## 3 Physical and Mathematical Model of a Single-Degree-of-Freedom Seat Suspension
### Results of the Theoretical and Experimental Analysis  
A model of the forcing acting on the seat suspension base, \(y_{f0}\), \(\dot{y}_{f0}\), was developed based on the actual forcing characteristics (Figure 4a). The shape of the forcing was chosen to have a significant negative impact on operator comfort and not to be susceptible to passive vibration. Given the above assumptions, the forcing shown in Figure 4 was selected for analysis. The shape of the developed forcing model slightly deviates from the actual forcing obtained during experimental testing. The responses of the system to the specified forcing obtained during numerical simulation correspond with sufficient accuracy to the responses of the real system. Therefore, it can be assumed that the behaviour of the virtual model corresponds to the real system. The parameters of the mathematical forcing model are presented in Table 1. The full list of simulation parameters is provided in Appendix B Table A3. Figure 6 illustrates the outcomes of the experiment conducted with initial pressure values of \(p\) = 5.0 bar and \(p\) = 5.5 bar. To compare how the system processes displacement signals at different frequencies for the simulation experiment with the real one, frequency characteristics are presented (Figures 7 and 8). The data obtained allow the graphs to be analysed for changes in amplitude as a function of frequency. In contrast, the logarithmic characteristics show changes in the low band (1-15 Hz), where the signal amplitude is significantly attenuated. An estimation of the seat suspension parameters was conducted for initial pressures of 4.0 bar, 4.5 bar, 5.0 bar and 5.5 bar. Table 2 presents a summary of the standard deviation \(\delta_{y_{1}-y_{70}}\) of the variation of relative displacement as a function of time for selected pressures in the pneumatic bellows. From the value determined from the approximation of the standard deviation of the displacement of the inertial element as a function of pressure, it can be inferred that the error made is of the order of 9.4%. An additional method of assessing the quality of the performed estimation is to compare the course of the displacement variation as a \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(p\)[bar] & **3.75** & **4.30** & **5.0** & **6.25** \\ \hline \(\delta_{y_{1}-y_{70}}\)[m] & 0.0537 & 0.0455 & 0.0310 & 0.0355 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Overview of standard deviation \(\delta_{y_{1}-y_{70}}\). Figure 6: Displacement obtained from simulation \(y_{71}-y_{70}\) and experimental \(y-y_{0}\) for initial pressure values of (**a**) \(p\) = 5.0 bar and (**b**) \(p\) = 5.5 bar. Figure 7: Frequency responses of vertical acceleration; (**a**) pre\(-\)pressure \(p\) = 4.0 bar; (**b**) pre\(-\)pressure \(p\) = 4.5 bar. function of time obtained from the performed motion simulation of the formulated model (Figures 5 and 7) with the result obtained from the experimental tests of the real system. The course of the variation in stiffness values as a function of pressure for characteristic moments is illustrated in Figure 9.  
Figure 8: Frequency responses of vertical acceleration; (**a**) pre\(-\)pressure \(p\) = 5.0 bar; (**b**) pre\(-\)pressure \(p\) = 5.5 bar. Figure 9: Stiffness \(k_{\mathrm{\gamma}1}\) of the seat suspension as a function of pressure. The course of the variation in the damping value as a function of time for the initial pressure values of \(p=\{3.75,\,4.0,\,4.3,\,4.5,\,5.0,\,5.5,\,6.25\}\) bar is shown in Figure 9(a). The course of the variation in the static displacement of the inertial element as a function of pressure is shown in Figure 9(b). The subsequent stages of the investigation, which seek to ascertain the bellows pressure at which the operator's work is conducted, are presented. The value of the determined pressure is of paramount importance from the standpoint of human safety and comfort. The resulting excursions caused by the overrun of a single hump are exceedingly challenging to rectify.