# Susceptibility Modeling of a Rubber Track for Lightweight Mobile Robots
## 5 Model Validation
### Track Modeling Using Variable Stiffness Constraints  
In the variable stiffness model, the torque occurring in each constraint was applied depending on the rotation angle value. The constraint torque-links rotation angle characteristic plot was assumed after many simulation attempts. They were able to shape the results of the simulation tests by changing the torque values in the different angle ranges. The determined function, divided into modified ranges, is presented in Figure 9.  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Simulation Variant** & \(\mathbf{k_{z1}}\) & \(\mathbf{k_{z2}}\) & \(\mathbf{c_{z}}\) & **Hs** & **Ws** & **Hd** & **Wd** & **T [s]** & **\(\mathbf{\Lambda}\)** \\  & \([\frac{\mathbf{N}\cdot\mathbf{mm}}{\mathbf{rad}}]\) & \([\frac{\mathbf{N}\cdot\mathbf{mm}}{\mathbf{rad}}]\) & \([\frac{\mathbf{N}\cdot\mathbf{mm}\cdot\mathbf{s}}{\mathbf{rad}}]\) & **[mm]** & **[mm]** & **[mm]** & **[mm]** & **[mm]** & **T [s]** & **\(\mathbf{\Lambda}\)** \\ \hline E38 & 125,000 & 8600 & 200 & 322 & 1045 & 249 & 1073 & 0.82 & 0.74 \\ \hline E56 & 190,000 & 12,600 & 260 & 324 & 1054 & 255 & 1071 & 0.81 & 0.78 \\ \hline E76 & 258,000 & 17,200 & 360 & 327 & 1055 & 257 & 1086 & 0.8 & 0.69 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Simulations results from bivalent stiffness constraint model (II) in variants E38, E56, E76.  
Figure 8: The track oscillations plots comparison: experimental measurement and simulation results from bivalent stiffness constraint model in variants E38, E56 and E76.  
The negative part of the function plot was divided into three ranges: Z1, Z2, and Z3 (Figure 9). The constraint torque in the Z1 range determined the maximum rotation angle that occurred in the simulation. Shaping the function plot in this range allowed to set the height Hd. In order to achieve the expected Hs value, it was necessary to modify the Z2 angle range (Figure 9). By reducing the curve inclination, the difference between Hd and Hs values increased. At the same time, this procedure reduced the oscillation time period. In order to balance the phenomenon for the Z3 small angles, the torque function was shaped more inclined. Increased stiffness in the Z3 range allowed for the proper time period to occur and also maintained the correct overall geometric shape of the deflected track. The torque values in the positive angle range (Figure 10) were determined to be more inclined than in the range of Z2 and Z3, but the difference between the positive and negative angle ranges of function inclination was not as high as in the bivalent stiffness simulation method (II).  
Using these steps, stiffness characteristics and damping values for three variants of the track model were assumed. They are presented in Figure 10. The oscillation plots for all three variants of simulations are presented in Figure 11. Simulation static and dynamic test parameters and assumed damping coefficients are presented in Table 6, and the obtained rotation angle ranges are described in Table 7.  
Figure 10: Comparison of constraint torque in the segment rotation angle function plots assumed for variants E38, E56 and E76.  
Figure 9: The constraint torque-links rotation angle characteristics determined for E38 (III) track simulation variant.  
Using the variable stiffness model, the simulation results were compatible with the experiments. The maximum difference between the values obtained in simulations and the mean value of the experiment results does not exceed 10% of the experimental value in any of the parameters.