Passive Vibration-Driven Locomotion

TL;DR

This work investigates passive vibration-driven locomotion in a capsule containing an internal pendulum that is excited by vertical environmental vibrations. By incorporating asymmetric damping and employing two complementary asymptotic analyses, it characterizes two distinct propulsion mechanisms: a $2:1$ parametric resonance leading to oscillatory pendulum motion and forward drift, and a $1:1$ rotatory resonance yielding sustained pendulum rotation with forward propulsion. The authors derive a slow-flow model near the $2:1$ resonance via a multi-scale expansion and an averaged-flow model near the $1:1$ resonance, uncovering bifurcation structures and regions that guarantee robust progressive motion as well as regimes with bistability. Numerical simulations validate the reduced models and demonstrate how initial conditions and forcing parameters determine the propulsion regime, with potential implications for compact, robust energy-rectifying locomotion and energy-harvesting devices.

Abstract

We investigate a concept of passive, vibration-driven locomotion, in which a mechanical system achieves horizontal self-propulsion by resonantly harvesting energy from vertical environmental excitations (e.g. ambient vibrations of underwater pipelines), without a direct propulsive actuation. The system consists of a capsule containing an internal pendulum attached to its base mounted on a vertically vibrating substrate. The underlying locomotion mechanism relies on resonant energy transfer from the vertically vibrating substrate to the internal oscillatory element. Under appropriate forcing conditions and in the presence of asymmetric dissipative interactions, this internal oscillator induces a net unidirectional motion of the capsule. The analysis focuses on regimes of progressive motion arising in the vicinity of parametric resonances. Two asymptotic limits are considered: small-amplitude parametric excitation leading to a (2:1) resonant oscillatory motion of the pendulum, and large-amplitude excitation leading to a (1:1) resonant unidirectional rotational motion of the pendulum. Given the asymmetry of the dissipative force acting on the capsule, both resonant regimes result in a progressive motion of the capsule system. To identify optimal locomotion regimes in both cases, we employ tailored asymptotic approaches based on multi-scale expansions and direct averaging analysis. The resulting slow-flow and averaged-flow models reveal the full bifurcation structure of steady-state solutions associated with forward capsule motion for both low- and high- amplitude excitations. Analytical predictions are shown to be in good agreement with direct numerical simulations of the full capsule-pendulum system.

Figures (14)

• Figure 1: Schematic of the passive capsule-pendulum - model.
• Figure 2: (Case 1) Time histories illustrating the pendulum and capsule response under (2:1) resonance conditions, where the pendulum exhibits an oscillatory motion. System parameters: $\omega=2$, $\varepsilon=0.01$, $A=0.01$, $\zeta=0.01$, $\mu_1=0.01$, and $\mu_2=0.02$. Initial conditions: $x(0)=0$, $\theta(0)=2$, $x'(0)=0$, and $\theta'(0)=0$. (a) The time history of the capsule's velocity $v=x'$ versus time $t$, (b) the time history of the pendulum's angle $\theta$ versus time, and (c) the time histories of capsule's position $x$ versus time.
• Figure 3: (Case 2) Time histories illustrating the pendulum and capsule response under (2:1) resonance conditions, where the pendulum exhibits an oscillatory motion. System parameters: $\omega=2$, $\varepsilon=0.01$, $A=0.08$, $\zeta=0.01$, $\mu_1=0.01$, and $\mu_2=0.02$. Initial conditions: $x(0)=0$, $\theta(0)=0.001$, $x'(0)=0$, and $\theta'(0)=0$. (a) The time history of the capsule's velocity $v=x'$ versus time $t$, (b) the time history of the pendulum's angle $\theta$ versus time, and (c) the time histories of capsule's position $x$ versus time. The insets in panels (a) and (b) show the magnified views of the corresponding graphs, and the orange curve in panel (a) depicts the running average of the velocity.
• Figure 4: (Case 3) Time histories illustrating the pendulum and capsule response under (2:1) resonance conditions, where the pendulum exhibits an oscillatory motion. System parameters: $\omega=1.94$, $\varepsilon=0.01$, $A=0.08$, $\zeta=0.01$, $\mu_1=0.01$, and $\mu_2=0.02$. Initial conditions: $x(0)=0$, $\theta(0)=2$, $x'(0)=0$, and $\theta'(0)=0$. (a) The time history of the capsule's velocity $v=x'$ versus time $t$, (b) the time history of the pendulum's angle $\theta$ versus time, and (c) the time histories of capsule's position $x$ versus time.
• Figure 5: (Case 4) Time histories illustrating the pendulum and capsule response under (2:1) resonance conditions, where the pendulum exhibits an oscillatory motion. System parameters: $\omega=1.94$, $\varepsilon=0.01$, $A=0.08$, $\zeta=0.01$, $\mu_1=0.01$, and $\mu_2=0.02$. Initial conditions: $x(0)=0$, $\theta(0)=0.5$, $x'(0)=0$, and $\theta'(0)=0$. (a) The time history of the capsule's velocity $v=x'$ versus time $t$, (b) the time history of the pendulum's angle $\theta$ versus time, and (c) the time histories of capsule's position $x$ versus time.