Abstract Summary
The dynamic analysis of continuous beams crossed by moving loads is a topic of great interest in the field of structural engineering. Indeed, the transversal deflection and stresses due to moving loads could be considerably higher than those observed under static loads. Usually, the damping of beams is assumed to be of the viscous type. This leads to a frequency dependence of the viscous mechanism, while experimental tests indicate that the damping forces are nearly independent of frequency [1]. Recently, it has been widely recognized that the most realistic model to describe the dissipation of the energy of materials is the viscoelastic one involving fractional derivatives [2]. The present study addresses the dynamic analysis of fractional viscoelastic Euler-Bernoulli beams crossed by moving loads. The motion of such beams is governed by a fractional differential equation. Specifically, since the structural systems can be considered quiescent before the motions appear, the fractional constitutive law involves Caputo’s derivative [3]. The main purpose of this paper is to propose a step-by-step procedure for the numerical integration of the fractional differential equation. To this aim, first, the Grünwald–Letnikov approximation of the fractional Caputo’s derivative is adopted [4]. Then, the so-called improved pseudo-force approach, introduced by Muscolino [5] for evaluating the response of linear structural systems governed by classical ordinary differential equations, is extended to fractional differential equations. Numerical results show that the proposed procedure reduces the computational burden and increases the accuracy with respect to numerical integration procedures proposed in the literature. References [1] Clough R.W., Penzien J. 1993. Dynamics of Structures. MacGraw-Hill, New York. [2] Di Paola M., Pirrotta A., Valenza A. 2011. Visco-elastic behavior through fractional calculus: an easier method for best fitting experimental results. Mechanics of materials 43: 799-806. [3] Podlubny I. 1999. On Solving Fractional Differential Equations by Mathematics. Science and Engineering, vol. 198. Academic Press. [4] R. Scherer, S.L. Kalla, Y.Tang, J. Huang. 2011. The Grünwald–Letnikov method for fractional differential equations. Computers & Mathematics with Applications 62: 902–917. [5] Muscolino G. 1996. Dynamically modified linear structures: deterministic and stochastic response. Journal Engineering Mechanics (ASCE) 122: 1044-1051.
