# Locomotion of Self-Excited Vibrating and Rotating Objects in Granular Environments
## 1 Introduction  
Granular Materials (GMs) are common in nature. A GM is defined as an assembly of many discrete macroscopic solid particles that interact via a hard-core potential and dissipate energy on collisions [1]. GMs can behave like a solid, supporting loads, or flow like a fluid [2]. However, full continuum theories for the solid and fluid [3] behaviors are still to emerge. Therefore, understanding GM in general is still a challenging problem.  
Some sand lizards, such as _Scincus scincus_, are well known to'swim' through sand beds by effecting a solid-to-fluid and back transition of the media around them [4, 5]. Such lizards can both dive into desert sand, to hide from heat and predators, and travel beneath the dry sand surface with considerable speed over reasonable distances [6, 7, 8]. These capabilities are enabled: by physiological attributes: the spatula-shaped snout, the streamlined body shape, the smooth integument, and fringed digits [4, 9, 10], which are well adapted to life in the desert, and by the rapid lateral oscillations that loosen the sand surrounding the swimmer's body, leading to vertical burrowing. Another mechanism of moving in GM is by helical rotation, such as employed by _Pelargonium_ and _Erodium_ seeds for digging into cohesionless soils [11, 12]. This mechanism is reminiscent of a common propulsion method of many bacterial species, such as _E. coli_, that propel themselves byrotating bundles of helical filaments [13; 14]. Thus, discovering mechanism of locomotion for organisms in GMs is of great engineering significance, not least to the development of walking robots in sandy terrains.  
Recently, [PERSON] et al. [7] have studied the swimming of sandfish _Scincus scincus_ in sand using high-speed \(X\)-ray imaging. [PERSON] et al. [15] have constructed a simplified model of periodic contraction and extension of large intruders in granular beds, using an event-driven simulation. A number of researchers applied the resistive force theory (RFT) [16; 17] in fluid mechanics to model the sandfish swimming dynamics [18; 19] and concluded that RFT predicts reasonably well the motion on granular surfaces [20]. Legged sand walking robots and sand swimming robots have been developed, based on experiments and simulations [21; 22]. However, the aforementioned studies focused mainly on the in-plane motion the walking or swimming, less is known about the mechanics of large-scale vertical motion within the GM.  
The main aim of this work is to investigate the vertical behavior of a self-energized SO within a dense granular medium. In this paper, we report three-dimensional DEM simulations of the locomotion of vibrating and/or rotating SOs, studying their trajectories under various conditions. To the best of our knowledge, this is the first work to study such a coupled mode of motion, which could prove very efficient for subsurface locomotion. Specifically, we studied the effect of three parameters: \(A\), \(\Omega\), and \(f\), on the rise/sink of SO in this mode of motion. The results of this study have the potential to advance the design of a bio-inspired robot to rapidly rise to the surface or dive into sand to a significant depth. The rest of the paper is organized as follows. In Section 2 we describe the numerical simulation method and the system. In Section 3 we present all the simulation results and includes our analysis and discussion. In Section 4, we present our conclusions and suggest future work.