Fractional differential equations under stochastic input processes handled by the improved pseudo-force approach

Fractional differential equations under stochastic input processes handled by the improved pseudo-force approach

Alba Sofi, Giuseppe Muscolino, Mario Di Paola

Abstract. This paper presents a step-by-step procedure for the numerical integration of the fractional differential equation governing the response of a single-degree-of-freedom (SDOF) system with fractional derivative damping. The procedure is developed by extending the improved pseudo-force method proposed by the second author for the numerical integration of classical differential equations. To this aim, the Grünwald–Letnikov approximation of the fractional derivative is adopted. The proposed numerical procedure is exploited to compute response statistics of a SDOF system subjected to stochastic excitation by applying classical Monte Carlo Simulation.

Keywords
Fractional Differential Equations, Stochastic Processes, Step-By-Step Integration

Published online 3/17/2022, 6 pages
Copyright © 2023 by the author(s)
Published under license by Materials Research Forum LLC., Millersville PA, USA

Citation: Alba Sofi, Giuseppe Muscolino, Mario Di Paola, Fractional differential equations under stochastic input processes handled by the improved pseudo-force approach, Materials Research Proceedings, Vol. 26, pp 549-554, 2023

DOI: https://doi.org/10.21741/9781644902431-89

The article was published as article 89 of the book Theoretical and Applied Mechanics

Content from this work may be used under the terms of the Creative Commons Attribution 3.0 license. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.

References
[1] I. Podlubny, Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of their Solution and Some of their Applications, Academic Press, San Diego, CA, 1999.
[2] R. Scherer, S.L. Kalla, Y.Tang, J. Huangc, The Grünwald–Letnikov method for fractional differential equations, Comput. Math. Appl. 62 (2011) 902–917. https://doi.org/10.1016/j.camwa.2011.03.054
[3] G. Muscolino, Dynamically modified linear structures: deterministic and stochastic response. J. Eng. Mech. 122(11) (1996), 1044-1051. https://doi.org/10.1061/(ASCE)0733-9399(1996)122:11(1044)
[4] J. Xu, J. Li, Stochastic dynamic response and reliability assessment of controlled structures with fractional derivative model of viscoelastic dampers, Mech. Syst. Sig. Process. 72-73(2016) 865–896. https://doi.org/10.1016/j.ymssp.2015.11.016
[5] M.P. Singh, T.-S. Chang, H. Nandan, Algorithms for seismic analysis of MDOF systems with fractional derivatives, Eng. Struct. 33(8) (2011) 2371 – 2381. https://doi.org/10.1016/j.engstruct.2011.04.010