## 3 Results  
### Modeling Results  
The mathematical model that describes the motion of an object on an inclined plane has been solved numerically using MATLAB R2023a software. The Runge-Kutta ordinary differential equation solver ode45 was used to solve the non-dimensional differential equation. Figure 5 represents the non-dimensional average velocity depending on the phase shift \(\phi\) at various inclination angles. The results have shown that the directional motion of the moving particle can be controlled by changing the phase shift \(\phi\). The average velocity was significantly increased at \(\alpha=0.035\) rad between \(\phi=0\) and \(\phi=\pi/2\) and reached its maximum at \(\lambda=\pi\), and then the average velocity was gradually decreased between \(\phi=\pi/2\) and \(\phi=3\pi/2\), as shown in Figure 5a. However, as the inclination angle was increased to \(\alpha=0.1\) rad, the non-dimensional average velocity was significantly decreased between  
Figure 4: Oscillogram of the piezoelectric actuator and harmonic oscillations signals (\(\phi=\pi/3\), \(\lambda=\pi\) and \(\omega=62.83\) rad/s). Figure 3: Experimental setup for transportation of an object on an inclined plane by employing dynamic dry friction manipulations. (**a**) Principle diagram: (1) inclined plane; (2) piezoelectric actuator; (3) transportation surface; (4) electrodynamic vibrator; (5) vibration sensor; (6) power amplifier; (7) piezo linear amplifier; (8) waveform generator; (9) digital oscilloscope; (10) camera; (11) PC. (**b**) General view of the experimental setup. \(\phi=\pi/2\) and \(\phi=\pi\) due to the potential energy that is exerted on the moving particle, as demonstrated in Figure 5b, where \(\mu_{0}=0.2\) and \(\left\langle\mu_{m}\right\rangle/\mu_{0}=1/5\). Therefore, \(\phi\) is an essential parameter to control the direction of the moving object. Since \(\lambda\) is a parameter that corresponds to the level of the frictional asymmetry, at \(\alpha=0.035\) rad, which is almost equivalent to 2 degrees of inclination angle, the average velocity was gradually increased and reached a maximum between \(\lambda=\pi\) and \(\lambda=2\pi/3\) at \(\phi=\pi/4\). Then, the average velocity was sharply decreased, and the particle moved downward in the opposite direction at \(\lambda=2\pi\), as shown in Figure 6a. The average velocity was slightly increased at \(\alpha=0.1\) rad and reached a maximum between \(\lambda=\pi\) and \(\lambda=3\pi/2\), then it drastically decreased at \(\lambda=2\pi\) and then, the object started to move backward in a negative direction, as presented in Figure 6b. Therefore, \(\lambda\) is a crucial parameter to control the velocity of the moving particle on an inclined plane. The modeling has demonstrated that the ratio of the modified effective dry friction coefficient \(\langle\mu_{m}\rangle\) to the nominal dry friction coefficient \(\mu_{0}\) has yielded a notable influence on the non-dimensional average velocity of moving objects. The average velocity was significantly decreased as \(\langle\mu_{m}\rangle/\mu_{0}\) increased at \(\alpha=0.035\) rad. However, the average velocity of the moving object reached a maximum at \(\mu_{0}=0.15\) and a minimum at \(\mu_{0}=0.55\), as presented in Figure 7a. Nevertheless, the average velocity gradually decreased when the inclination angle increased to \(\alpha=0.1\) rad. The average velocity was dramatically reduced when this ratio increased from \(\langle\mu_{m}\rangle/\mu_{0}=0.1\) to \(\langle\mu_{m}\rangle/\mu_{0}=0.5\), as shown in Figure 7b. The results have revealed that the directional motion of the object is also affected by \(\gamma\) ratio, which represents the gravity divided by amplitude multiplied by squared angular frequency as shown in Equation (10). The average velocity was gradually decreased between \(\gamma=1\) and \(\gamma=6\). The highest values were observed at \(\lambda=2\pi/3\), \(\phi=\pi/2\), as demonstrated in Figure 8a.