Asymptotic analysis of an axially viscoelastic string constituted by a fractional differentiation law

The non-linear creep vibration of an axially moving string constituted by a fractional differentiation law is investigated in this paper. A non-linear partial fractional order differential equation governing the transverse vibration of the string is derived from Newton's second law, the fractional derivative Kelvin constitutive relationship and the Lagrangian strain. Unlike the classic models, the derived governing equation takes creep behavior into account. By using the multi-scale method, the non-linear frequency of the axially moving string is obtained. The results show that the amplitude predicted by the fractional model is larger than that predicted by the viscoelastic solid models. Crown Copyright (C) 2012 Published by Elsevier Ltd. All rights reserved.