Chaotic Flexural Vibrations in Biomimetic Scale Substrates

TL;DR

This study demonstrates that biomimetic scale substrates can exhibit deterministic chaos in flexural vibrations, controllable through geometry and asymmetry, with validated models and simulations revealing complex nonlinear dynamics.

Contribution

The paper introduces a reduced-order nonlinear oscillator model for biomimetic scale substrates, linking geometry to chaotic behavior, validated by finite element simulations.

Findings

Biomimetic scale substrates develop chaos at modest amplitudes due to unilateral contact and jamming.

Asymmetry in scale distribution delays chaos onset and fragments the response into intermittent windows.

Geometry controls chaos programmability, independent of large deflections or material nonlinearity.

Abstract

Overlapping fish-scale architectures are among nature's most distinctive surface adaptations, combining protection, contact regulation, hydrodynamics, optical and directional mechanical response within a thin textured integument. Here, we show that their biomimetic structural analogues can host deterministic chaos. Biomimetic scale substrates develop chaotic flexural vibrations at modest amplitudes because bending activates unilateral contact and progressive jamming, while built-in asymmetry from unequal texturing biases the restoring response and shifts the onset of chaos. From continuum mechanics, we derive a singular reduced-order model (sROM) that reduces the scale-covered beam to a nonlinear oscillator whose parameters map directly to overlap, scale inclination, damping, forcing, and substrate stiffness. Finite element (FE) simulations validate the model in quasi-static bending and long-time forced response. Stroboscopic regime maps reveal a period-doubling cascade from period-1 to period-2 and period-4, ultimately chaos. Overlap and inclination determine the strength of post-engagement nonlinearity, whereas damping bounds the chaotic operating window. Unequal top-bottom scale distributions break the antisymmetry of the restoring response, generating offset force-displacement laws. This reduced symmetry does not accelerate instability; instead, it delays the onset of chaos and fragments the response into intermittent periodic windows, whereas restoring symmetry can paradoxically widen the chaotic regime. When the texture is sufficiently sparse or steep on one side, it remains dynamically inactive, and the beam behaves as a fully asymmetric one-sided system. The results identify biomimetic scale substrates as a distinct class of contact-rich architectured metasurfaces in which chaos is programmable through geometry rather than large deflection or constitutive nonlinearity.