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

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.