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

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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Nayfeh, A. H. , and Mook, D. T. , 1979, Nonlinear Oscillations, Wiley, New York.
Ali Akbar, M. , Shamsul Alam, M. , and Sattar, M. A. , 2006, “ KBM Unified Method for Solving an nth Order Non-Linear Differential Equation Under Some Special Conditions Including the Case of Internal Resonance,” Int. J. Non-Linear Mech., 41(1), pp. 26–42. [CrossRef]
Dunne, J. F. , 2006, “ Subharmonic-Response Computation and Stability Analysis for a Nonlinear Oscillator Using a Split-Frequency Harmonic Balance Method,” ASME J. Comput. Nonlinear Dyn., 1(3), pp. 221–229. [CrossRef]
Akgün, D. , and Çankaya, İ. , 2010, “ Frequency Response Investigations of Multi-Input Multi-Output Nonlinear Systems Using Automated Symbolic Harmonic Balance Method,” Nonlinear Dyn., 61(4), pp. 803–818. [CrossRef]
Liao, S. J. , 2004, “ An Analytic Approximate Approach for Free Oscillations of Self-Excited Systems,” Int. J. Non-Linear Mech., 39(2), pp. 271–280. [CrossRef]
Zhen, Y. X. , and Fang, B. , 2015, “ Nonlinear Vibration of Fluid-Conveying Single-Walled Carbon Nanotubes Under Harmonic Excitation,” Int. J. Non-Linear Mech., 76, pp. 48–55. [CrossRef]
Lim, C. W. , and Wu, B. S. , 2005, “ Accurate Higher-Order Approximations to Frequencies of Nonlinear Oscillators With Fractional Powers,” J. Sound Vib., 281(3–5), pp. 1157–1162. [CrossRef]
Chung, K. W. , He, Y. B. , and Lee, B. H. K. , 2009, “ Bifurcation Analysis of a Two-Degree-of-Freedom Aeroelastic System With Hysteresis Structural Nonlinearity by a Perturbation-Incremental Method,” J. Sound Vib., 320(1–2), pp. 163–183. [CrossRef]
Howcroft, C. , Lowenberg, M. , Neild, S. , Krauskopf, B. , and Coetzee, E. , 2015, “ Shimmy of an Aircraft Main Landing Gear With Geometric Coupling and Mechanical Freeplay,” ASME J. Comput. Nonlinear Dyn., 10(5), p. 051011. [CrossRef]
Yang, J. Y. , Peng, T. , and Lim, T. C. , 2014, “ An Enhanced Multi-Term Harmonic Balance Solution for Nonlinear Period Dynamic Motions in Right-Angle Gear Pairs,” Nonlinear Dyn., 76(2), pp. 1237–1252. [CrossRef]
Kim, Y. B. , and Noah, S. T. , 1991, “ Stability and Bifurcation Analysis of Oscillators With Piecewise-Linear Characteristics: A General Approach,” ASME J. Appl. Mech., 58(2), pp. 545–553. [CrossRef]
Villa, C. , Sinou, J. J. , and Thouverez, F. , 2008, “ Stability and Vibration Analysis of a Complex Flexible Rotor Bearing System,” Commun. Nonlinear Sci. Numer. Simul., 13(3), pp. 804–821. [CrossRef]
Zhang, Z. Y. , and Chen, Y. S. , 2014, “ Harmonic Balance Method With Alternating Frequency/Time Domain Technique for Nonlinear Dynamical System With Fractional Exponential,” Appl. Math. Mech., 35(4), pp. 423–436. [CrossRef]
Zhang, Z. Y. , Chen, Y. S. , and Cao, Q. J. , 2015, “ Bifurcations and Hysteresis of Varying Compliance Vibrations in the Primary Parametric Resonance for a Ball Bearing,” J. Sound Vib., 350(18), pp. 171–184. [CrossRef]
Chen, Y. M. , Meng, G. , and Liu, J. K. , 2011, “ A New Method for Fourier Series Expansions: Applications in Rotor-Seal Systems,” Mech. Res. Commun., 38(5), pp. 399–403. [CrossRef]
Liu, J. K. , Chen, F. X. , and Chen, Y. M. , 2012, “ Bifurcation Analysis of Aeroelastic Systems With Hysteresis by Incremental Harmonic Balance Method,” Appl. Math. Comput., 219, pp. 2398–2411. [CrossRef]
Davis, P. J. , and Rabinowitz, P. , 2007, Methods of Numerical Integration, 2nd ed., Dover Publications, Mineola, NY.
Shen, J. H. , Lin, K. C. , Chen, S. H. , and Sze, K. Y. , 2008, “ Bifurcation and Route-to-Chaos Analyses for Mathieu–Duffing Oscillator by the Incremental Harmonic Balance Method,” Nonlinear Dyn., 52(4), pp. 403–414. [CrossRef]
Lu, C. J. , and Lin, Y. M. , 2011, “ A Modified Incremental Harmonic Balance Method for Rotary Periodic Motions,” Nonlinear Dyn., 66(4), pp. 781–788. [CrossRef]
Chen, Y. M. , Liu, J. K. , and Meng, G. , 2010, “ Relationship Between the Homotopy Analysis Method and Harmonic Balance Method,” Commun. Nonlinear Sci. Numer. Simul., 15(8), pp. 2017–2025. [CrossRef]
Gottlieb, D. , and Shu, C. W. , 1997, “ On the Gibbs Phenomenon and Its Resolution,” SIAM Rev., 39(4), pp. 644–668. [CrossRef]
Eckhoff, K. S. , 1995, “ Accurate Reconstruction of Functions of Finite Regularity From Truncated Fourier Series Expansions,” Math. Comput., 64(210), pp. 671–690. [CrossRef]


Grahic Jump Location
Fig. 1

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

Grahic Jump Location
Fig. 2

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

Grahic Jump Location
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

Grahic Jump Location
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

Grahic Jump Location
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.

Grahic Jump Location
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.

Grahic Jump Location
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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