Skip to main content
SpringerLink
Log in

Optimization analysis of Duffing oscillator with fractional derivatives

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

The improved version of the constrained optimization harmonic balance method is presented to solve the Duffing oscillator with two kinds of fractional order derivative terms. The analytical gradients of the objective function and nonlinear quality constraints with respect to optimization variables are formulated and the sensitivity information of the Fourier coefficients can also obtained. A new stability analysis method based on the analytical formulation of the nonlinear equality constraints is presented for the nonlinear system with fractional order derivatives. Furthermore, the robust stability boundary of periodic solution can be determined by the interval eigenvalue problem. In addition, the sensitivity information mixed with the interval analysis method is used to quantify the response bounds of periodic solution. Numerical examples show that the proposed approach is valid and effective for analyzing fractional derivative nonlinear system in the presence of uncertainties. It is illustrated that the bifurcation solution in the fractional nonlinear systems may not be sensitive to the variation of the influence parameters.

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. Podlubny, I.: Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. Academic press, London (1998). 198.

    Google Scholar 

  2. Petras, I.: Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation. Higher Education Press, Beijing (2011)

    Book  Google Scholar 

  3. He, G.T., Luo, M.K.: Dynamic behavior of fractional order Duffing chaotic system and its synchronization via singly active control. Appl. Math. Mech. 33, 567–582 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  4. Leung, A.Y.T., Guo, Z., Yang, H.X.: Fractional derivative and time delay damper characteristics in Duffing–van der Pol oscillators. Commun. Nonlinear Sci. Numer Simul. 18(10), 2900–2915 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  5. Xiao, M., Zheng, W.X., Cao, D.: Approximate expressions of a fractional order Van der Pol oscillator by the residue harmonic balance method. Math. Comput. Simul. 89, 1–12 (2013)

    Article  MathSciNet  Google Scholar 

  6. Cao, J.Y., et al.: Nonlinear dynamics of duffing system with fractional order damping. ASME J. Comput. Nonlinear Dyn. 5(4), 041012 (2010)

    Article  Google Scholar 

  7. Cao, J.Y., et al.: Nonlinear dynamic analysis of a cracked rotor-bearing system with fractional order damping. ASME J. Comput. Nonlinear Dyn 8(3), 031008 (2013)

    Article  Google Scholar 

  8. Kovacic, I., Miodrag, Z.: Oscillators with a power-form restoring force and fractional derivative damping: application of averaging. Mech. Res. Commun. 41, 37–43 (2012)

    Article  Google Scholar 

  9. Shen, Y.J., et al.: Primary resonance of Duffing oscillator with two kinds of fractional-order derivatives. Int. J. Non-linear Mech. 47(9), 975–983 (2012)

    Article  Google Scholar 

  10. Shen, Y.J., et al.: Primary resonance of Duffing oscillator with fractional-order derivative. Commun. Nonlinear Sci. Numer Simul. 17(7), 3092–3100 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  11. Gotz, V.G., Ewins, J.: The harmonic balance method with arc-length continuation in rotor/stator contact problems. J. Sound Vib. 241(2), 223–233 (2001)

    Article  Google Scholar 

  12. Peletan, L., et al.: A comparison of stability computational methods for periodic solution of nonlinear problems with application to rotordynamics. Nonlinear Dyn. 72(3), 671–682 (2013)

    Article  MathSciNet  Google Scholar 

  13. Wang, Z.H., Hu, H.Y.: Stability of a linear oscillator with damping force of the fractional-order derivative. Sci. China Phys. Mech. Astron. 53(2), 345–352 (2010)

    Article  Google Scholar 

  14. Li, C.P., Zhang, F.R.: A survey on the stability of fractional differential equations. Eur. Phys. J. Spec. Top. 193(1), 27–47 (2011)

    Article  Google Scholar 

  15. Elishakoff, I., Ohsaki, M.: Optimization and Anti-optimization of Structures Under Uncertainty. Imperial College Press, London (2010)

    Book  MATH  Google Scholar 

  16. Emmanuelle, S., Dessombz, O., Sinou, J.J.: Stochastic study of a non-linear self-excited system with friction. Eur. J. Mech.-A/Solids 40, 1–10 (2013)

    Article  MathSciNet  Google Scholar 

  17. David, M., Vandepitte, D.: A survey of non-probabilistic uncertainty treatment in finite element analysis. Comput. Methods Appl. Mech. Eng. 194(12), 1527–1555 (2005)

    MATH  Google Scholar 

  18. Liao, H.T., Sun, W.: A new method for predicting the maximum vibration amplitude of periodic solution of non-linear system. Nonlinear Dyn. 71(3), 569–582 (2013)

    Article  MathSciNet  Google Scholar 

  19. Tseng, C.C., Pei, S.C., Hsia, S.C.: Computation of fractional derivatives using Fourier transform and digital FIR differentiator. Signal Process. 80(1), 151–159 (2000)

    Article  MATH  Google Scholar 

  20. Arnaud, L., Thomas, O.: A harmonic-based method for computing the stability of periodic solutions of dynamical systems. Comptes Rendus Mécanique 338(9), 510–517 (2010)

  21. Ma, Y.H., Cao, P., Wang, J., Chen, M., Hong, J.: Interval analysis method for rotor dynamics with Uncertain parameters. In: Proceedings of ASME Turbo Expo 2011, Vancouver, Canada, pp. 307–314(2011)

  22. Qiu, Z.P., Wang, X.J.: Parameter perturbation method for dynamic responses of structures with uncertain-but-bounded parameters based on interval analysis. Int. J. Solids Struct. 42(18), 4958–4970 (2005)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

This study has been financially supported by Natural Science Foundation of China (Project No. 10904178).

Author information

Authors and Affiliations

  1. Chinese Aeronautical Establishment, Beijing, 100012, China

    Haitao Liao

Authors
  1. Haitao Liao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Haitao Liao.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liao, H. Optimization analysis of Duffing oscillator with fractional derivatives. Nonlinear Dyn 79, 1311–1328 (2015). https://doi.org/10.1007/s11071-014-1744-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-014-1744-z

Keywords

Search

Navigation