## 3 Results
### The Effect of Tractor Dampers on the Vibrations Transmitted to the Operator's Seat  
In this section, the results of several analyses are compared in order to highlight the effect of each tractor damper on the vibrations that are transmitted to the operator's seat. In the 6 DoFs tractor model, suspension dampers were those indicated in Figure 1c as \(c_{1}\), \(c_{3}\), \(c_{5}\), and \(c_{6}\), along with \(c_{2}\) and \(c_{4}\), which represent the tire damping. Figure 15a,b shows the effect of each damping value listed in Table 1 on the transmissibility between the wheels  
Figure 14: The effect of the forward speed on the root mean square value of the vertical acceleration \(a_{wz}\) of the seat.  
Figure 13: Transmissibility between the wheels of the 6-DoFs tractor and the operator’s seat \(z_{1}\) with rear-mounted (RMM) operating machine and front-mounted (FMM) operating machine: (**a**) rear wheels \(z_{1}/z_{R}\); (**b**) front wheels \(z_{1}/z_{F}\).  
and the operator's seat. Note that the undamped system gives rise to very high levels of transmissibility (dotted black line); therefore, the presence of the dampers is essential to preserve the health of the operator. A single damper affects each modal damping in a different way and thus the transmissibility in relation to the relative mode frequency. For example, in Figure 15b the effect of the damper of the seat suspension (\(c_{1}\)) is clearly visible: it reduces the transmissibility of the undamped system at the resonance frequencies of modes one (at 1.19 Hz), three (at 2.77 Hz), and four (at 4.09 Hz); but the transmissibility remains the same of the undamped system at the resonance frequencies of modes two (at 1.28 HZ), five (at 4.42 Hz), and six (at 4.82 Hz). Moreover, the transmissibility plots show that the combined effects of the six dampers (red line) attenuate the vibrations of the operator's seat due to the vertical displacement of the front wheels (red line in Figure 15b). The vertical displacement of rear wheels is amplified at the operator's seat for frequencies below 2 Hz and is attenuated for higher frequencies (red line in Figure 15a).  
To better understand the effect of a given damper on the vibration isolation of the operator's seat, Table 2 compares the modal damping ratios \(\zeta_{i}\) of the 6-DoFs tractor modes obtained accounting for the effects of each damper \(c_{i}\). For each column, the values in bold indicate the modes with the most relevant variations of the modal damping ratio \(\zeta_{i}\) compared to \(\zeta_{0}\), accounting only for the stiffness proportional damping coefficient \(\beta\). The modal damping ratio \(\zeta\) accounts for the combined effect of the six dampers of the 6 DoFs tractor. The values in Table 2 are in agreement with the effects of each damper \(c_{i}\) on the transmissibility in Figure 15.  
Figure 15: The effects of tractor dampers on the transmissibility between the tractor wheels and the operator’s seat \(z_{1}\): (**a**) rear wheels \(z_{1}/z_{R}\); (**b**) front wheels \(z_{1}/z_{F}\).  
The weighted r.m.s. value \(a_{wz}\) of the vertical acceleration of the operator's seat is shown in the last row of Table 2. It was evaluated for the 6-DoFs tractor with an operative speed of \(v=7.0\) km/h on the rougher track defined in [42].  
The r.m.s results highlight that dampers \(c_{1}\), \(c_{2}\), and \(c_{6}\) are the most effective on the attenuation of the vibration transmitted to the operator. In fact, the damping ratios \(\zeta_{i}\) of modes one, three, and four, are significantly affected by the presence of dampers \(c_{1}\), \(c_{2}\), and \(c_{6}\), respectively.