Research Papers

Superharmonic Resonance of Fractional-Order Mathieu–Duffing Oscillator

[+] Author and Article Information
Jiangchuan Niu

School of Mechanical Engineering,
Shijiazhuang Tiedao University,
Shijiazhuang 050043, China
e-mail: menjc@163.com

Xiaofeng Li

School of Mechatronical Engineering,
Beijing Institute of Technology,
Beijing 100081, China
e-mail: lixiaofeng@bit.edu.cn

Haijun Xing

State Key Laboratory of Mechanical Behavior in
Traffic Engineering Structure and System Safety,
Shijiazhuang 050043, China
e-mail: xinghj@stdu.edu.cn

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received September 10, 2018; final manuscript received April 9, 2019; published online May 13, 2019. Assoc. Editor: Brian Feeny.

J. Comput. Nonlinear Dynam 14(7), 071005 (May 13, 2019) (10 pages) Paper No: CND-18-1398; doi: 10.1115/1.4043523 History: Received September 10, 2018; Revised April 09, 2019

The superharmonic resonance of fractional-order Mathieu–Duffing oscillator subjected to external harmonic excitation is investigated. Based on the Krylov–Bogolubov–Mitropolsky (KBM) asymptotic method, the approximate analytical solution for the third superharmonic resonance under parametric-forced joint resonance is obtained, where the unified expressions of the fractional-order term with fractional order from 0 to 2 are gained. The amplitude–frequency equation for steady-state solution and corresponding stability condition are also presented. The correctness of the approximate analytical results is verified by numerical results. The effects of the fractional-order term, excitation amplitudes, and nonlinear stiffness coefficient on the superharmonic resonance response of the system are analyzed in detail. The results show that the KBM method is effective to analyze dynamic response in a fractional-order Mathieu–Duffing system.

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Li, X. H. , Hou, J. Y. , and Chen, J. F. , 2016, “ An Analytical Method for Mathieu Oscillator Based on Method of Variation of Parameter,” Commun. Nonlinear Sci. Numer. Simul., 37, pp. 326–353. [CrossRef]
Choudhury, A. G. , and Guha, P. , 2014, “ Damped Equations of Mathieu Type,” Appl. Math. Comput., 229, pp. 85–93.
Abou-Rayan, A. M. , Nayfeh, A. H. , Mook, D. T. , and Nayfeh, M. A. , 1993, “ Nonlinear Response of a Parametrically Excited Buckled Beam,” Nonlinear Dyn., 4(5), pp. 499–525. [CrossRef]
Luo, A. C. J. , and O'Connor, D. M. , 2014, “ On Periodic Motions in a Parametric Hardening Duffing Oscillator,” Int. J. Bifurcation Chaos, 24(1), p. 1430004. [CrossRef]
Hou, D. X. , Zhao, H. X. , and Liu, B. , 2013, “ Bifurcation and Chaos in Some Relative Rotation Systems With Mathieu-Duffing Oscillator,” Acta Phys. Sin., 62(23), p. 234501.
Zounes, R. S. , and Rand, R. H. , 2002, “ Subharmonic Resonance in the Non-Linear Mathieu Equation,” Int. J. Non-Linear Mech., 37(1), pp. 43–73. [CrossRef]
Ng, L. , and Rand, R. , 2002, “ Bifurcations in a Mathieu Equation With Cubic Nonlinearities,” Chaos, Solitons Fractals, 14(2), pp. 173–181. [CrossRef]
Shen, J. H. , Lin, K. C. , Chen, S. H. , and Sze, K. Y. , 2008, “ Bifurcation and Route-to-Chaos Analyses for Mathieu-Duffing Oscillator by the Incremental Harmonic Balance Method,” Nonlinear Dyn., 52(4), pp. 403–414. [CrossRef]
Petras, I. , 2011, Fractional-Order Nonlinear Systems, Higher Education Press, Beijing, China.
Podlubny, I. , 1999, Fractional Differential Equations, Mathematics in Science and Engineering, Academic Press, New York.
Li, C. P. , and Deng, W. H. , 2007, “ Remarks on Fractional Derivatives,” Appl. Math. Comput., 187(2), pp. 777–784.
Cao, J. X. , Ding, H. F. , and Li, C. P. , 2013, “ Implicit Difference Schemes for Fractional Diffusion Equations,” Commun. Appl. Math. Comput., 40(4), pp. 61–74.
Sun, H. G. , Zhang, Y. , Baleanu, D. , Chen, W. , and Chen, Y. Q. , 2018, “ A New Collection of Real World Applications of Fractional Calculus in Science and Engineering,” Commun. Nonlinear Sci. Numer. Simul., 64, pp. 213–231. [CrossRef]
Makris, N. , and Constantinou, M. C. , 1991, “ Fractional-Derivative Maxwell Model for Viscous Dampers,” J. Struct. Eng., 117(9), pp. 2708–2724. [CrossRef]
Mainardi, F. , 2010, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, World Scientific Publishing, Singapore.
Xu, Z. , and Chen, W. , 2013, “ A Fractional-Order Model on New Experiments of Linear Viscoelastic Creep of Hami Melon,” Comput. Math. Appl., 66(5), pp. 677–681. [CrossRef]
Cai, W. , Chen, W. , and Xu, W. , 2017, “ Fractional Modeling of Pasternak-Type Viscoelastic Foundation,” Mech. Time-Depend. Mater., 21(1), pp. 119–131. [CrossRef]
Chen, J. H. , and Chen, W. C. , 2008, “ Chaotic Dynamics of the Fractionally Damped Van Der Pol Equation,” Chaos, Solitons Fractals, 35(1), pp. 188–198. [CrossRef]
Song, C. , Cao, J. D. , and Liu, Y. Z. , 2015, “ Robust Consensus of Fractional-Order Multi-Agent Systems With Positive Real Uncertainty Via Second-Order Neighbors Information,” Neurocomputing, 165, pp. 293–299. [CrossRef]
Podlubny, I. , 1999, “ Fractional-Order Systems and PIλDμ–Controllers,” IEEE Trans. Autom. Control, 44(1), pp. 208–214. [CrossRef]
Tavazoei, M. S. , and Haeri, M. , 2008, “ Synchronization of Chaotic Fractional-Order Systems Via Active Sliding Mode Controller,” Physica A, 387(1), pp. 57–70. [CrossRef]
Yang, N. N. , and Liu, C. X. , 2013, “ A Novel Fractional-Order Hyperchaotic System Stabilization Via Fractional Sliding-Mode Control,” Nonlinear Dyn., 74(3), pp. 721–732. [CrossRef]
Shen, Y. J. , Wei, P. , and Yang, S. P. , 2014, “ Primary Resonance of Fractional-Order Van Der Pol Oscillator,” Nonlinear Dyn., 77(4), pp. 1629–1642. [CrossRef]
Rossikhin, Y. A. , and Shitikova, M. V. , 1997, “ Application of Fractional Derivatives to the Analysis of Damped Vibrations of Viscoelastic Single Mass Systems,” Acta Mech., 120(1–4), pp. 109–125. [CrossRef]
Xu, Y. , Li, Y. G. , Liu, D. , Jia, W. T. , and Huang, H. , 2013, “ Responses of Duffing Oscillator With Fractional Damping and Random Phase,” Nonlinear Dyn., 74(3), pp. 745–753. [CrossRef]
Shen, Y. J. , Yang, S. P. , Xing, H. J. , and Gao, G. S. , 2012, “ Primary Resonance of Duffing Oscillator With Fractional-Order Derivative,” Commun. Nonlinear Sci. Numer. Simul., 17(7), pp. 3092–3100. [CrossRef]
Wang, Z. H. , and Zheng, Y. G. , 2009, “ The Optimal Form of the Fractional-Order Difference Feedbacks in Enhancing the Stability of a Sdof Vibration System,” J. Sound Vib., 326(3–5), pp. 476–488. [CrossRef]
Guo, Z. J. , Leung, A. Y. T. , and Yang, H. X. , 2011, “ Oscillatory Region and Asymptotic Solution of Fractional Van Der Pol Oscillator Via Residue Harmonic Balance Technique,” Appl. Math. Modell., 35(8), pp. 3918–3925. [CrossRef]
Leung, A. Y. T. , Yang, H. X. , and Zhu, P. , 2014, “ Periodic Bifurcation of Duffing-Van Der Pol Oscillators Having Fractional Derivatives and Time Delay,” Commun. Nonlinear Sci. Numer. Simul., 19(4), pp. 1142–1155. [CrossRef]
Chen, L. C. , Zhao, T. L. , Li, W. , and Zhao, J. , 2016, “ Bifurcation Control of Bounded Noise Excited Duffing Oscillator by a Weakly Fractional-Order PID Feedback Controller,” Nonlinear Dyn., 83(1–2), pp. 529–539. [CrossRef]
Hamamci, S. E. , 2007, “ Stabilization Using Fractional-Order PI and PID Controllers,” Nonlinear Dyn., 51(1–2), pp. 329–343. [CrossRef]
Wen, S. F. , Shen, Y. J. , Wang, X. N. , Yang, S. P. , and Xing, H. J. , 2016, “ Dynamical Analysis of Strongly Nonlinear Fractional-Order Mathieu-Duffing Equation,” Chaos, 26(8), pp. 446–451. [CrossRef]
Wen, S. F. , Shen, Y. J. , Yang, S. P. , and Wang, J. , 2017, “ Dynamical Response of Mathieu-Duffing Oscillator With Fractional-Order Delayed Feedback,” Chaos, Solitons Fractals, 94, pp. 54–62. [CrossRef]
Yang, J. H. , Sanjuán, M. A. F. , and Liu, H. G. , 2015, “ Bifurcation and Resonance in a Fractional Mathieu-Duffing Oscillator,” Eur. Phys. J. B, 88(11), p. 310. [CrossRef]
Awrejcewicz, J. , 2014, Ordinary Differential Equations and Mechanical Systems, Springer International Publishing, Cham, Switzerland.
Shen, Y. J. , Yang, S. P. , and Sui, C. Y. , 2014, “ Analysis on Limit Cycle of Fractional-Order Van Der Pol Oscillator,” Chaos, Solitons Fractals, 67, pp. 94–102. [CrossRef]
Rand, R. H. , Sah, S. M. , and Suchorsky, M. K. , 2010, “ Fractional Mathieu Equation,” Commun. Nonlinear Sci. Numer. Simul., 15(11), pp. 3254–3262. [CrossRef]
Niu, J. C. , Gutierrez, H. , and Ren, B. , 2018, “ Resonance Analysis of Fractional-Order Mathieu Oscillator,” ASME J. Comput. Nonlinear Dyn., 13(5), p. 051003. [CrossRef]


Grahic Jump Location
Fig. 1

Comparison of superharmonic resonance for p =0.5

Grahic Jump Location
Fig. 2

Comparison of superharmonic resonance for p =1

Grahic Jump Location
Fig. 3

Comparison of superharmonic resonance for p =1.5

Grahic Jump Location
Fig. 4

Peak amplitude value of superharmonic resonance withK1

Grahic Jump Location
Fig. 5

Frequency of the superharmonic resonance with K1

Grahic Jump Location
Fig. 6

Peak amplitude value of superharmonic resonance withp

Grahic Jump Location
Fig. 7

Frequency of the superharmonic resonance with p

Grahic Jump Location
Fig. 8

Amplitude of superharmonic resonance with differentβ1

Grahic Jump Location
Fig. 9

Amplitude of superharmonic resonance with different F

Grahic Jump Location
Fig. 10

Amplitude of superharmonic resonance with differentα1



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