Self-excited oscillations in multi-degree-of-freedom systems subjected to discontinuous forcing

TL;DR

This paper analytically investigates self-excited oscillations and limit cycles in multi-degree-of-freedom systems with discontinuous forcing, revealing a universal SAF bifurcation mechanism affecting stability and modal responses.

Contribution

It introduces the analytical characterization of the SAF bifurcation in multi-degree-of-freedom systems, extending the understanding of limit cycle stability and modal interactions.

Findings

Stable limit cycles can exist in all natural modes.

The SAF bifurcation governs stability exchange between modes.

The framework applies to systems with higher degrees of freedom.

Abstract

This study investigates the existence and stability of limit cycles resulting from self-excited oscillations in linear multi-degree-of-freedom systems subjected to discontinuous, state-dependent forcing. Using the method of averaging and slow-flow phase-plane analysis, analytical expressions are derived for the amplitudes and stability boundaries of limit cycles in a two-degree-of-freedom system. The analysis demonstrates that stable limit cycles may exist in all natural modes, with the steady-state response governed by initial conditions in regimes of multistability. A central contribution of this work is the identification and analytical characterization of the stability-axis-flipping (SAF) bifurcation, which serves as the governing mechanism for the exchange of stability between modes. The framework is then systematically extended to systems with higher degrees of freedom, confirming that the SAF bifurcation remains a universal feature, even under varying feedback configurations. The steady-state dynamics, summarized through stability maps and validated by numerical simulations, delineate the existence and stability regions of modal limit cycles as functions of key system parameters. These results provide efficient criteria for guiding optimization studies to mitigate or generate limit cycles at targeted frequencies in flexible mechanical structures.