# Locomotion of Self-Excited Vibrating and Rotating Objects in Granular Environments
## 3 Results and Analysis
### Coupling Oscillatory Translation and Oscillatory Rotation Speed  
The above understanding of the oscillations in the \(z\) displacement during one cycle suggests changing the rotation direction in phase with the translational one,  
\[\omega=\Omega cos(2\pi ft)\, \tag{6}\]  
with \(\Omega\) the amplitude of the rotational angular velocity and \(f\) its oscillating frequency, which was the same as that of the translational vibration in the \(y\)-direction. The idea was that the direction of the displacement,  
\[Sign\{\Delta\vec{z}(t)\}\equiv\vec{y}(t)\times\vec{\omega}(t)\, \tag{7}\]  
would then be either always negative or always positive, resulting in a steady rise or sink during the cycle. The results for the \(z\) displacement of the SO for \(A=0.03\) m, \(\Omega=\pm 80\) rad/s, and \(f=3\) Hz, shown in Figure 4, confirmed the above expectation--the SO sank steadily with positive \(\Omega\) and rose steadily with negative \(\Omega\).  
We tracked the trajectories and show in Figure 5 examples of slices of the system in the \(y-z\) plane through the SO at times 0 s, 2.5 s, and 5 s for both the rising and sinking. In Tables 2 and 3 we summarize the observed dynamics during a cycle for \(\Omega=-80\) rad/s.  
Figure 3: The forces on the SO in the first quarter cycle.  
The motion in the \(z\) direction, both up and down, is due to the imbalance in the friction forces on the SO induced by the rotation. When \(\Omega=-80\) rad/s, the SO is subject to a steady upward friction force, given in Tables 2 and 3, and consequently rises steadily, as shown in Figure 4 (black line). When \(\Omega=80\) rad/s, the friction force on the SO is downward, leading to steady sinking, also shown in Figure 4 (red line). As can be observed in Figure 4, the steady rising and sinking were accompanied by fluctuations that were much smaller than those shown in Figure 2. These fluctuations, which appeared also at frequency \(f\), were caused by a combined effect of particles falling into the cavity, which the moving SO left behind, and of the stress release in the compressed medium ahead of the SO as it changes direction. Thus, the in-phase coupling of the rotation to the vibration  
Figure 4: The displacement-time graph describing the locomotion of the SO in the z direction at two opposite rotation angular velocities.  
Figure 5: Snapshots during the rising and sinking processes. Only the relevant part of the system is shown. Top: the initial, intermediate and final positions of the SO, for \(\Omega=-80\) rad/s, \(A=0.03\) m, and \(f=3\) Hz. The relevant simulation ended when the SO broke the top surface of the granular bed. Bottom: the initial, intermediate and final position of the SO, for \(\Omega=80\) rad/s, \(A=0.03\) m and \(f=3\) Hz. The SO settles at about \(z=-0.3\) m due to the interaction with the bottom boundary.  
eliminates the large oscillations, observed the constant rotation mode, and the steady rise and sink are faster in this mode. It follows that this mode of locomotion is a more efficient technique for desert robots and most likely for burrowing desert animals as well.  
To determine the conditions for the fastest locomotion, we turn next to investigate the effects of \(A\), \(\Omega\), and \(f\) on the rising or sinking rates in this mode.