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Research Papers

Multimode Dynamics of Inextensional Beams on the Elastic Foundation With Two-to-One Internal Resonances

[+] 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

Minghui Yang

e-mail: Yamih@hnu.edu.cn

Lifeng Li

e-mail: Lilifeng@hnu.edu.cn

Yueyu Zhao

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

1Corresponding author.

Manuscript received December 13, 2012; final manuscript received January 23, 2013; accepted manuscript posted February 19, 2013; published online August 21, 2013. Assoc. Editor: Alexander F. Vakakis.

J. Appl. Mech 80(6), 061016 (Aug 21, 2013) (11 pages) Paper No: JAM-12-1554; doi: 10.1115/1.4023694 History: Received December 13, 2012; Revised January 23, 2013; Accepted February 19, 2013

The modal interactions and nonlinear responses of inextensional beams resting on elastic foundations with two-to-one internal resonances are investigated and the primary resonance excitations are considered. The multimode discretization and the method of multiple scales are applied to obtain the modulation equations. The equilibrium and dynamic solutions of the modulation equations are examined by the Newton–Raphson, shooting, and continuation methods. Numerical simulations are performed to investigate the chaotic dynamics of the beam. It is shown that the nonlinear responses may undergo different bifurcations and exhibit rich nonlinear phenomena. Finally, the effects of the foundation models on the nonlinear interactions of the beam are examined.

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References

Figures

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Fig. 1

The beam resting on the elastic foundation and the foundation models

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Fig. 2

Variation of the in-plane natural frequencies of the clamped-free beam with K0

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Fig. 3

Pareto chart of the effective nonlinear coefficients

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Fig. 4

Frequency-response curves of the stiff-soil beam with f1 = 0.002 when Ω≈ω1

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Fig. 5

Periodic solution branch of the modulation equations when σ2 ∈ (-0.058,0.014)

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Fig. 6

Force-response curves of the stiff-soil beam with σ2 = 0.04 when Ω≈ω1

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Fig. 7

Frequency-response curves of the soft-soil beam with f1 = 0.002 when Ω≈ω1

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Fig. 8

Periodic solution branch of the modulation equations when σ2 ∈ (-0.002,0.024)

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Fig. 9

Force-response curves of the soft-soil beam with σ2 = 0.04 when Ω≈ω1

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Fig. 10

Frequency-response curves of the stiff-soil beam with f2 = 0.002 when Ω≈ω2

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Fig. 11

Periodic solution branch of the modulation equations when σ2 ∈ (-0.0391,-0.0042)

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Fig. 12

The period T of the periodic solutions

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Fig. 13

The chaotic attractors of the modulation equations: (a) σ2 = -0.0382, (b) σ2 = -0.0073, and (c) σ2 = -0.0065

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Fig. 14

Force-response curves of the stiff-soil beam with σ2 = 0.04 when Ω≈ω1

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Fig. 15

Frequency-response curves of the soft-soil beam with f2 = 0.002 when Ω≈ω2

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Fig. 16

Periodic solution branch of the modulation equations when σ2 ∈ (-0.0182,0.0145)

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Fig. 17

Force-response curves of the soft-soil beam with σ2 = 0.04 when Ω≈ω1

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Fig. 18

The shear force of (a) the stiff-soil beam, and (b) the soft-soil beam; the kinetic energy of the soil medium when (c) Ω = 14.452, and (d) Ω = 37.742

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