Transient vibrations of a simply supported viscoelastic beam of a fractional derivative type under the transient motion of the supports

Abstract

Transient vibrations analysis of a simply supported beam whose viscoelastic properties are expressed in terms of a fractional Kelvin–Voigt model are presented. The Riemann–Liouville fractional derivative of order 0 < γ < 1 is used. The Bernoulli–Euler beam excited by transient motion of the supports is considered. An oscillating function with linearly time-varying frequency is applied as an excitation. The forced-vibration solution of the beam is determined using the mode superposition method. A convolution integral of the fractional Green’s function and forcing function is used to achieve the beam response. The Green’s function is formulated by two terms. The first term describes damped vibrations around the drifting equilibrium position, while the second term describes the drift of the equilibrium position. The dynamic responses are numerically calculated. A comparison between results obtained using the fractional and integer (classical) viscoelastic material models is presented. In the analysed system, the influence the term describing the drift of the equilibrium position on the beam deflection is relatively low and may be neglected in some cases. The employed procedure widens the methods applied in damping modelling of structural elements.