On the dynamics of vibro-impact systems with ideal and non-ideal excitation


This study is concerned with modelling and analyses of a vibro-impact system consisting of a crank-slider mechanism and one oscillator attached to it, where the system is exposed to a non-ideal excitation. The impact occurs during the motion of the oscillator when it fits a base, and the excitation of the driving source is affected by this behaviour. The aim is to determine the interaction between a driving torque and the motion of the oscillator. To achieve this aim in a methodologically sound manner, both vibrating and vibro-impact systems with an ideal and non-ideal excitation are analysed. Analytical and numerical solutions are obtained for the vibrating system with the ideal excitation. Numerical analyses of the vibrating system with the non-ideal excitation is then conducted, where the characteristic curves for this system are found analytically. Numerical simulations are also carried out for other two systems and the results obtained are shown in terms of frequency–response diagrams, time-displacement diagrams and basins of attraction. The results found for different systems are compared mutually, and the differences between them are pointed out. Impact solutions for different regions of the excitation frequency are shown. For the vibro-impact system with the non-ideal excitation, the average value of its frequency is used.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15


  1. 1.

    Babitsky VI (1998) Theory of vibro- impact systems and applications. Springer, Berlin

    Google Scholar 

  2. 2.

    Karayannis I, Vakakis AF, Georgiades F (2008) Vibro-impact attachments as shock absorbers. Proc Inst Mech Eng Part C J Mech Eng Sci 222:1899–1908

    Article  Google Scholar 

  3. 3.

    Rong H, Wang X, Xu W, Fang T (2010) Resonant response of a non-linear vibro-impact system to combined deterministic harmonic and random excitations. Int J Non-Linear Mech 45(5):474–481

    Article  Google Scholar 

  4. 4.

    Liu Y, Wiercigroch M, Pavlovskaia E, Yu H (2013) Modelling of a vibro-impact capsule system. Int J Mech Sci 66:2–11

    Article  Google Scholar 

  5. 5.

    Tosic P (2018) Forced oscillations of vibro-impact system (In Serbian). Master thesis, University of Novi Sad, Serbia

  6. 6.

    Marzbanrad J, Shahsavar M, Beyranvand B (2017) Analysis of force and energy density transferred to barrier in a single degree of freedom vibro-impact system. J Cent South Univ 24(6):1351–1359

    Article  Google Scholar 

  7. 7.

    Raouf AI (2009) Vibro-impact dynamics—modeling, mapping and applications. Springer, Berlin

    Google Scholar 

  8. 8.

    Batako AD, Babitsky VI, Halliwell NA (2004) Modelling of vibro-impact penetration of self-exciting percussive-rotary drill bit. J Sound Vib 271:209–225

    Article  Google Scholar 

  9. 9.

    Yuri VM, Reshetnikova SN (2006) Dynamical interaction of an elastic system and a vibro-impact absorber. Math Probl Eng 2006:1–15

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Kononenko VO (1969) Vibrating systems with a limited power supply. Iliffe, London

    Google Scholar 

  11. 11.

    Zukovic M, Cveticanin L, Maretic R (2012) Dynamics of the cutting mechanism with flexible support and non-ideal forcing. Mech Mach Theory 58:1–12

    Article  Google Scholar 

  12. 12.

    Cveticanin L, Zukovic M, Balthazar JM (2018) Dynamics of mechanical systems with non-ideal excitation. Mathematical Engineering. Springer, Berlin

    Google Scholar 

  13. 13.

    Balthazar JM, Mook DT, Weber HI, Fenili A, Belato D, Felix JLP (2003) An overview on non-ideal vibrations. Meccanica 38:613–621

    Article  Google Scholar 

  14. 14.

    Cveticanin L, Zukovic M (2015) Motion of a motor-structure non-ideal system. Eur J Mech A Solids 53:229–240

    MathSciNet  Article  Google Scholar 

  15. 15.

    Cveticanin L, Zukovic M (2013) Non-ideal mechanical system with an oscillator with rational nonlinearity. J Vib Control 21(11):2149–2164

    MathSciNet  Article  Google Scholar 

  16. 16.

    Karthikeyan M, Bisoi A, Samantaray AK, Bhattacharyya R (2015) Sommerfeld effect characterization in rotors with non-ideal drive from ideal drive response and power balance. Mech Mach Theory 91:269–288

    Article  Google Scholar 

  17. 17.

    Kovacic I, Zukovic M (2016) Coupled purely nonlinear oscillators: normal modes and exact solutions for free and forced responses. Nonlinear Dyn 87(1):713–726

    Article  Google Scholar 

  18. 18.

    Samantaray AK, Dasgupta SS, Bhattacharyya RR (2010) Sommerfeld effect in rotationally symmetric planar dynamics systems. Int J Eng Sci 48(1):21–36

    Article  Google Scholar 

  19. 19.

    Warminski J, Balthazar JM (2003) Vibrations of a parametrically and self-excited system with ideal and non-ideal energy sources. J Braz Soc Mech Sci Eng 25(4):413–420

    Article  Google Scholar 

  20. 20.

    Zukovic M, Cveticanin L (2009) Chaos in non-ideal mechanical system with clearance. J Vib Control 15(8):1229–1246

    MathSciNet  Article  Google Scholar 

  21. 21.

    Lampart M, Zapomel J (2013) Dynamics of the electromechanical system with impact element. J Sound Vib 332:701–713

    Article  Google Scholar 

  22. 22.

    Lampart M, Zapomel J (2014) Dynamic properties of the electromechanical system damped by an impact element with soft stops. Int J Appl Mech 06(02):1450016

    Article  Google Scholar 

  23. 23.

    Moraes FH, Pontes BR Jr, Silveira M, Balthazar JM (2013) Influence of ideal and non-ideal excitation sources on the dynamics of a nonlinear vibro-impact system. J Theor Appl Mech 51:763–774

    Google Scholar 

  24. 24.

    Souza SLT, Caldas IL, Viana RL, Balthazar JM (2008) Control and chaos for vibro-impact and non-ideal oscillators. J Theor Appl Mech 46:641–664

    Google Scholar 

  25. 25.

    Moraes FH, Pontes BR Jr, Silveira M, Balthazar JM (2005) Impact dampers for controlling chaos in systems with limited power supply. J Sound Vib 279:955–967

    Article  Google Scholar 

  26. 26.

    Dormand JR, Prince PJ (1980) A family of embedded Runge–Kutta formulae. J. Comp. Appl. Math. 6:19–26

    MathSciNet  Article  Google Scholar 

  27. 27.

    Shampine LF, Reichelt MW (1997) The MATLAB ODE suite. SIAM J Sci Comput 18:1–22

    MathSciNet  Article  Google Scholar 

Download references


The first and the third author acknowledge support of the Ministry of Education and Science of the Republic of Serbia.

Author information



Corresponding author

Correspondence to Dzanko Hajradinovic.

Ethics declarations

Conflict of interest

:The Authors declare that there is no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Zukovic, M., Hajradinovic, D. & Kovacic, I. On the dynamics of vibro-impact systems with ideal and non-ideal excitation. Meccanica 56, 439–460 (2021). https://doi.org/10.1007/s11012-020-01280-5

Download citation


  • Vibro-impact systems
  • Frequency–response diagram
  • Impact solutions
  • Non-ideal excitation
  • Limited power supply