Journal of Mechanical Science and Technology

, Volume 31, Issue 4, pp 1551–1560 | Cite as

Vibrational behavior of MDOF oscillators subjected to multiple contact constraints



Vibrational behavior of harmonically excited MDOF oscillators subjected to multiple contact constraints is investigated in this paper using the combination of the Newmark integration scheme and the Linear complementarity problem (LCP) formulation. An oscillator with gap-activated non-smooth spring constraints exhibits various complex behavior such as sub-harmonic resonances, bifurcations and chaos, which are effectively predicted using the proposed method. Numerical results were obtained and presented for SDOF and 5-DOF systems with frequency and stiffness parameters varying in wide ranges to validate the Newmark-LCP method and to demonstrate its effectiveness in dealing with MDOF systems with multiple contact constraints.


Bifurcation Chaos Contact Linear complementarity problem Piecewise linear nonlinearity Sub-harmonic resonance Super-harmonic resonance 


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  1. [1]
    B. C. Wen, Y. N. Li, Y. M. Zhang and Z. W. Song, Vibration utilization engineering, Science Publisher, Beijing, China (2005).Google Scholar
  2. [2]
    S. W. Shaw, The dynamics of a harmonically excited system having rigid amplitude constraints, part 1: subharmonic motions and local bifurcations, Journal of Applied Mechanics, 52 (1985) 453–458.MathSciNetCrossRefGoogle Scholar
  3. [3]
    J. S. Walker and T. Soule, Chaos in a simple impact oscillator: The bender bouncer, American Journal of Physics, 64 (4) (1996) 397–409.CrossRefGoogle Scholar
  4. [4]
    S. W. Shaw and P. J. Holmes, A periodically forced piecewise linear oscillator, Journal of Sound and Vibration, 90 (1983) 129–155.MathSciNetCrossRefMATHGoogle Scholar
  5. [5]
    G. S. Whiston, The vibro-impact response of a harmonically excited and preloaded one-dimensional linear oscillator, Journal of Sound and Vibration, 115 (1987) 303–319.MathSciNetCrossRefMATHGoogle Scholar
  6. [6]
    A. V. Dyskin, E. Pasternak and E. Pelinovsky, Periodic motions and resonances of impact oscillators, Journal of Sound and Vibration, 331 (12) (2012) 2856–2873.CrossRefGoogle Scholar
  7. [7]
    M. Wiercigroch, Modelling of dynamical systems with motion dependent discontinuities, Chaos Solutions and Fractals, 11 (2000) 2429–2442.MathSciNetCrossRefMATHGoogle Scholar
  8. [8]
    V. W. T. Sin and M. Wiercigroch, A symmetrically piecewise linear oscillator: Design and measurement, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 213 (3) (1999) 241–249.Google Scholar
  9. [9]
    P. Ing, E. Pavlovskaia, M. Wiercigroch and S. Banerjee, Experimental study of impact oscillator with one-sided elastic constraint, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 366 (2008) 679–704.CrossRefMATHGoogle Scholar
  10. [10]
    Z. K. Peng, Z. Q. Lang, S. A. Billings and Y. Ku, Analysis of bilinear oscillators under harmonic loading using nonlinear output frequency response functions, International Journal of Mechanical Sciences, 49 (2007) 1213–1225.CrossRefGoogle Scholar
  11. [11]
    G. W. Luo, Period-doubling bifurcations and routes to chaos of the vibratory systems contacting stops, Physics Letters A, 323 (2004) 210–217.MathSciNetCrossRefMATHGoogle Scholar
  12. [12]
    G. W. Luo and Y. Zhang, Analyses of impact motions of harmonically excited systems having rigid amplitude constraints, International Journal of Impact Engineering, 34 (11) (2007) 1883–1905.CrossRefGoogle Scholar
  13. [13]
    S. H. Doole and S. J. Hogan, A piecewise linear suspension bridge model: nonlinear dynamics and orbit continuation, International Journal of Impact Engineering, 11 (1996) 19–47.MathSciNetMATHGoogle Scholar
  14. [14]
    S. D. Yu, An efficient computational method for vibration analysis of unsymmetric piecewise-linear dynamical systems with multiple degrees of freedom, Nonlinear Dynamics, 71 (3) (2013) 493–504.MathSciNetCrossRefGoogle Scholar
  15. [15]
    M. Fadaee and S. D. Yu, Two-dimensional stick-slip motion of Coulomb friction oscillators, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science (2015).Google Scholar
  16. [16]
    A. B. Nordmark, Non-periodic motion caused by grazing incidence in impact oscillators, Journal of Sound and Vibration, 145 (2) (1991) 279–297.CrossRefGoogle Scholar
  17. [17]
    F. Peterka, Laws of impact motion of mechanical system with one degree of freedom, Part I-theoretical analysis of nmultiple (1/n)-impact motion, Acta Technica CSAV, 4 (1974) 462–473.MATHGoogle Scholar
  18. [18]
    F. Peterka, T. Kotera and S. Cipera, Explanation of appearance and characteristics of intermittency chaos of the impact oscillator, Chaos, Solitons and Fractals, 19 (5) (2004) 1251–1259.CrossRefMATHGoogle Scholar
  19. [19]
    S. S. Rao, Mechanical vibrations, 5th ed., Pearson Prentice Hall, Upper Saddle River, New Jersey, USA (2010).Google Scholar

Copyright information

© The Korean Society of Mechanical Engineers and Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Department of Mechanical and Industrial EngineeringRyerson UniversityTorontoCanada

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