# Inertial Forces and Friction in Propulsion of a Rigid Body
## 4 Numerical Studies of the System  
The dynamical system described by Equation (7) is discontinuous due to the friction model. Unfortunately, this feature is the source of numerical difficulties during the time integration of ordinary differential equations (ODEs). To overcome this problem, we apply a simple smoothing technique that was discussed, for example, in [20,21]. Consider the dimensionless counterpart of the friction force (4):  
\[\hat{T}(V)=\mu\hat{N}\;\text{sgn}(V), \tag{13}\]  
where \(V=\dot{X}\). This discontinuous function can be replaced with a smooth approximation of the following form [20,21]:  
\[\hat{T}_{1}(V)=\frac{2}{\pi}\mu\hat{N}\;\text{arctan}(\eta V), \tag{14}\]  
where \(\eta\) is the accuracy parameter (or steepness parameter). Such smoothed friction characteristics for different values of \(\eta\) are shown in Figure 6a. As can be seen, the largerthe value of the parameter, the better approximation of Equation (13) and the steeper slope at \(V=0\). Obviously, the classical Coulomb model is a considerable idealization, i.e., it assumes that the force of kinetic friction (in absolute value) is constant and equal to the maximum static friction force. More realistic approaches reflect the Stribeck effect, which consists in a decrease in the kinetic friction force (in absolute value) with increasing \(\mid\!V\!\mid\). In such a case, the idealized law Equation (13) should be replaced, for example, with the following smoothed model [20,21]:  
\[\hat{T}_{2}(V)=\frac{2}{\pi}\cdot\frac{\mu\hat{N}}{1+\delta|V|}\arctan(\eta V), \tag{15}\]  
where the new parameter \(\delta\) is a positive number that affects the rate of decrease in the kinetic friction force. Friction characteristics of this type for various values of \(\eta\) and \(\delta\) are depicted in Figure 6b,c. It should be emphasized that models Equations (14) and (15) work well for the slip phase of motion (\(\mid\!V\!\mid>0\)), while in the stick phase (\(\mid\!V\!\mid=0\)), the friction force becomes zero. This is the main drawback of the basic smoothing approach. However, for the sake of clarity, the preliminary dynamic studies illustrating the robot's concept have been conducted using this simplification. Let us start with numerical solutions computed using the Coulomb friction model smoothed with \(\eta=10^{4}\). The results have been obtained for the basic dataset with varying \(\Omega\) and for zero initial conditions: \(X(0)=V(0)=0\). The time histories of \(X\) and \(V\) are presented in Figure 7. Three solutions (blue lines: solid, dashed, and dotted) illustrate the cases shown in Figure 5b-d, whereas the basic solution (solid grey) is connected with Figure 3. Hence, the active phase including both forward motion (\(V>0\)) and backward motion (\(V<0\)) occurs only for high values of the angular speed (\(\Omega=\Omega_{\max}\) and \(\Omega=1.7\)). The rest phase, in turn, is indicated by the intervals of constant position and zero velocity. The steady-state motion of the system is achieved after several periods of the excitation, that is, for \(\tau>k\tau_{p}\), where \(\tau_{p}=2\pi/\Omega\) and \(k\) is a small natural number. In the cases shown in Figure 7, we can estimate that \(k=2\). Evidently, the differences between the ranges of functions \(X(\tau)\) and \(V(\tau)\) for various values of \(\Omega\) are significant. For comparison purposes, we can estimate the total displacement per period and the average velocity per period:  
\[X_{p}=X\big{(}(k+1)\tau_{p}\big{)}-X\big{(}k\tau_{p}\big{)},\qquad V_{p}= \frac{X_{p}}{\tau_{p}}. \tag{16}\]  
Figure 6: Friction models approximated using the smoothing technique: (a) the Coulomb model smoothed with various values of \(\eta\); (b) the Stribeck model for \(\delta=3\), smoothed with various values of \(\eta\); (c) the Stribeck model for various values of \(\delta\), smoothed with \(\eta=10^{4}\).