###### Abstract  
This paper presents the development of empirical mathematical models of draught force, \(F_{x}\), and vertical force, \(F_{y}\), acting on duckfoots attached to the times with different stiffness and working in various soil conditions. The models consider technical variables such as stiffness, \(k\), tool depth-to-width ratio, \(d/w\), tool movement speed, \(v\), and soil moisture content, \(MC\), which have not been thoroughly analysed in the literature. The correlation coefficients for predicting \(F_{x}\) and \(F_{y}\) values were 0.4996 and 0.6227, respectively. Statistical analysis confirmed the significant effect of these parameters on the forces acting on the tools, with the variables \(d/w\) and \(v\) having the most critical impact on \(F_{x}\) and \(F_{y}\). The SLSQP (sequential least squares programming) optimisation method was used to determine the optimal values of technical variables. The maximum value of \(F_{x}\) was 438.55 N, and the minimum was 98.98 N, with variable values at the edges of the studied ranges. Similarly, \(F_{y}\) values of 135.25 N and \(-\)84.55 N, respectively, were obtained. The optimisation results showed good fitness with experimental results, and the negative relative errors (from \(-\)1.72% do \(-\)4.81%), indicating overestimating, confirmed the accuracy of the model's predictions. The justification of the research results allowed us to conclude that there is no basis for rejecting the explanatory hypotheses. The developed models have a generalisable value in the analysed ranges, and further research should focus on creating more universal, theoretical models of soil-tool interactions. soil loosening element; control variables; cutting forces; mathematical model +
Footnote †: journal: Applied sciences  
0000-0001-895X/0000-0001-895X/0000-0001-895X/00000-0001-895X/00000-0001-895X/000001-895X/000001-895X/00 soils increases with clay content, affecting soil strength and limiting plant growth [3]. High wheel pressure on the soil can lead to deep substrate compaction, resulting in a permanent reduction in yield [3]. This condition worsens as agricultural machinery's weight increases [4]. Soil compaction changes soil structure, reducing aeration and root development and contributing to environmental problems [5], limiting access to water for the proper development of plants [6]. To improve compacted soil structure, especially in row crops, new tools combine classic knives and vertical minidiscs to remove weeds and prevent the plants from being buried in the rows. These mechanical improvements in inter-row crops are critical to chemical-free weed control [7]. Mechanical inter-row weeding tools have effectively reduced herbicide use in sugarcane cultivation, significantly impacting weed coverage and yields [8]. In addition, the experiments showed that weed cover was reduced considerably with treatments combining mechanical and chemical weed control compared to the crop itself, and cereal yields were slightly reduced only under certain conditions [9]. These results highlight the importance of cultivator design and working speed and the effectiveness of combining mechanical and chemical weed control methods for effective weed management in row crops. Combining different inter-row tillage tools can increase the effectiveness of weed removal. However, more research is needed to reduce crop damage while maintaining high crop protection efficiency [10]. The interrelated phenomena related to the technology of treatments, soil compaction, and the ecological aspect of inter-row cultivation require the search for technical solutions to limit the forces needed to overcome soil resistance with the most favourable cultivation parameters. Understanding the forces acting on tillage tools is critical to optimising soil tillage processes and improving energy efficiency in agriculture.