Study on Mechanical Vibration Control of Limit Cycle Oscillations in the Van der Pol Oscillator by means of Nonlinear Energy Sink.

Purpose: Van der Pol (VDP) oscillator is a typical self-excited system with observable limit cycle oscillations (LCOs). To suppress LCOs, a VDP oscillator coupled nonlinear energy sink (NES) is proposed in this paper. To achieve the optimal vibration reduction effect, the sensitive parameters of the VDP-NES system are investigated and its vibration reduction mechanism is revealed. Methods: First, the dynamic model of the coupled NES system is established by utilizing Newton's law. Then we reveal the vibration reduction mechanism based on stability and bifurcation theory, which may be a new angle of view to investigate the VDP oscillator with NES. Finally, the influence of different parameters on the vibration reduction effect is discussed based on numerical simulation. Results: The LCOs of VDP oscillator are caused by Hopf bifurcation, and its amplitude is closely related to the eigenvalue of the equilibrium point. When the NES is coupled, the unstable equilibrium point becomes stable, then the LCOs are completely suppressed. Although the stability of the equilibrium point of the system with NES keeps invariant, the real part of the eigenvalue decreases. As a result, the LCOs are partially suppressed, and the amplitude decreases by about 70%. In addition, the self-excited intensity and damping ratio are sensitive to the vibration reduction effect. Conclusion: Coupling NES can have a better vibration reduction effect on the LCOs, and vibration reduction is closely related to Hopf bifurcation and the equilibrium point of the system. The research shows that putting off the Hopf bifurcation to widen the parameter range of the stable equilibrium point can obtain the optimal vibration control effect. At the same time, decreasing the real part of the eigenvalue also can reduce the amplitude of LCOs. It may provide some ideas for vibration control strategies in practical engineering. [ABSTRACT FROM AUTHOR]