# Investigation of the Motion Characteristics of Parts on a Platform Subjected to Planar Oscillations
## 2 Dynamic and Mathematical Models of the Motion of a Part on a Horizontally Oscillating Platform  
In order to carry out an investigation on the motion characteristics of a part on a platform subjected to planar oscillations along circular and elliptical trajectories for positioning applications, dynamic and mathematical models were developed. Figure 1 shows the dynamic model of the positioning of a part on a platform subjected to planar oscillations along circular and elliptical trajectories. The system consists of a stationary base (Figure 1, (1)), on which a horizontal movable platform (Figure 1, (2)) is attached. The origin of the base's coordinate system \(\Sigma\) is at the intersection of two horizontal stationary \(\xi\)- and \(\eta\)-axes. The coordinate system of the moving platform is named \(\Sigma_{pl}\). This coordinate system moves with the platform. The mobile platform is excited in two perpendicular horizontal directions by harmonic sinusoidal excitation. When the amplitude of the harmonic excitation is the same for both directions and the frequency is also the same, any point on the platform traces a circular trajectory (Figure 1, (4)). The trajectory of the platform movement is elliptical when the excitation amplitudes are not equal in different directions. As the platform is excited with the same amplitude in both directions, the equations of motion of the platform are written as follows:  
\[\left\{ \begin{array}{l}\xi=\xi_{0}+A_{x}\sin\omega t,\\ \eta=\eta_{0}+A_{y}\sin(\omega t+\varepsilon),\end{array} \right. \tag{1}\]  
where \(t\) is time, \(A_{x}=A_{y}\) are the excitation amplitudes (\(A_{x}\) is the excitation amplitude in the \(\xi\)-axis, and \(A_{y}\) is the excitation amplitude in the \(\eta\)-axis), \(\omega\) is the frequency of excitation, and \(\varepsilon\) is the phase shift between the harmonic waves along the \(\xi\)- and \(\eta\)-axes (it is equal \(\varepsilon=\pi/2\) for circular motion). The horizontal platform can be subjected to reversed planar oscillations. Then, the equations are written as follows:  
\[\left\{ \begin{array}{l}\xi=\xi_{0}-A_{x}\sin\omega t,\\ \eta=\eta_{0}-A_{y}\sin(\omega t+\varepsilon).\end{array} \right. \tag{2}\]  
The excitation can also be shifted with respect to time. If it is shifted by \(\pi/2\), the equations can be expressed as follows:  
\[\left\{ \begin{array}{l}\xi=\xi_{0}+A_{x}\cos(\omega t+\varepsilon),\\ \eta=\eta_{0}+A_{y}\cos\omega t.\end{array} \right. \tag{3}\]  
In this case, the equations for the reversed planar oscillations are written as follows:  
\[\left\{ \begin{array}{l}\xi=\xi_{0}-A_{x}\cos(\omega t+\varepsilon),\\ \eta=\eta_{0}-A_{y}\cos\omega t.\end{array} \right. \tag{4}\]  
By changing the phase shift between the perpendicular excitation directions, circular, elliptical, and linear shapes of the oscillations of the platform can be obtained, and the principle of changing orientation is identical to the principle of the Lissajous curve. When there is no phase shift, the platform oscillates in a linear trajectory. To formulate the equations of motion of the part to be positioned (Figure 1, (3)) that is placed on the platform, it is assumed that the part is flat, and the mass of the part is concentrated in the center of mass, and the coefficient of dry friction between the part and  
Figure 1: Dynamic model of the positioning on a platform subjected to planar oscillations along circular and elliptical trajectories: (1) stationary base; (2) mobile platform; (3) part to be positioned; (4) trajectory of the platform’s motion; (5) trajectory of the part’s transient sliding motion; (6) trajectory of the part’s steady-state motion; (7) traveled path \(\Delta d\). the platform is constant. When the part slides relative to the platform, the projections of the acceleration of the center of mass are written as follows:  
\[\left\{ \begin{array}{l}a_{\xi}=\tilde{x}+\bar{\xi}=\tilde{x}-A_{x}\omega^{2} \sin\omega t,\\ a_{\eta}=\tilde{y}+\tilde{\eta}=\tilde{y}-A_{y}\omega^{2}\sin(\omega t+\varepsilon ).\end{array} \right.