# Investigation of the Motion Characteristics of Parts on a Platform Subjected to Planar Oscillations
## 3 Modeling Results  
To investigate the influence of the input parameters \(\mu\), \(\omega\) and \(A_{e}\) on the motion characteristics of the part, modeling was carried out using software developed for this purpose. The software was written using the MATLAB (Mathsoft, Cambridge, MA, USA) programming language. The motion equations were solved using the ode45 solver based on the Runge-Kutta (4, 5) formula and the Dormand-Prince pair. The value of \(g=9.81\) m/s\({}^{2}\) was set for all of the modeled cases. When a horizontal platform is subjected to planar oscillations, the part may initially move from the start point \(P_{0}\) at a certain distance over a spiral or wavy trajectory (Figure 2, (1)). The characteristics of motion and the nature of the trajectory depend on the parameters of the system, such as the amplitude and frequency of the excitation, the dry friction conditions, or the phase shift between the harmonic waves along the \(\xi\) and \(\eta\) axes. However, the part to be positioned eventually starts to oscillate around the equilibrium point \(P_{c}\) (Figure 2, (2)), and the nature of the steady trajectory depends mainly on the amplitude and the phase shift. It was observed that the part on the platform moves at a certain angle \(\alpha\). The displacement angle is measured from the starting point \(O\) to the center of the steady trajectory \(P_{c}\). As can be seen in Figure 2, the part still slides around a certain point after a certain number of oscillation cycles of the platform. This oscillation around the equilibrium position is no longer necessary for positioning, since the part is positioned; therefore, the excitation of the platform can be terminated at this time. To determine the average velocity of the part, it is important to determine how long it takes for the part to settle in the equilibrium position. To do that, the developed software was registering the centers of the arcs and circles of each oscillation cycle and was checking the differences between the coordinates of two adjacent points. When the difference between the center points of adjacent circles was \(\Delta p<2\cdot 10^{-4}\) m, the code was terminated, and the final positioning time was determined. In this paper, the motion characteristics of parts on the platform are analyzed in the following four cases:  
1. When the platform is excited in two perpendicular directions by sinusoidal waves of the same amplitude (\(A_{x}=A_{y}\)), and the phase shift between the perpendicular components is \(\pi/2\). In this case, the platform moves in a circular trajectory. 2. When the platform is excited in two perpendicular directions by sinusoidal waves of different amplitudes (\(A_{x}\
eq A_{y}\)), and the phase shift between the perpendicular components is \(\pi/2\). In this case, the platform moves in an elliptical trajectory. 3. When the platform is excited in two perpendicular directions by sinusoidal waves of the same amplitude (\(A_{x}=A_{y}\)), and the phase shift between the perpendicular components varies in the range of \(\pi/10\leq\varepsilon<9\pi/10\). These phase shift values are chosen since the part moves in a straight path at phase shift values of 0 and \(\pi\), and that is not suitable for positioning. When the phase shift varies in the range of \(\pi/10\leq\varepsilon<\pi/2\), the motion of the platform follows inclined elliptical trajectories; then, at \(\varepsilon=\pi/2\), the trajectory of the platform becomes circular, and when the phase shift increases even more (\(\pi/2<\varepsilon\leq 9\pi/10\)), the trajectory becomes elliptical again, but the angle of inclination of the ellipse changes. 4. When the platform is excited in two perpendicular directions by sinusoidal waves of different frequencies (\(\omega_{x}\
eq\omega_{y}\)).