# Depth Optimization Analysis of Articulated Steering Hinge Position Based on Genetic Algorithm
## 2 Mechanism Analysis of Stroke Difference and Arm of Force Difference  
In the steering process of wheel loaders, the relation between the front and rear frame around the hinge joint is realized by the elongation of the outer cylinder and the synchronous shortening of the inner cylinder. A schematic diagram of the steering mechanism is shown in Figure 1. Points A and B are the hinge points of the steering cylinder with the front frame, points C and D are the hinge points of the steering cylinder with the rear frame, and point O is the hinge point of the front and rear frames. For simplification, we represent \(\frac{\overline{AB}}{2}\) as \(x_{f}\) and \(\frac{\overline{CD}}{2}\) as \(x_{r}\). The distance from the center of the connection between the two hinge points of the front frame to the hinge point O is represented as \(y_{f}\), and the distance between the center of the connection of the two hinge points of the rear frame to the hinge point O of the rear frame is represented as \(y_{r}\). With the change of the steering angle, the extension of AC on the left cylinder and the shortening of BD on the right cylinder are shown in Formulas (1) and (2), respectively. The difference between them is the stroke difference. \[\Delta L_{l}=\overline{A^{\prime\prime}C}-\overline{AC}=\sqrt{\left(-x_{f}\cos \theta+y_{f}\sin\theta+x_{r}\right)^{2}+\left(x_{f}\sin\theta+y_{f}\cos\theta- y_{r}\right)^{2}} \tag{1}\]  
\[\Delta L_{r}=\overline{B^{\prime\prime}D}-\overline{BD}=\sqrt{\left(x_{f}\cos \theta+y_{f}\sin\theta-x_{r}\right)^{2}+\left(y_{f}\cos\theta+x_{f}\sin\theta -y_{r}\right)^{2}} \tag{2}\]  
According to [PERSON]'s formula [19], the force arms \(\overline{OM}\)/ and \(\overline{ON}\)/ of the two cylinders are obtained respectively, and the difference between them is the arm of force difference. \[\overline{OM^{\prime}}=\frac{2\sqrt{P(P-\overline{OA^{\prime}})(P-\overline{ OA^{\prime}C})}}{\overline{AC^{\prime}C}} \tag{3}\]  
\[\overline{ON^{\prime}}=\frac{2\sqrt{S(S-\overline{OB})(S-\overline{OD})(S- \overline{BD})}}{\overline{B^{\prime}D}} \tag{4}\]  
where:  
\[\frac{P=\overline{OC+\overline{OA^{\prime}}+\overline{A^{\prime}C}}}{A^{ \prime}C},\frac{S=\overline{OD+\overline{OB^{\prime}}+\overline{B^{\prime}D} }}{\overline{B^{\prime}D}},\overline{OA^{\prime}}=\overline{OA},\overline{ OB^{\prime}}=\overline{OB}, \tag{5}\]  
The calculation of steering resistance distance is generally based on the empirical formula, but the empirical formula has its limitations [19]. In this paper, the steering resistance distance was calculated. The analysis showed that the steering resistance distance consisted of three parts: torque when the tire pivots around the center of its contact patch \(M_{m}\), the resistant torque caused by the opposite steering of the left and right wheels \(M_{S}\), and the torque generated by the tangential force on the rear axle \(F_{r}\). As shown in Figure 2, the formulas for calculating the steering resistance moment based on the virtual displacement principle are as follows:  
\[\left\{ \begin{array}{l}\frac{L_{l}}{\sin\beta}=\frac{L_{l}+L_{g}}{\sin\theta }\\ \theta=\alpha+\beta\end{array} \right.\Rightarrow\left\{ \begin{array}{l} \partial\alpha=\left(1-\frac{k\cos\theta}{\sqrt{1-k^{2}\sin^{2}\theta}} \right)\cdot\partial\theta\\ \partial\beta=\frac{k\cos\theta}{\sqrt{1-k^{2}\sin^{2}\theta}}\cdot\partial \theta\end{array} \right. \tag{6}\]  
Figure 1: Schematic diagram of the steering mechanism. where \(T\) is the steering torque, \(M_{F},M_{R}\) are the front and rear resistance torques, \(k=\frac{L_{1}}{L_{1}+L_{2}}=0.5\). Through coordinate system transformation, \(A^{\prime\prime}\) coordinates of \(A^{\prime}\) in the coordinate system XOY and \(B^{\prime\prime}\) coordinates of \(B^{\prime}\) in the coordinate system XOY were obtained.