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Journal of Computational and Nonlinear Dynamics | Volume 13 | Issue 1 | research-article
Research Papers

Feedback Control of Grazing-Induced Chaos in the Single-Degree-of-Freedom Impact Oscillator

[+] Author and Article Information
Yongkang Shen, Shan Yin

State Key Laboratory of Advanced Design
and Manufacture for Vehicle Body,
College of Mechanical and Vehicle Engineering,
Hunan University,
Changsha, Hunan 410082, China

Guilin Wen

State Key Laboratory of Advanced Design
and Manufacture for Vehicle Body,
College of Mechanical and Vehicle Engineering,
Hunan University,
Changsha, Hunan 410082, China
e-mail: glwen@hnu.edu.cn

Huidong Xu

College of Mechanics,
Taiyuan University of Technology,
Taiyuan, Shanxi 030024, China

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received December 11, 2016; final manuscript received August 23, 2017; published online October 31, 2017. Assoc. Editor: Katrin Ellermann.

J. Comput. Nonlinear Dynam 13(1), 011012 (Oct 31, 2017) (9 pages) Paper No: CND-16-1619; doi: 10.1115/1.4037924 History: Received December 11, 2016; Revised August 23, 2017
Article
References
Figures

Based on the special dynamical property of continuous transition at certain degenerate grazing points in the single-degree-of-freedom impact oscillator, the control problem of the grazing-induced chaos is investigated in this paper. To design degenerate grazing bifurcations, we show how to obtain the degenerate grazing points of the 1/n impact periodic motions by the existence and stability analysis first. Then, a discrete-in-time feedback control strategy is used to suppress the grazing-induced chaos into the 1/n impact periodic motions precisely by the desired degenerate grazing bifurcation. The feasibility of the control strategy is verified by numerical simulations.
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References

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Figures

Grahic Jump Location
Fig. 1

Schematic model of impact system

+Schematic model of impact system
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Schematic model of impact system
Grahic Jump Location
Fig. 2

Grazing-induced chaos at ω=3, r=0.86, ζ=4. (a) Bifurcation diagram obtained by Ps on section Π1. (b) Bifurcation diagram obtained by Pg on section Π2. (c) Bifurcation diagram by direct numerical simulations on section Π2. (d, e) Phase diagram of grazing-induced chaos at d=d∗−2×10−6. (f) Poincaré phase diagram at d=d∗−2×10−6 on the section Π2.

+Grazing-induced chaos at ω=3, r=0.86, ζ=4. (a) Bifurcation diagram obtained by Ps on section Π1. (b) Bifurcation diagram obtained by Pg on section Π2. (c) Bifurcation diagram by direct numerical simulations on section Π2. (d, e) Phase diagram of grazing-induced chaos at d=d∗−2×10−6. (f) Poincaré phase diagram at d=d∗−2×10−6 on the section Π2.
View Large  |  Save Figure  |  Download Slide (.ppt)
Grazing-induced chaos at ω=3, r=0.86, ζ=4. (a) Bifurcation diagram obtained by Ps on section Π1. (b) Bifurcation diagram obtained by Pg on section Π2. (c) Bifurcation diagram by direct numerical simulations on section Π2. (d, e) Phase diagram of grazing-induced chaos at d=d∗−2×10−6. (f) Poincaré phase diagram at d=d∗−2×10−6 on the section Π2.
Grahic Jump Location
Fig. 3

Control parameters diagram. The black dashed line segment represents the n=1 case. The red solid line segment represents the n=2 case. And the blue dashed line segment stands for the case where the inequality condition in Eq. (44) fails. Inequalities in Eq. (45) hold in the whole region. For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.

+Control parameters diagram. The black dashed line segment represents the n=1 case. The red solid line segment represents the n=2 case. And the blue dashed line segment stands for the case where the inequality condition in Eq. (44) fails. Inequalities in Eq. (45) hold in the whole region. For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.
View Large  |  Save Figure  |  Download Slide (.ppt)
Control parameters diagram. The black dashed line segment represents the n=1 case. The red solid line segment represents the n=2 case. And the blue dashed line segment stands for the case where the inequality condition in Eq. (44) fails. Inequalities in Eq. (45) hold in the whole region. For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.
Grahic Jump Location
Fig. 4

Suppressing the grazing-induced chaos by degenerate grazing bifurcation. (a) Bifurcation diagram on section Π1 for n=1 case. (b) Bifurcation diagram on section Π2. (c) Direct numerical simulations on section Π2. (d) Phase diagram of 1/1 impact periodic motion. (e, f) The drawing of partial enlargement of (d) in vicinity of the impact surface and the section Π2. (g–l) n=2 case. (m) Control the grazing-induced chaos to 1/1 and 1/2 impact periodic motions.

+Suppressing the grazing-induced chaos by degenerate grazing bifurcation. (a) Bifurcation diagram on section Π1 for n=1 case. (b) Bifurcation diagram on section Π2. (c) Direct numerical simulations on section Π2. (d) Phase diagram of 1/1 impact periodic motion. (e, f) The drawing of partial enlargement of (d) in vicinity of the impact surface and the section Π2. (g–l) n=2 case. (m) Control the grazing-induced chaos to 1/1 and 1/2 impact periodic motions.
View Large  |  Save Figure  |  Download Slide (.ppt)
Suppressing the grazing-induced chaos by degenerate grazing bifurcation. (a) Bifurcation diagram on section Π1 for n=1 case. (b) Bifurcation diagram on section Π2. (c) Direct numerical simulations on section Π2. (d) Phase diagram of 1/1 impact periodic motion. (e, f) The drawing of partial enlargement of (d) in vicinity of the impact surface and the section Π2. (g–l) n=2 case. (m) Control the grazing-induced chaos to 1/1 and 1/2 impact periodic motions.

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