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

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.

Figures (11)

• Figure 1: Schematic representation of a one-dimensional spring-mass model subjected to an implicit terminal forcing
• Figure 2: Functional plots of $F(\delta)$ (Eq. \ref{['eq:5']}, top row) and slow-flow phase portraits (second row) for varying values of $q$: (a, d) $q = 0.39$; (b, e) $q = 0.75$; and (c, f) $q = 3.4$. Note that $F(\delta)$ is discontinuous at $\delta=1$
• Figure 3: (a) Locus of the mixed-mode critical point $(\bar{A}_{1}, \bar{A}_{2})$ in the first quadrant of the slow-flow phase plane. (b) Bifurcation sequence for $q \in [0.5, 2]$, illustrating the SAF bifurcation
• Figure 4: Stability characteristics of the three single-mode limit cycles in the $q_1$--$q_2$ parameter plane. S and U denote stable and unstable limit cycles, respectively
• Figure 5: Slow-flow stability characteristics of the mixed-mode solutions in the $q_1$--$q_2$ parameter plane. Symbols $\alpha_{\pm}$, $\beta_{\pm}$, and $\gamma_{\pm}$ denote the planar saddle points at their inception, where the subscripts $-$ and $+$ identify attracting and repelling saddles, respectively. The birth of these points determines the global stability of the axial limit cycles, while the mixed-mode solutions themselves are always unstable in the 3DOF system