# Propulsion via flexible flapping in granular media

## TL;DR

This paper investigates how flexibility affects propulsion in granular media using a simplified model, revealing optimal conditions for maximum thrust and comparing dynamics with viscous fluids.

## Contribution

It introduces a combined numerical and asymptotic analysis of a flexible flapper in granular media, highlighting the role of actuation amplitude and spring stiffness in propulsion.

## Key findings

- Maximum propulsive force occurs at finite spring stiffness and large actuation.
- Flexible flappers can achieve steady propulsion with optimal parameters.
- Dynamics differ from those in viscous fluids, showing unique propulsion characteristics.

## Abstract

Biological locomotion in nature is often achieved by the interaction between a flexible body and its surrounding medium. The interaction of a flexible body with granular media is less understood compared with viscous fluids partially due to its complex rheological properties. In this work we explore the effect of flexibility on granular propulsion by considering a simple mechanical model where a rigid rod is connected to a torsional spring that is under a displacement actuation using a granular resistive force theory. Through a combined numerical and asymptotic investigation, we characterize the propulsive dynamics of such a flexible flapper in relation to the actuation amplitude and spring stiffness and compare these dynamics with those observed in a viscous fluid. In addition, we demonstrate that the maximum possible propulsive force can be obtained in the steady propulsion limit with a finite spring stiffness and large actuation amplitude. These results may apply to the development of synthetic locomotive systems that exploit flexibility to move through complex terrestrial media.