## 3 Results and Discussion
### Mathematical Empirical Models
#### 3.2.1 Detailed Model Description for Draught Force, \(F_{x}\)  
The empirical model developed for draught force, \(F_{x}\), considered the influence of several independent variables, such as stiffness, \(k\), tool depth-to-width ratio, \(d/w\), tool movement speed, \(v\), and soil moisture, \(MC\). The model also included interactions between these variables, including double, triple, and single quadrupole interactions. These were crucial for correctly mapping the effect of different combinations of factors on the value of draught force. A vital element of this model was to consider non-linear relationships, which allowed for more accurate prediction of draught forces under varying soil conditions. The regression analysis results show that the speed of tool movement, \(v\), and its square, \(v^{2}\), are among the regression coefficients. The regression coefficient for the square of speed was 17.8, and the value of the Student's \(t\)-test for this variable reached 7.81, indicating the importance of this variable in the model. The speed of the implement's movement was one of the main factors determining the draught force, which is confirmed by previous studies indicating that an increase in speed leads to an increase in draught forces, especially in soils with higher cohesion and higher moisture content [39]. The soil flowing through the surface of the sweep was cracked, thus creating a loose soil effect. This result differs from the findings regarding narrow tools with a larger tooth angle [46] and a wide tool with a dimension of 200 mm [24]. At higher speeds, there was a more significant increase in tractive force with an increasing height (thickness) of the tool's cutting edge, but this had little effect on vertical force [48]. In the study of three types of tines (S, Vibroflex, and with a double coil of a coiled spring) concerning the working speed on the draught force of the tines working on different soils, good correlations between the values of forces and velocities were found [49]. These results are consistent with soil deformation mechanics, which suggest that increasing the speed of the implement leads to an increase in dynamic forces, which, in turn, increases the pulling resistance. Another essential element of the model was the tool depth-to-width ratio, \(d/w\). The regression coefficient for this linear variable was \(-502\), and the \(t\)-test value was \(-2.04\), indicating a robust and negative effect of the ratio \(d/w\) on the value of the draught force. This confirms previous observations that a greater working depth concerning a smaller tool width increases pulling resistance [30]. In this context, increasing the ratio of \(d/w\) can increase resistance, which is especially important for tools of different widths. The negative regression coefficient may suggest that under certain conditions, an increase in the tool depth-to-width ratio leads to a decrease in tractive effort, confirming the results of previous studies [30]. The negative regression coefficient given \(d/w\) should be considered in the context of interaction with other model variables, such as implement speed and soil moisture. The combined interaction of these parameters can modulate the effect of the variable \(d/w\) pulling power; therefore, increasing the ratio of \(d/w\) does not always lead directly to a reduction in resistance, and this effect can vary depending on the width of the tool and other operating conditions. Interactions between variables played a vital role in the \(F_{x}\) model. Among the dual interactions, the product of the \(d/w\) ratio and the movement speed \(v\) (\(d/w\times v\)) significantly affected the value of the draught force, with a regression coefficient of 1073 and a \(t\)-test value of 6.65. These results suggest that the combination of higher movement speed and the \(d/w\) ratio of the implement increased soil resistance, which is in line with expectations for the behaviour of agricultural implements in the soil.