Latch, Spring and Release: The Efficiency of Power-Amplified Jumping

TL;DR

This analysis suggests that it is possible to control jumps even once the latch has been set, and shows that the rate at which this adhesion is lost is crucial in determining the efficiency of jumping and, indeed, whether jumping occurs at all.

Abstract

Many small animals, particularly insects, use power-amplification to generate rapid motions such as jumping, which would otherwise not be possible with the standard power density of muscle. A common framework for understanding this power amplification is Latch-Mediated, Spring Actuated (or LaMSA) jumping, in which a spring is slowly compressed, latched in its compressed state and the latch released to allow jumping. Motivated by the jumps of certain insect larvae, we consider a variant of this in which the latching occurs externally via adhesion to a substrate that is quickly released for jumping. We show that the rate at which this adhesion is lost is crucial in determining the efficiency of jumping and, indeed, whether jumping occurs at all. As well as showing how release rate should be chosen to facilitate optimal jumping, our analysis suggests that it is possible to control jumps even once the latch has been set.

Figures (4)

• Figure 1: Release rate control of externally-latched jumping. (a) Jumping behavior in larval beetles (Laemophloeus biguttatus). Adapted from BertoneGibson22 (equally spaced frames show instants between $t=0,100~\text{ms}$ after the initation of jumping.) (b-c) A curved elastic strip is initially indented at its center by a pressing finger (b) and a hanging weight (c). A low release rate does not result in jumping ((b), equally timed frames between up to $867~\text{ms}$ after first motion), while quick release does ((c) with equally timed frames up to $64~\text{ms}$ after first motion). Unlatching times in the time composite images (a-c) are $t_u=(5.5, 683, 1)~\text{ms}$. Inset in panel (b): Schematic representation of the shell in the $xz$-plane at three different configurations; in absence of external forces (other than its own weight) in dashed blue, when it is loaded at the center (in solid blue), and when the loading is partially released (orange).
• Figure 2: (a) Phase diagram of the jumping behavior of a curved shell on a flat surface at different values of the damping $H$. The jumping behavior of the shell is characterized by the triad $(\beta,\tau,H)$: there is a jumping phase (in green) and a non-jumping phase (in orange). (b) Phase boundaries at different values of parameter $H$ and small $(\beta,\tau)$. The colour scheme of the boundaries in (a) is the same as in (b).
• Figure 3: Latch force required for an indented, naturally curved, shell with dimensions from Laemophloeus biguttatusBertoneGibson22. Latch force as a function of the shell's curvature given by the linearized theory (in red) and the non-linear model (in blue). Line styles denote different Young's modulus: $E\sim 10~\mathrm{kPa}$ (estimated value from the time scale of curling of an elastic object Callan-JonesBrun12, solid lines); $E\sim 9.25~\mathrm{kPa}$ (estimated value from the attachment force of 2nd instar of G. viridulaZurekGorb15, dashed lines); $E\sim 14.9~\mathrm{kPa}$ (estimated value from the attachment force of 3rd instar of G. viridulaZurekGorb15, dot--dashed lines). (Here 'instar' refers to different developmental stages of insects, between moulting of the exoskeleton.)
• Figure 4: Energy efficiency of jumping of a curved shell given by the ratio of translational kinetic energy at $T=T_{\text{off}}$ to stored energy at $T=0$ as a function of the latch lifting time scale. Theoretical predictions from linearized theory are depicted in solid lines. Scaled values of the efficiency for L. biguttatusBertoneGibson22, jumping hoops Yang2012, and our experiments of a curved strip are also included as symbols. Solid lines for $\beta=1$ and dashed lines for $\beta=10$. Colors range from dark brown (small $H$) to yellow (large $H$) for $H=0.1,1,2,10$,