# Investigation of the Motion Characteristics of Parts on a Platform Subjected to Planar Oscillations
## 3 Modeling Results
Figure 2: Trajectories of transient (1) and steady motion (2) of the part on the platform subjected to planar oscillations, when \(\mu=0.2\), \(A_{x}=A_{y}=0.002\) m, \(\omega=70\) rad\(/\)s, \(\varepsilon=\pi/2\): (a) displacement of the part with respect to time in the \(x\) and \(y\) directions; (b) trajectory of the part to be positioned, where \(P_{0}\) is the start point, and \(P_{c}\) is the center of the steady trajectory. The first two cases might be best suited for applications related to feeding systems or similar tasks, where there is no need to rapidly change the direction of motion. The third case is very well suited for 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 fourth case is best suited for applications related to positioning tasks or for the implementation of the search phase of automatic assembly, as, in this case, the platform moves along a complex trajectory. Nevertheless, all four techniques can be applied for the positioning of parts and non-prehensile manipulation tasks, as parts can be moved in any direction. The trajectories of the part after 10 circular oscillation cycles are presented in Figure 3. As can be seen in Figure 3a, to position the part over a greater distance, it is necessary to increase the amplitude of the excitation of the platform in the direction of travel (in the \(x\)-direction, in this case). Different trajectories of the part's motion can be obtained by subjecting the platform to harmonic sinusoidal oscillations in two perpendicular directions with different values of phase shift. Linear, elliptical, or circular trajectories of the platform can be obtained by changing the phase shift, which results in a similar trajectory of the part placed on the platform. When the phase shift is equal to 0 or \(\pi\), the part moves from the starting point along a line inclined at an angle of 45\({}^{\circ}\) or \(-45^{\circ}\) with respect to the \(x\)-axis, respectively. Positioning along a straight line is only possible at these angles; therefore, these values of the phase shift were rejected in the investigation on the influence of the system parameters on the average velocity of the part. Figure 3b demonstrates the influence of the phase shift between the perpendicular components of the excitation on the nature and direction of the motion of the part. If the phase shift is in the ranges of \(\pi/10\leq\varepsilon<\pi/2\) and \(\pi/2<\varepsilon\leq 9\pi/10\), then the part moves along an inclined and stretched trajectory during the transient motion stage, and the trajectory of the steady motion stage takes the shape of an inclined ellipse. The trajectory of the steady motion stage is circular when the phase shift is \(\pi/2\). It was also observed that the part moves a greater distance from the starting point when \(\varepsilon\
eq\pi/2\). Figure 4 shows the positioning area achieved in the case when the platform is excited in two perpendicular directions by sinusoidal waves of the same amplitude (\(A_{x}=A_{y}\)) according to Equations (1)-(4), and the phase shift between the perpendicular components varies in the range of \(\pi/10\leq\varepsilon<9\pi/10\). Figure 3: Trajectories of the part: (**a**) when the platform is excited in two perpendicular directions with a phase shift of \(\varepsilon=\pi/2\), where \(\mu=0.2\), \(\omega=70\) rad/s; (**b**) when the platform is excited in two perpendicular directions under various values of the phase shift, where \(\mu=0.2\), \(\omega=70\) rad/s, \(A_{x}=A_{y}=0.002\) m.  
The analysis of the motion trajectories showed that the trajectories of the part cover a triangular area when the phase shift changes in the range of \(\pi/10<\epsilon\leq 9\pi/10\). This means that the part can be positioned at any location in this triangle. Furthermore, the part can be positioned at any location in a certain square by changing the direction of the initial excitation (Figure 4).