# Research on the Influence of Tractor Parameters on Shift Quality, Based on Uniform Design
## 2 Powertrain Model
### Hydraulic Component Actuation
#### 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. Figure 3: Simplified hydraulic actuation system. 1. Spool mechanical systems 1. 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: \[m_{1}\frac{{dx_{1}}^{2}}{{dt}^{2}}+B_{1}\frac{{dx_{1}}}{{dt}}=F_{mag,NC1}+F_{1, flow,NC1}-P_{NC1,X}A_{NC1,X}\] (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, m\({}^{2}\); \(x_{1}\) is the spool displacement, m; and \(F_{1,flow,NC1}\) is the net steady flow force acting on the spool, N. 2. 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: \[m_{2}\frac{{dx_{2}}^{2}}{{dt}^{2}}+B_{2}\frac{{dx_{2}}}{{dt}}+K_{NC1}x_{2}=P_{ NC1,C}A_{NC1,C}-P_{NC1,A}A_{NC1,A}-F_{2,flow,NC1}\] (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 \(C\), m\({}^{2}\); \(x_{2}\) 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)  
Figure 4: The structure of \(NC1\). The given voltage current equation of the proportional electromagnet was as follows:  
\[U_{g}=\frac{1}{K_{e}}\bigg{(}\Big{(}R_{L}+K_{e}K_{fi}\Big{)}i+L_{h}\frac{di}{dt} \bigg{)} \tag{20}\]  
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, \(\Omega\); \(L_{h}\) 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, \(Q_{Line}\).