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Research Papers

Refined Modeling and Free Vibration of Inextensional Beams on the Elastic Foundation

[+] Author and Article Information
Lianhua Wang

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

Jianjun Ma

e-mail: Majianjun@hnu.edu.cn

Yueyu Zhao

e-mail: Yyzhao@hnu.edu.cn

Qijian Liu

e-mail: Q.Liu@hnu.edu.cn
College of Civil Engineering,
Hunan University,
Changsha,
Hunan 410082, China

1Corresponding author.

Manuscript received April 12, 2012; final manuscript received November 14, 2012; accepted manuscript posted November 19, 2012; published online May 23, 2013. Assoc. Editor: Alexander F. Vakakis.

J. Appl. Mech 80(4), 041026 (May 23, 2013) (11 pages) Paper No: JAM-12-1147; doi: 10.1115/1.4023032 History: Received April 12, 2012; Revised November 14, 2012; Accepted November 19, 2012

In this study, the nonlinear equation of motion of the beam on the elastic foundation is obtained via the Newton's second law of motion, and its free vibration nature is investigated. Considering the inextensional condition, the planar model of the beam accounting for the effects of the rotary inertia is derived. Then, the linear vibration and nonlinear vibration of the beam on the elastic foundation are examined. It is shown that the cut-off frequency can be observed in the frequency spectrum of the beam response. The effects of the rotary inertia on the natural frequencies are systematically investigated. Finally, the frequency differences, due to the different foundation models, and the possible modal interaction of the beam are discussed.

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Figures

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Fig. 1

Beam resting on the elastic foundation and the foundation models

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Fig. 4

Variation of the in-plane natural frequencies of the free-free beam with K0: K1 = 0, K1 = 50

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Fig. 5

The mode shapes ϕ1(x) of the clamped-free beam with (a) K0 = 500 and (b) K0 = 700; of the hinged-free beam with (c) K0 = 500 and (d) K0 = 700

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Fig. 7

The zero-velocity configurations (thick line) and evolution of the second-order configurations (thin line) of the beam over half nonlinear period TN1 with K1 = 50 and a1 = 0.01 (a) clamped-free beam, (b) hinged-free beam, (c) free-free beam

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Fig. 8

The Pareto chart of the frequencies ω1 and ωN1 of the beam with a1 = 0.05 (a) clamped-free beam, (b) hinged-free beam, (c) free-free beam

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Fig. 2

Variation of the in-plane natural frequencies of the clamped-free beam with K0: K1 = 0, K1 = 50

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Fig. 3

Variation of the in-plane natural frequencies of the hinged-free beam with K0: K1 = 0, K1 = 50

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Fig. 6

The backbone curves of the first mode of the beam on the elastic foundation (a) clamped-free beam, (b) hinged-free beam, (c) free-free beam

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