Phase Space Reduction in Stochastic Dynamics
Publication: Journal of Engineering Mechanics
Volume 126, Issue 6
Abstract
In the field of structural engineering, various analytical procedures have been developed for the analysis of nonlinear structural systems subjected to random dynamic loading. To date, the Monte Carlo simulation (MCS) seems to be the most generally applicable approach for the reliability analysis of large nonlinear multi-degree-of-freedom systems. In this paper, a method that allows for reduction of the computational effort when using the MCS method is presented. First, the method requires the application of digital or analytical techniques (such as equivalent linearization) to obtain the response covariance matrix. Then, by means of the well-known Karhunen-Loéve expansion, the dimension of the system is reduced and MCS is applied. Moreover, in the transformed space the variables with smaller variance are approximated by Gaussian variables. It is shown that this technique allows for a considerable reduction of the computational effort without a significant loss of accuracy. As numerical examples, a 9-degree-of-freedom hysteretic structure and a conservative system (i.e., a 10-degree-of-freedom Duffing structure), respectively, are examined.
Get full access to this article
View all available purchase options and get full access to this article.
References
1.
Askar, A., Koyluoglu, H. U., Cakmak, A. S., and Nielsen, S. R. K. ( 1995). “Faster simulation methods for the non-stationary random vibration of non-linear MDOF systems.” P. D. Spanos, ed., Computational stochastic mechanics.
2.
Atalik, T. S., and Utku, S. (1976). “Stochastic linearization of multi-degree-of-freedom systems.” Earthquake Engrg. and Struct. Dyn., 4, 411–420.
3.
Bergman, L. A., and Spencer, B. F., Jr. (1992). “Robust numerical solution of the transient Fokker Planck equation for nonlinear dynamical systems.” Proc., IUTAM Symp. on Nonlinear Stochastic Mech., N. Bellomo and F. Casciati, eds., Springer, Berlin Heidelberg, Germany.
4.
Caughey, T. K. (1963). “Equivalent linearization techniques.” J. Acoust. Soc. Am., 35, 1706–1711.
5.
Crandall, S. H. (1963). “Perturbation techniques for random vibration of nonlinear systems.” J. Accoust. Soc. Am., 35, 1700–1705.
6.
Crandall, S. H. (1978). “Heuristic and equivalent linearization techniques for random vibration of nonlinear oscillators.” Proc., 6th Int. Conf. Nonlinear Oscillations (ICNO).
7.
Di Paola, M., Falsone, G., and Pirrotta, A. (1992). “Stochastic response analysis of non-linear systems under Gaussian inputs.” Probabilistic Engrg. Mech., 7, 15–21.
8.
Emam, H. H., Pradlwarter, H. J., and Schuëller, G. I. (1998). “On the computational implementation of EQL in FE-analysis.” Proc., 4th Int. Conf. on Stochastic Struct. Dyn., SSD '98, B. F. Spencer and E. Johnson, eds., Balkema, Rotterdam, The Netherlands.
9.
Ghanem, R., and Spanos, P. (1991). Stochastic finite elements: A spectral approach. Springer.
10.
Honerkamp, J. (1994). Stochastic dynamical systems. VCH, New York.
11.
Langley, R. S. (1985). “A finite element method for the statistics of non-linear random vibration.” Sound and Vibration, 101(1), 41–54.
12.
Lin, Y. K., and Cai, G. Q. (1988). “On exact stationary solutions of equivalent non-linear stochastic systems.” Int. J. Non-Linear Mech., 23(5/6), 409–420.
13.
Lin, Y. K., and Cai, G. Q. (1995). Probabilistic structural dynamics. McGraw-Hill International Editions.
14.
Loéve, M. (1978). Probability theory II. Springer, New York, Heidelberg, Germany, and Berlin.
15.
Ma, F. (1987). “Extension of second moment analysis to vector-valued and matrix-valued functions.” Int. J. Non-Linear Mech., 22, 251–260.
16.
Naess, A., and Johnsen, J. M. (1993). “Response statistics of nonlinear, compliant offshore structures by the path integral solution method.” Probabilistic Engrg. Mech., 8(2), 91–106.
17.
Pradlwarter, H. J., and Schuëller, G. I. ( 1992). “A practical approach to predict the stochastic response on many-DOF-systems modeled by finite elements.” Nonlinear Stochastic Mech. IUTAM Symp., N. Bellomo and F. Casciati, eds.
18.
Pradlwarter, H. J., and Schuëller, G. I. (1993). “Equivalent linearization—a suitable tool for analyzing MDOF systems.” Probabilistic Engrg. Mech., 8, 115–126.
19.
Pradlwarter, H. J., and Schuéller, G. I. (1995). “On advanced Monte Carlo simulation procedures in stochastic structural dynamics.” Int. J. Non-Linear Mech., 32, 735–744.
20.
Pradlwarter, H. J., Schuëller, G. I., and Chen, X. W. (1988). “Consideration of non-Gaussian response properties by use of stochastic equivalent linearization.” Proc., 3rd Int. Conf. on Recent Adv. in Struct. Dyn., 737–752.
21.
Roberts, J. B., and Spanos, P. D. (1990). Random vibration and statistical linearization. Wiley, New York.
22.
Rubinstein, R. Y. (1981). Simulation and Monte Carlo method. Wiley, New York.
23.
Schuëller, G. I., and Pradlwarter, H. J. (1996). “Methods of non-linear stochastic dynamics in view of practical applications.” Proc., EURODYN '96, G. Augusti et al., eds., Vol. 1, Balkema, Rotterdam, The Netherlands, 31–40.
24.
Shinozuka, M. (1972). “Monte Carlo solution of structural dynamics.” Comp. and Struct., 5/6, 855–874.
25.
Stratonovich, R. L. (1963). Topics in the theory of random noise. Vol. 1, Gordon and Breach, New York.
26.
Wu, W. F., and Lin, Y. K. (1984). “Cumulant closure for non-linear oscillator under random parametric and external excitation.” Int. J. Non-Linear Mech., 19, 349–362.
27.
Yamazaki, F., and Shinozuka, M. (1990). “Simulation of stochastic fields by statistical preconditioning.”J. Engrg. Mech., ASCE, 116(2), 268–287.
28.
Yamazaki, F., Shinozuka, M., and Dasgupta, G. (1988). “Neumann expansion for stochastic finite-element analysis.”J. Engrg. Mech., ASCE, 114(8), 1335–1354.
29.
Yu, J. S., Cai, G. Q., and Lin, Y. K. (1997). “A new path integration procedure based on Gauss-Legendre scheme.” Int. J. Non-Linear Mech., 32(4), 759-768.
Information & Authors
Information
Published In
Journal of Engineering Mechanics
Volume 126 • Issue 6 • June 2000
Pages: 626 - 632
History
Received: Oct 2, 1998
Published online: Jun 1, 2000
Published in print: Jun 2000
Authors
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.