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Research Papers

Nonlinearity of Finite-Amplitude Sloshing in Rectangular Containers

[+] Author and Article Information
Mohammed F. Daqaq

Nonlinear Vibrations and Energy
Harvesting Laboratory,
Department of Mechanical Engineering,
Clemson University,
Clemson, SC 29634
e-mail: mdaqaq@clemson.edu

Yawen Xu

Nonlinear Vibrations and Energy
Harvesting Laboratory,
Department of Mechanical Engineering,
Clemson University,
Clemson, SC 29634

Walter Lacarbonara

Department of Structural and
Geotechnical Engineering,
University of Rome, LaSapienza,
Rome 0318, Italy
e-mail: walter.lacarbonara@uniroma1.it

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received September 29, 2016; final manuscript received December 1, 2016; published online December 22, 2016. Editor: Yonggang Huang.

J. Appl. Mech 84(3), 031002 (Dec 22, 2016) (13 pages) Paper No: JAM-16-1482; doi: 10.1115/1.4035363 History: Received September 29, 2016; Revised December 01, 2016

This paper investigates the influence exerted by small surface tension on the nonlinear normal sloshing modes of a two-dimensional irrotational, incompressible fluid in a rectangular container. To this end, the influence of surface tension on the modal frequencies is investigated by assuming pure slipping at the contact line and a 90 deg contact angle between the fluid surface and the walls. The regions of possible nonlinear internal resonances up to the fifth mode are highlighted. Away from the highlighted regions, the influence of surface tension on the effective nonlinearity of the lowest four modes is studied and used to shed light onto its effect on the softening/hardening behavior of the uncoupled nonlinear modes. Subsequently, the response of the sloshing waves near two-to-one internal resonances is studied. It is shown that, in the vicinity of such internal resonance, the steady-state sloshing response can either contain a contribution from the two interacting modes (coupled-mode response) or only the high-frequency mode (high-frequency uncoupled mode response). The regions where the coupled mode uniquely exists are shown to depend on the surface tension. Moreover, it is demonstrated that such regions may be underestimated considerably when neglecting the influence of the cubic nonlinearities.

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References

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Figures

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Fig. 1

A schematic representation of the fluid sloshing in a rectangular container

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Fig. 2

(a) Variation of the lowest five sloshing frequencies with the height-to-length ratio, h/L, when β= 0 and (b) The associated mode shapes

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Fig. 3

Variation of the lowest five sloshing frequencies with the height-to-length ratio, h/L for (a) β=1.9×10−3 and (b) β=0.013

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Fig. 4

Loci in the (h/L, β)-plane on which internal resonance may be activated. Refer to Table 1 for details.

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Fig. 5

Variation of the effective nonlinearity with the height-to-length ratio, h/L for (a) β= 0, (b) β=0.0016, and (c) β=0.013

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Fig. 6

Variation of the effective nonlinearity with the surface tension coefficient, β, and h/L=0.2. Shaded areas represent regions where a two-to-one internal resonance can be activated.

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Fig. 7

Critical values of h/L at which Γk=0

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Fig. 8

Loci of the two-to-one internal resonances in the design parameters' space

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Fig. 9

Variation of the steady-state amplitudes with the frequency detuning σ2. Results are obtained using n = 1, m = 2, h/L=0.2, β=0.001, and (a) am0=0.0005 and (b) am0=0.005.

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Fig. 10

Regions where the coupled 2:1 internally resonant modes can be uniquely realized (shaded region) in the (σ2, a20)-plane. Results are obtained using n = 1, m = 2, and (a) β=0.001, (b) β=0.0025, and (c) β=0.005.

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Fig. 11

Regions where the 2:1 coupled mode can be uniquely realized (shaded region) in the (σ2a20)-plane. Results are obtained using n = 1, m = 2, and (a) β=0.001, (b) β=0.0025, and (c) β=0.005.

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Fig. 12

Steady-state coupled-mode surface waves obtained using am0=0.1, βm0=−π/2, n = 1, m = 2, β=0.001, and h/L=0.3. Response amplitudes are obtained for one cycle (2π) at π/8 steps normalized by am0. Solid line represents quadratic approximations while dashed lines represent cubic approximations.

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