# Inertial Forces and Friction in Propulsion of a Rigid Body
## 3 Analytical Studies of the System  
In order to present the results, it is necessary to assume some values of the system parameters. In the computational studies of the considered system, the following values of the model parameters are treated as the basic dataset:  
\[r=0.2\ [\text{m}],\ \ \ \ \mu=0.4,\ \ \ \ \beta=0.3,\ \ \ \ \Omega=1.7. \tag{8}\]  
Figure 2: The main idea of the inertial drive of the robot represented by different configurations of the internal masses: (a) \(0<\alpha<\pi/2\); (b) \(\pi/2<\alpha<\pi\); (c) \(\pi<\alpha<3\pi/2\); (d) \(3\pi/2<\alpha<2\pi\). These sample values ensure the robot's movement. In the further part of the paper, a detailed qualitative and quantitative analysis of the robot's behaviour for different values of particular parameters is presented. Let us start with the determination of conditions for motion of the robot. First of all, the propelling force \(F_{x}\) must exceed the friction force. More precisely, we require \(F_{x}>T_{\max}\), where \(T_{\max}=\mu N\text{sgn}(F_{x})\) is the maximum force of static friction. In the non-dimensional formulation, the corresponding quantities are given as follows:  
\[\hat{F}_{X}=\beta\Omega^{2}[\sin(\alpha)+\cos(\alpha)], \tag{9a}\] \[\hat{N}=1+2\beta-\beta\Omega^{2}[\sin(\alpha)+\cos(\alpha)],\] (9b) \[\hat{T}_{\max}=\mu\hat{N}\text{sgn}\big{(}\hat{F}_{X}\big{)}. \tag{9c}\]  
The first and the third force as the functions of angle \(\alpha\) are presented in Figure 3. The shaded areas in the plot correspond to two phases of the robot dynamics: the active phase (yellow regions), and the rest phase (grey regions). The active phase includes both the desired forward motion (FM) and the undesired backward motion (BM). The latter one could be eliminated, for example, by decreasing the speed \(\Omega\). However, it would flatten the curve \(\hat{F}_{X}(\alpha)\) and reduce the active phase too. Extrema of the propelling force occur at \(\alpha_{1}=\pi/4\) (maximum) and \(\alpha_{2}=5\pi/4\) (minimum). In order to ensure that the system moves forward without the return phase, we consider the conditions  
\[\hat{F}_{X}(\alpha_{1})>\hat{T}_{\max}(\alpha_{1}),\hskip 14.226378 pt\hat{F}_{X}( \alpha_{2})>\hat{T}_{\max}(\alpha_{2}) \tag{10}\]  
that lead to the conclusions that \(\Omega_{1}<\Omega<\Omega_{2}\) with  
\[\Omega_{1}=\sqrt{\frac{\mu(1+2\beta)}{\sqrt{2}(1+\mu)\beta}},\hskip 14.226378 pt \Omega_{2}=\sqrt{\frac{\mu(1+2\beta)}{\sqrt{2}(1-\mu)\beta}}. \tag{11}\]  
Figure 3: Comparison of the dimensionless propelling force and the maximum friction force: the regions of active phase (yellow) and the rest phase (grey) of the robot dynamics; FM—forward motion, BM—backward motion. Moreover, the maximal allowed angular speed can be determined based on the restriction that the box should not lift off the ground, which means that the normal reaction (9b) must be positive. This condition leads to \(\Omega<\Omega_{\text{max}}\) with  
\[\Omega_{\text{max}}=\sqrt{\frac{1+2\beta}{\sqrt{2}\beta}}. \tag{12}\]  
For dataset (8), the angular speed limits are \(\Omega_{1}\approx 1.04\), \(\Omega_{2}\approx 1.59\), and \(\Omega_{\text{max}}\approx 1.94\). The three characteristic speeds as functions of the friction coefficient are shown in Figure 2:0 The shaded area (\(\Omega_{1}<\Omega<\Omega_{\text{max}}\)) indicates the working region of the robot. The curves \(\Omega_{2}(\mu)\) and \(\Omega_{\text{max}}(\mu)\) intersect exactly at \(\mu=1/2\) for all values of \(\beta\). Although \(\Omega_{2}\) tends towards infinity as \(\mu\) tends to 1, the condition \(\Omega<\Omega_{\text{max}}\) is more restrictive for \(\mu>1/2\). Ultimately, backward motion is eliminated if \(\Omega_{1}<\Omega<\text{min}(\Omega_{2},\Omega_{\text{max}})\). Figure 5 contains graphs of the non-dimensional forces for various values of the angular speed, including the characteristic ones. Apart from the quantities \(\hat{F}_{X}\) and \(\hat{T}_{\text{max}}\), the normal reaction \(\hat{N}\) and the dimensionless total weight \(\hat{G}=1+2\beta\) are plotted.