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Research Papers

Periodic Response of a Duffing Oscillator Under Combined Harmonic and Random Excitations

[+] Author and Article Information
Hai-Tao Zhu

Associate Professor
Mem. ASME
Department of Civil Engineering,
Key Laboratory of Coast Civil Structure Safety
Tianjin University,
Ministry of Education,
Tianjin 300072, China
e-mail: htzhu@tju.edu.cn

Siu-Siu Guo

International Center for Applied Mechanics,
State Key Laboratory for Strength
and Vibration of Mechanical Structures,
Xi'an Jiaotong University,
Xi'an 710049, China
e-mail: siusiuguo@mail.xjtu.edu.cn

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received August 29, 2014; final manuscript received January 31, 2015; published online April 15, 2015. Assoc. Editor: Michael Leamy.

J. Vib. Acoust 137(4), 041015 (Aug 01, 2015) (10 pages) Paper No: VIB-14-1321; doi: 10.1115/1.4029993 History: Received August 29, 2014; Revised January 31, 2015; Online April 15, 2015

This paper presents a solution procedure to investigate the periodic response of a Duffing oscillator under combined harmonic and random excitations. The solution procedure consists of an implicit harmonic balance method and a Gaussian closure method. The implicit harmonic balance method, previously developed for the case of harmonic excitation, is extended to the present case of combined harmonic and random excitations with the help of the Gaussian closure method. The amplitudes of the periodic response and the steady variances can be automatically found by the proposed solution procedure. First, the response process is separated into the mean part and the random process part. Then the Gaussian closure method is adopted to reformulate the original equation into two coupled differential equations. One is a deterministic equation about the mean part and the other is a stochastic equivalent linear equation. In terms of these two coupled equations, the implicit harmonic balance method is used to obtain a set of nonlinear algebraic equations relating to amplitudes, frequency, and variance. The resulting equations are not explicitly determined and they can be implicitly solved by nonlinear equation routines available in most mathematical libraries. Three illustrative examples are further investigated to show the effectiveness of the proposed solution procedure. Furthermore, the proposed solution procedure is particularly convenient for programming and it can be extended to obtain the periodic solutions of the response of multi degrees-of-freedom systems.

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Figures

Grahic Jump Location
Fig. 3

Comparison between the case of unique harmonic excitation and the case of harmonic plus random excitations (the softening case and 2πK=0.001): (a) comparison on amplitude and (b) comparison on variance

Grahic Jump Location
Fig. 5

Comparison between the case of unique harmonic excitation and the case of harmonic plus random excitations (the softening case and 2πK=0.01): (a) comparison on amplitude and (b) comparison on variance

Grahic Jump Location
Fig. 1

Comparison between the proposed method and the Iyengar's method: (a) comparison on amplitude and (b) comparison on variance

Grahic Jump Location
Fig. 2

Comparison between the case of unique harmonic excitation and the case of harmonic plus random excitations (the hardening case and 2πK=0.01): (a) comparison on amplitude and (b) comparison on variance

Grahic Jump Location
Fig. 4

Comparison between the case of unique harmonic excitation and the case of harmonic plus random excitations (the softening case and 2πK=0.008): (a) comparison on amplitude and (b) comparison on variance

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