# Multi-Body Model of Agricultural Tractor for Vibration Transmission Investigation
## 3 Results and Discussion
### Model Validation

Table 4: Correlation of frequencies for the most significant peaks. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **LEFT CABIN BUSHING** & \multicolumn{2}{c}{**RIGHT CABIN BUSHING**} \\ \hline Time [s] & Correlation [\%] & Time shift [s] & Time [s] & Correlation [\%] & Time shift [s] \\
7-March & 93.74 & 0.07 & 7-March & 76.57 & 0.07 \\
7-May & 93.25 & 0.06 & 7-May & 85.82 & 0.03 \\
8-August & 89.56 & 0 & 8-August & 92.71 & 0.05 \\
9 & 70.32 & 0 & 9 & 63.54 & 0.01 \\
12-July & 77.93 & 0.02 & 12-July & 53.12 & 0.01 \\
12-September & 78.94 & 0.01 & 12-September & 90.09 & 0.01 \\
14-March & 79.15 & 0 & 14-March & 93.85 & 0.06 \\
14-May & 64.76 & 0 & 14-May & 67.34 & 0 \\ \hline Mean values & 80.96 & 0.02 & Mean values & 77.88 & 0.03 \\ \hline \hline \multicolumn{4}{c}{**LEFT SPRING-DAMPER**} & \multicolumn{2}{c}{**RIGHT SPRING-DAMPER**} \\ \hline Time [s] & Correlation [\%] & Time shift [s] & Time [s] & Correlation [\%] & Time shift [s] \\
7-March & 63.29 & 0.03 & 7-March & 68.45 & 0.01 \\
7-May & 82.13 & 0 & 7-May & 82.32 & 0 \\
8-August & 85.65 & 0 & 8-August & 82.22 & 0 \\
9 & 75.35 & 0 & 9 & 77.31 & 0.01 \\
12-July & 87.88 & 0.03 & 12-July & 80.97 & 0.06 \\
12-September & 49.76 & 0.08 & 12-September & 65.13 & 0.05 \\
14-March & 86.07 & 0.03 & 14-March & 94.60 & 0 \\
14-May & 92.52 & 0.03 & 14-May & 79.60 & 0.03 \\ \hline Mean values & 77.83 & 0.025 & Mean values & 78.83 & 0.02 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Correlation of maximum values of the most significant peaks. where N is the total number time of samples from the experiment. The results are summarised in Table 5.  
The correlation coefficient values are relatively high except for the spring-dampers' location when the front axle passes the big bump. The cruise speed of the tractor plays an enormous role here. The speed is not constant. While passing the bump, there was a specific decrease in speed, but the cruise speed was not measured in the experiment. To determine the effect of velocity in the model simulation, in Figure 13, there are vertical acceleration curves for the constant angular velocity of the tractor rear wheels and two different angular velocity curves for the tractor rear wheels. A big bump with the tractor's front axle occurs in the highlighted area. The conclusion is that the accurate cruise speed needs to be measured in the following experiment. The effect of damping on amplitude values can be seen in the time domain graphs (Figures 9-12). The curves from the experiment and simulation are very similar except for the location of the right bushing. At the location of the left front cabin bushing, the vertical acceleration curve from the simulation has the same number of amplitudes as the curve that represents vertical acceleration from the experiment. A frequency shift can be seen right after the tractor's front wheels pass the bump in both cases: small bump and big bump. On the other hand, when the tractor's rear wheels pass the bump, the oscillation frequency becomes very similar for simulation and experiment; the correlation between these frequencies is 93.63 for the small bump and 87.84 for the big bump. The reason for the frequency shift in the beginning would require further investigation. When passing the small bump, the most significant difference in acceleration value is 2.05 m\(\cdot\)s\({}^{-2}\) at time 9.02  
Figure 13: Effect of wheels’ angular velocity on the vertical acceleration of the body at the location of the left spring-damper. \begin{table}
\begin{tabular}{c|c c c c} \hline \hline \multicolumn{5}{c}{**CORRELATION COEFFICIENT FOR THE WHOLE PERIOD**} \\  & **Small Bump** & & **Big Bump** & \\  & **Front Axle** & **Rear Axle** & **Front Axle** & **Rear Axle** \\ \hline
**LEFT CABIN BUSHING** & 95 & 91 & 88 & 87 \\
**RIGHT CABIN BUSHING** & 78 & 85 & 82 & 88 \\
**LEFT SPRING-DAMPER** & 86 & 91 & 43 & 92 \\
**RIGHT SPRING-DAMPER** & 91 & 93 & 66 & 94 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Correlation coefficient values. s (see Figure 9--left).