Abstract
The global analysis method in parameter-state space for achieving both the distribution and the transition rule of periodic motions and the distribution rule of the multiple motions coexistence of the vibro-impact system is developed. A nonlinear dynamic model in a system of vibro-impact with asymmetric clearances is researched by the new method. Three grazing motions and relevant conditions have been discussed. The distribution and the transition rule of periodic motions are analyzed. The influence of the grazing and saddle-node bifurcation during the change in the left and the right gaps are demonstrated. The distribution of subharmonic motions is illustrated. The coexistence of multiple motions which exist at the junction of periodic motions within the motion-sensitive areas is researched by the global analysis method in parameter-state space, and the distributions of periodic motions and the multiple motions coexistence are illustrated. The transition of the multiple motions coexistence with the guide of the global distribution of the motions is further analyzed by the evolution of the attractors and the corresponding attracting domains. The results contribute a lot to the study of the transition and the control of the motions.
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs11071-019-04996-6/MediaObjects/11071_2019_4996_Fig1_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs11071-019-04996-6/MediaObjects/11071_2019_4996_Fig2_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs11071-019-04996-6/MediaObjects/11071_2019_4996_Fig3_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs11071-019-04996-6/MediaObjects/11071_2019_4996_Fig4_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs11071-019-04996-6/MediaObjects/11071_2019_4996_Fig5_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs11071-019-04996-6/MediaObjects/11071_2019_4996_Fig6_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs11071-019-04996-6/MediaObjects/11071_2019_4996_Fig7_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs11071-019-04996-6/MediaObjects/11071_2019_4996_Fig8_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs11071-019-04996-6/MediaObjects/11071_2019_4996_Fig9_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs11071-019-04996-6/MediaObjects/11071_2019_4996_Fig10_HTML.png)
Similar content being viewed by others
References
Shaw, S.W., Holmes, P.: Periodically forced linear oscillator with impacts: chaos and long-period motions. Phys. Rev. Lett. 51(8), 623–626 (1983)
Thompson, J.M.T., Ghaffari, R.: Chaos after period-doubling bifurcations in the resonance of an impact oscillator. Phys. Lett. A 91(1), 5–8 (1982)
Janin, O., Lamarque, C.H.: Stability of singular periodic motions in a vibro-impact oscillator. Nonlinear Dyn. 28(3), 231–241 (2002)
Li, Q.H., Lu, Q.S.: Coexisting periodic orbits in vibro-impacting dynamical systems. Appl. Math. Mech. 24(3), 234–244 (2003)
Ding, W., Li, G., Luo, G., et al.: Torus T-2 and its locking, doubling, chaos of a vibro-impact system. J. Frankl. Inst. 349(1), 337–348 (2012)
Yue, Y., **e, J., Gao, X.: Determining Lyapunov spectrum and Lyapunov dimension based on the Poincaré map in a vibro-impact system. Nonlinear Dyn. 69(3), 743–753 (2012)
Li, G., Ding, W., Wu, S.: Global behavior of vibro-impact system with multiple non-smooth mechanical Factors. J. Comput. Nonlinear Dyn. 12(6), 061004-1–061004-11 (2017)
Li, G., Ding, W.: Global Behavior of a vibro-impact system with asymmetric clearances. J. Sound Vib. 423, 180–194 (2018)
**ao-Juan, W., Ning-Zhou, L., Wang-Cai, D., et al.: Model-free chaos control based on AHGSA for a vibro-impact system. Nonlinear Dyn. 94, 845–855 (2018)
Czolczynski, K., Kapitaniak, T.: On the existence of a stable periodic solution of an impacting oscillator with two fenders. Int. J. Bifurc. Chaos 14(9), 3115–3134 (2004)
Nordmark, A.B.: Non-periodic motion caused by grazing incidence in an impact oscillator. J. Sound Vib. 145(2), 279–297 (1991)
Whiston, G.S.: Singularities in vibro-impact dynamics. J. Sound Vib. 152(3), 427–460 (1992)
Chin, W., Ott, E., Nusse, H.E., et al.: Universal behavior of impact oscillators near grazing incidence. Phys. Lett. A 201(2–3), 197–204 (1995)
Lamba, H., Budd, C.J.: Scaling of Lyapunov exponents at non-smooth bifurcations. Phys. Rev. E. 50(1), 84–90 (1994)
Virgin, L.N., Begley, C.J.: Grazing bifurcations and basins of attraction in an impact-friction oscillator. Phys. D Nonlinear Phenom. 130(1–2), 43–57 (1999)
Pavlovskaia, E., Ing, J., Wiercigroch, M., et al.: Complex dynamics of bilinear oscillator close to grazing. Int. J. Bifurc. Chaos. 20(11), 3801–3817 (2010)
Shan, Y., **chen, J., Shuning, D., et al.: Neimark–Sacker bifurcations near degenerate grazing point in a two degree-of-freedom impact oscillator. J. Comput. Nonlinear Dyn. 13(6), 111007-1–111007-8 (2018)
Chillingworth, D.R.J.: Discontinuity geometry for an impact oscillator. Dyn. Syst. 17(4), 389–420 (2002)
Humphries, N., Piiroinen, P.T.: A discontinuity-geometry view of the relationship between saddle-node and grazing bifurcations. Phy. D 241(22), 1911–1918 (2012)
Jiang, H., Wiercigroch, M.: Geometrical insight into non-smooth bifurcations of a soft impact oscillator. J. Appl. Math. 81(4), 662–678 (2016)
Luo, A.C.J., Chen, L.D.: Arbitrary periodic motions and grazing switching of a forced piecewise-linear, impacting oscillator. ASME J. Vib. Acoust. 129, 276–284 (2007)
Wagg, D.J.: Periodic sticking motion in a two-degree-freedom impact oscillator. Int. J. Bifurc. Chaos 40(8), 1076–1087 (2005)
Nordmark, A.B., Piiroinen, P.T.: Simulation and stability analysis of impacting systems with complete chattering. Nonlinear Dyn. 58(1–2), 85–106 (2009)
Ma, Y., Agarwal, M., Banerjee, S.: Border collision bifurcations in a soft impact system. Phys. Lett. A 354(4), 281–287 (2006)
Gritli, Hassène, Belghith, S.: Diversity in the nonlinear dynamic behavior of a one-degree-of-freedom impact mechanical oscillator under OGY-based state-feedback control law: order, chaos and exhibition of the border-collision bifurcation. Mech. Mach. Theory 124, 1–41 (2018)
Du, Z., Zhang, W.: Melnikov method for homoclinic bifurcation in nonlinear impact oscillators. Comput. Math. Appl. 50(3–4), 445–458 (2005)
Xu, W., Feng, J., Rong, H.: Melnikov’s method for a general nonlinear vibro-impact oscillator. Nonlinear Anal. 71(1–2), 418–426 (2009)
Hsu, C.S., Guttalu, R.S.: An unraveling algorithm for global analysis of dynamical systems: an application of cell-to-cell map**s. J. Appl. Mech. 47(4), 940–948 (1980)
Hsu, C.S.: A generalized theory of cell-to-cell map** for nonlinear dynamical systems. ASME J. Appl. Mech. 48(3), 634–642 (1981)
Hsu, C.S.: Global analysis by cell map**. Int. J. Bifurc. Chaos 2(4), 727–771 (1992)
Zufiria, P.J., Guttalu, R.S.: The adjoining cell map** and its recursive unraveling, part I: description of adaptive and recursive algorithms. Nonlinear Dyn. 4(3), 207–226 (1993)
Guttalu, R.S., Zufiria, P.J.: The adjoining cell map** and its recursive unraveling, part II: application toselected problems. Nonlinear Dyn. 4(4), 309–336 (1993)
Guder, R., Dellnitz, M., Kreuzer, E.: An adaptive method for the approximation of the generalized cell map**. Chaos Solitons Fractals 8(4), 525–534 (1997)
** method for global analysis of high-dimensional nonlinear dynamical systems. J. Appl. Mech. 82(11), 111010-1–111010-12 (2015)
Liu, X., Hong, L., Jiang, J., et al.: Global dynamics of fractional-order systems with an extended generalized cell map** method. Nonlinear Dyn. 83(3), 1419–1428 (2016)
Gao, X.J., Li, Y.H., Yue, Y., et al.: Symmetric/asymmetric bifurcation behaviours of a bogie system. J. Sound Vib. 332(4), 936–951 (2013)
Liu, Y., Pavlovskaia, E., Wiercigroch, M., et al.: Forward and backward motion control of a vibro-impact capsule system. Int. J. Nonlinear Mech. 70, 30–46 (2015)
Luo, A.C.J., O’Connor, D.: Periodic motions and chaos with impacting chatter and stick in a gear transmission system. Int. J. Bifurc. Chaos 19(6), 0902385 (2014)
Huang, Z.L., Liu, Z.H., Zhu, W.Q.: Stationary response of multi-degree-of-freedom vibro-impact systems under white noise excitations. J. Sound Vib. 275(1), 223–40 (2004)
**e, J.: A mathematical model for the impact hammer and it’s global bifurcations. Acta Mech. Sin. 13(4), 456–463 (1997)
Ibrahim, R.A.: Vibro-Impact Dynamics: Modeling, Map** and Applications. Lecture Notes in Applied and Computational Mechanics, vol. 43. Springer, Berlin (2009)
Awrejcewicz, J., Lamarque, C.H.: Bifurcation and Chaos in Nonsmooth Mechanical Systems, vol. 45. World Scientific, Singapore (2003)
Luo, A.C.J., Yu, G.: Vibro-Impact Dynamics. Wiley, London (2013)
Leine, R.I., Campen, D.H.V., Vrande, B.L.V.D.: Bifurcations in nonlinear discontinuous systems. Nonlinear Dyn. 23(2), 105–164 (2000)
Leine, R.I., Nijmeijer, H.: Dynamics and Bifurcations of Non-Smooth Mechanical Systems. Lecture Notes in Applied and Computational Mechanics, vol. 18. Springer, Berlin (2004)
Hsu, C.S.: Cell-to-Cell Map**: A Method of Global Analysis for Nonlinear Systems. Springer, Berlin (1987)
Sun, J.Q., ** Methods Algorithmic Approaches and Applications. Springer, Berlin (2019)
Acknowledgements
We acknowledge the editors and the anonymous reviewers for insightful suggestions on this work. This work is supported by the National Natural Science Foundation of China (11462011, 11732014), the Gansu Provincial Natural Science Foundation of China (17JR5RA098).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
We declare that we have no conflict of interest concerning the publication of this manuscript and the paper is in compliance with ethical standards.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Li, G., Sun, J. & Ding, W. Dynamics of a vibro-impact system by the global analysis method in parameter-state space. Nonlinear Dyn 97, 541–557 (2019). https://doi.org/10.1007/s11071-019-04996-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-019-04996-6