Skip to main content
SpringerLink
Log in

Analytical period-1 motions to chaos in a two-degree-of-freedom oscillator with a hardening nonlinear spring

  • Published:
International Journal of Dynamics and Control Aims and scope Submit manuscript

Abstract

In this paper, obtained are analytical solutions for period-1 motions to chaos in a two-degree-of-freedom (2-DOF) oscillator with a nonlinear hardening spring. From the finite Fourier series transformation, a dynamical system of coefficients of the finite Fourier series is developed to determine existence, stability and bifurcations of periodic motions in such a 2-DOF nonlinear oscillator. The equilibriums of such a dynamical system of coefficients give the analytical solutions of period-m motions, and the corresponding stability and bifurcations of period-m motions are determined through the eigenvalue analysis of equilibriums. Analytical bifurcation trees of period-1 motions to chaos are presented through frequency–amplitude curves. From the frequency–amplitude curves, the harmonic effects on the periodic motions can be discussed and nonlinear behaviors of periodic motions can be determined. Displacements, velocity, and trajectories of periodic motions in the 2-DOF nonlinear oscillator are used to illustrate motion complexity, and harmonic amplitude spectrums give harmonic effects on periodic motions of the 2-DOF nonlinear oscillator. Through the analytical solutions, the complex dynamics of the 2-DOF nonlinear oscillator is studied. The analytical solutions presented herein for periodic motions in such 2-DOF systems can be used to further discuss the corresponding nonlinear behaviors, and can also be applied to engineering for design.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

References

  1. Luo ACJ, Yu B (2015) Bifurcation trees of period-1 motions to chaos in a two-degree-of-freedom, nonlinear oscillator. Int J Bifurc Chaos (in press)

  2. Moochhala YE, Raynor S (1972) Free vibration of multi-degree-of-freedom non-linear systems. Int J Non-Linear Mech 7(6):651–661

    Article  MATH  Google Scholar 

  3. Masri SF (1972) Forced vibration of a class of non-linear two-degree-of-freedom oscillator. Int J Non-Linear Mech 7(6):663–674

    Article  Google Scholar 

  4. Vakakis AF, Rand RH (1992) Normal modes and global dynamics of a two-degree-of-freedom non-linear system-I. Low energies. Int J Non-linear Mech 27(5):861–874

    Article  MATH  Google Scholar 

  5. Vakakis AF, Rand RH (2004) Nonlinear dynamics of a system of coupled oscillators with essential stiffness nonlinearities. Int J Non-linear Mech 39:1079–1091

    Article  MATH  Google Scholar 

  6. Zhu SJ, Zheng YF, Fu YM (2004) Analysis of non-linear dynamics of a two-degree-of-freedom vibration with nonlinear damping spring. J Sound Vib 271:15–24

    Article  Google Scholar 

  7. Childs DW (1982) Fractional-frequency rotor motion due to nonsymmetric clearance effects. ASME J Energy Power 104:533–541

    Article  Google Scholar 

  8. Choi YS, Noah ST (1987) Nonlinear steady-state responses of a rotor-support system. ASME J Vib Acoust Stress Reliab Des 109:255–261

    Article  Google Scholar 

  9. Day WB (1987) Asymmetric expansion in nonlinear rotordynamics. Q Appl Math 44:779–792

    Article  MATH  Google Scholar 

  10. Enrich FF (1988) Higher order subharmonic responses of high speed rotors in bearing clearance. ASME J Vib Acoust Stress Reliab Des 110:9–16

    Article  Google Scholar 

  11. Kim YB, Noah YB (1990) Bifurcation analysis for a modified Jeffcott rotor with bearing clearances. Nonlinear Dyn 1:221–241

    Article  Google Scholar 

  12. Choi SK, Noah ST (1994) Mode-locking and chaos in a Jeffcott rotor with bearing clearances. ASME J Appl Mech 61:131–138

    Article  MATH  Google Scholar 

  13. Chu F, Zhang Z (1998) Bifurcation and chaos in a rub-impact Jeffcott with bearing effects. J Sound Vib 210:1–18

    Article  Google Scholar 

  14. Luo ACJ (2012) Continuous dynamical systems. Higher Education Press/L&H Scientific Publishing, Beijing/Glen Carbon

    MATH  Google Scholar 

  15. Luo ACJ, Huang JZ (2012) Approximate solutions of periodic motions in nonlinear systems via generalized harmonic balance. J Vib Control 18:1661–1671

    Article  MathSciNet  Google Scholar 

  16. Luo ACJ, Huang JZ (2012) Analytical dynamics of period-m flows and chaos in nonlinear systems. Int J Bifurc Chaos 22(4):29

    Article  MathSciNet  MATH  Google Scholar 

  17. Luo ACJ, Huang JZ (2012) Analytical routes of period-1 motions to chaos in a periodically forced Duffing oscillator with a twin-well potential. J Appl Nonlinear Dyn 1:73–108

    Article  MATH  Google Scholar 

  18. Luo ACJ, Huang JZ (2012) Unstable and stable period-\(m\) motions in a twin-well potential Duffing oscillator. Discontin Nonlinearity Complex 1:113–145

    Article  MATH  Google Scholar 

  19. Luo ACJ, Yu B (2013) Analytical solutions for stable and unstable period-1 motions in a periodically forced oscillator with quadratic nonlinearity. ASME J Vib Acoust 135(3):5

    Article  Google Scholar 

  20. Luo ACJ, Yu B (2015) Complex period-1 motions in a periodically forced, quadratic nonlinear oscillator. J Vib Control 21:896–906

    Article  MathSciNet  Google Scholar 

  21. Luo ACJ, Yu B (2013) Period-m motions and bifurcation trees in a periodically excited, quadratic nonlinear oscillator. Discontin Nonlinearity Complex 2(3):263–288

    Article  MATH  Google Scholar 

  22. Luo ACJ, Laken AB (2013) Analytical solutions for period-m motions in a periodically forced van del Pol oscillator. Int J Dyn Control 1(2):99–155

    Article  Google Scholar 

  23. Luo ACJ, Yu B (2014) Bifurcation trees of periodic motion to chaos in a parametric, quadratic nonlinear oscillator. Int J Bifurc Chaos 24:28

    MathSciNet  MATH  Google Scholar 

  24. Luo ACJ, Yu B (2014) On effects of excitation amplitude on bifurcation trees of periodic motions in a parametric, quadratic nonlinear oscillator. Nonlinear Dyn Mob Robot 1(2):231–263

    Google Scholar 

  25. Luo ACJ, Yu B (2014) Analytical routes of period-m motions to chaos in a parametric, quadratic nonlinear oscillator. Int J Dyn Control. doi:10.1007/s40435-014-0112-7

    Google Scholar 

  26. Huang JZ, Luo ACJ (2014) Analytical periodic motions and bifurcations in a nonlinear rotor system. Int J Dyn Control 2:425–459

    Article  Google Scholar 

  27. Huang JZ, Luo ACJ (2015) Periodic motions and bifurcation trees in a buckled, nonlinear Jeffcott rotor system. Int J Bifurc Chaos 28(1):34

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanical and Industrial Engineering, Southern Illinois University Edwardsville, Edwardsville, IL, 62026-1805, USA

    Bo Yu & Albert C. J. Luo

Authors
  1. Bo Yu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Albert C. J. Luo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Albert C. J. Luo.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yu, B., Luo, A.C.J. Analytical period-1 motions to chaos in a two-degree-of-freedom oscillator with a hardening nonlinear spring. Int. J. Dynam. Control 5, 436–453 (2017). https://doi.org/10.1007/s40435-015-0216-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40435-015-0216-8

Keywords

Search

Navigation