Journal Journal of Mechanical Engineering – Problemy Mashynobuduvannia
Publisher A. Podgorny Institute for Mechanical Engineering Problems
National Academy of Science of Ukraine
ISSN 0131-2928 (Print), 2411-0779 (Online)
Issue Vol. 20, no. 4, 2017 (December)
Pages 24-30
Cited by J. of Mech. Eng., 2017, vol. 20, no. 4, pp. 24-30



B. V. Uspenskiy, A. Podgorny Institute of Mechanical Engineering Problems of NASU (2/10, Pozharsky St., Kharkiv, 61046, Ukraine)

K. V. Avramov, A. Podgorny Institute of Mechanical Engineering Problems of NASU (2/10, Pozharsky St., Kharkiv, 61046, Ukraine), e-mail:, ORCID: 0000-0002-8740-693X

O. Ya. Nikonov, Kharkiv National Automobile and Highway University (25, Yaroslava Mudrogo St, Kharkiv, 61002, Ukraine)



A method for calculating forced oscillations of essentially non-linear piecewise linear systems at super-harmonic resonances is proposed. The basis of this method is a combination of non-linear normal forms and the Rauscher method, by which a non-autonomous dynamic system is reduced to an equivalent autonomous system. With the help of the proposed method, super-harmonic oscillations are investigated in the power train section of an internal combustion engine. Properties of resonant oscillations are considered in detail.


Keywords: super-harmonic resonances, Rauscher method, non-linear normal forms, configuration space



  1. Avramov, K. V. (2009). Nonlinear modes of parametric vibrations and their applications to beams dynamics. Journal of Sound and Vibration, no. 322, iss. 5, pp. 476–489.
  2. Avramov, K. V. (2008). Analysis of forced vibrations by nonlinear modes. Nonlinear Dynamics, vol. 53, iss. 1-2,  pp. 117–127.
  3. Shaw, S. W., Pierre, C., & Pesheck, E. (1999). Modal analysis-based reduced-order models for nonlinear structures – an invariant manifolds approach. The Shock and Vibration Digest, vol. 31, pp. 3–16.
  4. Avramov, K. & Mihlin, Yu. (2013). Review of applications of nonlinear normal modes for vibrating mechanical systems. Appl. Mech. Reviews, vol. 65, iss. 2, pp. 4–25.
  5. Ostrovsky, L. A. & Starobinets, I. M. (1995). Transitions and statistical characteristics of vibrations in a bimodal oscillator. Chaos, vol. 5, iss. 3, pp. 496–500.
  6. Bishop, R. S. (1994). Impact oscillators. Philosophy Transactions of Royal Society, no. A347, pp. 347–351.
  7. Avramov, K. V. (2001). Bifurcation analysis of a vibropercussion system by the method of amplitude surfaces. Intern. Appl. Mech., vol. 38, iss. 9, pp. 1151–1156.
  8. Avramov, K. & Raimberdiyev, T. (2017). Bifurcations behavior of bending vibrations of beams with two breathing cracks. Eng. Fracture Mech., no. 178, pp. 22–38.
  9. Avramov, K. & Raimberdiyev, T. (2017). Modal asymptotic analysis of sub-harmonic and quasi-periodic flexural vibrations of beams with fatigue crack. Nonlinear Dynamics, vol. 88, iss. 2, pp. 1213–1228.
  10. Bovsunovsky, A. P. & Surace, C. (2005). Considerations regarding superharmonic vibrations of a cracked beam and the variation in damping caused by the presence of the crack. Journal of Sound and Vibrations, vol. 288, iss. 4–5, pp. 865–886.
  11. Ji, J. C. & Hansen, H. (2005). On the approximate solution of a piecewise nonlinear oscillator under superharmonic resonance. Journal of Sound and Vibrations, vol. 283, iss. 1–2, pp. 467–474.
  12. Chen, S. C. & Shaw, S. W. (1996). Normal modes for piecewise linear vibratory systems. Nonlinear dynamics, vol. 10, iss. 2, pp. 135–164.
  13. Jiang, D., Pierre, C., & Shaw, S. W. (2004). Large amplitude non-linear normal modes of piecewise linear systems. Journal of Sound and Vibration, vol. 272, iss. 3-5,  pp. 869–891.
  14. Uspensky, B. V. & Avramov, K. V. (2014). On the nonlinear normal modes of free vibration of piecewise linear systems. Journal of Sound and Vibration, vol. 333, iss. 14,  pp. 3252–3265.
  15. Uspensky, B. & Avramov, K. (2014). Nonlinear modes of piecewise linear systems under the action of periodic excitation. Nonlinear dynamics, vol. 76, iss. 2,  pp. 1151–1156.
  16. Vakakis, A., Manevich, L. I., Mikhlin, Yu. V., Pilipchuk, N., & Zevin, A. A. (1996). Normal modes and localization in nonlinear systems. New York:  Wiley Interscience, 780 p.
  17. Nayfeh, A. H., & Mook, D. T. (1995). Nonlinear oscillations. New York:  John Wiley and Sons, 720 p.
  18. Parlitz, U. (1993). Common dynamical features of periodically driven strictly dissipative oscillators. Intern. J. Bifurcation and Chaos, vol. 3, no. 3, pp. 703–715.


Received 23 October 2017

Published 30 December 2017