## 3 Results and Discussion
### Optimisation of Parameters and Values of Objective Functions \(F_{x}\) and \(F_{y}\)
These values align with the research [40], suggesting that tools working at greater depths and higher speeds generate higher vertical forces due to the more significant amount of soil being moved upwards. The minimisation of vertical force (\(-84.55\) N) was achieved at lower values of \(k,d/w\), and \(v\), suggesting that reducing the tool's working depth, speed, and stiffness leads to lower vertical forces. Minimum values are consistent with the results [30] indicating that tools that work shallowly and slower generate less vertical force because less soil is moved upwards. A critical rake angle has already been calculated, related to the angle of friction of the soil with the metal, which is \(22.5^{\circ}\), relative to which the vertical force changes the sign from positive to negative [52]. When working duckfoot at the lowest depth, the gravitational force of the soil resting on the tool and the vertical component of soil resistance was smaller than the vertical component of the soil breakout force by the tool's cutting edge [50]. Therefore, the vertical force value was negative at the lowest working depth. The negative vertical force can be assigned to natural sandy soil [40]. Thus, for very small angles of attack, the proportion of vertical force from the weight of the soil wedge was less than from the adhesion force. The optimal values of the variables were at the ends of the considered ranges, suggesting that in the analysed conditions, the best results are achieved at the extremes of the technical variables. This is visible in all eight surface plots (Figure 2). The runs of these charts allow us to assume that a combination of \(d/w\) and \(v\) values can be expected for optimal forces \(F_{x}\) and \(F_{y}\) with maximum values. It is, therefore, advisable to carry out tests at higher velocities where a more significant influence of the square of the speed (\(v^{2}\)) can be expected and with a higher depth-to-width ratio, \(d/w\), e.g., by increasing the working depth. This would only be appropriate for cognitive purposes because expanding the depth of loosening excessively, e.g., in agricultural practice, would not be advisable, due to the risk of covering plants with soil. In addition, a more significant \(d/w\) relationship would go beyond a wide tool's scope and apply to narrow tools. Particular attention should be paid to the same value of soil moisture \(MC=14\%\), which was optimal for both maximum and minimum draught and vertical force values. This result indicates that soil with this moisture content offers optimal resistance to the tool, and changes in moisture outside this range can destabilise the tool's work process. This shows that with this moisture, the soil offers stable resistance to the implementation, which reduces variability in the results. Soil moisture affects soil adhesion and friction, crucial for soil displacement during tool operation [41]. The developed mathematical, empirical models are characterised by minor relative errors, which, for optimal values, range from \(-1.72\%\) to \(-4.81\%\) (negative values indicate an overestimation of the model concerning actual values). Optimisation of the draught \(F_{x}\) and vertical \(F_{y}\) forces using partial derivatives and the SLSQP method provided detailed information on the influence of technical and environmental variables on the value of these forces. Introducing the variable \(k\) (stiffness) into the empirical model's \(F_{x}\) and \(F_{y}\) has enabled a more precise representation of the actual operating conditions of agricultural implements, which is essential for optimisingmachine operation and energy savings. The SLSQP optimisation method provided tools to accurately determine the values of variables that minimise or maximise fraught and vertical forces, which is of great importance for agricultural practice. The lower values of the correlation coefficient R\({}^{2}\) result from dynamic changes and confounding factors that affected the pulling forces \(F_{x}\) and the vertical force component \(F_{y}\).