Open Access Research

Application of fractional calculus in the dynamics of beams

D Dönmez Demir1*, N Bildik1 and BG Sinir2

Author Affiliations

1 Department of Mathematics, Faculty of Art & Science, Celal Bayar University, Manisa, 45047, Turkey

2 Department of Civil Engineering, Faculty of Engineering, Celal Bayar University, Manisa, 45140, Turkey

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Boundary Value Problems 2012, 2012:135 doi:10.1186/1687-2770-2012-135

Published: 15 November 2012

Abstract

This paper deals with a viscoelastic beam obeying a fractional differentiation constitutive law. The governing equation is derived from the viscoelastic material model. The equation of motion is solved by using the method of multiple scales. Additionally, principal parametric resonances are investigated in detail. The stability boundaries are also analytically determined from the solvability condition. It is concluded that the order and the coefficient of the fractional derivative have significant effect on the natural frequency and the amplitude of vibrations.

Keywords:
perturbation method; fractional derivative; method of multiple scales; linear vibrations