DYNAMIC ANALYSIS OF NONLINEAR STRUCTURAL SYSTEMS BY MEANS OF FRACTIONAL CALCULUS D

Abstract

In this article the vibrational analysis of

non-linear systems by means of fractional

calculus is presented. This kind of non-linear

problems appears in many areas of mechanical

engineering, such as the contact in bearings,

complex behavior materials, great displacements,

vibrations in buildings, etc. The aim of this

study is to find an alternative to methods like

Runge-Kutta, which duplicate the number of

equations to solve. The aim is, therefore, to apply

traditional methods in structural mechanics

that mantain the size of the system, like, for

instance, the finite central difference method.

For that it is necessary to linearize the system

of equations. The proposed method transforms

the original non-linear problem into fractional

linear integro-differential equations, where

the fractional operator represents a variable

stiffness. Two typical application examples in

mechanical engineering are presented: the

elastic impact of two spheres (system of one

degree of freedom) and the transient dynamic

response of a building (system of multiple

degrees of freedom) under the effect of a

wind load. From the obtained results it can be

concluded that the proposed method describes

correctly the time response of the studied

systems, reducing the computational effort.