# Investigation of the Motion Characteristics of Parts on a Platform Subjected to Planar Oscillations
## 4 Conclusions  
Dynamic and mathematical models of the motion of a part on a platform subjected to planar oscillations were developed to investigate the motion characteristics. The research showed that when the platform was excited in two perpendicular directions by sinusoidal waves, different trajectories of the part's motion can be obtained by controlling the excitation parameters, such as the frequencies and amplitudes of the waves and  
Figure 11: Average displacement velocity depending on phase shift: (**a**) at different excitation frequencies, where \(\mu=0.15\), \(A_{x}=0.0025\) m; (**b**) at different excitation amplitudes, where \(\mu=0.15\), \(\omega=\text{rad}/\text{s}\); (**c**) at different values of the friction coefficient \(\mu\), where \(\omega=200\) rad\(/\text{s}\), \(A_{x}=0.002\) m.  
Figure 12: Displacement angle \(\alpha\) depending on the phase shift, where \(\mu=0.15\), \(A_{x}=0.0025\) m.  
the phase shift between the waves. When the platform is excited in two perpendicular directions by sinusoidal waves of the same amplitude, and the phase shift between the perpendicular components is \(\pi/2\), the platform follows a circular trajectory, while at other values of phase shift, the platform follows an elliptical trajectory. The case in which the platform is excited in two perpendicular directions by sinusoidal waves with the same amplitude, and the phase shift between the perpendicular components varies in the range of \(\pi/10\leq\epsilon<9\pi/10\), is very well suited to applications related to non-prehensile manipulation tasks, as the phase shift employed as an additional control parameter makes it easier and faster to change the direction of motion of the part. The trajectory of the motion of the platform and the part on the platform becomes complex when the platform is excited in two perpendicular directions by sinusoidal waves of different frequencies. Various complex trajectories of the part can be achieved by controlling the ratio between the frequencies of the waves. Such complex trajectories, when the center of the part to be positioned performs loops, i.e., it moves very densely over a certain area, are perfectly suited for stages of automatic assembly such as the positioning and the search. It was found that choosing a higher frequency of the platform excitation yields a denser positioning trajectory. In all of the analyzed cases, the frequency and amplitude of excitation have the greatest influence on the average displacement velocity of the part, with higher values of these parameters leading to higher velocities of the part. When the platform is excited in two perpendicular directions by sinusoidal waves of the same amplitude, the minimum displacement velocity is observed at a phase shift value of \(\pi/2\). In the case where the platform is excited in two perpendicular directions by sinusoidal waves of different amplitudes, the values of the average displacement velocity are the highest when the amplitude is higher in the direction of the motion of the part. The results demonstrated that parts on the platform can be moved in any direction at a given velocity and can be subjected to dense complex positioning trajectories by controlling excitation parameters such as the frequencies and amplitudes of the excitation waves and the phase shift between the waves. Therefore, such a platform can be applied in feeding, positioning, and manipulation tasks. In future work, we plan to develop an experimental setup and carry out experimental tests of the functionality of the proposed methods. Conceptualization, S.K. and K.L. ; methodology, S.K. and K.L. ; software, S.K. and K.L. ; validation, S.K. and K.L. ; formal analysis, S.K. and K.L. ; investigation, S.K., K.L., R.C., A.K. and R.M. ; resources, S.K., K.L. and R.C. ; data curation, S.K. and K.L. ; writing--original draft preparation, S.K. and K.L. ; writing--review and editing, S.K., K.L., R.C., A.K. and R.M. ; visualization, S.K. and K.L. ; supervision, S.K. ; funding acquisition, R.C.