# Inertial Forces and Friction in Propulsion of a Rigid Body
## 4 Numerical Studies of the System
when dealing with different combinations of model parameters, the value of \(k\) may be increased (e.g., \(k=5\) or \(k=10\)) to ensure that the transient response of the system is omitted. For the analyzed solutions, the results are given in Table 1. The displacement and velocity increase nearly linearly with the angular speed. Figure 8 contains the phase portraits for these four cases. More precisely, the steady motion of the system is illustrated on the plane (\(X^{\star}\), \(V\)) with \(X^{\star}=\text{mod}(X,X_{p})\). Thus, the horizontal ranges of the curves are equal just to \(X_{p}\). The horizontal segments result from the specific definition of \(X^{\star}\). The loops near \(V=0\), in turn, are related to backward motion of the robot. The time histories presented in Figure 7 indicates that a stick-slip motion appears for \(\Omega<\Omega_{2}\). This looks similar to the self-induced oscillations, although the physics of the phenomenon is different. In order to gain a deeper insight into the effect of various parameters on the total displacement and average velocity per period, series of dynamic simulations have been conducted for numerous datasets. Table 2 contains the ranges and step sizes of particular parameters varied. All numerical solutions have been analyzed with \(k=5\) to estimate \(X_{p}\) and \(V_{p}\). The resulting filled contour plots on particular parameter planes are presented in Figure 9.  
As results from the maps, the three considered parameters can have a significant effect on the displacement and velocity. It may seem, for example, that for a given value of \(\mu\) we can choose a new value of \(\Omega\) or \(\beta\) to obtain two or three times larger \(X_{p}\) or \(V_{p}\) when compared to the results in Table 1. However, the working regions of the mechanical system are restricted by the conditions for motion. These constraints are indicated by the dashed lines and curves overlaid on the maps. On the plane (\(\mu\), \(\Omega\)), one can see the three limit functions derived analytically: \(\Omega_{1}(\mu\), \(\beta)\), \(\Omega_{2}(\mu\), \(\beta)\), and \(\Omega_{\text{max}}(\beta)\). Hence, the applicability of the contour plots in Figure 8c,d is limited to the region filled with yellow in \[\hat{T}_{1}(V)=\frac{2}{\pi}\mu\hat{N}\;\text{arctan}(\eta V), \tag{14}\]0 When \begin{table}
\begin{tabular}{c c c c} \hline \hline \(\Omega\) & \(\tau_{p}\) & \(X_{p}\) & \(V_{p}\) \\ \hline
1.30 & 4.83 & 0.242 & 0.050 \\
1.59 & 3.96 & 0.629 & 0.159 \\
1.70 & 3.70 & 0.756 & 0.205 \\
1.94 & 3.24 & 0.979 & 0.303 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The total displacement per period and the average velocity per period evaluated for the numerical solutions shown in Figure 7.  
Figure 7: Numerical solutions for various values of the angular speed: (a) displacement of the box; (b) velocity of the box. Results obtained for the Coulomb friction model. it comes to the planes (\(\Omega\), \(\beta\)) and (\(\mu\), \(\beta\)), the corresponding limits of \(\beta\) are visible: \(\beta_{1}(\mu,\Omega)\), \(\beta_{2}(\mu,\Omega)\), and \(\beta_{\max}(\Omega)\). Consequently, in Figure 9a,b,e,f, we should focus on the subregions bounded by the curves/lines labelled as \(\beta_{1}\) and \(\beta_{\max}\). Figure 10 contains the time histories \(X(\tau)\) and \(V(\tau)\) obtained for modified values of selected parameters. As can be seen, the basic solution (solid grey) can be compared to the cases when \(\Omega\) (dashed blue and dash-dotted blue) or \(\beta\) (solid blue and dotted blue) is varied. Although the active phase includes the return effect (rather weak), the near maximal values of the parameters (\(\Omega_{\max}\approx 1.94\), \(\beta_{\max}\approx 0.48\)) result in greater ranges of motion per period, as well as higher average velocities of the system: \(X_{p}\approx 0.943\), \(V_{p}\approx 0.285\) and \(X_{p}\approx 1.255\), \(V_{p}\approx 0.339\), respectively.