In this paper, the analysis of the time evolution of a levitated droplet is proposed. The analysis is composed of two parts: in the first part, a nonlinear dynamics approach was considered to calculate quantities characterizing time series data such as attractor dimension or largest Lyapunov exponent. The number of degrees of freedom in the system was also assessed. Based on the results obtained in the first part, Floquet theory was applied in the second part of the analysis to investigate the stability of the system. Data acquired from a levitation instrument developed by Space Power Institute at Auburn University was used to perform the analysis. [S0739-3717(00)01903-6]
Issue Section:
Technical Papers
Keywords:
magnetic levitation,
time series,
drops,
nonlinear dynamical systems,
Lyapunov methods,
flow instability
Topics:
Attractors,
Chaos,
Dimensions,
Drops,
Levitation,
Signals,
Stability,
Time series,
Instrumentation,
Degrees of freedom
1.
Mestel
, A. J.
, 1981
, “Magnetic levitation of liquid metals
,” J. Fluid Mech.
, 117
, pp. 27
–43
.2.
Busse
, F. H.
, 1984
, “Oscillations of a rotating liquid drop
,” J. Fluid Mech.
, 142
, pp. 1
–8
.3.
Cummings
, D. L.
, and Blackburn
, D. A.
, 1989
, “Oscillations of magnetically levitated aspherical droplets
,” J. Fluid Mech.
, 224
, pp. 395
–416
.4.
Chen, S.-F., and Overfelt, A., 1997, “Effects of Sample Size on Surface Tension Measurements of Nickel in Reduced Gravity Parabolic Flight,” in 13th Symposium on Thermophysical Properties, June 22–27, Boulder, CO.
5.
Cusumano
, J. P.
, and Moon
, F. C.
, 1995
, “Chaotic Non-Planar Vibrations of the Thin Elastica, Part I: Experimental Observation of Planar Instability
,” J. Sound Vib.
, 179
, No. 2
, pp. 185
–208
.6.
Kang
, Y.
, Chang
, Y. P.
, and Jen
, S. C.
, 1998
, “Strongly Non-Linear Oscillations of Winding Machines, Part I: Mode-Locking Motion and Routes to Chaos
,” J. Sound Vib.
, 209
, No. 3
, pp. 473
–492
.7.
Nayfeh, A. H., and Balachandran, B., 1995, Applied Nonlinear Dynamics, Wiley, New York.
8.
Farmer
, J. D.
, Ott
, E.
, and Yorke
, J. A.
, 1983
, “The dimension of chaotic attractors
,” Physica D
, 7
, pp. 153
–180
.9.
Abarbanel, H. D. I., 1996, Analysis of Observed Chaotic Data, Springer-Verlag, New York.
10.
Packard
, N. H.
, Crutchfield
, J. P.
, Farmer
, J. D.
, and Shaw
, R. S.
, 1980
, “Geometry from a time series
,” Phys. Rev. Lett.
, 45
, pp. 712
–716
.11.
Albano
, A. M.
, Muench
, J.
, Schwartz
, C.
, Mees
, A. I.
, and Rapp
, P. E.
, 1988
, “Singular-value decomposition and the Grassberger-Procaccia algorithm
,” Phys. Rev. A
, 38
, pp. 3017
–3034
.12.
Provanzale
, A.
, Smith
, L. A.
, Vio
, R.
, and Murante
, G.
, 1992
, “Distinguishing between low-dimensional dynamics and randomness in measured time series
,” Physica D
, 58
, pp. 31
–49
.13.
Cambel, A. B., 1993, Applied Chaos Theory, Academic Press, Boston.
14.
Smith
, L. A.
, 1988
, “Intrinsic limits on dimensional calculation
,” Phys. Lett. A
, 133
, No. 6
, pp. 283
–288
.15.
Grassberger
, P.
, and Procaccia
, I.
, 1984
, “Dimensions and entropies of strange attractors from a fluctuating dynamics approach
,” Physica D
, 13
, pp. 34
–54
.16.
Mandelbrot, B., 1983, The Fractal Geometry of Nature, Freeman, San Francisco.
17.
Grassberger
, P.
, and Procaccia
, I.
, 1983
, “Characterization of strange attractors
,” Phys. Rev. Lett.
, 50
, No. 5
, pp. 346
–349
.18.
Wolf
, A.
, Swift
, J. B.
, Swinney
, H. L.
, and Vastano
, J. A.
, 1985
, “Determining Lyapunov exponents from a time series
,” Physica D
, 16
, pp. 285
–317
.19.
Eckermann
, J. P.
, Kamphorst
, S. O.
, Ruelle
, D.
, and Ciliberto
, S.
, 1986
, “Lyapunov exponents from time series
,” Phys. Rev. A
, 34
, No. 6
, pp. 4971
–4979
.20.
Kapitaniak, T., 1991, Chaotic Oscillations in Mechanical Systems, Manchaster University Press, New York.
21.
Moon, F. C., 1992, Chaotic and Fractal Dynamics, Wiley, New York.
22.
Theiler
, J.
, 1991
, “Some comments on the correlation dimension of 1/fα noise
,” Phys. Lett. A
, 155
, Nos. 8–9
, pp. 480
–493
.23.
Theiler
, J.
, Eubank
, S.
, Longtin
, A.
, and Galdrikian
, B.
, 1992
, “Testing for nonlinearity in time series: The method of surrogate data
,” Physica D
, 58
, pp. 77
–94
.24.
Chennaoui
, A.
, Pawelzik
, K.
, Liebert
, W.
, Schuster
, H. G.
, and Pfister
, G.
, 1990
, “Attractor reconstruction from filtered chaotic time series
,” Phys. Rev. A
, 41
, No. 8
, pp. 4151
–4159
.25.
Agguirre
, L. A.
, Mendes
, E. M.
, and Billings
, S. A.
, 1996
, “Smoothing data with local instabilities for the identification of chaotic systems
,” Int. J. Control
, 63
, No. 3
, pp. 483
–505
.26.
Broomhead
, D. S.
, and King
, G. P.
, 1986
, “Extracting qualitative dynamics from experimental data
,” Physica D
, 20
, pp. 217
–236
.27.
Vautard
, R.
, and Ghil
, M.
, 1989
, “Singular spectrum analysis in nonlinear dynamics with applications to paleoclimatic time series
,” Physica D
, 35
, pp. 395
–424
.28.
Casdagli
, M.
, 1989
, “Nonlinear prediction of chaotic time series
,” Physica D
, 35
, pp. 335
–356
.29.
Abarbanel
, D. I.
, Frison
, T. W.
, and Tsimring
, L. S.
, 1998
, “Obtaining order in a world of chaos
,” IEEE Signal Proc. Mag.
, 15
, No. 3
, pp. 49
–65
.30.
Marghitu
, D. B.
, 1997
, “Quantitative Measure of Vehicle Stability
,” Comput. Model. Simul. Eng.
, 2
, No. 2
, pp. 163
–176
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