TECHNICAL PAPERS
May 1, 1986

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.

Get full access to this article

View all available purchase options and get full access to this article.

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.

Information & Authors

Information

Published In

Go to Journal of Engineering Mechanics
Journal of Engineering Mechanics
Volume 112Issue 5May 1986
Pages: 465 - 484

History

Published online: May 1, 1986
Published in print: May 1986

Permissions

Request permissions for this article.

Authors

Affiliations

Gabriel R. Toro, S. M. ASCE
Sr. Engr., Risk Engineering, Inc., 5255 Pine Ridge Rd., Golden, CO 80401; formerly, Grad. Research Asst., Stanford Univ.
C. Allin Cornell, M. ASCE
Prof. (Research), Dept. of Civ. Engrg., Stanford Univ., Stanford, CA 94305

Metrics & Citations

Metrics

Citations

Download citation

If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

Cited by

View Options

Get Access

Access content

Please select your options to get access

Log in/Register Log in via your institution (Shibboleth)
ASCE Members: Please log in to see member pricing

Purchase

Save for later Information on ASCE Library Cards
ASCE Library Cards let you download journal articles, proceedings papers, and available book chapters across the entire ASCE Library platform. ASCE Library Cards remain active for 24 months or until all downloads are used. Note: This content will be debited as one download at time of checkout.

Terms of Use: ASCE Library Cards are for individual, personal use only. Reselling, republishing, or forwarding the materials to libraries or reading rooms is prohibited.
ASCE Library Card (5 downloads)
$105.00
Add to cart
ASCE Library Card (20 downloads)
$280.00
Add to cart
Buy Single Article
$35.00
Add to cart

Get Access

Access content

Please select your options to get access

Log in/Register Log in via your institution (Shibboleth)
ASCE Members: Please log in to see member pricing

Purchase

Save for later Information on ASCE Library Cards
ASCE Library Cards let you download journal articles, proceedings papers, and available book chapters across the entire ASCE Library platform. ASCE Library Cards remain active for 24 months or until all downloads are used. Note: This content will be debited as one download at time of checkout.

Terms of Use: ASCE Library Cards are for individual, personal use only. Reselling, republishing, or forwarding the materials to libraries or reading rooms is prohibited.
ASCE Library Card (5 downloads)
$105.00
Add to cart
ASCE Library Card (20 downloads)
$280.00
Add to cart
Buy Single Article
$35.00
Add to cart

Media

Figures

Other

Tables

Share

Share

Copy the content Link

Share with email

Email a colleague

Share