FORCED TRANSVERSE VIBRATIONS OF FRACTIONAL VISCOELASTIC EULER-BERNOULLI BEAM USING THE MODAL ANALYSIS METHOD

Abstract

In the paper, the forced transverse vibration of fractional viscoelastic Euler-Bernoulli is studied. Based on the fractional relationship of stress and strain, the partial differential equation describing transverse vibration of Euler-Bernoulli viscoelastic beam is considered. The Riemann-Liouville fractional derivative of the order  and  is used. Using the Ritz-Galerkin method, the fractional partial derivative equation describing the vibration of the beam is transformed into a system of differential equations containing fractional derivatives. The dynamic response of a simply supported fractional viscoelastic beam to a harmonic concentrated force is calculated in detail. The forced vibration solution of the beam is determined using the harmonic balancing method. The solution to the vibration equations is determined analytically, while dynamic responses are calculated numerically. The effects of fractional–order parameters on the vibration amplitude-time curves are investigated. From the calculation results, we can see that the lower the parameter is, the larger the vibration amplitude. This is consistent with our logic thinking. A comparison between the approximately analytical solution and the numerical one shows a good agreement, and the correctness of the obtained results is therefore verified.