Characteristics of Response to Nonstationary White Noise: Theory
Publication: Journal of Engineering Mechanics
Volume 115, Issue 9
Abstract
New methods of analysis are developed to study characteristics of the response of linear systems to nonstationary stochastic input. An envelope of the response based on the Hilbert transformation of the process is used, and derivations are made with respect to stationary response properties. The analysis begins with a derivation of the statistics of the response of general linear systems to modulated white‐noise input. Then, the important case of oscillator systems is examined and, by use of functional operator concepts, approximations are obtained for the second‐order statistics of the oscillator response. It is found that the original expressions for these statistics, involving lengthy double integrals, become reduced to simple, closed‐form expressions, which are stated in terms of the oscillator properties and one convolution integral of the modulation function. The results provide physically meaningful insight into the characteristics of nonstationary response and its envelope. The relationships between the nonstationary and stationary resonse characteristics are explored in detail.
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References
1.
Caughey, T. K., and Stumpf, H. J. (1961). “Transient response of a dynamic system under random excitation.” J. Appl. Mech., American Society of Mechanical Engineers, 28(4), 563–566.
2.
Corotis, R. B., Vanmarcke, E. H., and Cornell, C. A. (1972). “First passage of nonstationary processes.” J. Engrg. Mech., ASCE, 98(2), 401–414.
3.
Cramer, H., and Leadbetter, M. R. (1967). Stationary and related stochastic processes. John Wiley and Sons, Inc., New York, N.Y.
4.
Der Kiureghian, A. (1980). “Structural response to stationary excitation.” J. Engrg. Mech., ASCE, 106(6), 1195–1213.
5.
Gasparini, D. A. (1980). “Response of MDOF systems to nonstationary random excitation.” J. Engrg. Mech., ASCE, 105(1), 13–27.
6.
Igusa, T. (1989). “Characteristics of response to nonstationary white noise: applications.” J. Engrg. Mech., ASCE, 115(9), 1920–1935.
7.
Krenk, S. (1979). “Nonstationary narrow‐band response and first‐passage probability.” J. Appl. Mech., American Society of Mechanical Engineers, 46(4), 919–924.
8.
Krenk, S., Madsen, H. O., and Madsen, P. H. (1983). “Stationary and transient response envelopes,” J. Engrg. Mech., ASCE, 109(1), 263–278.
9.
Lin, Y. K. (1976). Probabilistic theory of structural dynamics. Krieger Publishing Co., Inc., Huntington, N.Y.
10.
Madsen, P. H., and Krenk, S. (1984). “An integral method for the first‐passage problem in random vibration.” J. Appl. Mech., American Society of Mechanical Engineers, 51(3), 674–679.
11.
Madsen, H. O., Krenk, S., and Lind, N. C. (1986). Methods of structural safety. Prentice‐Hall, Inc., Englewood Cliffs, N.J.
12.
Nigam, N. C. (1983). Introduction to random vibrations. The MIT Press, Cambridge, Mass.
13.
Rice, S. O. (1944, 1945). “Mathematical analysis of random noise.” Bell Sys. Tech. J., 23, 282–332;
24, 46–156.
14.
Solomos, G. P., and Spanos, P. D. (1984). “Oscillator response to nonstationary excitation.” J. Appl. Mech., American Society of Mechanical Engineers, 51(4), 907–912.
15.
Spanos, P. D. (1980). “Probabilistic earthquake energy spectra equations,” J. Engrg. Mech., ASCE, 106(1), 147–159.
16.
Spanos, P. D. (1982). “Numerics for a common first‐passage problem.” J. Engrg. Mech., ASCE, 108(5), 864–882.
17.
Spanos, P. D., and Roberts, J. B. (1986). “Stochastic averaging: an approximate method of solving random vibration problems.” Int. J. Non Linear Mech., 21(2), 111–134.
18.
Spanos, P. D., and Solomos, G. P. (1983). “Markov approximation to nonstationary random vibration.” J. Engrg. Mech., ASCE, 109(4), 1134–1150.
19.
Stratanovich, R. L. (1963). Topics in the theory of random noise, Vol. II, Gordon and Breach, Science Publishers, Inc., New York, N.Y.
20.
Vanmarcke, E. H. (1972). “Properties of spectral moments with applications to random vibraion.” J. Engrg. Mech., ASCE, 98(2), 425–446.
21.
Vanmarcke, E. H. (1975). “On the distribution of the first‐passage time for normal stationary random processes.” J. Appl. Mech., American Society of Mechanical Engineers, 42, 215–220.
22.
Yang, J. N., and Shinozuka, M. (1971). “On the first excursion probability in stationary narrow‐band random vibration.” J. Appl. Mech., American Society of Mechanical Engineers, 38(4), 1017–1022.
23.
Yang, J. N. (1973). “First excursion probability in nonstationary random vibration.” J. Sound Vib., 27(2), 165–182.
Information & Authors
Information
Published In
Journal of Engineering Mechanics
Volume 115 • Issue 9 • September 1989
Pages: 1904 - 1918
Copyright
Copyright © 1989 ASCE.
History
Published online: Sep 1, 1989
Published in print: Sep 1989
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