# 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
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. 1. Pilot valve The flow through chamber \(X\) of the \(NC1\) was defined by applying the following continuity equation to the chamber: \[Q_{NC1,X,in}-Q_{NC1,X,ex}=\frac{V_{NC1,X}}{\beta}\frac{dP_{NC1,X}}{dt}\] (22) \[Q_{NC1,\ X\&C}=Q_{NC1,X,in}+Q_{NC1,C}\] (23) \[\left\{ \begin{array}{c}Q_{NC1,X\&C}=C_{d}(\lambda)A^{\prime}_{\ O11}\sqrt{ \frac{\beta}{\rho}|\Delta P|}\text{sgn}(\Delta P)\\ \Delta P=P_{NC1,in}-P_{NC1,X}\end{array} \right.\] (24) where \(Q_{NC1,X\&C}\) is the total flow rate into chambers \(X\) and \(C\), m\({}^{3}\)/s; \(C_{d}(\lambda)\) is the flow coefficient; \(A^{\prime}_{\ O11}\) is the constant area of sharp-edged orifice 11, m\({}^{2}\); \(\rho\) is the density of the transmission fluid, kg/m\({}^{3}\); \(\beta\) is the transmission fluid bulk modulus, Pa; \(Q_{NC1,X,ex}\) is the flow rate at the outlet of chamber \(X\), m\({}^{3}\)/s; and \(V_{NC1,X}\) is the volume of chamber \(X\), m\({}^{3}\). 2. 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: \[Q_{NC1,in}-Q_{NC1,control}=\frac{V_{NC1,B}}{\beta}\frac{dP_{NC1,B}}{dt}\] (25) \[Q_{NC1,control}-Q_{NC1,ex}=\frac{V_{NC1,B}}{\beta}\frac{dP_{NC1,B}}{dt}\] (26) where \(V_{NC1,B}\) is the volume of chamber \(B\), m\({}^{3}\); \(P_{NC1,B}\) 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:  
\[Q_{\text{NCL},A}+A_{\text{NCL},A}\cdot\dot{x}_{2}=\frac{V_{\text{NCL},A}}{\beta} \frac{dP_{\text{NCL},A}}{dt} \tag{27}\]  
\[Q_{\text{NCL},C}-A_{\text{NCL},C}\cdot\dot{x}_{2}=\frac{V_{\text{NCL},C}}{\beta }\frac{dP_{\text{NCL},C}}{dt} \tag{28}\]  
\[\left\{ \begin{array}{c}Q_{\text{NCL},A}=C_{d}(\lambda)\,A^{\prime}{}_{O12} \sqrt{\frac{2}{\rho}|\Delta P|}\text{sgn}(\Delta P)\\ \Delta P=P_{\text{NCL},control}-P_{\text{NCL},A}\end{array} \right. \tag{29}\]  
where \(Q_{\text{NCL},A}\) and \(Q_{\text{NCL},C}\) are the flow rates into or out of chambers \(A\) and \(C\), m\({}^{3}\)/s; \(\dot{x}_{2}\) is the spool speed, m/s; and \(V_{\text{NCL},A}\) and \(V_{\text{NCL},C}\) are the volumes in chambers \(A\) and \(C\), m\({}^{3}\). 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: \[m_{\text{C1}}\ddot{x}_{\text{C1}}+B_{\text{C1}}\dot{x}_{\text{C1}}+K_{\text{ C1}}x_{\text{C1}}=A_{\text{C1}}P_{\text{C1}}-F_{initial,C1}\] (30) where \(m_{\text{C1}}\) is the mass of the clutch piston, kg; \(B_{\text{C1}}\) is the viscous damping coefficient, N/(m/s); \(K_{\text{C1}}\) is the spring constant, N/mm; \(A_{\text{C1}}\) is the pressure acting area of the clutch piston, m\({}^{2}\); \(P_{\text{C1}}\) is the pressure applied to the clutch piston, Pa; \(F_{initial,C1}\) is the spring preload, and N; \(x_{\text{C1}}\), is defined as piston displacement and measured from the unloaded static equilibrium, m.
2.