# Investigation of the Motion Characteristics of Parts on a Platform Subjected to Planar Oscillations
## 3 Modeling Results
The trajectory of the motion of the platform becomes complex when the platform is excited in two perpendicular directions by sinusoidal waves of different frequencies. Therefore, the part that is sliding on the platform also moves along a complex trajectory. Various complex trajectories of the part can be achieved by controlling the ratio between the frequencies of the waves \(k\) (\(\omega_{y}=k\omega_{x}\)). The trajectories that are the most suitable for positioning are presented in Figure 5.  
Figure 4: Positioning area, where \(\mu=0.2\), \(\omega=100\) rad/s, \(A_{x}=A_{y}=0.002\) m: (1) trajectories when the platform is excited according to Equation (1), as the phase shift changes in the range of \(\pi/10<\epsilon\leq 9\pi/10\); (2) trajectories when the platform is excited according to Equation (2), as the phase shift changes in the range of \(\pi/10<\epsilon\leq 9\pi/10\); (3) trajectories when the platform is excited according to Equation (3), as the phase shift changes \(\pi/10<\epsilon\leq 9\pi/10\); (4) trajectories when the platform is excited according to Equation (4), as the phase shift changes in the range of \(\pi/10<\epsilon\leq 9\pi/10\). Figure 5: _Cont._  
Such complex trajectories, in which the center of the part to be positioned makes loops, i.e., it moves very densely over a certain area, are perfectly suited to stages of automatic assembly such as the positioning and the search. The analysis showed that when the frequency difference is equal to one (\(\,|\,\omega_{x}-\omega_{y}\,|\,=1\)), the positioning trajectory of the part is arranged in a regular square (Figure 6). The size of the positioning square depends on the excitation amplitude of the platform. It was observed that choosing a higher frequency of the platform excitation yields a denser positioning trajectory. Motion Characteristics of the Part 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 \(\pi/2\)  
As mentioned earlier, in order to determine the influence of the system parameters on the average displacement velocity of the part, the positioning time was recorded up until the instant at which the part starts to repeatedly rotate around one point. It was found that when the excitation frequency increases, the average displacement velocity increases linearly (Figure 7a). A similar linear relation was observed between the excitation amplitude and the average displacement velocity (Figure 7b). However, an increase in the friction coefficient between the platform and the part results in a decrease in the average velocity of the part (Figure 7c). Thus, when the horizontal platform is excited in two perpendicular directions by sinusoidal waves of the same amplitude, the frequency and amplitude of the excitation have the greatest influence on the part's average displacement velocity. Figure 5: Trajectories of the sliding part when the platform is excited in two perpendicular directions by sinusoidal waves of different frequencies, where \(g=9.81\) m/s\({}^{2}\), \(\mu=0.2\), \(\omega_{x}=100\) rad/s, \(A_{x}=A_{y}=0.001\) m, \(\varepsilon=\pi/2\): (**a**) \(k=1.2\); (**b**) \(k=1.3\); (**c**) \(k=1.6\); (**d**) \(k=1.8\); (**e**) \(k=2.2\); (**f**) \(k=2.4\). Figure 8 shows the dependencies of the displacement angle on the system parameters. It was found that the displacement angle of the sliding part also depends on all these parameters. The excitation frequency has the least influence on the displacement angle (Figure 8a). As it increases, the angle increases slightly (the angle varies within a range of about \(\pi/36\)). The excitation amplitude of the platform has a greater influence on the displacement angle of the part. When the excitation amplitudes \(A_{x}\) and \(A_{y}\) are increased to 0.003 m, the displacement angle increases more rapidly (Figure 8b). The friction coefficient has the greatest influence on the displacement angle: when it increases, the sliding angle decreases (Figure 8c).