## 2 Methodology  
### Mathematical Model Using Lagrangian Mechanics  
Figure 1 depicts the dynamic model of the transportation of objects on an inclined plane harmonically oscillating in its longitudinal direction. The inclined plane is subjected to harmonic oscillations along its longitudinal direction:  
\[\eta(t)=A\text{sin}(\omega t), \tag{1}\]  
where \(\eta\) is the displacement of the platform, \(A\) is the amplitude, \(\omega\) is the angular frequency of the harmonic excitation along the longitudinal direction, and \(t\) is the time. The Lagrange equation represents the difference between kinetic and potential energy, as shown in Equation (2). \[L=T+F_{r}\dot{x}-U, \tag{2}\]  
where \(T\) corresponds to the translational kinetic energy of the object being transported, \(m\) is the object's mass, \(F_{r}\) is the dry friction force between the object and the inclined plane, \(x\) is the displacement of the object along the inclined plane, and \(U\) is the potential energy. \[T=\frac{1}{2}\;m\big{(}\dot{x}+\dot{\eta}\big{)}^{2}=\frac{1}{2}m\dot{x}^{2}+ mA\omega\text{cos}(\omega t)\dot{x}+\frac{1}{2}mA^{2}\omega^{2}\text{cos}^{2}( \omega t), \tag{3}\]  
\[U=mgx\text{sin}\,\alpha, \tag{4}\]  
\[F_{r}\dot{x}=\mu(\omega t)mg\text{cos}\,(\alpha)\big{(}\dot{x}+\dot{\eta} \big{)}\text{sign}\big{(}\dot{x}\big{)}, \tag{5}\]  
where \(g\) is the gravity, \(\alpha\) is the inclination angle of the oscillating platform, and \(\mu(\omega t)\) is the effective dry friction coefficient, where, in this study, it is considered as a periodic function of the same period as the longitudinal harmonic oscillations of the platform. The Euler-Lagrange equation was written as shown in Equation (6). \[\frac{d}{dt}\frac{\partial L}{\partial\dot{x}}-\frac{\partial L}{\partial x}+ \frac{\partial\big{(}F_{r}\dot{x}\big{)}}{\partial\dot{x}}=0. \tag{6}\]  
Equation (7) shows the partial derivative of the Lagrange equation that describes the momentum of the moving particle on an inclined plane. \[\frac{d}{dt}\frac{\partial L}{\partial\dot{x}}=m\ddot{x}-mA\omega^{2}\text{ sin}(\omega t). \tag{7}\]  
The rate of change of momentum can be affected and changed due to the potential energy and the friction forces exerted on the moving object on an inclined plane, as shown in Equations (8) and (9), respectively. Figure 1: Dynamic model of transportation of objects on an inclined plane oscillating in its longitudinal direction. \[\frac{\partial L}{\partial x}=-mg\text{sin}(\alpha), \tag{8}\] \[\frac{\partial\left(\bar{r}\,\dot{x}\right)}{\partial\dot{x}}=\mu( \omega t)mg\text{cos}(\alpha)\text{sign}(\dot{x}),\] (9) \[\text{where}\left\{ \begin{aligned} F_{r}&=mg \text{cos}\left(\alpha\right)\mu(\omega t)\text{sign}(\dot{x}),\dot{x}\
eq 0\\ -mg\text{cos}\left(\alpha\right)\mu(\omega t)<F_{r}&<mg \text{cos}\left(\alpha\right)\mu(\omega t),\ \dot{x}=0\end{aligned} \right.\]  
Non-dimensionalization was applied to identify and determine the characteristic control parameters:  
\[\tau=\omega t,\ \xi=x/A,\ \ \gamma=g/A\omega^{2}, \tag{10}\]  
The following non-dimensional equation can describe the relative body motion on the inclined plane:  
\[\xi^{\prime\prime}=\text{sin}(\tau)-\gamma\big{(}\text{sin}\left(\alpha\right) +\mu(\tau)\text{cos}\left(\alpha\right)\text{sign}\big{(}\xi^{\prime}\big{)} \big{)}, \tag{11}\]  
The average non-dimensional velocity of the moving object is presented in Equation (12):  
\[\big{\langle}\xi^{\prime}\big{\rangle}=\frac{1}{2\pi}\int_{0}^{2\pi}\xi^{ \prime}d\tau, \tag{12}\]  
To achieve the additional asymmetry needed for the upward motion of the object, the dry friction coefficient magnitude is periodically manipulated in each cycle of oscillations of the inclined plane. With respect to the period harmonic excitations, the dry friction is being manipulated by a rectangular function, as shown in Figure 2.  
The rectangular function that represents the dynamic dry friction manipulations with respect to the longitudinal harmonic oscillations can be described by the following equation:  
\[\mu(\tau)=\left\{