## 3 Results and Discussion
### Case-Study: Simulation of the Test of a Farm Tractor
#### 3.7.2 Quasi-Static Test (Without Tyres Deflection)  
In a second test (quasi-static tests at the inclination \(\alpha\) = 20.00\({}^{\circ}\) and 37.73\({}^{\circ}\)), the turntable rotates counter-clockwise by 360\({}^{\circ}\) around its axis. Under the same assumption of the previous paragraph (i.e., considering that the tyres are not deformable), the locus of \(CoG^{\prime}\) described during a complete revolution of the turntable in the two above-referred cases is represented in Figure 17; it has a radius of 309 mm (at \(\alpha\) = 20.00\({}^{\circ}\)) or 658 mm (at \(\alpha\) = 37.73\({}^{\circ}\)) respectively. In the latter case, the distance between the circumference traveled by \(CoG^{\prime}\) and the support polygon is zero in the left side of the tractor (the circumference is tangent to the support polygon), and this means that in that angular position (corresponding, in this case, to \(\beta\) = 0\({}^{\circ}\)) the vehicle is in a condition of incipient rollover. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Quantity** & **Meas. Unit** & \multicolumn{3}{c}{\(\alpha\)} \\ \cline{3-5}  & **0.00\({}^{\circ}\)** & **20.00\({}^{\circ}\)** & **37.73\({}^{\circ}\)** \\ \hline \(W_{FR}\) (\(\equiv W_{2}\)) & N & 7717 & 3840 & 0 \\ \hline \(W_{FL}\) (\(\equiv W_{3}\)) & N & 7905 & 10,840 & 12,356 \\ \hline \(W_{RR}\) (\(\equiv W_{1}\)) & N & 10,788 & 5368 & 0 \\ \hline \(W_{RL}\) (\(\equiv W_{4}\)) & N & 11,051 & 15,154 & 17,273 \\ \hline Normal force (\(\sum W_{i}\)) & N & 37,461 & 35,202 & 29,629 \\ \hline \(W_{A}\) (\(=W_{2}+W_{3}\); front part of the tractor) & N & 15,622 & 14,680 & 12,356 \\ \hline \(W_{B}\) (\(=W_{1}+W_{4}\); rear part of the tractor) & N & 21,839 & 20,522 & 17,273 \\ \hline \(W_{C}\) (\(=W_{1}+W_{2}\); right part of the tractor) & N & 18,506 & 9209 & 0 \\ \hline \(W_{D}\) (\(=W_{3}+W_{4}\); left part of the tractor) & N & 18,956 & 25,994 & 29,629 \\ \hline \(x(CoG^{\prime})_{SP}\) & mm & 0.0 & 0.0 & 0.0 \\ \hline \(y(CoG^{\prime})_{SP}\) & mm & \(-\)8.0 & \(-\)317.4 & \(-\)665.6 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Results from the simulated static test (at three different inclinations: 0.00\({}^{\circ}\), 20.00\({}^{\circ}\) and 37.73\({}^{\circ}\)). Figure 17: Locus of the positions of \(CoG^{\prime}\) in the frame of reference fixed to the turntable, when the turntable does a complete counter-clockwise rotation around its rotation axis (i.e., the origin of the reference frame) for (**a**) \(\alpha\) = 20.00\({}^{\circ}\) and (**b**) \(\alpha\) = 37.73\({}^{\circ}\). The blue arrow indicates the direction of rotation of the support platform (the \(y_{SP}\) axis is oriented as in the figures only at the beginning, then rotates accordingly to the blue arrow); the red arrow instead indicates the direction of travel of the locus of the positions of \(CoG^{\prime}\) (the initial point is the one evidenced, then it moves clockwise along the red circumference). The red cross indicates the position of \(CoG^{\prime}\) (NB: not coincident with the origin). The normal forces (absolute values, percentage of the total weight of the tractor) which can be recorded on each quadrant during such test are visible in Figure 18 (for an inclination \(\alpha=20.00^{\circ}\)) and Figure 19 (for \(\alpha=37.73^{\circ}\)). Note that the same condition of incipient rollover is evidenced by two of the four curves reaching a condition of null force, i.e., touching the abscissa axis as in Figure 19a,b (for \(\beta=0^{\circ}\)). From the graphs shown here, it is possible to see that the safest stability condition is, as expectable, the one at \(\beta=90^{\circ}\), i.e., corresponding to a situation where the front of the tractor is heading downstream. The safety margin is in fact the minimum vertical distance of the curves from the abscissae axis, in this case maximum for \(\beta=90^{\circ}\).