The Role of Roots of Orthogonal Polynomials in the Dynamic Response of Stochastic Systems
E Jacquelin (Universite de Lyon, France), S Adhikari, MI Friswell (Swansea University) & J-J Sinou (Ecole Centrale de Lyon, France)
ASCE Journal of Engineering Mechanics, Vol. 142, No. 8, August 2016, paper 06016004
This note 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 showed that, in a particular case of uncertainties and with Hermite and Legendre polynomials, the PC-eigenfrequencies are related to the roots of the underlying polynomials, which belongs to the polynomial chaos set used to derive the polynomial chaos expansion. When a Legendre polynomial is used, the PC-eigenfrequencies remain in a bounded interval closed to the deterministic eigenfrequencies, as they are related to the roots of a Legendre polynomial. Higher the PC order, higher is the density of PC-eigenfrequencies close to the bounds of the interval and this tends to smooth the frequency response quickly. On the contrary, when Hermite polynomials are used, the PC-eigenfrequencies spread from the deterministic eigenfrequencies (the highest root of the Hermite polynomials tend to infinity when the order tend to infinity). Consequently, when the PC number increases, the smoothing effect becomes inefficient.
This material has been published in the ASCE Journal of Engineering Mechanics, Vol. 142, No. 8, August 2016, paper 06016004. Unfortunately the copyright agreement with ASCE does not allow for the PDF file of the paper to be available on this website.
Link to paper using doi: 10.1061/(ASCE)EM.1943-7889.0001102
Link to the Journal of Engineering Mechanics website