Abstract
Grazing bifurcation might bring dramatic and abrupt qualitative changes of system responses in non-smooth systems. The control problem of near-grazing dynamics in a two-degree-of-freedom vibro-impact system with a clearance is addressed in this paper. The discontinuity mapping is built to derive a normal form map near the grazing point. Based on the stability criterion of grazing orbit formulated through the normal form map, a discrete-in-time feedback method is developed to control the stability of grazing orbit in the vibro-impact system. Numerical simulations reveal a discontinuous grazing bifurcation with a jump between the transitions of periodic motions and show the feasibility of the proposed control strategy.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Luo, G.W., Xie, J.H.: Hopf bifurcations of a two-degree-of-freedom vibro-impact system. J. Sound Vib. 213, 391–480 (1998)
Wen, G.L., Chen, S.J., Jin, Q.T.: A new criterion of period-doubling bifurcation in maps and its application to an inertial impact shaker. J. Sound Vib. 311, 212–223 (2008)
Xie, J.H., Ding, W.C., Dowell, E.H., Virgin, L.N.: Hopf-flip bifurcation of high dimensional maps and application to vibro-impact systems. Acta Mech. Sin. 21, 402–410 (2005)
Luo, G.W., Zhang, Y.L., Zhang, J.G.: Dynamical behavior of a class of vibratory systems with symmetrical rigid stops near the point of codimension two bifurcation. J. Sound Vib. 297, 17–36 (2006)
Xie, J.H., Ding, W.C.: Hopf–Hopf bifurcation and invariant torus \(T^{2}\) of a vibro-impact system. Int. J. Non Linear Mech. 40, 531–543 (2005)
Wen, G.L., Xie, J.H., Xu, D.L.: Onset of degenerate Hopf bifurcation of a vibro-impact oscillator. ASME J. Appl. Mech. 71, 579–581 (2004)
Shaw, S.W., Holmes, P.J.: A periodically forced piecewise linear oscillator. J. Sound Vib. 90, 129–155 (1983)
Whiston, G.S.: Singularities in vibro-impact dynamics. J. Sound Vib. 152, 427–460 (1992)
Nordmark, A.B.: Non-periodic motion caused by grazing incidence in an impact oscillator. J. Sound Vib. 145, 279–297 (1991)
Nordmark, A.B.: Universal limit mapping in grazing bifurcations. Phys. Rev. E 55, 266–270 (1997)
Molenaar, J., de Weger, J.G., van de Water, W.: Mappings of grazing impact oscillators. Nonlinearity 14, 301–321 (2001)
Chin, W., Ott, E., Nusse, H.E., Grebogi, C.: Grazing bifurcations in impact oscillators. Phys. Rev. E 50, 4427–4444 (1994)
di Bernardo, M., Budd, C.J., Champneys, A.R., Kowalczyk, P.: Piecewise-Smooth Dynamical Systems: Theory and Applications. Springer, London (2008)
di Bernardo, M., Budd, C.J., Champneys, A.R.: Normal form maps for grazing bifurcations in n-dimensional piecewise-smooth dynamical systems. Phys. D 160, 222–254 (2000)
Kryzhevich, S., Wiercigroch, M.: Topology of vibro-impact systems in the neighborhood of grazing. Phys. D 241, 1919–1931 (2012)
Du, Z.D., Li, Y.R., Shen, J., Zhang, W.N.: Impact oscillators with homoclinic orbit tangent to the wall. Phys. D 245, 19–33 (2013)
Dankowicz, H., Zhao, X.P.: Local analysis of co-dimension-one and co-dimension-two grazing bifurcations in impact microactuators. Phys. D 202, 238–257 (2005)
Piiroinen, P.T., Virgin, L.N., Champneys, A.R.: Chaos and period adding: Experimental and numerical verification of the grazing bifurcation. J. Nonlinear Sci. 14, 383–404 (2004)
Wilcox, B., Svahn, F., Dankowicz, H., Jerrelind, J.: Transient growth rates of near-grazing impact velocities: theory and experiments. J. Sound Vib. 325, 950–958 (2009)
Wen, G.L., Xu, H.D., Xiao, L., Xie, X.P., Chen, Z., Wu, X.: Experimental investigation of a two-degree-of-freedom vibro-impact system. Int. J. Bifurc. Chaos 22, 1250110–1250117 (2012)
Qin, W.Y., Chen, G., Ren, X.M.: Grazing bifurcation in the response of cracked Jeffcott rotor. Nonlinear Dyn. 35, 147–157 (2004)
Fredriksson, M.H., Nordmark, A.B.: Bifurcations caused by grazing incidence in many degrees of freedom impact oscillators. Proc. R. Soc. Lond. Ser. A 453, 1261–1276 (1997)
Thota, P., Dankowicz, H.: Continuous and discontinuous grazing bifurcation in impacting oscillators. Phys. D 214, 187–197 (2006)
Luo, A.C.J.: On grazing and strange attractors fragmentation in non-smooth dynamical systems. Commun. Nonlinear Sci. Numer. Simul. 11, 922–933 (2006)
Dankowicz, H., Jerrelind, J.: Control of near-grazing dynamics in impact oscillators. Proc. R. Soc. Lond. Ser. A 461, 3365–3380 (2005)
Acknowledgments
This work was supported by the National Science Fund for Distinguished Young Scholars in China (No. 11225212), the National Natural Science Foundation of China (Nos. 11002052, 11072074, 11372101 and 61364001), the Hunan Provincial Natural Science Foundation for Creative Research Groups of China (Grant No. 12JJ7001), the Young Teacher Development Plan of Hunan University, and the Collaborative Innovation Center of Intelligent New Energy Vehicle.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Xu, H., Yin, S., Wen, G. et al. Discrete-in-time feedback control of near-grazing dynamics in the two-degree-of-freedom vibro-impact system with a clearance. Nonlinear Dyn 87, 1127–1137 (2017). https://doi.org/10.1007/s11071-016-3103-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-016-3103-8