DYNAMIC ANALYSIS OF BEAMS ON FRACTIONAL VISCOELASTIC FOUNDATION SUBJECT TO A VARIABLE SPEED MOVING LOAD

The present paper investigates the dynamic response of beams resting on fractional viscoelastic foundation subjected to a moving load with variable speeds. The Galerkin with finite difference methods are used to deal with the governing equation of motion. The effect of various parameters, like fractional order derivative, foundation stiffness and damping, speed of moving load on the response of the beam are investigated and discussed.


Introduction
Recently, the dynamic response of beams on linear nonlinear viscoelastic foundations became one of important fields of research in the world. Specifically, when the Stephenson's bridge across river Dee Chester in England in 1947 collapsed, it encourages the engineers for research on moving load problems. Many problems of considerable practical importance can be related to the solution of beams on foundation. Reinforced concrete pavements of highways and airport runways and foundation slabs of buildings are well-known direct applications. Moving loads have a great effect on the bodies or structures over which it travels. Moreover, high-speed trains are becoming increasingly popular. For instance, the passage of the load with high speed will increase the deflection, due to this fact; we see that the study of moving loads is important for engineers. Michaltsos [1], investigated dynamic response of beam on linear viscoelastic foundation subjected to moving load varying with time. Froio [2], analyzed the vibration of beams on nonlinear elastic foundations under harmonic moving load using finite element method and HHTimplementation (see Hilber et al. [3]) to deal with this problem. The fractional derivative was adopted in free vibration of beam by Ansari [4].
The paper focuses on the solution of a finite Euler-Bernoulli beam on a fractional viscoelastic foundation subjected to a variable speed moving load. Accent is on the fractional foundation and their effects. The Galerkin method is used to solve the initial boundary value problem that governs the transverse vibration of the beam. Time response histories of the beam are graphically presented for various speeds of force and frequency. The effects of speed parameters and foundation stiffness are also investigated. As well as fractional derivative.

Problem formulation
Where , , and are the flexural rigidity, the density and the cross-sectional of the beam, respectively. ( , ) is a moving load with velocity V that is varying with time with the form = 0 ± , where 0 is initial velocity at = 0 of the load when enters the beam and the constant acceleration or deceleration. Thus, the load moves at time a distance 0 defined as: 0 = 0 ± ( 2 /2).
denotes the Dirac delta function.
( , ) is the foundation taken as a linear Winkler foundation with linear stiffness and viscous damping that is generalized by the term containing fractional derivative as: In which , , are the linear and damping parameters of the foundation, respectively and is the Caputo fractional derivative of order ∈ (0,1) which is defined by the following convolution integrals [6].

Solution of the problem
For simply supported beams the deflection ( , ) is defined in a series form as: where ( ) is the amplitude of the nth mode. Substituting Eqs. (2) and (6) To deal with fractional derivative term of Eq. (7) we use a discrete approximation method for the computation of Caputo's fractional derivative given by the expression: 2 )] (9) This system is solved using an explicit time evaluation. In the numerical computation, the initial conditions are (0) =̇(0) = 0, the displacement of beam is obtained at any time t and coordinate x. A MATLAB code is elaborated to compute and plot the deflection. The methodology presented in section 3 is verified here for the special case of non-fractional foundation when ( = 1), = 0 and = , for which there are results in the literature for the case of moving load, Maximum deflection occurs at the mid-span of the beam, L/2. The results are presented at this point. Fig. 2 shows the comparison between results of Ref [7] and the present work for a linear foundation for time response of beam. The results show a perfect agreement with those in Ref [7], and this verifies the accuracy of presented method. Another point is that for = 1, the transverse displacement of the beam is close to be symmetrical but as the fractional order derivative used, the symmetry of the displacement is destroyed.

Numerical results
To discuss the accuracy and effects of different parameters of system, we will take a practical type of beams with data properties listed in Table. 1. The effect of fractional order derivative on the vertical deflection is illustrated in Fig. 3. It can be seen that the fractional derivative order has a remarkable effect on the response. The transverse vibration amplitude decreases with increasing fractional order derivative.

Conclusion
In this study, the dynamic response of beams on a Winkler fractional viscoelastic foundation subjected to moving load with variable speed is studied. A numerical code for analyzing the problem considering the fractional derivative has been elaborated. The numerical results of the proposed method are verified for the case of non-fractional derivative and compared with results in the literatures and a good agreement was observed. The effects of fractional derivative order on the vertical displacement of beam on Winkler viscoelastic foundation subjected to moving load are discussed.