Applied Mathematics and Mechanics

, Volume 40, Issue 1, pp 97–110 | Cite as

Suppression of grazing-induced instability in single degree-of-freedom impact oscillators

  • Shan Yin
  • Guilin WenEmail author
  • Xin Wu


As a typical non-smooth bifurcation, grazing bifurcation can induce instability of elementary near-grazing impact periodic motion in impact oscillators. In this paper, the stability for near-grazing period-one impact motion to suppress grazing-induced instabilities is analyzed, based on which, a control strategy is proposed. The commonly-used leading order zero time discontinuity mapping is extended to a higher order one to aid the perturbation analysis of the characteristic equation. It is shown that the degenerate grazing bifurcation can eliminate the singular term in the characteristic equation, leading to bounded eigenvalues. Based on such a precondition, the bounded eigenvalues are further restricted inside the unit circle, and a continuous transition between non-impact and controlled impact motion is observed. One discrete feedback controller that changes the velocity of the oscillator based on the selected Poincaré sections is adopted to demonstrate the control procedure.

Key words

grazing-induced instability higher order discontinuity mapping eigenvalue perturbation degenerate grazing bifurcation 

Chinese Library Classification

O322 TH113.1 

2010 Mathematics Subject Classification

34H20 37L10 


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Copyright information

© Shanghai University and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mechanical and Electric EngineeringGuangzhou UniversityGuangzhouChina
  2. 2.State Key Laboratory of Advanced Design and Manufacture for Vehicle BodyHunan UniversityChangshaChina
  3. 3.School of Mechanical EngineeringHunan Institute of EngineeringXiangtanChina

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