## 3 Results and Discussion
### Mathematical Empirical Models
#### 3.2.2 Model Description for Vertical Force, \(F_{y}\)  
The empirical model for vertical force, \(F_{y}\), considered vital independent variables, as did the model for draught force, \(F_{x}\), i.e., stiffness, \(k\), tool depth-to-width ratio, \(d/w\), movement speed, \(v\), and soil moisture, \(MC\). However, due to other physical phenomena related to the vertical impact of the implement on the soil, not all variables and interactions had the same significant impact on the vertical force as in the case of the draught force. Regression analysis revealed that the tool depth to the width ratio, \(d/w\), had the most significant effect on \(F_{y}\). The regression coefficient for this variable was 1705, and the value of the Student's \(t\)-test reached 33.34 (Table 4). A high regression factor value meant that the greater the ratio of the depth to the width of the tool, the greater the vertical force acting on the tool. This variable, therefore, had a dominant influence on the \(F_{y}\), which is in line with the results of previous studies that highlight that the working depth of the tool concerning its width plays a crucial role in generating vertical forces when working in soil [30]. As with the \(F_{x}\), tool movement speed, \(v\), also significantly affected the value of the vertical force \(F_{y}\), but its effect was negative. The regression coefficient for this variable was \(-176\), and the \(t\)-test value was \(-13.36\). This result suggests that an increase in the tool movement speed resulted in a decrease in vertical force. This aligns with the results of previous studies showing that soil can be ejected more efficiently at higher working speeds, reducing the vertical forces exerted on the implement [51]. In this context, the higher speed could have been conducive to more efficient lifting of the soil from the implement, which reduced vertical force. The interactions between the variables were less meaningful in the model for vertical force \(F_{y}\) than in the model \(F_{x}\), but a few were statistically significant. One crucial interaction was the combination of the depth-to-width ratio of the tool, \(d/w\), and soil moisture, \(MC\), which had a regression coefficient of \(-56.3\) and a \(t\)-test value of \(-13.56\). This interaction indicates that at higher soil moisture values, the effect of the \(d/w\) was weakened to vertical force. This result is consistent with the results of studies on soil-tool interactions, which suggest that under higher moisture conditions, soil tends to reduce the vertical resistance of the tool, which may be due to the lower soil cohesion associated with moisture [41]. In the \(F_{y}\), as in the \(F_{x}\), the squares of the variables were included, but not all of them were statistically significant. For example, the square of the movement speed, \(\vec{v}^{2}\), was rejected from the model because its effect on the vertical force was statistically insignificant. On the other hand, the square of the depth-to-width ratio \((d/w)^{2}\) had a significant negative impact on the vertical force, with a regression coefficient value of \(-776\). This result suggests that at very high ratios \(d/w\), the vertical force began to decrease, which may be because wide implements no longer move the soil upwards efficiently, and most of the forces are directed horizontally [30]. The linear influence of the stiffness \(k\) and its square \(k^{2}\) were rejected from the model as statistically insignificant. This result may be because the stiffness primarily affects the horizontal draught forces, not the vertical forces, which depend more on the implement geometry and soil conditions [40]. The most critical factors in the \(F_{y}\) were the tool's geometrical parameters, \(d/w\), and movement speed, \(v\). In conclusion, the empirical model for vertical force, \(F_{y}\), showed that the most significant factor influencing vertical force was the depth-to-tool width ratio, \(d/w\), which had a dominant influence on the vertical forces generated.