[Paper Review] Self-oscillation and Synchronisation Transitions in Elasto-Active Structures

Ellen Zheng, Martin Brandenbourger+4 more|arXiv (Cornell University)|Jun 10, 2021

Micro and Nano Robotics4 citations

TL;DR

This study introduces a macroscopic, centimeter-scale model system of one-dimensional elasto-active chains to experimentally investigate self-oscillations and synchronization. By tuning elasticity and coupling stiffness, the authors demonstrate transitions to flagellar motion, self-snapping, and 1:1 frequency synchronization via nonlinear feedback between active forces and elastic deflections, with synchronization governed by a critical coupling stiffness of κ = 1.1.

ABSTRACT

The interplay between activity and elasticity often found in active and living systems triggers a plethora of autonomous behaviors ranging from self-assembly and collective motion to actuation. Amongst these, spontaneous self-oscillations of mechanical structures is perhaps the simplest and most wide-spread type of non-equilibrium phenomenon. Yet, we lack experimental model systems to investigate the various dynamical phenomena that may appear. Here, we report self-oscillation and synchronization transitions in a centimeter-sized model system for one-dimensional elasto-active structures. By combining precision-desktop experiments of elastically coupled self-propelled particles with numerical simulations and analytical perturbative theory, we demonstrate that the dynamics of single chain follows a Hopf bifurcation. We show that this instability is controlled by a single non-dimensional elasto-active number that quantifies the interplay between activity and elasticity. Finally, we demonstrate that pairs of coupled elasto-active chains can undergo a synchronization transition: the oscillations phases of both chains lock when the coupling link is sufficiently stiff. Beyond the canonical case considered here, we anticipate our work to open avenues for the understanding and design of the self-organisation and response of active artificial and biological solids, e.g. in higher dimensions and for more intricate geometries.

Motivation & Objective

• To develop a controllable, macroscopic experimental platform for studying dynamical transitions in active solids, particularly self-oscillations and synchronization.
• To investigate how the interplay between activity, elasticity, and viscous damping drives transitions to flagellar motion and self-snapping in one-dimensional elasto-active structures.
• To experimentally verify the emergence of synchronization in elastically coupled chains and determine the critical coupling stiffness for synchronization.
• To validate the observed transitions with simple models of coupled pendula with follower forces, providing a quantitative framework for understanding the dynamics.

Proposed method

• Design and fabricate centimeter-sized active particles with tunable active forces, connected by flexible rubber chains to form one-dimensional elasto-active chains.
• Control the elasto-active number σ by varying particle width W, which governs the ratio of active force to elastic stiffness.
• Use pinned boundary conditions (one or two ends) to induce flagellar motion or self-snapping, respectively.
• Couple two chains via a stiff elastic link with tunable stiffness κ to study synchronization dynamics.
• Measure time series of particle angles and compute instantaneous phase Φ(t) to analyze frequency mismatch δν and phase alignment.
• Compare experimental results with numerical simulations and analytical models based on coupled pendula with follower forces, using the equation dΨ/dt = dν − (ε/cosΨ₀)sin(Ψ − Ψ₀).

Experimental results

Research questions

• RQ1 What are the dynamical transitions from overdamped to self-oscillatory motion in one-dimensional elasto-active chains, and what controls the onset of flagellar motion?
• RQ2 How does the coupling stiffness between two elasto-active chains influence the emergence of synchronization, and what is the critical stiffness for 1:1 frequency locking?
• RQ3 To what extent can the synchronization behavior of elasto-active chains be described by a classical nonisochronous synchronization model with a constant phase shift?
• RQ4 How does the elasto-active number σ influence the synchronization regime, particularly near the bifurcation point?
• RQ5 Can a simple model of coupled pendula with follower forces quantitatively describe the observed self-oscillations and synchronization in macroscopic elasto-active systems?

Key findings

• Flagellar motion emerges when the elasto-active number σ exceeds a threshold, with self-oscillations observed at σ = 0.695 and above, while overdamped motion occurs at σ = 0.166.
• Self-snapping behavior is observed when the chain is pinned at both ends, resulting in large-amplitude, asymmetric oscillations due to nonlinear elastic feedback.
• Synchronization between two elasto-active chains occurs when the coupling stiffness exceeds a critical value of κ = 1.1, confirmed by a dip in frequency mismatch δν and phase alignment in both experiments and simulations.
• The synchronization regime exhibits an Arnold tongue centered on the 1:1 frequency ratio, with lower coupling stiffness required for synchronization when chains have similar elasto-active numbers or are closer to the bifurcation point.
• The analytical model dΨ/dt = dν − (ε/cosΨ₀)sin(Ψ − Ψ₀) accurately predicts the synchronization threshold, with the condition |ε/cosΨ₀| > |dν| determining the onset of stable synchronization.
• Numerical simulations confirm that the synchronization transition is governed by a square root singularity in the phase dynamics, consistent with classical nonisochronous synchronization theory.