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.
Issue Section:
Technical Brief
Keywords:
Algorithms,
Fractional calculus ,
Nonlinear dynamical systems ,
Numerical,
Integration methods
Topics:
Algorithms
References
1.
Bagley
, L.
, and Torvik
, J.
, 1983
, “A Theoretical Basis for the Application of Fractional Calculus to Viscoelasticity
,” J. Rheol.
, 27
(3
), pp. 201
–210
.2.
Meral
, C.
, Royston
, J.
, and Magin
, R.
, 2010
, “Fractional Calculus in Viscoelasticity: An Experimental Study
,” Commun. Nonlinear Sci. Numer. Simul.
, 15
(4
), pp. 939
–945
.3.
Chen
, W.
, 2008
, “An Intuitive Study of Fractional Derivative Modeling and Fractional Quantum in Soft Matter
,” J. Vib. Control
, 14
(9–10
), pp. 1651
–1657
.4.
Nicolle
, S.
, Vezin
, P.
, and Palierne
, F.
, 2010
, “A Strain-Hardening Bi-Power Law for the Nonlinear Behaviour of Biological Soft Tissues
,” J. Biomech.
, 43
(5
), pp. 927
–932
.5.
Metzler
, R.
, and Klafter
, J.
, 2000
, “The Random Walk's Guide to Anomalous Diffusion: A Fractional Dynamics Approach
,” Phys. Rep.
, 339
(1
), pp. 1
–77
.6.
Zhokh
, A.
, Trypolskyi
, I.
, and Strizhak
, E.
, 2017
, “An Investigation of Anomalous Time-Fractional Diffusion of Isopropyl Alcohol in Mesoporous Silica
,” Int. J. Heat Mass Transfer
, 104
, pp. 493
–502
.7.
Podlubny
, I.
, 1999
, “Fractional-Order Systems and PI-Lambda-D-Mu-Controllers
,” IEEE Trans. Autom. Control
, 44
(1
), pp. 208
–214
.8.
Monje
, A.
, Chen
, Q.
, Vinagre
, M.
, Xue
, D.
, and Feliu-Batlle
, V.
, 2010
, Fractional-Order Systems and Controls: Fundamentals and Applications
, Springer Science & Business Media
, London.9.
Rossikhin
, A.
, and Shitikova
, V.
, 2010
, “Application of Fractional Calculus for Dynamic Problems of Solid Mechanics: Novel Trends and Recent Results
,” ASME Appl. Mech. Rev
, 63
(1
), p. 010801
.10.
Baleanu
, D.
, Asad
, H.
, and Petras
, I.
, 2015
, “Numerical Solution of the Fractional Euler-Lagrange's Equations of a Thin Elastica Mode
,” Nonlinear Dyn.
, 81
(1–2
), pp. 97
–102
.11.
Diethelm
, K.
, Ford
, J.
, and Freed
, D.
, 2002
, “A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations
,” Nonlinear Dyn.
, 29
(1
), pp. 3
–22
.12.
Diethelm
, K.
, and Ford
, J.
, 2004
, “Multi-Order Fractional Differential Equations and Their Numerical Solution
,” Appl. Math. Comput.
, 154
(3
), pp. 621
–640
.13.
Asl
, M. S.
, and Javidi
, M.
, 2017
, “An Improved PC Scheme for Nonlinear Fractional Differential Equations: Error and Stability Analysis
,” J. Comput. Appl. Math.
, 324
, pp. 101
–117
.14.
Soares
, D. S.
, Jr., Großeholz
, G.
, and Estorff
, O. V.
, 2015
, “An Efficient Staggered Time-Marching Procedure for Porodynamics
,” Comput. Methods Appl. Mech. Eng.
, 297
, pp. 1
–17
.15.
Simsek
, M.
, 2010
, “Dynamic Analysis of an Embedded Microbeam Carrying a Moving Microparticle Based on the Modified Couple Stress Theory
,” Int. J. Eng. Sci.
, 48
(12
), pp. 1721
–1732
.16.
Deü
, F.
, and Matignon
, D.
, 2010
, “Simulation of Fractionally Damped Mechanical Systems by Means of a Newmark-Diffusive Scheme
,” Comput. Math. Appl.
, 59
(5
), pp. 1745
–1753
.17.
Zhang
, W.
, and Shimizu
, N.
, 1998
, “Numerical Algorithm for Dynamic Problems Involving Fractional Operators
,” JSME Int. J. C
, 41
(3
), pp. 364
–370
.18.
Bucher
, C.
, and Pirrotta
, A.
, 2015
, “Dynamic Finite Element Analysis of Fractionally Damped Structural Systems in the Time Domain
,” Acta Mech.
, 226
(12
), p. 3977
.19.
Spanos
, D.
, and Evangelatos
, I.
, 2010
, “Response of a Non-Linear System With Restoring Forces Governed by Fractional Derivatives-Time Domain Simulation and Statistical Linearization Solution
,” Soil Dyn. Earthquake Eng.
, 30
(9
), pp. 811
–821
.20.
Diethelm
, K.
, Ford
, N. J.
, and Freed
, A. D.
, 2004
, “Detailed Error Analysis for a Fractional Adams Method
,” Numer. Algorithm
, 36
(1
), pp. 31
–52
.21.
Yuan
, L.
, and Agrawal
, O. P.
, 2002
, “A Numerical Scheme for Dynamic Systems Containing Fractional Derivatives
,” ASME J. Vib. Acoust.
, 124
(2
), pp. 321
–324
.22.
Singh
, S. J.
, and Chatterjee
, A.
, 2006
, “Galerkin Projections and Finite Elements for Fractional Order Derivatives
,” Nonlinear Dyn.
, 45
(1–2
), pp. 183
–206
.23.
Singh
, S. J.
, and Chatterjee
, A.
, 2011
, “Unified Galerkin- and DAE-Based Approximation of Fractional Order Systems
,” ASME J. Comput. Nonlinear Dyn.
, 6
(2
), p. 021010
.24.
Das
, S.
, and Chatterjee
, A.
, 2013
, “Simple Recipe for Accurate Solution of Fractional Order Equations
,” ASME J. Comput. Nonlinear Dyn.
, 8
(3
), p. 031007
.Copyright © 2018 by ASME
You do not currently have access to this content.
