Suppression of limit cycle oscillations in the van der Pol oscillator by means of passive non-linear energy sinks

• Published in: Structural control and health monitoring Vol. 13; no. 1; pp. 41 - 75
• Main Authors: Lee, Young S., Vakakis, Alexander F., Bergman, Lawrence A., Michael McFarland, D.
• Format: Journal Article
• Language: English
• Published: Chichester, UK John Wiley & Sons, Ltd 01.01.2006
• Subjects: LCO suppression; limit cycle oscillation (LCO); non-linear energy sink (NES); numerical continuation; resonance capture; van der Pol oscillator
• ISSN: 1545-2255; 1545-2263
• DOI: 10.1002/stc.143

We study suppression of the limit cycle oscillations (LCOs) of a van der Pol (VDP) oscillator, utilizing two configurations of non‐linear energy sink (NES)—grounded and ungrounded. Through computational parameter studies we verify LCO elimination for certain ranges of NES parameters. Then, we establish the slow‐flow equations to perform numerical and analytical studies of the transient and steady‐state dynamics of the configuration studied. The transient dynamics turns out to involve resonance captures, through which irreversible, passive energy transfer (called non‐linear energy pumping) is achieved from the VDP oscillator to the NES. We also perform computational bifurcation analysis by means of numerical continuation to gain some understanding of the dynamics of generation and elimination of non‐linear energy pumping. The steady‐state dynamics shows that the system may possess either subcritical or supercritical LCOs, leading to different degrees of robustness of suppression. The results reported in this work support the applicability of the NES concept to suppression of instabilities in self‐excited systems. Copyright © 2005 John Wiley & Sons, Ltd.