0
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.
FIGURES IN THIS ARTICLE
<>
Copyright © 2018 by ASME
Your Session has timed out. Please sign back in to continue.

References

1
Peterka, F. , and Vacík, J. , 1992, “ Transition to Chaotic Motion in Mechanical Systems With Impacts,” J. Sound Vib., 154(1), pp. 95–115. [CrossRef]
2
Xie, J. H. , and Ding, W. C. , 2005, “ Hopf–Hopf Bifurcation and Invariant Torus of a Vibro-Impact System,” Int. J. Non Linear Mech., 40(4), pp. 531–543. [CrossRef]
3
Ding, W. C. , and Xie, J. H. , 2006, “ Torus T2 and its Routes to Chaos of a Vibro-Impact System,” Phys. Lett. A, 349(5), pp. 324–330. [CrossRef]
4
Luo, G. W. , and Lv, X. H. , 2009, “ Controlling Bifurcation and Chaos of a Plastic Impact Oscillator,” Nonlinear Anal.: Real World Appl., 10(4), pp. 2047–2061. [CrossRef]
5
Mason, J. F. , and Piiroinen, P. T. , 2012, “ Saddle-Point Solutions and Grazing Bifurcations in an Impacting System,” Chaos Interdiscip. J. Nonlinear Sci., 22(1), p. 013106. [CrossRef]
6
Wang, L. , Xu, W. , and Li, Y. , 2008, “ Impulsive Control of a Class of Vibro-Impact Systems,” Phys. Lett. A, 372(32), pp. 5309–5313. [CrossRef]
7
Wen, G. L. , Xu, H. D. , and Chen, Z. , 2010, “ Anti-Controlling Quasi-Periodic Impact Motion of an Inertial Impact Shaker System,” J. Sound Vib., 329(19), pp. 4040–4047. [CrossRef]
8
de Souza, S. L. T. , Wiercigroch, M. , Caldas, I. L. , and Balthazar, J. M. , 2008, “ Suppressing Grazing Chaos in Impacting System by Structural Nonlinearity,” Chaos Solitons Fractals, 38(3), pp. 864–869. [CrossRef]
9
Nordmark, A. B. , 1991, “ Non-Periodic Motion Caused by Grazing Incidence in an Impact Oscillator,” J. Sound Vib., 145(2), pp. 279–297. [CrossRef]
10
Nordmark, A. B. , 1997, “ Universal Limit Mapping in Grazing Bifurcations,” Phys. Rev. E, 55(1), pp. 266–270. [CrossRef]
11
Fredriksson, M. H. , and Nordmark, A. B. , 1997, “ Bifurcations Caused by Grazing Incidence in Many Degrees of Freedom Impact Oscillators,” Proc. R. Soc. A, 453(1961), pp. 1261–1276. [CrossRef]
12
Chillingworth, D. R. J. , 2002, “ Discontinuity Geometry for an Impact Oscillator,” Dyn. Syst., 17(4), pp. 389–420. [CrossRef]
13
Chillingworth, D. R. J. , 2010, “ Dynamics of an Impact Oscillator Near a Degenerate Graze,” Nonlinearity, 23(11), pp. 2723–2748. [CrossRef]
14
Molenaar, J. , de Weger, J. G. , and van de Water, W. , 2001, “ Mappings of Grazing Impact Oscillators,” Nonlinearity, 14(2), pp. 301–321. [CrossRef]
15
Piiroinen, P. T. , Virgin, L. N. , and Champneys, A. R. , 2004, “ Chaos and Period Adding: Experimental and Numerical Verification of the Grazing Bifurcation,” J. Nonlinear. Sci., 14(4), pp. 383–404. [CrossRef]
16
Thota, P. , and Dankowicz, H. , 2006, “ Continuous and Discontinuous Grazing Bifurcations in Impacting Oscillators,” Phys. D, 214(2), pp. 187–197. [CrossRef]
17
Wilcox, B. , Svahn, F. , Dankowicz, H. , and Jerrelind, J. , 2009, “ Transient Growth Rates of Near-Grazing Impact Velocities: Theory and Experiments,” J. Sound Vib., 325(4), pp. 950–958. [CrossRef]
18
Wen, G. L. , Yin, S. , Xu, H. D. , Zhang, S. J. , and Lv, Z. Y. , 2016, “ Analysis of Grazing Bifurcation From Periodic Motion to Quasi-Periodic Motion in Impact-Damper Systems,” Chaos Solitons Fractals, 83, pp. 112–118. [CrossRef]
19
Mehran, K. , Zahawi, B. , and Giaouris, D. , 2012, “ Investigation of the Near-Grazing Behavior in Hard-Impact Oscillators Using Model-Based TS Fuzzy Approach,” Nonlinear Dyn., 69(3), pp. 1293–1309. [CrossRef]
20
Dankowicz, H. , and Svahn, F. , 2007, “ On the Stabilizability of Near-Grazing Dynamics in Impact Oscillators,” Int. J. Robust Nonlinear Control, 17(15), pp. 1405–1429. [CrossRef]
21
Misra, S. , and Dankowicz, H. , 2010, “ Control of Near-Grazing Dynamics and Discontinuity Induced Bifurcations in Piecewise-Smooth Dynamical Systems,” Int. J. Robust Nonlinear Control, 20(16), pp. 1836–1851.
22
Bernardo, M. , Budd, C. J. , Champneys, A. R. , and Kowalczyk, P. , 2008, Piecewise-Smooth Dynamical Systems: Theory and Applications, Springer, London.
23
Dankowicz, H. , and Jerrelind, J. , 2005, “ Control of Near-Grazing Dynamics in Impact Oscillators,” Proc. R. Soc. Lond. A, 461(2063), pp. 3365–3380.
24
Kundu, S. , Banerjee, S. , and Giaouris, D. , 2011, “ Vanishing Singularity in Hard Impact Systems,” Discrete Contin. Dyn. Syst.-Ser. B, 16(1), pp. 319–332. [CrossRef]
25
Xu, J. Q. , Chen, P. , and Li, Q. H. , 2015, “ Theoretical Analysis of Co-Dimension-Two Grazing Bifurcation in n-Degree-of-Freedom Impact Oscillator With Symmetrical Constrains,” Nonlinear Dyn., 82(4), pp. 1641–1657. [CrossRef]
26
Foale, S. , 1994, “ Analytical Determination of Bifurcations in an Impact Oscillator,” Philos. Trans. R. Soc. London, Ser. A, 347(1683), pp. 353–364. [CrossRef]
27
Peterka, F. , 1996, “ Bifurcations and Transition Phenomena in an Impact Oscillator,” Chaos Solitons Fractals, 7(10), pp. 1635–1647.
28
Thota, P. , Zhao, X. P. , and Dankowicz, H. , 2006, “ Co-dimension-Two Grazing Bifurcations in Single-Degree-of-Freedom Impact Oscillators,” ASME J. Comput. Nonlinear Dynam., 1(4), pp. 328–335.
29
Zhao, X. P. , and Dankowicz, H. , 2006, “ Unfolding Degenerate Grazing Dynamics in Impact Actuators,” Nonlinearity, 19(2), pp. 399–418.
30
Dankowicz, H. , and Zhao, X. P. , 2005, “ Local Analysis of Co-Dimension-One and Co-Dimension-Two Grazing Bifurcations in Impact Microactuators,” Phys. D (Amsterdam, Neth.), 202(3), pp. 238–257.
31
Wen, G. L. , and Xu, D. L. , 2004, “ Implicit Criteria of Eigenvalue Assignment and Transversality for Bifurcation Control in Four-Dimensional Maps,” Int. J. Bifurcation Chaos, 14(10), pp. 3489–3503.

Figures

Grahic Jump Location
Fig. 1

Schematic model of impact system

+Schematic model of impact system
View Large  |  Save Figure  |  Download Slide (.ppt)
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.

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
On the Evaluation of Thermal and Mechanical Factors in Low-Speed Sliding
Tribology of Mechanical Systems> Chapter 11
Dynamic Behavior in a Singular Delayed Bioeconomic Model
International Conference on Instrumentation, Measurement, Circuits and Systems (ICIMCS 2011)
QP Based Encoder Feedback Control
Robot Manipulator Redundancy Resolution
Feedback-Aided Minimum Joint Motion
Robot Manipulator Redundancy Resolution
Key Technology of Interactive Music Game Installation based on Arduino
International Conference on Control Engineering and Mechanical Design (CEMD 2017)
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
This site uses cookies. By continuing to use our website, you are agreeing to our privacy policy. | Accept
Sign In