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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

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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