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Research Papers

Three-to-One Resonant Responses of Inextensional Beams on the Elastic Foundation

[+] Author and Article Information
Lianhua Wang

College of Civil Engineering,
Hunan University,
Changsha, Hunan 410082, PRC;
Key Laboratory of Building Safety
and Energy Efficiency,
Ministry of Education,
Hunan 410082, PRC
e-mail: Lhwang@hnu.edu.cn

Jianjun Ma

e-mail: Majianjun@hnu.edu.cn

Lifeng Li

e-mail: Lilifeng@hnu.edu.cn

Jian Peng

e-mail: Pengjian@hnu.edu.cn
College of Civil Engineering,
Hunan University,
Changsha, Hunan 410082, PRC

1Corresponding author.

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received January 9, 2012; final manuscript received May 13, 2012; published online February 4, 2013. Assoc. Editor: Walter Lacarbonara.

J. Vib. Acoust 135(1), 011015 (Feb 04, 2013) (10 pages) Paper No: VIB-12-1011; doi: 10.1115/1.4007019 History: Received January 09, 2012; Revised May 13, 2012

The large amplitude vibrations of the inextensional beam resting on the elastic foundation under three-to-one internal resonance are investigated. The inextensional condition and multimodal discretization are used to obtain the equation of in-plane motion and modulation equations. Due to the second-order moment of the subgrade reaction, the quadratic nonlinearity is included in the present model. Moreover, this moment destroys the conservative character of the system. The nonlinear response and the associated stability are examined by means of frequency (force)-response curves, and the shooting method is applied to investigate the dynamic solutions. Particular attention is placed on the effects of the cut-off frequency and boundary conditions. The results show that the cut-off frequency and boundary conditions do not significantly affect the contribution of nonresonant modes to the nonlinear coefficients. Moreover, the effects of foundation models on the three-to-one resonant dynamics of the beam are discussed.

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Figures

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Fig. 1

The beam resting on the elastic foundation

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Fig. 2

Planar natural frequency spectrum of the clamped-free beam for different K1

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Fig. 3

Modal contributions to nonlinear coefficients of C-F beam: (a) stiff-soil beam; (b) soft-soil beam

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Fig. 4

Pareto chart of the effective nonlinear coefficients

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Fig. 5

Frequency-response curves of the beam on the elastic foundation with f1=0.006 when Ω≈ω1: (a) stiff-soil beam; (b) soft-soil beam

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Fig. 6

Force-response curves of the beam on the elastic foundation with σ2=0.04 when Ω≈ω1: (a) stiff-soil beam; (b) soft-soil beam

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Fig. 7

Periodic solution branch of the modulation equations when Ω≈ω1: (a) stiff-soil beam; (b) soft-soil beam

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Fig. 8

The time-period of the periodic solutions

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Fig. 9

Frequency-response curves of the beam on the elastic foundation with f2=0.010 when Ω≈ω2: (a) stiff-soil beam; (b) soft-soil beam

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Fig. 10

Force-response curves of the beam on the elastic foundation with σ2=0.10 when Ω≈ω2: (a) stiff-soil beam; (b) soft-soil beam

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Fig. 11

Periodic solution branch of the modulation equations of soft-soil beam when Ω≈ω2

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Fig. 12

The quasi-periodic solution of the modulation equations with σ2=0.154: (a) time history; (b) Fourier spectrum; (c) Poincaré section

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Fig. 13

The chaotic solution of the modulation equations with f2=0.011: (a) phase portrait; (b) Fourier spectrum; (c) Poincaré section

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Fig. 14

The shear force of the clamped-free beams and kinetic energy of the soil medium when (a) (c): Ω≈ω1, (b) (d): Ω≈ω2

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Fig. 15

Modal contributions to nonlinear coefficients: (a) hinged-free beam; (b) free-free beam

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