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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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Figures

Grahic Jump Location
Fig. 1

Comparison of superharmonic resonance for p =0.5

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Fig. 2

Comparison of superharmonic resonance for p =1

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Fig. 3

Comparison of superharmonic resonance for p =1.5

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Fig. 4

Peak amplitude value of superharmonic resonance withK1

Grahic Jump Location
Fig. 5

Frequency of the superharmonic resonance with K1

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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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