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Journal of Computational and Nonlinear Dynamics | Volume 7 | Issue 2 | Research Paper
Research Papers

Nonlinear Vibrations and Chaos in Floating Roofs

[+] Author and Article Information
R. Shabani1

Mechanical Engineering Department,  Urmia University, Urmia 15311-57561, Iranr.shabani@urmia.ac.ir

S. Tariverdilo

Civil Engineering Department,  Urmia University, Urmia 15311-57561, Irans.tariverdilo@urmia.ac.ir

H. Salarieh

Mechanical Engineering Department,  Sharif University of Technology, Tehran 11155-9567, Iransalarieh@sharif.edu

1

Corresponding author.

J. Comput. Nonlinear Dynam 7(2), 021012 (Jan 10, 2012) (13 pages) doi:10.1115/1.4005437 History: Received May 16, 2011; Revised October 24, 2011; Accepted October 25, 2011; Published January 10, 2012; Online January 10, 2012
Article
References
Figures
Tables

Variational principle is used to derive the nonlinear response of the floating roof of cylindrical liquid storage tanks due to harmonic base excitations. The formulation accounts for nonlinearity due to large deflections of the floating roof. The derived nonlinear governing equation for the sloshing response of the floating roof has a cubic nonlinear stiffness term similar to the well known Duffing equation. It is shown that accounting for large deflections could substantially reduce the wave elevation for near resonance harmonic excitations. Evaluating the response of the nonlinear model for increasing amplitudes of near resonance harmonic excitations gives rise to the appearance of sub and super harmonics in the response. The broadband structure of the frequency spectrum, the fractal structure of the Poincare maps, and the bifurcation diagram as qualitative criteria and Lyapunov exponent evolution as quantitative criterion are used to investigate the emergence of chaotic response.
Copyright © 2012 by American Society of Mechanical Engineers
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References

1
Chang, J. I., and Lin, C.-C., 2006, “Study of Storage Tank Accidents,” J. Loss Prev. Process Ind., 19 (1), pp. 51–59.  [CrossRef]
2
Hatayama, K., Zama, S., Nishi, H., Yamada, M., Hirokawa, M., and Inoue, R., 2005, “The Damages of Oil Storage Tanks During the 2003 Tokachi-oki Earthquake and the Long Period Ground Motions,” "Proceedings of the JSCE-AIJ Joint Symposium on Huge Subduction Earthquakes—Wide Area Strong Ground Motion Prediction", pp. 7–18.
3
Jacobsen, L. S., 1949, “Impulsive Hydrodynamics of Fluid Inside a Cylindrical Tank and of Fluid Surrounding a Cylindrical Pier,” Bull. Seismol. Soc. Am., 39 (3), pp. 189–202.
4
Senda, K., and Nakagawa, K., 1954, “On the Vibration of an Elevated Water Tank I,” Technol. Rep. Osaka Univ., 4 (117), pp. 247–264.
5
MatsuiT., 2007, “Sloshing in a Cylindrical Liquid Storage Tank With a Floating Roof Under Seismic Excitation,” J. Pressure Vessel Technol., 129 , pp. 557–566.  [CrossRef]
6
Isshiki, H., and Nagata, S., 2001, “Variational Principles Related to Motion of an Elastic Plate Floating on a Water Surface,” "Proceedings of the 11th International Offshore and Polar Engineering Conference", Stavanger, Norway, pp. 190–197.
7
Nagata, S., Yoshida, H., Fujita, T., and Isshiki, H., 1997, “The Analysis of the Wave Induced Responses of an Elastic Floating Plate,” "Proceedings of the 16th International Conference on Offshore Mechanics and Arctic Engineering", vol. 6 .
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Ohmatsu, S., 1997, “Numerical Calculation of Hydro-Elastic Responses of Pontoon Type VLFS,” J. Soc. Nav. Archit. Jpn., 182 , pp. 329–340.  [CrossRef]
9
Sakai, F., Nishimura, M., and Ogawa, H., 1984, “Sloshing Behavior of Floating-Roof Oil Storage Tanks,” Comput. Struct., 19 (2), pp. 183–192.  [CrossRef]
10
Shabani, R., Tariverdilo, S., Salarieh, H., and Rezazadeh, G., 2010, “Importance of the Flexural and Membrane Stiffnesses in Large Deflection Analysis of Floating Roofs,” Appl. Math. model., 34 (9), pp. 2426–2436.  [CrossRef]
11
Frandsen, J. B., 2004, “Sloshing Motions in Excited Tanks,” J. Comput. Phys., 196 , pp. 53–87.  [CrossRef]
12
Cho, J. R., and Lee, H. W., 2004, “Numerical Study on Liquid Sloshing in Baffled Tank by Nonlinear Finite Element Method,” Comput. Methods Appl. Mech. Eng., 193 , pp. 2581–2598.  [CrossRef]
13
Hernandez-Barrios, H., Heredia-Zavoni, E., and Aldama-Rodrıguez, A., 2007, “Nonlinear Sloshing Response of Cylindrical Tanks Subjected to Earthquake Ground Motion,” Eng. Struct.29 , pp. 3364–3376.  [CrossRef]
14
Utsumi, M., Ishida, K., and Hizume, M., 2010, “Internal Resonance of a Floating Roof Subjected to Nonlinear Sloshing,” J. Appl. Mech., 77 (1), 011016.  [CrossRef]
15
Amibili, M., 2008, "Nonlinear Vibration and Stability of Shells and Plates,"Cambridge University Press, Cambridge, England.
16
Moon, F. C., 1992, "Chaotic and Fractal Dynamics: An Introduction for Applied Scientists and Engineers, John"Wiley, New York.
17
Guckenheimer, J., and Holmes, P. J., 1983, "Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields", Springer-Verlag, Berlin.
18
Wolf, A., Swift, J. B., Swinney, H. L., and VastanoJ. A., 1985, “Determining Lyapunov Exponents From a Time Series,” Physica D, 16 (3), pp. 285–317.  [CrossRef]

Figures

Grahic Jump Location
+Typical liquid storage tank with single deck floating roof
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Typical liquid storage tank with single deck floating roof
Figure 1

Typical liquid storage tank with single deck floating roof

Grahic Jump Location
+Response of a linear model for a ground motion acceleration amplitude of 1.42 m/s2 and an excitation frequency of 0.6 Hz. (a) Time history of wave elevation at roof edge, (b) time history of pressure at roof edge, and (c) FFT of wave elevation at roof edge.
View Large  |  Save Figure  |  Download Slide (.ppt)
Response of a linear model for a ground motion acceleration amplitude of 1.42 m/s2 and an excitation frequency of 0.6 Hz. (a) Time history of wave elevation at roof edge, (b) time history of pressure at roof edge, and (c) FFT of wave elevation at roof edge.
Figure 2

Response of a linear model for a ground motion acceleration amplitude of 1.42 m/s2 and an excitation frequency of 0.6 Hz. (a) Time history of wave elevation at roof edge, (b) time history of pressure at roof edge, and (c) FFT of wave elevation at roof edge.

Grahic Jump Location
+Response of a nonlinear model for a ground motion acceleration amplitude of 1.42 m/s2 and an excitation frequency of 0.6 Hz. (a) Time history of wave elevation at roof edge, (b) time history of pressure at roof edge, and (c) FFT of wave elevation at roof edge.
View Large  |  Save Figure  |  Download Slide (.ppt)
Response of a nonlinear model for a ground motion acceleration amplitude of 1.42 m/s2 and an excitation frequency of 0.6 Hz. (a) Time history of wave elevation at roof edge, (b) time history of pressure at roof edge, and (c) FFT of wave elevation at roof edge.
Figure 3

Response of a nonlinear model for a ground motion acceleration amplitude of 1.42 m/s2 and an excitation frequency of 0.6 Hz. (a) Time history of wave elevation at roof edge, (b) time history of pressure at roof edge, and (c) FFT of wave elevation at roof edge.

Grahic Jump Location
+Response of a linear model for a ground motion acceleration amplitude of 1.42 m/s2 and an excitation frequency of 0.4 Hz. (a) Time history of wave elevation at roof edge, (b) time history of pressure at roof edge, and (c) FFT of wave elevation at roof edge.
View Large  |  Save Figure  |  Download Slide (.ppt)
Response of a linear model for a ground motion acceleration amplitude of 1.42 m/s2 and an excitation frequency of 0.4 Hz. (a) Time history of wave elevation at roof edge, (b) time history of pressure at roof edge, and (c) FFT of wave elevation at roof edge.
Figure 4

Response of a linear model for a ground motion acceleration amplitude of 1.42 m/s2 and an excitation frequency of 0.4 Hz. (a) Time history of wave elevation at roof edge, (b) time history of pressure at roof edge, and (c) FFT of wave elevation at roof edge.

Grahic Jump Location
+Response of a nonlinear model for a ground motion acceleration amplitude of 1.42 m/s2 and an excitation frequency of 0.4 Hz. (a) Time history of wave elevation at roof edge, (b) time history of pressure at roof edge, and (c) FFT of wave elevation at roof edge.
View Large  |  Save Figure  |  Download Slide (.ppt)
Response of a nonlinear model for a ground motion acceleration amplitude of 1.42 m/s2 and an excitation frequency of 0.4 Hz. (a) Time history of wave elevation at roof edge, (b) time history of pressure at roof edge, and (c) FFT of wave elevation at roof edge.
Figure 5

Response of a nonlinear model for a ground motion acceleration amplitude of 1.42 m/s2 and an excitation frequency of 0.4 Hz. (a) Time history of wave elevation at roof edge, (b) time history of pressure at roof edge, and (c) FFT of wave elevation at roof edge.

Grahic Jump Location
+Response of a linear model for a ground motion acceleration amplitude of 1.42 m/s2 and an excitation frequency of 3 Hz. (a) Time history of wave elevation at roof edge, (b) time history of pressure at roof edge, and (c) FFT of wave elevation at roof edge.
View Large  |  Save Figure  |  Download Slide (.ppt)
Response of a linear model for a ground motion acceleration amplitude of 1.42 m/s2 and an excitation frequency of 3 Hz. (a) Time history of wave elevation at roof edge, (b) time history of pressure at roof edge, and (c) FFT of wave elevation at roof edge.
Figure 6

Response of a linear model for a ground motion acceleration amplitude of 1.42 m/s2 and an excitation frequency of 3 Hz. (a) Time history of wave elevation at roof edge, (b) time history of pressure at roof edge, and (c) FFT of wave elevation at roof edge.

Grahic Jump Location
+Response of a nonlinear model for a ground motion acceleration amplitude of 1.42 m/s2 and an excitation frequency of 3 Hz. (a) Time history of wave elevation at roof edge, (b) time history of pressure at roof edge, and (c) FFT of wave elevation at roof edge.
View Large  |  Save Figure  |  Download Slide (.ppt)
Response of a nonlinear model for a ground motion acceleration amplitude of 1.42 m/s2 and an excitation frequency of 3 Hz. (a) Time history of wave elevation at roof edge, (b) time history of pressure at roof edge, and (c) FFT of wave elevation at roof edge.
Figure 7

Response of a nonlinear model for a ground motion acceleration amplitude of 1.42 m/s2 and an excitation frequency of 3 Hz. (a) Time history of wave elevation at roof edge, (b) time history of pressure at roof edge, and (c) FFT of wave elevation at roof edge.

Grahic Jump Location
+Response of a nonlinear model for a ground motion acceleration amplitude of 2.84 m/s2 and an excitation frequency of 0.6 Hz. (a) Phase plane, (b) Poincare map, and (c) spectrum.
View Large  |  Save Figure  |  Download Slide (.ppt)
Response of a nonlinear model for a ground motion acceleration amplitude of 2.84 m/s2 and an excitation frequency of 0.6 Hz. (a) Phase plane, (b) Poincare map, and (c) spectrum.
Figure 8

Response of a nonlinear model for a ground motion acceleration amplitude of 2.84 m/s2 and an excitation frequency of 0.6 Hz. (a) Phase plane, (b) Poincare map, and (c) spectrum.

Grahic Jump Location
+Response of a nonlinear model for a ground motion acceleration amplitude of 5.26 m/s2 and an excitation frequency of 0.6 Hz. (a) Phase plane, (b) Poincare map (50000 points), and (c) spectrum.
View Large  |  Save Figure  |  Download Slide (.ppt)
Response of a nonlinear model for a ground motion acceleration amplitude of 5.26 m/s2 and an excitation frequency of 0.6 Hz. (a) Phase plane, (b) Poincare map (50000 points), and (c) spectrum.
Figure 9

Response of a nonlinear model for a ground motion acceleration amplitude of 5.26 m/s2 and an excitation frequency of 0.6 Hz. (a) Phase plane, (b) Poincare map (50000 points), and (c) spectrum.

Grahic Jump Location
+Response of a nonlinear model for a ground motion acceleration amplitude of 11.65 m/s2 and anexcitation frequency of 0.6 Hz. (a) Phase plane, (b) Poincare map (50000 points), and (c) spectrum.
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Response of a nonlinear model for a ground motion acceleration amplitude of 11.65 m/s2 and anexcitation frequency of 0.6 Hz. (a) Phase plane, (b) Poincare map (50000 points), and (c) spectrum.
Figure 10

Response of a nonlinear model for a ground motion acceleration amplitude of 11.65 m/s2 and anexcitation frequency of 0.6 Hz. (a) Phase plane, (b) Poincare map (50000 points), and (c) spectrum.

Grahic Jump Location
+Response of a nonlinear model for a ground motion acceleration amplitude of 12.08 m/s2 and an excitation frequency of 0.6 Hz. (a) Phase plane, (b) Poincare map (50000 points), and (c) spectrum.
View Large  |  Save Figure  |  Download Slide (.ppt)
Response of a nonlinear model for a ground motion acceleration amplitude of 12.08 m/s2 and an excitation frequency of 0.6 Hz. (a) Phase plane, (b) Poincare map (50000 points), and (c) spectrum.
Figure 11

Response of a nonlinear model for a ground motion acceleration amplitude of 12.08 m/s2 and an excitation frequency of 0.6 Hz. (a) Phase plane, (b) Poincare map (50000 points), and (c) spectrum.

Grahic Jump Location
+Bifurcation diagram of the roof edge displacement for increasing ground accelerations and an excitation frequency of 0.6 Hz
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Bifurcation diagram of the roof edge displacement for increasing ground accelerations and an excitation frequency of 0.6 Hz
Figure 12

Bifurcation diagram of the roof edge displacement for increasing ground accelerations and an excitation frequency of 0.6 Hz

Grahic Jump Location
+Largest Lyapunov exponent associated to the bifurcation diagram (Fig. 1) for an excitation frequency of 0.6 Hz
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Largest Lyapunov exponent associated to the bifurcation diagram (Fig. 1) for an excitation frequency of 0.6 Hz
Figure 13

Largest Lyapunov exponent associated to the bifurcation diagram (Fig. 1) for an excitation frequency of 0.6 Hz

Grahic Jump Location
+Bifurcation diagram of the roof edge displacement for increasing ground accelerations and an excitation frequency of 0.5 Hz
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Bifurcation diagram of the roof edge displacement for increasing ground accelerations and an excitation frequency of 0.5 Hz
Figure 14

Bifurcation diagram of the roof edge displacement for increasing ground accelerations and an excitation frequency of 0.5 Hz

Grahic Jump Location
+Largest Lyapunov exponent associated to the bifurcation diagram (Fig. 1) for an excitation frequency of 0.5 Hz
View Large  |  Save Figure  |  Download Slide (.ppt)
Largest Lyapunov exponent associated to the bifurcation diagram (Fig. 1) for an excitation frequency of 0.5 Hz
Figure 15

Largest Lyapunov exponent associated to the bifurcation diagram (Fig. 1) for an excitation frequency of 0.5 Hz

Grahic Jump Location
+Response of a nonlinear model for a ground motion acceleration amplitude of 1.77 m/s2 and an excitation frequency of 0.5 Hz. (a) Phase plane, (b) Poincare map, and (c) spectrum.
View Large  |  Save Figure  |  Download Slide (.ppt)
Response of a nonlinear model for a ground motion acceleration amplitude of 1.77 m/s2 and an excitation frequency of 0.5 Hz. (a) Phase plane, (b) Poincare map, and (c) spectrum.
Figure 16

Response of a nonlinear model for a ground motion acceleration amplitude of 1.77 m/s2 and an excitation frequency of 0.5 Hz. (a) Phase plane, (b) Poincare map, and (c) spectrum.

Grahic Jump Location
+Response of a nonlinear model for a ground motion acceleration amplitude of 7.10 m/s2 and an excitation frequency of 0.5 Hz. (a) Phase plane, (b) Poincare map (50000 points), and (c) spectrum.
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Response of a nonlinear model for a ground motion acceleration amplitude of 7.10 m/s2 and an excitation frequency of 0.5 Hz. (a) Phase plane, (b) Poincare map (50000 points), and (c) spectrum.
Figure 17

Response of a nonlinear model for a ground motion acceleration amplitude of 7.10 m/s2 and an excitation frequency of 0.5 Hz. (a) Phase plane, (b) Poincare map (50000 points), and (c) spectrum.

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