Role of Roots of Orthogonal Polynomials in the Dynamic Response of Stochastic Systems
Publication: Journal of Engineering Mechanics
Volume 142, Issue 8
Abstract
This paper investigates the fundamental nature of the polynomial chaos (PC) response of dynamic systems with uncertain parameters in the frequency domain. The eigenfrequencies of the extended matrix arising from a PC formulation govern the convergence of the dynamic response. It is shown that, in the particular case of uncertainties and with Hermite and Legendre polynomials, the PC eigenfrequencies are related to the roots of the underlying polynomials, which belong to the polynomial chaos set used to derive the polynomial chaos expansion. When Legendre polynomials are used, the PC eigenfrequencies remain in a bounded interval close to the deterministic eigenfrequencies because they are related to the roots of a Legendre polynomial. The higher the PC order, the higher the density of the PC eigenfrequencies close to the bounds of the interval, and this tends to smooth the frequency response quickly. In contrast, when Hermite polynomials are used, the PC eigenfrequencies spread from the deterministic eigenfrequencies (the highest roots of the Hermite polynomials tend to infinity when the order tends to infinity). Consequently, when the PC number increases, the smoothing effect becomes inefficient.
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Acknowledgments
J.-J. Sinou acknowledges the support of the Institut Universitaire de France.
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Published In
Journal of Engineering Mechanics
Volume 142 • Issue 8 • August 2016
Copyright
© 2016 American Society of Civil Engineers.
History
Received: Oct 16, 2015
Accepted: Feb 10, 2016
Published online: Mar 28, 2016
Published in print: Aug 1, 2016
Discussion open until: Aug 28, 2016
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