## 3 Results
### Setup of the Tractor Model  
A medium-size tractor, employed in the experimental tests in [41], was modeled by the lumped parameter model shown in Figure 1c. The parameters of the lumped model were initially estimated and subsequently adjusted to provide dynamic behavior close to that of the tractor. The final parameter values of the tractor model are listed and described in Table 1.  
By forcing with a given soil excitation, the experimental response of the medium-sized tractor and the response of the numerical model are compared. Numerical and experimental vertical accelerations of the operator's seat are compared in terms of their power spectral density (PSD). Figure 5 shows the results obtained by the authors using the excitation provided by the rougher track defined in the standard [42] when the forward speed of the tractor was \(v=6.3\) km/h.  
\begin{table}
\begin{tabular}{l r r l} \hline \hline
**Description** & **Symbol** & **Value** & **Unit** \\ \hline Mass of the seat with operator & \(m_{1}\) & 150 & kg \\ Mass of the tractor frame & \(m_{2}\) & 3350 & kg \\ Mass of the front axle & \(m_{3}\) & 700 & kg \\ Mass of the cabin & \(m_{4}\) & 350 & kg \\ \hline Moment of inertia of the tractor frame & \(l_{2}\) & 3959 & kg m\({}^{2}\) \\ Moment of inertia of the cabin & \(l_{4}\) & 77 & kg m\({}^{2}\) \\ \hline Stiffness of the seat suspension & \(k_{1}\) & 9.8 & kN/m \\ Stiffness of the rear wheels & \(k_{2}\) & 712.5 & kN/m \\ Stiffness of the front axle suspension & \(k_{3}\) & 85.6 & kN/m \\ Stiffness of the front wheels & \(k_{4}\) & 546.2 & kN/m \\ Stiffness of the front cabin suspension & \(k_{5}\) & 145.8 & kN/m \\ Stiffness of the rear cabin suspension & \(k_{6}\) & 54.3 & kN/m \\ \hline Damping coefficient of the seat suspension & \(c_{1}\) & 6065 & N s/m \\ Damping coefficient of the rear wheels & \(c_{2}\) & 28.5 & N s/m \\ Damping coefficient of the front axle suspension & \(c_{3}\) & 12.4 & N s/m \\ Damping coefficient of the front wheels & \(c_{4}\) & 21.8 & N s/m \\ Damping coefficient of the front cabin suspension & \(c_{5}\) & 5703 & N s/m \\ Damping coefficient of the rear cabin suspension & \(c_{6}\) & 3880 & N s/m \\ \hline Stiffness proportional viscous damping coefficient & \(\beta\) & \(5\times 10^{-5}\) & s\({}^{-1}\) \\ \hline Horizontal position of the seat suspension with respect to the CoG of the tractor frame & \(l_{1}\) & \(-0.50\) & m \\ Horizontal position of the rear axle with respect to the CoG of the tractor frame & \(l_{2}\) & \(-0.83\) & m \\ tractor frame & & \(l_{3}\) & 1.54 & m \\ Horizontal position of the CoG of the cabin with respect to the CoG of the tractor frame & \(l_{4}\) & \(-0.07\) & m \\ \hline Horizontal position of the front cabin suspension with respect to the CoG of the tractor frame & \(l_{5}\) & \(-0.31\) & m \\ Horizontal position of the rear cabin suspension with respect to the CoG of the tractor frame & & \(l_{6}\) & \(-0.83\) & m \\ Radius of the rear wheels & \(R_{r}\) & 0.83 & m \\ Radius of the front wheels & \(R_{f}\) & 0.62 & m \\ \hline \hline \end{tabular}
\end{table}
Table 1: Numerical parameters of the 6-DoFs tractor.  
The results show that the measured and simulated responses are similar and that the numerical model of the tractor can be considered representative of the actual machine for the purpose of this paper.  
In the following, models of simpler tractors are also considered: the 3-DoFs tractor in Figure 1a and the 4-DoFs tractor in Figure 1b. The 4-DoFs model was obtained by assuming that the cabin was rigidly connected to the tractor frame; instead, the 3-DoFs model was obtained by assuming that both the cabin and the front axle were rigidly connected to the tractor frame.