Higher-Order Transverse Discontinuity Mapping in Filippov Systems: Analysis and Experimental Validation using an Electronic Circuit

TL;DR

This paper develops a higher-order transverse discontinuity mapping for Filippov systems, improving impact prediction accuracy near soft barriers, validated through numerical simulations and an innovative electronic circuit model.

Contribution

It introduces a higher-order TDM for impact oscillators, enhancing prediction accuracy and stability analysis, validated by experiments with a novel electronic circuit model.

Findings

Higher-order TDM accurately predicts impact onset.

Experimental circuit validates theoretical impact predictions.

Discrepancies of linearized models near grazing impacts are addressed.

Abstract

This paper shows that linearizing the transverse discontinuity mapping (TDM) in Filippov systems can produce inaccurate predictions of the dynamics in impact oscillators operating near a pre-stressed soft barrier. This discrepancy arises from the limitations of the linearized saltation matrix, which inaccurately predicts impacts in the local neighborhood of the discontinuity boundary. To address this issue, a higher-order approximation of the TDM is derived, which accurately captures the onset of impacts and closely matches the results obtained from both numerical simulations and electronic experiments. The proposed higher-order TDM results in a quadratic estimation of flight time for impacts. Geometrically, real-valued impact events are only feasible when the discriminant of this quadratic equation is positive. The differences in the predicted higher-order flight times and mapping estimates become more pronounced for low-velocity impacts close to grazing. Subsequently, a numerical approximation of the higher-order saltation matrix and its consequent Floquet multipliers and Lyapunov exponents for stability analysis is proposed and demonstrated on a pre-stressed, forced, damped, soft impact oscillator. To validate the numerically observed discontinuity-induced bifurcations, an analog electronic circuit is proposed that models a soft mechanical prestressed oscillator. This inductor-less circuit overcomes the limitations of typical LCR-based circuits, which are used to design such oscillators but cannot accommodate low stiffness ratios. The experimentally obtained limit cycles, finger-shaped Poincar\'e sections, and bifurcation diagrams match the predictions of the higher-order TDM, validating that the proposed circuit accurately models the soft-impact oscillator for both low and high stiffness ratios.