Role of roots of orthogonal polynomials in the dynamic response of stochastic systems

Jacquelin, E., Adhikari, S. , Friswell, M. I. and Sinou, J. -J. (2016) Role of roots of orthogonal polynomials in the dynamic response of stochastic systems. Journal of Engineering Mechanics, 142(8), 06016004. (doi: 10.1061/(ASCE)EM.1943-7889.0001102)

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.

• Item Type: Articles
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Adhikari, Professor Sondipon
• Authors: Jacquelin, E., Adhikari, S., Friswell, M. I., and Sinou, J. -J.
• College/School: College of Science and Engineering > School of Engineering > Infrastructure and Environment
• Journal Name: Journal of Engineering Mechanics
• Publisher: American Society of Civil Engineers
• ISSN: 0733-9399
• ISSN (Online): 1943-7889
• Published Online: 28 March 2016