## 3 Results and Discussion
### Case-Study: Simulation of the Test of a Farm Tractor
#### 3.7.3 Quasi-Static Test (Considering Also the Tyres Deflection)
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \multirow{2}{*}{} & \multicolumn{6}{c}{**Wheels Loads (N)**} & \multicolumn{6}{c}{**Wheels Load Distribution (\%)**} \\ \cline{3-10}  & & \(W_{\text{FR}}\) & \(W_{\text{FL}}\) & \(W_{\text{FR}}\) & \(W_{\text{RL}}\) & \(W_{\text{FR},\text{\text{\text{\text{\text{\text{\text{\text{\textin [45], spanning from 0.21 and 0.75 with an average of 0.51 (even if referred to car types), it is reasonable to hypothesize a lateral stiffness equal to the 0.50 of the stiffness measured along the vertical direction respectively for both the types of the present case (129 N-mm\({}^{-1}\) for the 14.9-24 type, 181 N-mm\({}^{-1}\) for the 18.4-34 type). The locus of \(CoG^{\prime}\) described during a complete revolution of the turntable in the two above-referred cases (at the inclination \(\alpha=20.00^{\circ}\) and \(37.73^{\circ}\)), plus a third intermediate inclination (\(\alpha=33.95^{\circ}\)) is represented in Figure 20. As visible, the locus is no longer a perfect circumference, but appears to be widened and deformed by the effect of the tyre deflection. The minimum and maximum dimensions are: 626 and 706 mm for \(\alpha=20.00^{\circ}\), 1157 and 1315 mm for \(\alpha=33.95^{\circ}\), 1329 and 1511 mm for \(\alpha=37.73^{\circ}\). Moreover, with the highest inclination (\(\alpha=37.73^{\circ}\)), the locus travelled by \(CoG^{\prime}\) exceeds the support polygon in both the sides of the tractor, specifically for \(\beta=[0^{\circ};\!12^{\circ}]U[3-48^{\circ};\!360^{\circ}]\) and for \(\beta=[168^{\circ};\!192^{\circ}]\), and this means that, with the test rig at that inclination and the turntable in those angular positions, the vehicle would already be overturned unless it was held at the structure by some security system (see next paragraph for details about it). Indeed, under the described hypothesis (deformable tyres), the incipient overturning conditions occur at 33.95\({}^{\circ}\), i.e., at an angle that is the 10% smaller than the case where the deformation of the tyres is neglected. Figure 20: 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**) \(\alpha=20.00^{\circ}\), (**b**) \(\alpha=33.95^{\circ}\) and (**c**) \(\alpha=37.73^{\circ}\). 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 orange 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 orange circumference). It is reported for proper comparisons also the locus of the positions of \(CoG^{\prime}\) in case of non-deformable tyres. 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 a test are visible in Figure 21 (for an inclination \(\alpha=20.00^{\circ}\)) and Figure 22 (for \(\alpha=33.95^{\circ}\)). Note that the same condition of incipient rollover is evidenced by two of the four curves reaching a condition of the null force, i.e., touching the abscissa axis as in Figure 22a,b (for \(\beta=0^{\circ}\)). The considerations that can be drawn from the graphs of Figure 22 about the safest stability condition (at \(\beta=90^{\circ}\) or \(\beta=270^{\circ}\)) are the same of the case in which the deformability of the tyres is neglected.