Stochastic dynamic systems with complex-valued eigensolutions

International Journal for Numerical Methods in Engineering

Volumn 71, Issue 8, 2007, Pages 963-986

  Rahman, Sharif ^a  

^a UNIVERSITY OF IOWA   (United States)

Author keywords

Bivariate decomposition; Complex eigenvalue; Decomposition method; Disc brake system; Random eigenvalue; Random matrix; Univariate decomposition

Indexed keywords

APPROXIMATION THEORY; COMPUTER SIMULATION; EIGENVALUES AND EIGENFUNCTIONS; INTERPOLATION; LAGRANGE MULTIPLIERS; MONTE CARLO METHODS; PERTURBATION TECHNIQUES; PROBABILITY DENSITY FUNCTION;

FUNCTION DECOMPOSITION; LINEAR STOCHASTIC DYNAMICS; RANDOM MATRIX;

DYNAMICAL SYSTEMS;

APPROXIMATION THEORY; COMPUTER SIMULATION; DYNAMICAL SYSTEMS; EIGENVALUES AND EIGENFUNCTIONS; INTERPOLATION; LAGRANGE MULTIPLIERS; MONTE CARLO METHODS; PERTURBATION TECHNIQUES; PROBABILITY DENSITY FUNCTION;

EID: 34548042108     PISSN: 00295981     EISSN: 10970207     Source Type: Journal    
DOI: 10.1002/nme.1973     Document Type: Article

References (25)

1. Boyce WE. Probabilistic Methods in Applied Mathematics I. Academic Press: New York, NY, 1968.
2. Shinozuka M, Astill CJ. Random eigenvalue problems in structural analysis. AIAA Journal 1972; 10(4):456-462.
3. Zhang J, Ellingwood B. Effects of uncertain material properties on structural stability. Journal of Structural Engineering 1995; 121(4):705-714.
4. Mehlhose S, Vom Scheidt J, Wunderlich R. Random eigenvalue problems for bending vibrations of beams. Zeitschrift für Angewandte Mathematik und Mechanik 1999; 79:693-702.
5. Grigoriu M. A solution of the random eigenvalue problem by crossing theory. Journal of Sound and Vibration 1992; 158(1):69-80.
6. Nair PB, Keane AJ. An approximate solution scheme for the algebraic random eigenvalue problem. Journal of Sound and Vibration 2003; 260(1):45-65.
7. Adhikari S, Friswell MI. Random matrix eigenvalue problems in structural dynamics. International Journal for Numerical Methods in Engineering 2006, in press.
8. Ghosh D, Ghanem RG, Red-Horse J. Analysis of eigenvalues and modal interaction of stochastic systems. AIAA Journal 2005; 43(10):2196-2201.
9. Rahman S. A solution of the random eigenvalue problem by a dimensional decomposition method. International Journal for Numerical Methods in Engineering 2006; 67:1318-1340.
10. Caughey TK, O'Kelly MEJ. Classical normal modes in damped linear dynamic systems. ASME Journal of Applied Mechanics 1965; 32:583-588.
11. Adhikari S. Complex modes in stochastic systems. Advances in Vibration Engineering 2004; 3(1):1-11.
12. Xu H, Rahman S. A generalized dimension-reduction method for multi-dimensional integration in stochastic mechanics. International Journal for Numerical Methods in Engineering 2004; 61:1992-2019.
13. Xu H, Rahman S. Decomposition methods for structural reliability analysis. Probabilistic Engineering Mechanics 2005; 20:239-250.
14. Hoeffding W. A class of statistics with asymptotically normal distributions. Annals of Mathematical Statistics 1948; 19:293-325.
15. Efron B, Stein C. The Jackknife estimate of variance. Annals of Statistics 1981; 9:586-596.
16. Sobol IM. Theorems and examples on high dimensional model representations. Reliability Engineering and System Safety 2003; 79:187-193.
17. Rabitz H, Alis O. General foundations of high dimensional model representations. Journal of Mathematical Chemistry 1999; 25:197-233.
18. Li G, Wang SW, Rabitz H. Practical approaches to construct RS-HDMR component functions. Journal of Physical Chemistry A 2002; 106:8721-8733.
19. Li G, Rosenthal C, Rabitz H. High dimensional model representations. Journal of Physical Chemistry A 2001; 105:7765-7777.
20. Li G, Wang SW, Rosenthal C, Rabitz H. High dimensional model representations generated from low dimensional data samples - I. mp-Cut-HDMR. Journal of Mathematical Chemistry 2001; 30:1-30.
21. Puig B, Akian JL. Non-Gaussian simulation using hermite polynomial expansion and maximum entropy principle. Probabilistic Engineering Mechanics 2004; 19:293-305.
22. Newland DE. Mechanical Vibration Analysis and Computation. Wiley: New York, NY, 1989.
23. IMSL Numerical Libraries. User's Guide and Theoretical Manual. Visual Numerics Corporate Headquarters, San Ramon, CA, 2005.
24. ABAQUS. User's Guide and Theoretical Manual, Version 6.5. Hibbitt, Karlsson, and Sorenson, Inc.: Pawtucket, RI, 2005.
25. Bajer A, Belsky V, Zeng LJ. Combining a nonlinear static analysis and complex eigenvalue extraction in brake squeal simulation. Proceedings of 21st Annual Brake Colloquium and Exhibition, SAE Paper no. 2003-01-3349. Hollywood, FL, 2003.