Abstract
We investigate the presence of asymptotically stable periodic oscillations in a time-periodic impact oscillator close to an isochronous one. A new averaging method is developed to account for the position of the obstacle and for the impact restitution coefficient, which do not appear in the classical smooth situation.
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.References
Babitsky, V.I.: Theory of Vibro-Impact Systems and Applications. Translated from the Russian by N. Birkett and revised by the author. Foundations of Engineering Mechanics. Springer, Berlin, xvi+318 pp (1998)
Babitsky, V.I., Krupenin, V.L.: Vibrations of Strongly Nonlinear Discontinuous Systems. Springer, Berlin (2001)
Battelli, F., Fečkan, M.: Chaos in forced impact systems. Discrete Contin. Dyn. Syst. Ser. S 6(4), 861–890 (2013)
Bogoliubov, N.N., Mitropolsky, Y.A.: Asymptotic Methods in the Theory of Non-Linear Oscillations. Translated from the second revised Russian edition. International Monographs on Advanced Mathematics and Physics Hindustan Publishing Corp., Delhi, Gordon and Breach Science Publishers, New York, x+537 pp (1961)
Brogliato, B.: Nonsmooth Impact Mechanics. Models, Dynamics and Control. Lecture Notes in Control and Information Sciences, 220. Springer-Verlag London Ltd, London, xvi+400 pp (1996)
Buică, A., Llibre, J., Makarenkov, O.: Asymptotic stability of periodic solutions for nonsmooth differential equations with application to the nonsmooth van der Pol oscillator. SIAM J. Math. Anal. 40(6), 2478–2495 (2009)
Buică, A., Llibre, J., Makarenkov, O.: A note on forced oscillations in differential equations with jumping nonlinearities. Differ. Equ. Dyn. Syst. (2014). doi:10.1007/s12591-014-0199-5
Burd, V.S.: Resonance vibrations of impact oscillator with biharmonic excitation. Phys. D 241(22), 1956–1961 (2012)
Burd, V.S.: Resonant almost periodic oscillations in systems with slow varying parameters. Internat. J. Non-Linear Mech. 32(6), 1143–1152 (1997)
Cao, Q.J., Wiercigroch, M., Pavlovskaia, E., Yang, S.P.: Bifurcations and the penetrating rate analysis of a model for percussive drilling. Acta Mechanica Sinica 26(3), 467–475 (2010)
Chicone, C.: Lyapunov–Schmidt reduction and Melnikov integrals for bifurcation of periodic solutions in coupled oscillators. J. Differ. Equ. 112, 407–447 (1994)
Coombes, S., Thul, R., Wedgwood, K.C.A.: Nonsmooth dynamics in spiking neuron models. Phys. D 241(22), 2042–2057 (2012)
Dombovaria, Z., Barton, D.A.W., Wilson, R.E., Stepan, G.: On the global dynamics of chatter in the orthogonal cutting model. Int. J. Non-Linear Mech. 46(1), 330–338 (2011)
Fečkan, M., Pospíšil, M.: Persistence of periodic orbits in periodically forced impact systems. Math. Slovaca 64, 101–118 (2014)
Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Revised and corrected reprint of the 1983 original, vol. 42. Applied Mathematical Sciences. Springer, New York (1990)
Hos, C., Champneys, A.R.: Grazing bifurcations and chatter in a pressure relief valve model. Physica D 241(22), 2068–2076 (2012)
Huang, M., Liu, S., Song, X., Chen, L.: Periodic solutions and homoclinic bifurcation of a predator-prey system with two types of harvesting. Nonlinear Dyn. 73(1–2), 815–826 (2013)
Ibrahim, R.A., Babitsky, V.I., Okuma, M. (eds.): Vibro-Impact Dynamics of Ocean Systems and Related Problems. Springer, Berlin (2009)
Kamenskii, M., Makarenkov, O., Nistri, P.: An alternative approach to study bifurcation from a limit cycle in periodically perturbed autonomous systems. J. Dyn. Differ. Equ. 23(3), 425–435 (2011)
Klymchuk, S., Plotnikov, A., Skripnik, N.: Overview of V.A. Plotnikovs research on averaging of differential inclusions. Physica D 241(22), 1932–1947 (2012)
Kolmogorov, A.N., Fomin, S.V., S.V.: Elements of the Theory of Functions and Functional Analysis (Russian) 4th edn., revised. Izdat. “Nauka”, Moscow (1976)
Krantz, S.G., Parks, H.R.: The Implicit Function Theorem. History, Theory, and Applications. Birkhuser Boston Inc, Boston, MA, xii+163 pp (2002)
Leine, R.I., Heimsch, T.F.: Global uniform symptotic attractive stability of the non-autonomous bouncing ball system. Phys. D 241(22), 2029–2041 (2012)
Li, Z., Chen, L.: Periodic solution of a turbidostat model with impulsive state feedback control. Nonlinear Dyn. 58(3), 525–538 (2009)
Luo, A.C.J., O’Connor, D.: Periodic motions and chaos with impacting chatter and stick in a gear transmission system. Int. J. Bifur. Chaos Appl. Sci. Eng. 19(6), 1975–1994 (2009)
Makarenkov, O., Lamb, J.S.W.: Dynamics and bifurcations of nonsmooth systems: a survey. Phys. D 241(22), 1826–1844 (2012)
Makarenkov, O., Ortega, R.: Asymptotic stability of forced oscillations emanating from a limit cycle. J. Differ. Equ. 250(1), 39–52 (2011)
Mason, J.F., Piiroinen, P.T.: The effect of codimension-two bifurcations on the global dynamics of a gear model. SIAM J. Appl. Dyn. Syst. 8(4), 1694–1711 (2009)
Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations. Pure and Applied Mathematics. Wiley-Interscience [John Wiley & Sons], New York, xiv+704 pp (1979)
Okninski, A., Radziszewski, B.: Simple model of bouncing ball dynamics: displacement of the table assumed as quadratic function of time. Nonlinear Dyn. 67(2), 1115–1122 (2012)
Perestyuk, N.A., Plotnikov, V.A., Samoilenko, A.N., Skripnik, N.V.: Differential equations with impulse effects. Multivalued right-hand sides with discontinuities. de Gruyter Studies in Mathematics, vol. 40. Walter de Gruyter & Co., Berlin, xiv+307 pp (2011)
Pilipchuk, V., Ibrahim, R.A.: Dynamics of a two-pendulum model with impact interaction and an elastic support. Nonlinear Dyn. 21(3), 221–247 (2000)
Sartorelli, J.C., Lacarbonara, W.: Parametric resonances in a base-excited double pendulum. Nonlinear Dyn. 69(4), 1679–1692 (2012)
Srinivasan, M., Holmes, P.: How well can spring-mass-like telescoping leg models fit multi-pedal sagittal-plane locomotion data? J. Theor. Biol. 255(1), 1–7 (2008)
Stoker, J.J.: Nonlinear Vibrations in Mechanical and Electrical Systems. Reprint of the 1950 original. Wiley Classics Library. A Wiley-Interscience Publication. John Wiley & Sons Inc, New York, xx+273 (1992)
Tavakoli, A., Hurmuzlu, Y.: Robotic locomotion of three generations of a family tree of dynamical systems. Part II: impulsive control of gait patterns. Nonlinear Dyn. 73(3), 1991–2012 (2013)
Wang, T., Chen, L.: Nonlinear analysis of a microbial pesticide model with impulsive state feedback control. Nonlinear Dyn. 65(1–2), 1–10 (2011)
Zhang, Y., Zhang, Q., Zhang, X.: Dynamical behavior of a class of prey-predator system with impulsive state feedback control and Beddington–DeAngelis functional response. Nonlinear Dyn. 70(2), 1511–1522 (2012)
Zhuravlev, V.F., Klimov, D. M.: Applied Methods in the Theory of Oscillations. “Nauka”, Moscow, 327 pp (1988)
Acknowledgments
The research is supported by NSF Grant CMMI-1436856 and by RFBR Grant 13-01-00347. The reports of anonymous referees helped to improve the paper and are very appreciated.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Newman, J., Makarenkov, O. Resonance oscillations in a mass-spring impact oscillator. Nonlinear Dyn 79, 111–118 (2015). https://doi.org/10.1007/s11071-014-1649-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-014-1649-x