Periodic Motions, Bifurcation, and Hysteresis of the Vibro-Impact System^#

^#Communicated by S. Sinha.

Abstract

Stability periodic motions, saddle-node, grazing and periodic doubling bifurcation conditions for the single-degree-of-freedom impact oscillator are determined analytically and numerically. The regions for such condition are developed in parameter space. The Poincaré map of the system is established. The period-1 impact motion of the system and its stability are studied by analytical methods, and the physical origin of hysteresis is found as function of the drive amplitude. To understand the rich dynamical behavior of the system, some numerical methods are applied. The phase portraits of various period-1 orbits are exhibited in phase plane. The grazing bifurcation, period doubling bifurcation and periodic motions are illustrated at the Poincaré surface defined at constant drive phase as function of the drive amplitude. Using analytical results, the dependence of hysteretic region is determined in parameter space. In particular, for fixed value of coefficient of restitution (R) and viscous damping (ζ), the hysteretic region, in the amplitude-frequency (A − ω) space, increases with increasing A and with increasing ω above resonant frequency. The effect of increasing damping (increasing ζ or decreasing R) decreases the area of hysteretic region of in A − ω space.

Keywords:

• Bifurcation
• Hysteresis
• Periodic motions
• Poincaré map
• Stability
• Vibro-impact system

ACKNOWLEDGMENT

The authors gratefully acknowledge the support by the National Nature Science Foundation of China (No. 10372076).

Notes

^#Communicated by S. Sinha.