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Technical Brief

A Second-Order Scheme for Nonlinear Fractional Oscillators Based on Newmark-β Algorithm

[+] Author and Article Information
Q. X. Liu, J. K. Liu, Y. M. Chen

Department of Mechanics,
Sun Yat-Sen University,
No.135 Xingang Road West,
Guangzhou 510275, China

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received August 29, 2017; final manuscript received May 10, 2018; published online June 18, 2018. Assoc. Editor: Anindya Chatterjee.

J. Comput. Nonlinear Dynam 13(8), 084501 (Jun 18, 2018) (5 pages) Paper No: CND-17-1391; doi: 10.1115/1.4040342 History: Received August 29, 2017; Revised May 10, 2018

This paper presents an accurate and efficient hybrid solution method, based on Newmark-β algorithm, for solving nonlinear oscillators containing fractional derivatives (FDs) of arbitrary order. Basically, this method employs a quadrature method and the Newmark-β algorithm to handle FDs and integer derivatives, respectively. To reduce the computational burden, the proposed approach provides a strategy to avoid nonlinear algebraic equations arising routinely in the Newmark-β algorithm. Numerical results show that the presented method has second-order accuracy. Importantly, it can be applied to both linear and nonlinear oscillators with FDs of arbitrary order, without losing any precision and efficiency.

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Copyright © 2018 by ASME
Topics: Algorithms , Algebra
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Figures

Grahic Jump Location
Fig. 1

Convergence order of the presented method with γ = 1/2 and β = 1/4, and YA method with quadrature points N = 10, 20, 40, for different fractional orders α

Grahic Jump Location
Fig. 2

Comparison of results obtained by the presented method with γ = 1/2, β = 1/4 and Δt = 0.01, the Galerkin method solving Eq. (11) of fractional order α and (12) of α/2, respectively

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