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 modulating waveform asymmetry or 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- and asymmetry-modulated 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.

Appendices

Appendix

A.1 Closed-form expressions for PEA and SEA work loops

In Sect. 2.1, we defined general work-loop profiles: ${G}^{\pm }\left(x\right)$ and ${F}^{\pm }\left(x\right)$ for PEA systems; and ${X}^{\pm }\left(F\right)$ and ${U}^{\pm }\left(F\right)$ for SEA systems. These constructs can be defined analytically for a range of different systems. For a linear PEA system, with $D\left(\dot{x},\ddot{x}\right)=m\ddot{x}+c\dot{x}$, ${F}_{s}\left(x\right)=kx$, and undergoing simple-harmonic oscillation according to $x\left(t\right)=\widehat{x}\mathrm{cos}\left(\omega t\right)$, we have [25]:

$$ \begin{aligned} G^{ \pm } \left( x \right) & = - m\omega^{2} x \pm c\omega \sqrt {\hat{x}^{2} - x^{2} } , \\ F^{ \pm } \left( x \right) & = kx - m\omega^{2} x \pm c\omega \sqrt {\hat{x}^{2} - x^{2} } . \\ \end{aligned} $$

(A.1.1)

And for a quadratically damped PEA system, with $D\left(\dot{x},\ddot{x}\right)=m\ddot{x}+\mathrm{sgn}\left(\dot{x}\right)c{\dot{x}}^{2}$, undergoing the same simple-harmonic motion, we have [25]:

$$ \begin{aligned} G^{ \pm } \left( x \right) & = - m\omega^{2} x \pm c\omega^{2} \left( {x^{2} - \hat{x}^{2} } \right){,} \\ F^{ \pm } \left( x \right) & = kx - m\omega^{2} x \pm c\omega^{2} \left( {x^{2} - \hat{x}^{2} } \right). \\ \end{aligned} $$

(A.1.2)

These functions are defined for $x\in \left[-\widehat{x},\widehat{x}\right]$. For a linear SEA system, with $D\left(\dot{x},\ddot{x}\right)=m\ddot{x}+c\dot{x}$, ${F}_{s}\left(x\right)=kx$, and undergoing simple-harmonic oscillation according to $x\left(t\right)=\widehat{x}\mathrm{cos}\left(\omega t\right)$, the construction is more complex. For ${X}^{\pm }\left(F\right)$ and ${U}^{\pm }\left(F\right)$ we have:

$$ \begin{aligned} X^{ \pm } \left( F \right) & = \frac{1}{{m^{2} \omega^{2} + c^{2} }}\left( { - mF \pm \frac{c}{\omega }\sqrt {\hat{x}^{2} \omega^{2} \left( {m^{2} \omega^{2} + c^{2} } \right) - F^{2} } } \right), \\ U^{ \pm } \left( F \right) & = \frac{F}{k} + \frac{1}{{m^{2} \omega^{2} + c^{2} }}\left( { - mF \pm \frac{c}{\omega }\sqrt {\hat{x}^{2} \omega^{2} \left( {m^{2} \omega^{2} + c^{2} } \right) - F^{2} } } \right), \\ \end{aligned} $$

(A.1.3)

where these functions are defined over the force range $F\in \left[-\widehat{F},\widehat{F}\right]$, with $\widehat{F}=\omega \widehat{x}\sqrt{{m}^{2}{\omega }^{2}+{c}^{2}}.$

To compute the elastic-bound conditions, Eq. 6, one further profile is required: the derivatives ${{X}^{^{\prime}}}^{\pm }\left(F\right)$. These derivatives are given by:

$$ X^{\prime \pm } \left( F \right) = \frac{1}{{m^{2} \omega^{2} + c^{2} }}\left( { - m \mp \frac{cF}{{\omega \sqrt {\hat{x}^{2} \omega^{2} \left( {m^{2} \omega^{2} + c^{2} } \right) - F^{2} } }}} \right). $$

(A.1.4)

This completes the definition of work-loop profiles for these example systems.

A.2 Generalised triangle-wave function

To analyse frequency-band resonance in Sect. 4.1, we utilised a generalised triangle-like wave, allowing waveform alteration with associated frequency and offset modulation. This waveform was described via a freeplay-nonlinear time-invariant ODE. The time-domain solution to this ODE is:

$$ \begin{aligned} \ddot{x}\left( t \right) & = \left\{ {\begin{array}{*{20}l} { - A\left( \delta \right)\frac{{\hat{x}}}{{T^{2} }}\cos \left( {\frac{{\pi t}}{{2\delta T}}} \right)} \hfill & {\quad 0 \le t \le \delta T} \hfill \\ {A\left( \delta \right)\frac{{\hat{x}}}{{T^{2} }}\cos \left( {\frac{{\pi \left( {t - \frac{1}{2}T} \right)}}{{2\delta T}}} \right)} \hfill & {\quad \left( {\frac{1}{2} - \delta } \right)T \le t \le \left( {\frac{1}{2} + \delta } \right)T} \hfill \\ { - A\left( \delta \right)\frac{{\hat{x}}}{{T^{2} }}\cos \left( {\frac{{\pi \left( {t - T} \right)}}{{2\delta T}}} \right)} \hfill & {\quad T\left( {1 - \delta } \right) \le t \le T} \hfill \\ 0 \hfill & {\quad {\text{o.w.}}} \hfill \\ \end{array} } \right. \\ A\left( \delta \right) & = \frac{{2\pi ^{2} }}{{\left( {8 - 4\pi } \right)\delta ^{2} + \pi \delta }},~\;\delta = \delta \left( R \right) = \frac{1}{4}\frac{{\pi \left( {1 - R} \right)}}{{\left( {2 - \pi } \right)R + \pi }}, \\ \end{aligned} $$

(A.2.1)

with $\dot{x}\left(t\right)$ and $x\left(t\right)$ as:

$$ \dot{x}\left( t \right) = \int \limits_{0}^{t} \ddot{x}\left( \tau \right)\,{\text{d}}\tau , \;x\left( t \right) = \int \limits_{0}^{t} \dot{x}\left( \tau \right)\,{\text{d}}\tau + \hat{x}. $$

(A.2.2)

The parameter $\delta$ is a time-domain analogue of $R$, representing the time window over which the sinusoid velocity-reversal component acts, as per the piecewise component limits in Eq. A.2.1; and $T$ is the waveform period, $T=2\pi /\omega$. Analytical formulations of $x\left(t\right)$, via Eq. A.2.2, are available, but these formulations are not required for frequency-band acceleration-matching, and so we use numerical integration for convenience.