Stability approach to the fractional variational iteration method used for the dynamic analysis of viscoelastic beams

Abstract

Non-integer derivatives are frequently used to describe the constitutive behavior of viscoelastic materials. The dynamic analysis of a simply supported viscoelastic beam for a fractional Zener model is performed with the help of a modified variational iteration method. The structure is subjected to two loading scenarios: a uniformly distributed transverse load and a periodic concentrated force at the center of the beam. Using the properties of convolution product in Banach spaces L p ( R n ) and the fixed point theory, the stability temporal intervals of a nonlinear operator for a given fractional order ν are determined. The graphical representations present in the numerical examples show how the existence of fractional derivative in the selected rheological model influences the dynamic response of the structure compared to the classical Zener model. The results of this study prove that the presented algorithm is an efficient tool for solving the fractional integro-differential equations. • An algorithm based on the modified VIM and Laplace transformation techniques is presented. • Approach of the variable change in fractional integral calculus. • Stability study of a fractional nonlinear operator defined by VIM. • Using the fixed point theory are determined the stability intervals. • Numerical examples show the influence of fractional derivative.