regularity. Therefore, the clutch oil pressure had an irregular impact on the shift time and had a significant impact on the acceptable jerk.

4.

Orifice diameter

The simulation condition can be expressed as follows:$N_e$ = 2000 r/min, $F_r$ = 5 kN, $P_c$ = 2 MPa. Six tests were conducted at the orifice diameters of 3.75 mm, 3.5 mm, 3.25 mm, 3 mm, 2.75 mm, and 2.5 mm. The test results are shown in Figure 7d. The acceptable jerk value decreased with the decrease of the orifice diameter, but if the orifice diameter was too small, it caused an increase in the acceptable jerk value. The shift time increased with the decrease of the orifice diameter. Therefore, the orifice diameter had a significant effect on the shift time and the acceptable jerk.

3.4.2. The Shift Process under the Influence of Combined Parameters

In practice, the shift process was affected by combined parameters. Therefore, uniform design experimentation could be used for the research [23]. The uniform design experimentation method was proposed by the Chinese mathematicians Fang, K.T. and Wang, Y. in 1978. It is a test method that only considers the uniform distribution of test points in the test range. The uniform design table was used to arrange the experiment, and the regression analysis method was used to analyze the experimental data. The uniform design experimentation method is especially suitable for occasions with a large range of variables and many levels of factors (if the number of levels is not less than five).

The level values of all the factors are shown in Table 2. Firstly, according to the number of factors and levels, we selected a uniform design table, $U_6^{*}\left(6^4\right)$ or $U_7\left(7^4\right)$ [24]. Next, according to their usage table, when the factor number (m) is four, the deviations of the two tables are 0.2990 and 0.4750, respectively. Finally, based on the principle of minimum deviation, the uniform design table, $U_6^{*}\left(6^4\right)$, was used for the test, and the test scheme is shown in

Table 4. Design of shift test scheme influenced by multi-factor, $U_6^{*}\left(6^4\right)$.

Test	$${\mathit{N}}_{\mathit{e}}:\mathbf{Engine}\mathbf{Speed}(\mathbf{r}/\mathbf{min})$$	$${\mathit{F}}_{\mathit{r}}:\mathbf{Tractive}\mathbf{Resistance}\mathbf{of}\mathbf{Tractor}\left(\mathbf{kN}\right)$$	$${\mathit{P}}_{\mathit{c}}:\mathbf{Clutch}\mathbf{Oil}\mathbf{Pressure}\left(\mathbf{MPa}\right)$$	$${\mathit{D}}_{\mathit{o}}:\mathbf{Orifice}\mathbf{Diameter}\left(\mathbf{mm}\right)$$	Observed Response Value y
1	1000(1)	8(2)	1.4(3)	3.75(6)	y1
2	1200(2)	14(4)	2(6)	3.5(5)	y2
3	1400(3)	20(6)	1.2(2)	3.25(4)	y3
4	1600(4)	5(1)	1.8(5)	3(3)	y4
5	1800(5)	11(3)	1(1)	2.75(2)	y5
6	2000(6)	17(5)	1.6(4)	2.5(1)	y6

.

The test results that were obtained according to this scheme are shown in

Table 5. The test results of acceptable jerk and shift time.

Test	${\mathit{N}}_{\mathit{e}}:(\mathbf{r}/\mathbf{min})$	${\mathit{F}}_{\mathit{r}}:\left(\mathbf{kN}\right)$	${\mathit{P}}_{\mathit{c}}:\left(\mathbf{MPa}\right)$	${\mathit{D}}_{\mathit{o}}:\left(\mathbf{mm}\right)$	${\mathit{J}}_{\mathit{a}\mathit{j}\mathit{v}}:(\mathit{m}/{\mathit{s}}^3)$	${\mathit{t}}_{\mathit{e}}:\left(\mathbf{s}\right)$
1	1000(1)	8(2)	1.4(3)	3.75(6)	5.05	0.25
2	1200(2)	14(4)	2(6)	3.5(5)	11.44	0.34
3	1400(3)	20(6)	1.2(2)	3.25(4)	25.96	0.20
4	1600(4)	5(1)	1.8(5)	3(3)	3.82	0.28
5	1800(5)	11(3)	1(1)	2.75(2)	13.46	0.28
6	2000(6)	17(5)	1.6(4)	2.5(1)	19.23	0.41

. The analysis and processing of the test data will be presented later.

3.4.3. The Influence of Combined Parameters on Acceptable Jerk

Firstly, the multiple linear regression model was used for the analysis and processing of the test data. Secondly, the stepwise regression method was selected to select the variables, and then the variable with the greatest contribution was selected for inclusion in the regression equation. Two pre-determined thresholds, $F_{in}$ and $F_{out}$, were used to determine whether a variable should be selected or eliminated from the regression equation. Finally, the regression equation was as follows:

$$J_{ajv}=-4.842+1.440F_r$$

(35)

The value’s residual standard deviation $({\widehat{\sigma }}^{*})$ and multi-correlation index $(R^2)$ in the regression analysis were 2.741 and 0.916, respectively. Equation (35) shows that only one parameter, $F_r$, had a significant effect on $J_{ajv}$, while the other three parameters had no significant effect. This is not consistent with actual experience. Practical problems often need to consider high-order interaction, so the linear model does not necessarily meet the requirements. Therefore, a quadratic regression model was proposed, as follows:

$$Y={\beta }_0+{\displaystyle \sum }_{i=1}^m{\beta }_i{X}_i+{\displaystyle \sum }_{i=1}^m{\beta }_{ii}{X}_i^2+{\displaystyle \sum }_{i<j}{\beta }_{ij}{X}_i{X}_j+{\epsilon }_0$$

(36)

where ${\beta }_0,{\beta }_i,{\beta }_{ii},{\beta }_{ij}$ are the regression coefficients; ${\epsilon }_0$ is the random error. There are 15 terms in Equation (36). The regression equation can be obtained by using a stepwise regression technique, as follows:

$$J_{ajv}=9.969+0.054F_r\times F_r-0.14P_c\times D_o$$

(37)

The value’s residual standard deviation $({\widehat{\sigma }}^{*})$ and multi-correlation index $(R^2)$ in this regression analysis were 0.952 and 0.992, respectively, and $t_0=$5.918, $t_1=$17.336, $t_2=$−4.709. Obviously, the effect of Equation (37), was better than Equation (35).

Equation (37) shows that: parameters, $F_r$, and interaction, $P_c\times D_o$, had significant effects on $J_{ajv}$, because $\left|t_1\right|>\left|t_2\right|$, so $F_r$ was more important than $P_c\times D_o$. The larger the tractive resistance of the tractor, $F_r$, is, the more significant the impact on the acceptable jerk, $J_{ajv}$, is. The clutch oil pressure, $P_c$, and the orifice diameter, $D_o$, had a negative interaction with the acceptable jerk, $J_{ajv}$.

3.4.4. The Influence of Combined Parameters on Shift Time

The multiple linear regression model was used for the analysis and processing of the test data. Backward elimination was used to filter the variables. Firstly, all the variables were used to fit the regression equation, and then the variables that had no significant contribution to the equation were gradually eliminated, until all the variables in the equation had a significant contribution. The initial regression equation was as follows:

$$t_e=0.386+0.001F_r+0.013P_c-0.097D_o$$

(38)

The next step was to conduct an analysis of variance on equation 38. The result is shown in

Table 6. Analysis of variance table.

Source	Sum of Squares	df	Mean Square	F
Regression	0.018	3	0.006	1.328
Residual Error	0.009	2	0.004
Total	0.027	5

.

When $\mathsf{\alpha }=0.05$, the critical value, $F_{3,3}\left(0.05\right)$, of the F table is 9.23 > F = 1.328, so equation 38 was not credible. After this, the variables that had the least significant contribution to the equation were removed in turn, and the regression analysis was carried out. Finally, it was found that there was no regression relationship between $t_e$ and the four parameters ($N_e,F_r,P_c,D_o$). The test results showed that the four parameters ($N_e,F_r,P_c,D_o$) had no significant effect on the shift time, $t_e$.

3.5. Discussion

In previous studies, the indexes selected to evaluate the shift quality were transient indexes, chosen without considering the time process influence of shift jerk. Since shift is a continuous process, it is obviously unreasonable to only consider the transient maximum. In our research, the acceptable jerk and the shift time were selected as the evaluation indexes of shift quality. The acceptable jerk considered the effect of the two factors, peak jerk and root-mean-square jerk. The peak jerk refers to the maximum jerk during the shift. The jerk reflects the continuous impact of the jerk within the shift time. Therefore, the evaluation indexes of this paper were more comprehensive than the evaluation indexes in previous studies.

Previous research that has been conducted [15,16] only studied the influence of a single design parameter on shift quality, and did not further explore the influence of combined parameters closer to the actual engineering situation. The research [18,19] studied the influence of combined parameters on shift quality by an orthogonal test method, but did not consider the interaction between combined parameters, and the level number of each parameter was too small (only three). In our research, the uniform design experimentation method was used to explore the comprehensive influence of multi-level combined parameters on tractor shift quality. We designed a uniform design table with four parameters and six levels, and the complete experimental design goal was completed after six tests. The complex relationship between the four parameters was obtained by using regression analysis. In the same case, 49 tests would have been required if the orthogonal test method had been used, and further consideration of the interactions between parameters would have made the number of tests even greater. The results showed that the orthogonal test method was not suitable to study the influence of multi-level combined parameters on shift quality. The uniform design experimentation method used in this paper could not only effectively complete the test, but could also accurately obtain the complex interaction between the combined parameters, with the help of regression analysis.

4. Conclusions

In this paper, the kinetic model of tractor powertrain matching the HMCVT was established, and the shift quality of the tractor was analyzed. The simulation result of the tractor’s continuous shift process showed us that when shifting between adjacent ranges, the jerk produced is large. This played a decisive role in the shift quality of the tractor.

The test results of a single parameter showed that there was a positive correlation between the acceptable jerk and the four parameters (the engine speed, the tractive resistance of the tractor, the clutch oil pressure, and the orifice diameter). The higher the values of the four parameters, the higher the acceptable jerk value of the tractor. However, the shift time was only negatively correlated with the engine speed and the orifice diameter. The larger the engine speed and orifice diameter, the smaller the shift time.

The uniform design experimentation method was used to analyze the shift jerk and shift time of the tractor, under the influence of multiple combined parameters. The test data were processed by using regression analysis. The test results showed us that the influence degree of the combined parameters on the acceptable jerk degree in the process of shift was as follows: the tractive resistance of the tractor and the interaction between the clutch oil pressure and the orifice diameter. Furthermore, the four parameters had no significant effect on the shift time.