## 4 Conclusions  
In this paper, substructuring techniques were employed to investigate the vibrating of the operator's seat of an agricultural machine. The proposed approach allows getting the dynamic model of the whole agricultural machine by coupling together models (even of different types) of its components: the tractor, the three-point linkage, and the machine. Several models of tractors were considered and coupled, using the substructuring technique, with front or rear, mounted or semi-mounted devices. The results highlighted the effectiveness of the proposed approach when investigating a large number of different configurations and a wide range of operating conditions, given the reduced modeling burden. In fact, mathematical operations needed for the computation of frequency response functions and of transmissibilities have a complexity order of \(O(n^{3})\), where \(n\) is the number of degrees of freedom. Therefore, by splitting the complete system into \(m\) smaller subsystems, the computational burden is reduced approximately by \(m^{2}\). The subsequent step of analysis requires the coupling of the subsystem models and it can be performed using reduced-order models, including only coupling and boundary DoFs, instead of full models, thereby limiting the computational cost. Furthermore, some subsystems can be characterized experimentally when the numerical model is not very accurate, and modifications introduced to a single subsystem do not require reanalysis of the remaining subsystems. Finally, the use of reduced-order models that are usually expressed in the modal or in the frequency domain allows one to carry out the dynamic analysis of a complex agricultural machine without sharing sensitive design details of component subsystems.  
All authors have equally contributed to the conceptualization, methodology, and writing. All authors have read and agreed to the published version of the manuscript. This research was developed in the framework of the projects BRIC-ID26 (2016) and BRIC-ID14 (2019) funded by INAIL (National Institute for Insurance against Accidents at Work and Occupational Diseases). Not applicable. Not applicable. Not applicable. Not applicable.  
**Data Availability Statement:** Available data are contained within the article.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Mode** & \(f\) **[Hz]** & \(\zeta_{0}\) **[\%]** & \(\zeta_{1}\) **[\%]** & \(\zeta_{2}\) **[\%]** & \(\zeta_{3}\) **[\%]** & \(\zeta_{4}\) **[\%]** & \(\zeta_{5}\) **[\%]** & \(\zeta_{6}\) **[\%]** & \(\zeta\) **[\%]** \\ \hline
1 & 1.19 & 0.02 & **20.51** & 0.25 & 0.20 & 0.47 & 0.16 & 1.53 & 23.98 \\
2 & 1.28 & 0.02 & 0.52 & 0.03 & **52.46** & 1.86 & 0.02 & 0.67 & 50.82 \\
3 & 2.77 & 0.04 & 2.59 & **32.89** & 0.31 & 0.06 & 0.50 & 0.63 & 36.36 \\
4 & 4.09 & 0.06 & 6.06 & 3.32 & 0.07 & 0.07 & 0.15 & **82.63** & 89.78 \\
5 & 4.42 & 0.07 & 0.17 & 0.22 & **36.26** & 0.10 & **52.41** & 0.16 & 54.08 \\
6 & 4.82 & 0.08 & 0.08 & 0.08 & 0.07 & **51.43** & 0.10 & 0.08 & 91.26 \\ \hline \(a_{wz}\) & [m/s\({}^{2}\)] & 10.87 & 5.89 & 6.30 & 8.10 & 9.41 & 8.62 & 6.42 & 1.33 \\ \hline \hline \end{tabular}
\end{table}
Table 2: The effect of suspension damping on the modal damping ratio of the tractor. Highlighted in bold the modal damping ratios of modes that are more affected by each damper.  
**Acknowledgments:** The authors acknowledge the Italian National Institute for Insurance against Accidents at Work and Occupational Diseases (INAIL) for founding this research.  
**Conflicts of Interest:** The authors declare that they have no conflict of interest.