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# A probabilistic treatment of uncertainty in nonlinear dynamical systems

## Citation

Polidori, David Charles (1998) A probabilistic treatment of uncertainty in nonlinear dynamical systems. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/axk4-vz45. https://resolver.caltech.edu/CaltechThesis:03112014-113438332

## Abstract

In this work, computationally efficient approximate methods are developed for analyzing uncertain dynamical systems. Uncertainties in both the excitation and the modeling are considered and examples are presented illustrating the accuracy of the proposed approximations.

For nonlinear systems under uncertain excitation, methods are developed to approximate the stationary probability density function and statistical quantities of interest. The methods are based on approximating solutions to the Fokker-Planck equation for the system and differ from traditional methods in which approximate solutions to stochastic differential equations are found. The new methods require little computational effort and examples are presented for which the accuracy of the proposed approximations compare favorably to results obtained by existing methods. The most significant improvements are made in approximating quantities related to the extreme values of the response, such as expected outcrossing rates, which are crucial for evaluating the reliability of the system.

Laplace's method of asymptotic approximation is applied to approximate the probability integrals which arise when analyzing systems with modeling uncertainty. The asymptotic approximation reduces the problem of evaluating a multidimensional integral to solving a minimization problem and the results become asymptotically exact as the uncertainty in the modeling goes to zero. The method is found to provide good approximations for the moments and outcrossing rates for systems with uncertain parameters under stochastic excitation, even when there is a large amount of uncertainty in the parameters. The method is also applied to classical reliability integrals, providing approximations in both the transformed (independently, normally distributed) variables and the original variables. In the transformed variables, the asymptotic approximation yields a very simple formula for approximating the value of SORM integrals. In many cases, it may be computationally expensive to transform the variables, and an approximation is also developed in the original variables. Examples are presented illustrating the accuracy of the approximations and results are compared with existing approximations.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:uncertainty ; nonlinear dynamical systems
Degree Grantor:California Institute of Technology
Division:Engineering and Applied Science
Major Option:Applied Mechanics
Thesis Availability:Public (worldwide access)
• Beck, James L.
Thesis Committee:
• Beck, James L. (chair)
• Caughey, Thomas Kirk
• Franklin, Joel N.
• Iwan, Wilfred D.
Defense Date:2 October 1997
Other Numbering System:
Other Numbering System NameOther Numbering System ID
EERL97-09
Funders:
Funding AgencyGrant Number
NSFCMS-9503370
NSFCMS-9309149
Charles Lee Powell FoundationUNSPECIFIED
Harold Hellwig FoundationUNSPECIFIED
Record Number:CaltechThesis:03112014-113438332
Persistent URL:https://resolver.caltech.edu/CaltechThesis:03112014-113438332
DOI:10.7907/axk4-vz45
Related URLs:
URLURL TypeDescription
http://resolver.caltech.edu/CaltechEERL:1997.EERL-97-09Related DocumentTechnical Report EERL 97-09 in CaltechAUTHORS
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:8123
Collection:CaltechTHESIS
Deposited By: Kathy Johnson
Deposited On:12 Mar 2014 21:51

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