Extremes of Gaussian Processes with Bimodal Spectra
Publication: Journal of Engineering Mechanics
Volume 112, Issue 5
Abstract
An approximate solution is developed for the extreme‐value distribution of a stationary Gaussian process with a spectral density function that exhibits two well‐separated modes. Processes of this type arise in the analysis of the combined dynamic response to two loads or to one load that excites two of a structure's modes of vibration. Spectral moments of each of the modes taken separately are used to characterize the process; two envelope processes are then used to approximate the extreme value distribution. The proposed distribution is compared with other approximations and with results from Monte Carlo simulations.
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References
1.
Abramowitz, B., and Stegun, I., Handbook of Mathematical Functions, Dover, N.Y., 1972.
2.
Coleman, J. J., “Reliability of Aircraft Structures in Resisting Chance Failure,” Operations Research, Vol. 7, No. 5, 1959, pp. 639–645.
3.
Cramer, H., and Leadbetter, M. R., Stationary and Related Stochastic Processes, John Wiley and Sons, New York, N.Y., 1967.
4.
Crandall, S. H., “Zero Crossings, Peaks, and Other Statistical Measures of Random Responses,” Journal of the Acoustical Society of America, Vol. 35, No. 11, 1963, pp. 1693–1699.
5.
Crandall, S. H., “First‐Crossing Probabilities of the Linear Oscillator,” Journal of Sound and Vibration, Vol. 12, No. 3, 1970, pp. 285–299.
6.
Krenk, S., “A Double Envelope for Stochastic Processes,” Report No. 134, Structural Research Laboratory, Technical Univ. of Denmark, Feb., 1978.
7.
Lyon, R. H., “On the Vibration Statistics of a Randomly Excited Hard‐Spring Oscillator,” Journal of the Acoustical Society of America, Vol. 33, No. 10, 1961, p. 1395.
8.
Rice, S. O., “Mathematical Analysis of Random Noise,” Selected Papers on Noise and Stochastic Processes, Wax, Ed., Dover, N.Y., 1952.
9.
Shinozuka, M., “Probability of Structural Failure under Random Loading,” Journal of the Engineering Mechanics Division, ASCE, Vol. 90, No. EM5, 1964, pp. 147–170.
10.
Thoft‐Christensen, P., and Nielsen, “Bounds on the Probability of Failure in Random Vibration,” Journal of Structural Mechanics, Vol. 10, No. 1, 1982, pp. 67–91.
11.
Toro, G. R., “Probabilistic Analysis of Combined Dynamic Responses,” thesis presented to the Massachusetts Institute of Technology, at Cambridge, Mass., in 1984, in partial fulfillment of the requirements for the degree of Doctor of Philosophy (also, Report No. 65, John Blume Earthquake Engineering Center, Stanford Univ., 1984).
12.
Toro, G. R., and Cornell, C. A., “Gaussian Processes with Closely‐Spaced Modes,” submitted to the Journal of Engineering Mechanics, ASCE, 1983.
13.
Van Dyck, J. M., “Envelopes of Broad Band Processes,” thesis presented to the Massachusetts Institute of Technology, at Cambridge, Mass., in 1981, in partial fulfillment of the requirements for the degree of Master of Science.
14.
Vanmarcke, E. H., “Parameters of the Spectral Density Function: Their Significance in the Time and Frequency Domains,” Report R70‐58, MIT Department of Civil Engineering, 1970
(see also, “Properties of Spectral Moments with Application to Random Vibration,” Journal of the Engineering Mechanics Division, ASCE, Vol. 98, 1972, p. 425).
15.
Vanmarcke, E. H., “On the Distribution of the First‐Passage Time for Normal Stationary Gaussian Processes,” Journal of Applied Mechanics, Vol. 42, 1975, pp. 215–220.
16.
Veneziano, D., “Envelopes of Vector Random Processes and their Crossing Rates,” The Annals of Probability, Vol. 7, No. 1, 1979, pp. 62–74.
17.
Yang, J. N., and Shinozuka, M., “On the First‐Excursion Probability of Stationary Narrow‐Band Random Vibration,” Journal of Applied Mechanics, Transactions, ASME, Paper No. 71‐APM‐19, 1970, pp. 1017–1022.
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Copyright © 1986 ASCE.
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Published online: May 1, 1986
Published in print: May 1986
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