Abstract
We present the mechanical model of an array of elastic filaments and simulate the response with different mechanical couplings. This class of systems is inspired by robust and elegant solutions for locomotion mechanics that have emerged in several small-scale biological entities in the form of beating protrusions, such as cellular cilia and eukaryotic flagella. The collective dynamics of cilia arrays reveals important features such as array alignments, two-phase asymmetric beating of individual filaments, and the emergence of metachronal coordination, which make them suitable for bio-inspired terrestrial and aquatic locomotion. The model presented here is the basis for further developments towards the design of terrestrial and aquatic locomotion systems for general purpose robotic devices.
Highlights
Several small-scale biological entities have evolved highly adaptable, low specialization locomotion mechanics systems in the form of beating protrusions, such as cellular cilia and eukaryotic flagella [1,2]
The biomechanics of axonemal beating protrusions is extensively studied in the context of solid-fluid interaction, with the investigation often focused on the emergence of beating patterns induced by the coupling with an external flow [2,3,5,6,7]
Metachronal coordination in arrays of coupled protrusions consists of synchronization with constant phase shift between contiguous elements, resulting in travelling waves through the array, associated with locomotion mechanics
Summary
Several small-scale biological entities have evolved highly adaptable, low specialization locomotion mechanics systems in the form of beating protrusions, such as cellular cilia and eukaryotic flagella [1,2]. Recent work in [13] proposed that emerging metachronal waves in arrays of stiff filaments are fundamentally related to mechanical coupling through the base in addition to hydrodynamic coupling, suggesting that the emergence of metachronal waves can be achieved by pure basal mechanical coupling This has important implications for terrestrial locomotion, since from a design point of view, arrays of flexible protrusions are a robust solution as they can adapt and morph to a variety of terrains, and from the dynamics point of view, metachronal coordination and travelling waves are suitable to Mathematics 2020, 8, 1282; doi:10.3390/math8081282 www.mdpi.com/journal/mathematics. Each filament is considered as a stiff polymer with an active internal force, consistent with axonemal structures This class of mechanical objects is suitably modelled by using constrained Euler elastica, resulting in coupled nonlinear PDEs describing the evolution of a kinematic descriptor for the geometric shape, and a Lagrange multiplier that enforces inextensibility. Further developments are focused on general purpose robotic devices, which are attractive solutions for a variety of applications that include environmental monitoring and protection, civilian and military defence, and generally hazardous missions [14,15,16,17,18]
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