# Multi-Body Model of Agricultural Tractor for Vibration Transmission Investigation
## 3 Results and Discussion  
### Model Validation  
Several measurements of the agricultural tractor equipped with acceleration sensors were performed to validate the proposed multi-body model. The tractor node through an artificial test track of two bumps with different heights. The goal in the initial validation phase was to match the acceleration values on the tractor body at locations of cabin mountings (see Figure 8). The same test track was prepared for multi-body simulation. A speed of 6 km/h was chosen for this validation. The Figures 9-12 compare vertical acceleration values on the tractor body during a test run. The time domain records are divided into two parts: the first part shows the ride through the small bump, and the second part shows the ride through the big bump. Arrows in the graphs mark when the front and rear wheels encounter the bumps. The most significant peaks, when the front and rear axles pass the bump (see marked area in Figure 9), were compared regarding values for validation purposes. The results are in Table 3. The time shift between the maximum value in the simulation and experiment is included. The correlation for every location is circa 80 % when the values are averaged. Figure 8: Model validation test track (**left**) and the dimensions of the artificial bumps (right). Figure 7: Non-linearity in tractor type characteristics for different type inflation pressures. Figure 10: Vertical acceleration on the tractor body at the right front cabin bushing location. Figure 9: Vertical acceleration on the tractor body at the left front cabin bushing location. Figure 11: Vertical acceleration on the tractor body at the location of the left spring-damper (at the rear). Figure 12: Vertical acceleration on the tractor body at the location of the right spring-damper (at the rear). In the next step, the frequencies of these most significant peaks were calculated using the whole period to determine correlation; results can be seen in Table 4. The frequencies at the locations of spring-dampers have higher correlation values. Generally, the experiment's frequencies change more significantly when passing the bumps. The final step in validation was to calculate a correlation coefficient for the whole period between the curve of acceleration that was measured and the curve that was calculated while passing bumps. The correlation coefficient is defined as:  
\[R=1-\frac{\sum\bigl{(}ACC_{exp}-ACC_{sim}\bigr{)}^{2}}{\sum\bigl{(}ACC_{exp}- mean\bigl{(}ACC_{exp}\bigr{)}\bigr{)}^{2}}, \tag{4}\]  
where \(ACC_{exp}\) is the value of acceleration from the experiment, \(ACC_{sim}\) is the acceleration value from the simulation, and the mean function is defined as:  
\[mean\bigl{(}ACC_{exp}\bigr{)}=\frac{1}{N}\sum_{i=1}^{N}ACC_{exp}, \tag{5}\] \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & \multicolumn{2}{c}{**Small Bump**} & \multicolumn{4}{c}{**Big Bump**} & \multicolumn{2}{c}{**Rear Ade**} & \multicolumn{2}{c}{**Rear Ade**} & \multicolumn{2}{c}{**Rear Ade**} \\  & \multicolumn{2}{c}{**Front Ade**} & \multicolumn{2}{c}{**Sim**} & \multicolumn{2}{c}{**Exp**} & \multicolumn{2}{c}{**Sim**} & \multicolumn{2}{c}{**Exp**} & \multicolumn{2}{c}{**Sim**} & \multicolumn{2}{c}{**Exp**} & \multicolumn{2}{c}{**Sim**} \\ \hline
**LEFT** & February-34 & February-66 & February-51 & February-67 & 1.97 & 2.56 & 2.22 & 2.49 & frequency [Hz] \\
**BUSING** & & 86.32 & & 93.63 & & 70.05 & & 87.84 & correlation [\%] \\
**RIGHT** & February-13 & February-66 & February-44 & February-68 & 1.82 & 2.55 & 2.27 & 2.47 & frequency [Hz] \\
**CABINING** & & 75.12 & & 90.16 & & 59.89 & & 91.19 & [Hz] \\
**LEFT** & February-86 & February-89 & February-54 & February-92 & 2.33 & 2.39 & 2.35 & 2.83 & frequency [Hz] \\
**DAMPER** & & 98.05 & & 85.04 & & **97.42** & & 79.57 & correlation [\%] \\
**RICHT** & February-86 & February-87 & February-55 & February-91 & 2.18 & 2.38 & 2.32 & 2.79 & frequency [Hz] \\
**DRAMPER** & & 99.65 & & 85.88 & & 90.83 & & 79.74 & correlation [\%] \\ \hline \hline \end{tabular}
\end{table}