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Research Papers

Error Estimation of Fourier Series Expansion and Implication to Solution Accuracy for Nonlinear Dynamical Systems

[+] Author and Article Information
Y. M. Chen

Department of Mechanics,
Sun Yat-sen University,
No. 135 Xingang Road,
Guangzhou 510275, China
e-mail: chenymao@mail.sysu.edu.cn

Z. R. Lv

Department of Mechanics,
Sun Yat-sen University,
No. 135 Xingang Road,
Guangzhou 510275, China
e-mail: lvzr@mail.sysu.edu.cn

J. K. Liu

Professor
Department of Mechanics,
Sun Yat-sen University,
No. 135 Xingang Road,
Guangzhou 510275, China
e-mail: liujike@mail.sysu.edu.cn

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received August 31, 2015; final manuscript received July 3, 2016; published online September 1, 2016. Assoc. Editor: Bogdan I. Epureanu.

J. Comput. Nonlinear Dynam 12(1), 011002 (Sep 01, 2016) (6 pages) Paper No: CND-15-1266; doi: 10.1115/1.4034127 History: Received August 31, 2015; Revised July 03, 2016

Fourier series expansion (FSE) plays a pivotal role in frequency domain analysis of a wide variety of nonlinear dynamical systems. To the best of our knowledge, there are two general approaches for FSE, i.e., a collocation method (CM) previously proposed by the authors and the classical discrete FSE. Though there are huge applications of these methods, it still remains much less understood in their relationship and error estimation. In this study, we proved that they are equivalent if time points are uniformly chosen. Based on this property, more importantly, the error was analytically estimated for both discrete Fourier expansion (DFE) and CM. Furthermore, we revealed that the accuracy of frequency domain solutions cannot be improved by increasing the number of time points alone, whereas it absolutely depends upon the truncated number of harmonics. It indicates that an appropriate number of time points should be chosen in FSE if frequency domain solutions are targeted for nonlinear dynamical systems, especially those with complicated functions.

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Figures

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Fig. 2

Logarithmic errors of DFE and CM results with the exact one versus M when time points are randomly chosen

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Fig. 1

Logarithmic difference between the DFE and CM results when time points are uniformly chosen

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Fig. 3

Sketch of the accumulative error for the Fourier coefficients

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Fig. 4

The CM error when q[x(t)]=x3(t) with x(t) containing 33 harmonics

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Fig. 5

Logarithmic-averaged CM error and Fourier coefficients for the considered function versus M or n

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Fig. 6

Logarithmic-averaged CM error and Fourier coefficients for q[x(t)]=(1+x2(t))1/3 versus M or n

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Fig. 9

The residue of Eq. (26) versus number (N) of harmonics when M time points are chosen. The fitting curve is provided by 0.0625N−1.

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Fig. 7

The residue of Eq. (24) versus number (N) of harmonics when M time points are chosen. The fitting curve is provided by 1.459e−0.326N.

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Fig. 8

The residue of Eq. (25) versus the number of harmonics (N) when M time points are chosen. The fitting curve is provided by 0.087e−1.438N.

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