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Primary and subharmonic simultaneous resonance of fractional-order Duffing oscillator

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Abstract

The primary and subharmonic simultaneous resonance of Duffing oscillator with fractional-order derivative is studied. Firstly, the approximately analytical solution of the resonance is obtained by the method of multiple scales, and the correctness and satisfactory precision of the analytical solution are verified by numerical simulation. Then, the amplitude–frequency curve equation and phase–frequency curve equation are derived from the analytical solution. The stability condition of the steady-state response is obtained by Lyapunov’s first method, and the state switching between two stable periodic orbits is demonstrated. Finally, the effects of nonlinear factor on the system response are analyzed, and the difference between stiffness softening and stiffness hardening system is demonstrated. The influence of fractional-order term on the system is analyzed in depth, and the effect mechanism of fractional-order term is revealed, i.e., the focus and intensity of effect are determined by the order and coefficient of the fractional-order derivative, respectively.

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References

  1. Caputo, M., Mainardi, F.: A new dissipation model based on memory mechanism. Pure. Appl. Geophys. 91(1), 134–147 (1971)

    Article  MATH  Google Scholar 

  2. Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999)

    MATH  Google Scholar 

  3. Podlubny, I.: Fractional-order systems and PIλDμ-controllers. IEEE Trans. Autom. Control 44(1), 208–214 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  4. Shen, Y.J., Niu, J.C., Yang, S.P., Li, S.J.: Primary resonance of dry-friction oscillator with fractional-order Proportional-Integral-Derivative controller of velocity feedback. J. Comput. Nonlinear Dyn. 11(5), 051027 (2016)

    Article  Google Scholar 

  5. Niu, J.C., Shen, Y.J., Yang, S.P., Li, S.J.: Analysis of Duffing oscillator with time-delayed fractional-order PID controller. Int. J. Non-Linear Mech. 92, 66–75 (2017)

    Article  Google Scholar 

  6. Yang, B., Yu, T., Shu, H.C., Zhu, D.N., Zeng, F., Sang, Y.Y., Jiang, L.: Perturbation observer based fractional-order PID control of photovoltaics inverters for solar energy harvesting via Yin-Yang-Pair optimization. Energy Convers. Manag. 171, 170–187 (2018)

    Article  Google Scholar 

  7. Yaghi, M., Efe, M.O.: Fractional order PID control of a radar guided missile under disturbances. In: 9th International Conference on Information and Communication Systems (ICICS), IEEE, pp. 238–242 (2018)

  8. Lubich, C.: Discretized fractional calculus. SIAM J. Math. Anal. 17(3), 704–719 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  9. Diethelm, K., Ford, N.J., Freed, A.D.: A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dyn. 29(1), 3–22 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  10. Diethelm, K., Ford, N.J., Freed, A.D.: Detailed error analysis for a fractional Adams method. Numer. Algorithms 36(1), 31–52 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  11. Zhu, Z.Y., Li, G.G., Cheng, C.J.: A numerical method for fractional integral with applications. Appl. Math. Mech. 24(4), 373–384 (2003)

    Article  MATH  Google Scholar 

  12. Kumar, P., Agrawal, O.P.: An approximate method for numerical solution of fractional differential equations. Sig. Process. 86(10), 2602–2610 (2006)

    Article  MATH  Google Scholar 

  13. Cao, J.Y., Xu, C.J.: A high order schema for the numerical solution of the fractional ordinary differential equations. J. Comput. Phys. 238, 154–168 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  14. Cermak, J., Nechvatal, L.: Stability and chaos in the fractional Chen system. Chaos Solitons Fractals 125, 24–33 (2019)

    Article  MathSciNet  Google Scholar 

  15. Lei, Y.M., Fu, R., Yang, Y., Wang, Y.Y.: Dichotomous-noise-induced chaos in a generalized Duffing-type oscillator with fractional-order deflection. J. Sound Vib. 363, 68–76 (2016)

    Article  Google Scholar 

  16. Niu, J.C., Liu, R.Y., Shen, Y.J., Yang, S.P.: Chaos detection of Duffing system with fractional-order derivative by Melnikov method. Chaos 29(12), 123106 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  17. Leung, A.Y.T., Yang, H.X., Zhu, P.: Periodic bifurcation of Duffing-van der Pol oscillators having fractional derivatives and time delay. Commun. Nonlinear Sci. Numer. Simul. 19(4), 1142–1155 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  18. Leung, A.Y.T., Yang, H.X., Guo, Z.J.: The residue harmonic balance for fractional order van der Pol like oscillators. J. Sound Vib. 331(5), 1115–1126 (2012)

    Article  Google Scholar 

  19. 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(5), 567–582 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  20. Eshaghi, S., Ghaziani, R.K., Ansari, A.: Hopf bifurcation, chaos control and synchronization of a chaotic fractional-order system with chaos entanglement function. Math. Comput. Simul. 172, 321–340 (2020)

    Article  MathSciNet  Google Scholar 

  21. Eshaghi, S., Khoshsiar Ghaziani, R., Ansari, A.: Stability and chaos control of regularized Prabhakar fractional dynamical systems without and with delay. Math. Methods Appl. Sci. 42(7), 2302–2323 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  22. Du, L., Zhao, Y.P., Lei, Y.M., Hu, J., Yue, X.L.: Suppression of chaos in a generalized Duffing oscillator with fractional-order deflection. Nonlinear Dyn. 92(4), 1921–1933 (2018)

    Article  MATH  Google Scholar 

  23. Shen, Y.J., Yang, S.P., Xing, H.J., Gao, G.S.: Primary resonance of Duffing oscillator with fractional-order derivative. Commun. Nonlinear Sci. Numer. Simul. 17(7), 3092–3100 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  24. Shen, Y.J., Wei, P., Yang, S.P.: Primary resonance of fractional-order van der Pol oscillator. Nonlinear Dyn. 77(4), 1629–1642 (2014)

    Article  MathSciNet  Google Scholar 

  25. Van Khang, N., Chien, T.Q.: Subharmonic resonance of Duffing oscillator with fractional-order derivative. J. Comput. Nonlinear Dyn. 11(5), 051018 (2016)

    Article  Google Scholar 

  26. Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations. Wiley, New York (1979)

    MATH  Google Scholar 

  27. Yang, S.P., Nayfeh, A.H., Mook, D.T.: Combination resonances in the response of the Duffing oscillator to a three-frequency excitation. Acta Mech. 131(3), 235–245 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  28. Kacem, N., Baguet, S., Dufour, R., Hentz, S.: Stability control of nonlinear micromechanical resonators under simultaneous primary and superharmonic resonances. Appl. Phys. Lett. 98(19), 193507 (2011)

    Article  Google Scholar 

  29. Leung, A.Y.T., Yang, H.X., Zhu, P.: Bifurcation of a Duffing oscillator having nonlinear fractional derivative feedback. Int. J. Bifurcat. Chaos 24(03), 1450028 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  30. Zhao, G.Y., Raze, G., Paknejad, A., Deraemaeker, A., Kerschen, G., Collette, C.: Active nonlinear inerter damper for vibration mitigation of Duffing oscillators. J. Sound Vib. 473, 115236 (2020)

    Article  Google Scholar 

  31. Ding, H., Zhang, G.C., Chen, L.Q., Yang, S.P.: Forced vibrations of supercritically transporting viscoelastic beams. J. Vib. Acoust. 134(5), 051007 (2012)

    Article  Google Scholar 

  32. Ding, H., Huang, L.L., Mao, X.Y., Chen, L.Q.: Primary resonance of traveling viscoelastic beam under internal resonance. Appl. Math. Mech. 38(1), 1–14 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  33. Ertas, A., Chew, E.K.: Non-linear dynamic response of a rotating machine. Int. J. Non-Linear Mech. 25(2–3), 241–251 (1990)

    Article  MATH  Google Scholar 

  34. Pan, R., Davies, H.G.: Responses of a non-linearly coupled pitch-roll ship model under harmonic excitation. Nonlinear Dyn. 9(4), 349–368 (1996)

    Article  Google Scholar 

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

    Book  MATH  Google Scholar 

  36. Wang, Y. L., Yang, Z. B., Li, P. Y., Cao, D. Q., Huang, W. H., Inman, D.J.: Energy harvesting for jet engine monitoring. Nano Energy, 104853 (2020)

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Acknowledgements

This work is supported by the National Natural Science Foundation of China (Grant Nos. U1934201 and 11772206).

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Authors and Affiliations

  1. State Key Laboratory of Mechanical Behavior and System Safety of Traffic Engineering Structures, Shijiazhuang Tiedao University, Shijiazhuang, 050043, China

    Yongjun Shen & Shaopu Yang

  2. Department of Mechanical Engineering, Shijiazhuang Tiedao University, Shijiazhuang, 050043, China

    Yongjun Shen, Hang Li, Shaopu Yang, Mengfei Peng & Yanjun Han

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  1. Yongjun Shen
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  2. Hang Li
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Correspondence to Yongjun Shen.

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Shen, Y., Li, H., Yang, S. et al. Primary and subharmonic simultaneous resonance of fractional-order Duffing oscillator. Nonlinear Dyn 102, 1485–1497 (2020). https://doi.org/10.1007/s11071-020-06048-w

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