Nonlinear Dynamic Analysis of Fractional Damped Viscoelastic Beams

Abstract

The nonlinear dynamical response of a simply supported viscoelastic beam subjected to transverse harmonic excitations is investigated. The constitutive law of the viscoelastic beam is modeled in the fractional derivative Kelvin sense. The mathematical model is derived and discretized to a set of ordinary differential equations by Galerkin approximation method. The steady-state response of a single-mode system is obtained by the averaging method. Numerical results are obtained by an algorithm based on the fractional-order Grünwald–Letnikov definition, and compared with the analytical ones for verification. A parametric study and singularity analysis are carried out to determine the influence of the coefficients of the material’s constitutive equation on the responses. To study the effect of beam length and nonlinear coefficient on the nonlinear dynamic response, a numerical simulation is carried out. The periodic, multiple periodic, and chaotic responses are determined using Poincaré section bifurcation diagrams of the local maximum displacement. The above analysis allows us to optimize parametric design scheme for the viscoelastic beam.