# Study on the Control of Torque Distribution of 4 WD Corn Harvester Operation Drive
## 2 Design and Modeling
### Establishment of the Mathematical Model
#### 2.2.3 Mathematical Model of Inter-Axle and Inter-Wheel Torque Distribution Device  
The active torque distribution control between the front axle drive wheels used an active torque control differential, consisting of an electronically controlled clutch, a master clutch, a common symmetrical level gear differential, and a triplex compound gear, which could achieve torque distribution while taking the sensitive and fast control into account [31]. The active friction plate of the clutch was fixed to one half-axle of the differential, and the two sets of driven friction plates were connected to the housing of the symmetrical level gear differential through a composite gear. When the clutch on the left side was combined, as shown in Figure 3a, the output torque of the right half-axle of the rear axle differential decrease, and the output torque of the left half-axle increase; the situation was reversed when the clutch on the right side was combined [32]. Torque distribution characteristics of the front axle torque differential when the left clutch is combined:  
\[\left\{ \begin{array}{c}T_{\textit{ff}}=\frac{T_{\textit{ff}}}{2}+\frac{Z_{1 }Z_{\textit{d}}}{Z_{2}Z_{3}Z_{4}}T_{d}\\ T_{\textit{fp}}=\frac{T_{\textit{ff}}}{2}-\left(1-\frac{Z_{1}Z_{\textit{d}}}{Z_{ \textit{d}}Z_{3}Z_{4}}\right)T_{d}\end{array} \right. \tag{12}\]  
Figure 3: Torque distribution mechanism: (**a**) front axle torque active control differential, (**b**) electronically controlled central differential. similarly, when the right clutch is combined:  
\[\left\{ \begin{array}{c}T_{fl}=\frac{T_{6}}{2}-\frac{Z_{1}Z_{5}}{2Z_{2}Z_{4}}T_{cr }\\ T_{fp}=\frac{T_{6}}{2}+\left(1-\frac{Z_{1}Z_{5}}{2Z_{2}Z_{4}}\right)T_{cr}\end{array} \right. \tag{13}\]  
where \(T_{cl}\) and \(T_{cr}\) are the control torques between the left and right wheels of the front axle, respectively; \(T_{fp}\) is the input torque; \(T_{fp}\) is the torque output to the right front wheel; \(T_{fp}\) is the torque output to the left front wheel; \(Z_{1}\), \(Z_{2}\), \(Z_{3}\), \(Z_{4}\), \(Z_{5}\), \(Z_{6}\) are the number of teeth of the corresponding gears, respectively. The control torque of the control clutch is  
\[T_{c}=\frac{\pi\mu_{z}\mu_{c}\varepsilon_{o}\tau_{z}\tau_{k}n\eta q^{2}}{4r_{ t}}l^{2} \tag{14}\]  
where \(\varepsilon_{o}\) is the vacuum permeability; \(p\) is the number of active friction plate working surfaces of the main clutch of the inter-axle differential; \(n\) is the number of driven friction plate working surfaces of the main clutch of the inter-axle differential; \(r_{t}\) is the armature pole radius; \(q\) is the coil turns; \(r_{k}\) is the effective working radius of the control clutch of the inter-axle differential; \(r_{z}\) is the effective radius of the main clutch of the inter-axle differential; \(\mu_{z}\) and \(\mu_{c}\) are the friction coefficient of the main clutch friction plate and the control clutch friction plate, respectively. The input is the clutch control current \(l\), and the output is the generated limited-slip torque [33]. The active differential can actively distribute the drive torque to the drive wheels on both sides of the axle according to the harvester driving conditions and the road condition, thus improve the driving stability of the harvester [34; 35]. Friction was discarded in modeling the central differential with the drive wheels [33]. Mathematical model of the front and rear axles of the central differential:  
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