On the locomotion of the slider within a self-adaptive beam-slider system

TL;DR

The paper tackles the problem of passive locomotion of a slider on a self-adaptive beam by developing a multiscale framework that isolates slow, fast, and intermediate dynamics. It combines a reduced-order beam model with a single-term Harmonic Balance to derive the Super-Slow Invariant Manifold and backbone resonance conditions, then identifies three principal locomotion mechanisms: a pitching cycle that drives transport away from the beam center at low amplitudes, and slope- and rocking-induced sliding that dominate high-amplitude behavior. Validation against a numerically validated PCS model shows good agreement, revealing phase- and condition-dependent transport across three phases, including a dynamic equilibrium that halts net motion on the high-amplitude branch. The findings provide a rigorous foundation for designing and optimizing self-tuning vibration mitigation or energy harvesting systems based on slider–beam interactions, highlighting the roles of clearance, geometry, and contact conditions in governing locomotion regimes.

Abstract

A beam-slider system is considered whose passive self-adaption relies on an intricate locomotion process involving both frictional and unilateral contact. The system also exploits geometric nonlinearity to achieve broadband efficacy. The dynamics of the system take place on three distinct time scales: On the fast time scale of the harmonic base excitation are the vibrations and the locomotion cycle. On the slow time scale, the slider changes its position along the beam, and the overall vibration level varies. Finally, on an intermediate time scale, strong modulations of the vibration amplitude may take place. In the present work, first, an analytical approximation of the beam's response on the slow time scale is derived as function of the slider position, which is a crucial prerequisite for identifying the main drivers of the slider's locomotion. Then, the most important forms of locomotion are described and approximations of their individual contribution to the overall slider transport are estimated. Finally, the theoretical results are compared against numerical results obtained from an experimentally validated model.

Figures (14)

• Figure 1: Schematic of self-adaptive system: (a) two-dimensional model of clamped–clamped beam with attached slider, (b) slider detail.
• Figure 2: Illustration of the signature move and confrontation of model and experiment: (a) and (b) beam vibration level vs. time; (c) slider location vs. time.
• Figure 3: Super-Slow Invariant Manifold (sSIM): (a) overview of analytical approximation for parameters where self-adaptive behavior is expected; (b) the case of much lower excitation level; (c) approximation obtained with modal deflection shape of beam only vs. of beam with attached slider; (d) effect of dynamic contact interactions with slider. PCS: Pseudo-Constrained Slider. The analytical approximation in (a), (b) and (d) is obtained with the deflection shape of the beam with attached slider.
• Figure 4: Effect of non-ideal clamping and slider attached at $s=0.27$ on modal deflection shape.
• Figure 5: Comparison of super-slow dynamics resulting from PCS and FS models, blue squares indicate the reference situations (Case 1-3), representative of Phase 1-3, respectively: (a) beam's vibration level versus horizontal slider position, (b) horizontal slider transport per period. PCS: Pseudo-Constrained slider; FS: Free Slider