Abstract
In this paper a new efficient algorithm for numerical integration of the equation of motion of a non linear system with restoring forces governed by fractional derivatives in the time domain is devised. This approach is based on the Grunwald-Letnikov representation of a fractional derivative and on the well known Newmark numerical integration scheme for structural dynamic problems. A Taylor expansion is used at every time step to represent the near past terms of the solution; thus, a dual mesh of the time domain is introduced: the coarse mesh is used for the time integration and the fine mesh is used for the fractional derivative approximation. It is shown that with this formulation the problem yields an equivalent non linear system without fractional terms which involves effective values of mass, damping, and stiffness coefficients as a predictive approach and a correction on the excitation. The major advantage of this approach is that a rather small number of past terms are required for the numerical propagation of the solution; and that the calculation of the effective values of mass, damping, and stiffness is performed only once. Several examples of applications are included.
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Evangelatos, G.I., Spanos, P.D. (2011). An Accelerated Newmark Scheme for Integrating the Equation of Motion of Nonlinear Systems Comprising Restoring Elements Governed by Fractional Derivatives. In: Kounadis, A.N., Gdoutos, E.E. (eds) Recent Advances in Mechanics. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-0557-9_9
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DOI: https://doi.org/10.1007/978-94-007-0557-9_9
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-007-0556-2
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