# Inertial Forces and Friction in Propulsion of a Rigid Body
## 5 Conclusions  
The primary goal of this paper has been achieved. We have shown that the motion of a capsule system on a rough horizontal plane caused only by the rotation of two internal masses is possible.  
The mathematical model of the robot has been presented, and the working region of the system has been specified in terms of the model parameters. There are three characteristic values of angular velocities which define areas of qualitatively different robot movements (see Figure 4). To illustrate the idea of the inertial drive and the interplay between particular parameters, extensive numerical studies have been conducted for two models of friction.  
The numerical solutions to the dynamic problem have been analyzed with regard to the total displacement per period and the average velocity per period. The related maps on particular parameter planes allow for controlling the robot performance described by these two indicators. The quantitative analysis of the influence of three factors on the motion has been presented graphically (Figures 7-13). The analysis indicates that generally the displacement and velocity increase with the increase in dimensionless mass of the rotating points and with the increase in their angular velocity. The situation is different in the case of the coefficient of friction. Its increase causes a decrease in the displacement and average  
Figure 12: Numerical solutions for the Stribeck friction model: (**a**) displacement of the box for various values of the angular speed; (**b**) displacement of the box for modified values of \(\Omega\) or \(\beta\) compared to the basic case (solid grey).  
Figure 13: Comparison of the effect of parameters \(\Omega\) and \(\delta\) on the motion of the system for the Coulomb friction model (\(\delta\) = 0) and for the Stribeck friction model (\(\delta\) > 0): (**a**) total displacement per period; (**b**) average velocity per period.  
speed of the robot. It is worth noting that too low a value of \(\mu\) will result in an unfavourable reverse movement (see Figures 9c-f and 11c-f). Moreover, the more realistic friction model that includes the Stribeck effect turns out to be advantageous for efficient motion of the system when compared to the basic Coulomb model.  
It should be emphasized that the present study has not focused on detailed practical issues, such as the shape of the capsule, the physical realization of the inertial drive, or the power supply. For example, in a real system, the internal masses could be revolved around a common axis, which should reduce the tendency of the inertial forces to rotate and lift the box. This particular problem and many others will be considered in further steps, leading to the experimental validation of the theoretical results.  
Conceptualization, R.S.; methodology, R.S. and P.F.; software, R.S. and P.F.; validation, R.S.; formal analysis, R.S. and P.F.; investigation, P.F.; resources, R.S.; data curation, P.F.; writing--original draft preparation, P.F.; writing--review and editing, R.S.; visualization, P.F.; supervision, R.S.; project administration, R.S.; funding acquisition, R.S. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the Polish Ministry of Science and Higher Education, grant number 0612/SBAD/3628.  
Not applicable.  
Most datasets analyzed during the study are available in this article. Raw data supporting the conclusions of this article will be made available by the authors on request.  
The authors declare no conflicts of interest.