## 2 Materials and Methods
### Experimental Phase  
The experimental measurement of the vertical stiffness of the tyres was carried out on two different agricultural tyres (see Table 1) and in two pressure conditions (100 kPa and 160 kPa, i.e., 1.0 bar and 1.6 bar), for a total of 4 different tyre-loading scenarios. The main characteristics of the tested tyres, which are stamped on the sidewalls of tyres using a technical information format compliant with ETRTO standard [41], are shown in Table 1. The indicated tyres have very different sizes and they have been chosen because each one can be considered as representative of a specific class of tyres usually equipping 4 WD farm tractors (420/85 R24 for front tyres, 460/85 R34 for rear tyres).  
In order to replicate the geometry with CAD software, the two indicated tyres were removed from their own rims and the transversal section was measured. As the tested tyres were worn, some of their dimensions differed from the nominal ones:  
* the 420/85 R24 tyre presented a radius index of 600 mm at 1.0 bar and of 602 mm at 1.6 bar with 1240 kg\({}_{\text{f}}\) of load;
* the 460/85 R34 presented a radius index of 762 mm at 1.0 bar and of 765 mm at 1.6 bar with 1140 kg\({}_{\text{f}}\) of load.  
The chosen load values for the elastic characterization spanned from 0 N (corresponding to an _unladen tyre_) to 23 kN (_maximum load_). The maximum load was chosen to be the same for the four samples and in order to reach at least 80% of the nominal load capacity indicated for the 460/85 R34 tyre. Two repetitions of the loading sequence were carried out in all four experimental cases.  
The tyres were mounted on a _\"Laser 110 4 WD\"_ farm tractor by Same (Treviglio, Bergamo, Italy; [42]). The tractor was placed with the tyre to be tested over a weighing platform (M13 by Adda Bilance, Cassano d'Adda, MI, Italy) and a position transducer (PT1E by Celesco, Toronto, Canada) was connected with a cable to the wheel hub. The tractor axle with the tested tyre was lifted by a crane (ZKKE 16 t by Demag, Agrate Brianza, MB, Italy) in order to start in a condition of unloaded tyre. An acquisition system (DEWE-43A by Dewesoft, Roma, Italy) was set to start when the weight, measured by the platform, reached the value of 100 N. Once the tractor was completely released from the crane, the tractor rested on the floor (and hence, one of its wheel on the platform); an additional weight was then loaded on the axle of interest by pulling it down through a hydraulic cylinder (equipment developed by CREA-IT, Treviglio (BG), Italy) connected to the tractor hook. The load was applied with a medium speed of about 2 kN:s\({}^{-1}\). The frequency of acquisition was set to 100 Hz. The experimental setup is reported in Figure 2.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Tested Tyre** & **Load and Speed** & **Tread Width** & **Radius Index** & **Load Capacity** \\
**Denomination** & **Index (ETRTO)** & **(mm)** & **(mm)** & **(kg\({}_{\text{f}}\) @ 160 kPa)** \\ \hline
420/85 R24 & 137-A8/134-B & 429 & 625 & 2120 \\
460/85 R34 & 147-A8/144-B & 467 & 775 & 2800 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Nominal characteristics of the tested tyres.