Band-type resonance: non-discrete energetically-optimal resonant states

TL;DR

This paper introduces band-type resonance, a continuous range of energetically optimal resonant states, challenging the traditional notion that resonance must occur at discrete frequencies, with broad implications across various physical systems.

Contribution

It demonstrates the existence of non-discrete, band-type resonant states that maintain optimal energy efficiency, expanding the understanding of resonance phenomena in nonlinear dynamics.

Findings

Band-type resonance forms a continuous band of optimal states.

Small waveform modifications can generate these states.

Applicable to many linear and nonlinear oscillators across disciplines.

Abstract

Structural resonance involves the absorption of inertial loads by a tuned structural elasticity: a process playing a key role in a wide range of biological and technological systems, including many biological and bio-inspired locomotion systems. Conventional linear and nonlinear resonant states typically exist at specific discrete frequencies, and specific symmetric waveforms. This discreteness can be an obstacle to resonant control modulation: deviating from these states, by breaking waveform symmetry or modulating drive frequency, generally leads to losses in system efficiency. Here, we demonstrate a new strategy for achieving these modulations at no loss of energetic efficiency. Leveraging fundamental advances in nonlinear dynamics, we characterise a new form of structural resonance: band-type resonance, describing a continuous band of energetically-optimal resonant states existing around conventional discrete resonant states. These states are a counterexample to the common supposition that deviation from a linear (or nonlinear) resonant frequency necessarily involves a loss of efficiency. We demonstrate how band-type resonant states can be generated via a spectral shaping approach: with small modifications to the system kinematic and load waveforms, we construct sets of frequency-modulated and symmetry-broken resonant states that show equal energetic optimality to their conventional discrete analogues. The existence of these non-discrete resonant states in a huge range of oscillators - linear and nonlinear, in many different physical contexts - is a new dynamical-systems phenomenon. It has implications not only for biological and bio-inspired locomotion systems but for a constellation of forced oscillator systems across physics, engineering, and biology.