Stationary Two‐State Variable Problems in Stochastic Mechanics
Publication: Journal of Engineering Mechanics
Volume 116, Issue 2
Abstract
A numerical procedure is presented for the solution of stationary two‐state Markov‐process problems with a single source. The proposed solution method is an efficient splitting technique that combines finite element and finite difference methods to take advantage of the specific form of the governing differential equation. A standard one‐dimensional finite element technique using linear interpolation and weighting functions is used to determine the solution for each row, while a variable‐weighted finite difference scheme is used to step in the other direction. This method is then illustrated by determining the stationary response of the Duffing oscillator and the statistical moments of the time to reach a critical crack size for the stochastic fatigue crack growth problem. Excellent comparisons are shown between this method, previous analytical studies, and experimental results with a significant reduction in computer processing time and storage.
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References
1.
Bhandari, R. G., and Sherrer, R. E. (1968). “Random vibrations in discrete nonlinear dynamic systems.” J. Mech. Engrg. Sci., Vol. 10, 168–174.
2.
Dahlquist, G., and Bjorck, A. (1974). Numerical methods. N. Anderson, trans., Prentice‐Hall, Englewood Cliffs, N.J., 166–167.
3.
Fichera, G. (1960). “On a unified theory of boundary value problems for elliptic‐parabolic equations of second order.” Boundary problems in differential equations, R. E. Langer, ed., University of Wisconsin Press, Wis., 97–102.
4.
Langley, R. S. (1985). “A finite element method for the statistics of non‐linear random vibration.” J. Sound Vib., 101(1), 41–54.
5.
Lin, Y. K. (1967). Probabilistic theory of structural dynamics. McGraw‐Hill, New York, N.Y.
6.
Ortiz, K., and Kung, C. J. (1987). “Modeling errors in fatigue crack growth laws.” Materials and member behavior, Proc., Structures Congress '87, Orlando, Fla.
7.
Roberts, J. B. (1981a). “Response of non‐linear mechanical systems to random excitation. Part 1: Markov methods.” Shock Vib. Digest, 13(4), 17–28.
8.
Roberts, J. B. (1981b). “Response of non‐linear mechanical systems to random excitation. Part 2: Equivalent linearization and other methods.” Shock Vib. Digest, Vol. 13, 15–29.
9.
Spencer, B. F., Jr., and Bergman, L. A. (1985). “On the reliability of a simple hysteretic system.” J. Engrg. Mech., ASCE, 111(12), 1502–1514.
10.
Spencer, B. F., Jr., and Tang, J. (1988). “A Markov process model for fatigue crack growth.” J. Engrg. Mech., ASCE, 114(12), 2134–2157.
11.
Spencer, B. F., Jr., Tang, J., and Artley, M. E. (1989). “A stochastic approach to modeling fatigue crack growth.” AIAA Journal, 27(11), 1628–1635.
12.
Virkler, D. A., Hillberry, B. M., and Goel, P. K. (1979). “The statistical nature of fatigue crack propagation.” J. Engrg. Mater. Tech., 101, 148–152.
13.
Wen, Y. K. (1975). “Approximate method for non‐linear random vibration.” J. Engrg. Mech. Div., ASCE, 101(4), 389–401.
14.
Wen, Y. K. (1976). “Method for random vibration of hysteretic systems.” J. Engrg. Mech. Div., ASCE, 102(2), 249–263.
Information & Authors
Information
Published In
Journal of Engineering Mechanics
Volume 116 • Issue 2 • February 1990
Pages: 343 - 358
Copyright
Copyright © 1990 ASCE.
History
Published online: Feb 1, 1990
Published in print: Feb 1990
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