Research Papers

Vibration Control of Two-Degree-of-Freedom Structures Utilizing Sloshing in Nearly Square Tanks

[+] Author and Article Information
Takashi Ikeda

Department of Mechanical Systems Engineering,
Hiroshima University,
1-4-1, Kagamiyama, Higashi-Hiroshima,
Hiroshima 739-8527, Japan
e-mail: tikeda@hiroshima-u.ac.jp

Yuji Harata

Department of Mechanical Systems Engineering,
Hiroshima University,
1-4-1, Kagamiyama, Higashi-Hiroshima,
Hiroshima 739-8527, Japan
e-mail: harata@hiroshima-u.ac.jp

Shota Ninomiya

Olympus Corporation,
2951, Ishikawa-machi, Hachioji,
Tokyo 192-8507 Japan

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received August 3, 2016; final manuscript received March 12, 2017; published online May 4, 2017. Assoc. Editor: Mohammad Younis.

J. Comput. Nonlinear Dynam 12(5), 051012 (May 04, 2017) (18 pages) Paper No: CND-16-1367; doi: 10.1115/1.4036481 History: Received August 03, 2016; Revised March 12, 2017

This paper investigates the vibration control of a towerlike structure with degrees of freedom utilizing a square or nearly square tuned liquid damper (TLD) when the structure is subjected to horizontal, harmonic excitation. In the theoretical analysis, when the two natural frequencies of the two-degree-of-freedom (2DOF) structure nearly equal those of the two predominant sloshing modes, the tuning condition, 1:1:1:1, is nearly satisfied. Galerkin's method is used to derive the modal equations of motion for sloshing. The nonlinearity of the hydrodynamic force due to sloshing is considered in the equations of motion for the 2DOF structure. Linear viscous damping terms are incorporated into the modal equations to consider the damping effect of sloshing. Van der Pol's method is employed to determine the expressions for the frequency response curves. The influences of the excitation frequency, the tank installation angle, and the aspect ratio of the tank cross section on the response curves are examined. The theoretical results show that whirling motions and amplitude-modulated motions (AMMs), including chaotic motions, may occur in the structure because swirl motions and Hopf bifurcations, followed by AMMs, appear in the tank. It is also found that a square TLD works more effectively than a conventional rectangular TLD, and its performance is further improved when the tank width is slightly increased and the installation angle is equal to zero. Experiments were conducted in order to confirm the validity of the theoretical results.

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Fig. 2

Mode shapes of sloshing: (a) (1, 0) mode and (b) (0, 1) mode

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Fig. 3

Frequency response curves for amplitudes in: (a) structure xs, (b) structure ys, (c) (1,0) mode, and (d) (0,1) mode when μ0 = 0.94, μ1 = 0.06, kx = ky = 1.0, cx = cy = 0.013, w = 0.6, h = 0.6, ζij = 0.015, a0 = 0.0015, and α = 0 deg

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Fig. 4

Same as Fig. 3, but w = 1.0

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Fig. 5

Phase angle curves corresponding to Fig. 4

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Fig. 6

Stationary time histories including ηx at (x, y) = (1/2, 0) and ηy at (x, y) = (0, w/2) and orbits in the (xs, ys) and (ηx, ηy) planes in Figs. 4 and 5: (a) CW rotation at ω = 1.090 on branches HiDi, (b) CCW rotation at ω = 1.090 on branches H′jG′j, (c) regular AMMs swirl at ω = 1.070 on branches GiHi, (d) chaotic AMMs ω = 1.067 on branches GiHi, and (e) CCW rotation at ω = 0.935 on branches EiBi

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Fig. 7

Frequency response curves for amplitudes As and Amax and the orbits of the structure in the (xs, ys) plane: (a) frequency response curves for amplitudes As and Amax, (b) the orbit at ω = 1.090, and (c) the orbit at ω = 0.935

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Fig. 8

Same as Fig. 4, but α = 15 deg

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Fig. 9

Same as Fig. 4, but α = 45 deg

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Fig. 10

Bifurcation sets in the (ω, α) plane

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Fig. 11

Coordinate system in the case of bidirectional excitation

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Fig. 12

Same as Fig. 4, but w = 1.08

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Fig. 13

Same as Fig. 4, but w = 0.935

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Fig. 14

Bifurcation sets in the (ω, w) plane for α = 0 deg

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Fig. 15

Experiment setup: (a) schematic diagram and (b) photo

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Fig. 16

Frequency resonance curves for apparatus A

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Fig. 17

Frequency resonance curves for apparatus B

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Fig. 18

Stationary time histories and Lissajous curves in the (xs, ys) and (ηx, ηy) planes observed in Fig. 16: (a) ω = 2.971 Hz and (b) ω = 2.921 Hz

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Fig. 19

Frequency resonance curves for apparatus C

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Fig. 20

Frequency resonance curves for apparatus D




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