Response of Fractionally Damped Beams with General Boundary Conditions Subjected to Moving Loads

Abstract

This paper presents the transverse vibration of Bernoulli-Euler homogeneous isotropic damped beams with general boundary conditions. The beams are assumed to be subjected to a load moving at a uniform velocity. The damping characteristics of the beams are described in terms of fractional derivatives of arbitrary orders. In the analysis where initial conditions are assumed to be homogeneous, the Laplace transform cooperates with the decomposition method to obtain the analytical solution of the investigated problems. Subsequently, curves are plotted to show the dynamic response of different beams under different sets of parameters including different orders of fractional derivatives. The curves reveal that the dynamic response increases as the order of fractional derivative increases. Furthermore, as the order of the fractional derivative increases the peak of the dynamic deflection shifts to the right, this yields that the smaller the order of the fractional derivative, the more oscillations the beam suffers. The results obtained in this paper closely match the results of papers in the literature review.

Analytical responses of functionally graded beam under moving mass using Caputo and Caputo–Fabrizio fractional derivative models

“…Equation (11) can be solved by using the decomposition method with the Laplace transform, where the function Tn(t) is decomposed into an infinite series as (Abu-Mallouh et al, 2012; Alkhaldi et al, 2013)…”

“…Applying the fractional derivatives was proposed by many researchers to get better description for the viscoelastic behavior of such problems (Podlubny, 1999; Rossikhin and Shitikova, 1997). It is noted that the fractional derivative order has significant effect on the dynamic behavior of the beams (Abu-Mallouh et al, 2012; Alkhaldi et al, 2013; Freundlich, 2016). Many definitions for the fractional derivatives are presented; however, Caputo and Riemann–Liouville definitions are the most widely used (Di Lorenzo et al, 2014; Di Paola et al, 2013; Dönmez Demir et al, 2014; Freundlich, 2019; Rossikhin et al, 2017).…”

Dynamic Response and Parameter Analysis of Electromagnetic Railguns under Time Varying Moving Loads

“…One is a single moving load, and the other is a harmonic moving load. On the aspect of a single moving load, for instance, AlSaleh et al [19] investigated the dynamic response of Euler-Bernoulli beams under a traversing moving load based on Green's functions combined with a decomposition technique. Te load was assumed to be moving with diferent values of constant velocity.…”

Coupled bending torsional vibrations of viscoelastic rotors with fractional damper

“…A variety of models exist in the literature to study the viscoelastic behavior especially in rotors (Ganguly et al, 2016, 2018; Hendricks, 1986; Roy and Chandraker, 2018), viscoelastically supported stator experiencing rub, or in non-contacting rotor-bearing systems through the use of linear spring and viscous damper components (Patel et al, 2012; Smyth et al, 2016). Furthermore, the fractional calculus has been used by a number of studies for analyzing viscoelastically damped structures (Abu-Mallouh et al, 2012; Agrawal, 2004; Bagley and Torvik, 1983; Demir et al, 2012; Di Lorenzo et al, 2014; Liang and Tang, 2007; Ray et al, 2005; Sakakibara, 1997; Spanos and Malara, 2014). It is demonstrated that including viscoelasticity in the rotor supports leads to significant changes in the rotor critical speeds and an increased regime of stability (Shabaneh and Zu, 2000).…”