# Investigation of the Motion Characteristics of Parts on a Platform Subjected to Planar Oscillations
## 3 Modeling Results
The effect of friction is greater on the angle at low amplitudes. Figure 7: Average displacement velocity depending on: (**a**) excitation frequency \(\omega\), where \(\mu\) = 0.15, \(\varepsilon=\pi/2\); (**b**) excitation amplitude, where \(\mu\) = 0.15, \(\varepsilon=\pi/2\); (**c**) friction coefficient, where \(\omega\) = 100 rad/s, \(\varepsilon=\pi/2\). Motion Characteristics of the Part When the Platform Is Excited in Two Perpendicular Directions by Sinusoidal Waves of Different Amplitudes, and the Phase Shift between the Perpen-dicular Components Is \(\pi\)/2  
In this case, the following two options are possible:  
1. The sinusoidal wave in the \(x\)-direction has a higher amplitude than the sinusoidal wave in the \(y\)-direction;
2. The sinusoidal wave in the \(y\)-direction has a higher amplitude than the sinusoidal wave in the \(x\)-direction. In order to show the differences between these two options, the dependences of the average displacement velocity and the displacement angle on the system parameters are plotted. As in the case of circular excitation, an increase in the excitation frequency leads to an increase in the average displacement velocity (Figure 9a). When \(A_{x}>A_{y}\), the part tends to move much faster than when \(A_{x}<A_{y}\). This happens because the amplitude of \(A_{x}\) is oriented in the direction of the motion of the part. For the same reason, when the excitation amplitude of the platform in the \(x\)-direction increases, the average velocity of the part increases rapidly (Figure 9b). Meanwhile, the amplitude in the \(y\)-direction has no effect on the average sliding velocity (Figure 9c). The friction coefficient between the platform and the part has a low influence on the average displacement velocity. When the friction coefficient increases, the average displacement velocity decreases slightly (Figure 9c). The displacement angle increases slightly as the excitation frequency increases (Figure 10a). When the excitation amplitude in the \(x\)-direction \(A_{x}\) increases, the average velocity increases (Figure 10b). The response of the displacement angle to the excitation amplitude in the \(y\)-axis \(A_{y}\) when \(A_{x}>A_{y}\) is different from the response when \(A_{x}<A_{y}\)  
Figure 9: Average displacement velocity depending on: (**a**) excitation frequency \(\omega\), where \(\mu=0.15\), \(\varepsilon=\pi/2\); (**b**) excitation amplitude \(A_{x}\), where \(\mu=0.15\), \(\omega=150\) rad/s, \(\varepsilon=\pi/2\); (**c**) excitation amplitude \(A_{y}\), where \(\mu=0.15\), \(\omega=150\) rad/s, \(\varepsilon=\pi/2\); (**d**) friction coefficient \(\mu\), when \(\omega=150\) rad/s, \(\varepsilon=\pi/2\). (Figure 10c). When \(A_{x}<A_{y}\), \(A_{y}\) does not significantly affect the displacement angle; meanwhile, when \(A_{x}>A_{y}\), the displacement angle decreases slightly as \(A_{y}\) increases (Figure 10c). The friction coefficient has a greater influence on the displacement angle when \(A_{x}<A_{y}\). As the friction coefficient increases, the displacement angle decreases linearly (Figure 10d). 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 Varies in the Range of \(\pi\)/10 \(\leq\varepsilon<9\pi\)/10  
It is important to determine how the phase shift affects the average velocity and angle of the part's displacement on the platform when the platform is excited in two perpendicular directions by harmonic waves of the same amplitude with a phase shift. The average displacement velocity and the angle-related results are shown in Figures 11 and 12, respectively. It was found that the average displacement velocity decreases rapidly when the phase shift increases from \(\pi\)/10 to \(\pi\)/2, and the minimum displacement velocity is observed at \(\pi\)/2. A further increase in the phase shift results in an increase in the average displacement velocity (Figure 11).