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Research Papers: Structures and Safety Reliability

Extremes of Nonlinear Vibration: Comparing Models Based on Moments, L-Moments, and Maximum Entropy

[+] Author and Article Information
Steven R. Winterstein

Principal Engineer
Probability-Based Engineering,
Menlo Park, CA
e-mail: SteveWinterstein@alum.mit.edu

Cameron A. MacKenzie

Assistant Professor
Naval Postgraduate School,
Monterey, CA
e-mail: cmackenzie@ou.edu

Contributed by the Ocean Offshore and Arctic Engineering Division of ASME for publication in the JOURNAL OF Offshore MECHANICS AND ARCTIC ENGINEERING. Manuscript received July 14, 2011; final manuscript received April 13, 2012; published online February 25, 2013. Assoc. Editor: Bernt J. Leira.

J. Offshore Mech. Arct. Eng 135(2), 021602 (Feb 25, 2013) (7 pages) Paper No: OMAE-11-1065; doi: 10.1115/1.4007050 History: Received July 14, 2011; Revised April 13, 2012

Wind and wave loads on offshore structures show nonlinear effects, which require non-Gaussian statistical models. Here we critically review the behavior of various non-Gaussian models. We first survey moment-based models; in particular, the four-moment “Hermite” model, a cubic transformation often used in wind and wave applications. We then derive an “L-Hermite” model, an alternative cubic transformation calibrated by the response “L-moments” rather than its ordinary statistical moments. These L-moments have recently found increasing use, in part because they show less sensitivity to distribution tails than ordinary moments. We find here, however, that these L-moments may not convey sufficient information to accurately estimate extreme response statistics. Finally, we show that four-moment maximum entropy models, also applied in the literature, may be inappropriate to model broader-than-Gaussian cases (e.g., responses to wind and wave loads).

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References

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Najafian, G., 2010, “Comparison of Three Different Methods of Moments for Derivation of Probability Distribution Parameters,” Appl. Ocean Res., 32(3), pp. 298–307. [CrossRef]
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MacKenzie, C. A., and Winterstein, S. R., 2011, “Comparing L-Moments and Conventional Moments to Model Current Speeds in the North Sea,” Proceedings of the 2011 Industrial Engineering Research Conference, Inst. Industrial Eng. Annual Meeting, Reno, NV, May 21–25.
Winterstein, S. R., Lange, C. H., and Kumar, S., 1995, “FITTING: A Subroutine to Fit Four Moment Probability Distributions to Data,” Technical Report SAND94-3039, Sandia National Laboratories.
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Winterstein, S. R., and MacKenzie, C. A., 2011, “Extremes of Nonlinear Vibration: Models Based on Moments, L-Moments, and Maximum Entropy,” Proceedings of the 30th International Conference on Offshore Mechanics and Arctic Engineering, Rotterdam, The Netherlands, June 19–24, ASME Paper No. OMAE2011-49867. [CrossRef]

Figures

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Fig. 1

Weight functions Ln(u) contributing to the L-moment λn = E[Ln(U)] for a standard normal variable U. Note lesser weight to extreme outcomes (large |u|) for λn than for ordinary moment E[Un].

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Fig. 2

Mean upcrossing rates for various transformed Gaussian models, all calibrated to have kurtosis α4 = 5

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Fig. 3

Mean upcrossing rates for various transformed Gaussian models, all calibrated to have kurtosis α4 = 7

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Fig. 4

Mean upcrossing rates for various transformed Gaussian models, all calibrated to have L-kurtosis τ4 = 0.185

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Fig. 5

Mean upcrossing rates for various transformed Gaussian models, all calibrated to have L-kurtosis τ4 = 0.220

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Fig. 6

Moments versus L-moments fits to a lognormal process with coefficient of variation VX = 0.5

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Fig. 7

Moments versus L-moments fits to a lognormal process with coefficient of variation VX = 1.0

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Fig. 8

Maximum entropy PDF models for a lognormal process with coefficient of variation VX = 0.5

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Fig. 9

Maximum entropy PDF models for a lognormal process with coefficient of variation VX = 1.0

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Fig. 10

Moment-fit versus maximum entropy models of the wind response of a 1DOF oscillator

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Fig. 11

Mean damage rates for various transformed Gaussian models, all calibrated to have kurtosis α4 = 5

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Fig. 12

Mean damage rates for various transformed Gaussian models, all calibrated to have kurtosis α4 = 7

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Fig. 13

Mean damage rates for various transformed Gaussian models, all calibrated to have L-kurtosis τ4 = 0.185

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Fig. 14

Mean damage rates for various transformed Gaussian models, all calibrated to have L-kurtosis τ4 = 0.220

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