# Inertial Forces and Friction in Propulsion of a Rigid Body
## 4 Numerical Studies of the System
Thus, a change in the relative mass can give a better improvement in \(X_{p}\) and \(V_{p}\) than a change in the angular speed. For the lower values of \(\Omega\) and \(\beta\) (close to \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Symbol** & **Range** & **Step** \\ \hline Friction coefficient & \(\mu\) & \(\langle 0.02,1\rangle\) & 0.01 \\ Relative mass of the particles & \(\beta\) & \(\langle 0.05,1.5\rangle\) & 0.05 \\ Relative angular speed & \(\Omega\) & \(\langle 0.5,2.5\rangle\) & 0.05 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Ranges and step sizes for the dimensionless model parameters used in the numerical studies. Figure 8: Phase portraits of the steady-state solutions summarized in Table 1: (**a**) \(\Omega=1.3\); (**b**) \(\Omega=1.59\); (**c**) \(\Omega=1.7\); (**d**) \(\Omega=1.94\). \(\Omega_{2}\approx 1.59\) and \(\beta_{2}\approx 0.24\)), in turn, there is no backward motion, but the displacements and velocities are lower too: \(X_{p}\approx 0.622\), \(V_{p}\approx 0.156\) and \(X_{p}\approx 0.539\), \(V_{p}\approx 0.146\), respectively. Figure 9: The effect of parameters \(\mu\), \(\beta\), and \(\Omega\) on the displacement \(X_{p}\) (left column) and velocity \(V_{p}\) (right column): **(a,b)** plane (\(\Omega\), \(\beta\)); **(c,d)** plane (\(\mu\), \(\Omega\)); **(e,f)** plane (\(\mu\), \(\beta\)). The dashed lines and curves correspond to the conditions for motion of the system. Results obtained for the Coulomb friction model. A similar parametric analysis can be performed for the Stribeck model of friction. The results reported below have been obtained for \(\delta=2\) and \(\eta=10^{4}\). This time, let us start with the maps shown in Figure 11. The dependencies of \(X_{p}\) and \(V_{p}\) on particular pairs of parameters are of a similar nature, as in Figure 9. However, a careful comparison reveals that now the two quantities reach values higher by about 25% in the subregions bounded by the triples of the limit functions, i.e., \(\Omega_{1}\), \(\Omega_{2}\), and \(\Omega_{\max}\) or \(\beta_{1}\), \(\beta_{2}\), and \(\beta_{\max}\). As could be supposed, the decrease in the resistance to motion due to the Stribeck effect allows for the system to move more efficiently. By analogy to Figure 7a, the time histories \(X(\tau)\) obtained for selected values of the angular speed are shown in Figure 12a. When it comes to the modifications based on the discussed contour plots, the solutions are presented in Figure 12b that may be compared to Figure 10a. As before, the results exhibit the active phase of the robot with no backward motion when \(\Omega\leq\Omega_{2}\) or \(\beta\leq\beta_{2}\). However, the cases with \(\Omega_{2}<\Omega\leq\Omega_{\max}\) or \(\beta_{2}<\beta\leq\beta_{\max}\) may be treated as more beneficial despite the minor return effect. In order to more precisely confront the results related to the different friction models, we focused on just the effect of the angular speed on the total displacement per period and the average velocity. Figure 13 includes a straightforward comparison of the data given in Table 1 (the Coulomb model, i.e., \(\delta=0\)) and the analogous results obtained for the Stribeck model with \(\delta\in\{1,2,3\}\). Let us treat \(X_{p}(\Omega)\) and \(V_{p}(\Omega)\) for the Coulomb model as the reference values. When comparing the displacements and velocities in the cases with \(\delta=2\) and \(\delta=0\), for example, the relative differences vary from about 50% for \(\Omega=1.3\) to about 20% for \(\Omega\approx 1.94\). Certainly, the occurrence of the Stribeck effect in the friction characteristics plays a positive role in the improvement of the robot performance. Values of \(X_{p}(\Omega)\) and \(V_{p}(\Omega)\) increase with parameter \(\delta\). In the end, it is important to notice that the dynamics of the mechanical system depends on the initial conditions. In the reported simulations, we employed the zero conditions only, i.e., \(X(0)=X_{0}\) and \(V(0)=V_{0}\) with \(X_{0}=V_{0}=0\).