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Mathematical Problems in Engineering
Volume 2017, Article ID 1393954, 9 pages
https://doi.org/10.1155/2017/1393954
Research Article

Analytical Analysis on Nonlinear Parametric Vibration of an Axially Moving String with Fractional Viscoelastic Damping

1School of Arts, Anhui Polytechnic University, Wuhu 241000, China
2School of Mechanical and Automotive Engineering, Anhui Polytechnic University, Wuhu 241000, China

Correspondence should be addressed to Ye Tang; moc.361@tih_0102eygnat

Received 14 May 2017; Accepted 25 October 2017; Published 16 November 2017

Academic Editor: Jaromir Horacek

Copyright © 2017 Ying Li and Ye Tang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The nonlinear parametric vibration of an axially moving string made by rubber-like materials is studied in the paper. The fractional viscoelastic model is used to describe the damping of the string. Then, a new nonlinear fractional mathematical model governing transverse motion of the string is derived based on Newton’s second law, the Euler beam theory, and the Lagrangian strain. Taking into consideration the fractional calculus law of Riemann-Liouville form, the principal parametric resonance is analytically investigated via applying the direct multiscale method. Numerical results are presented to show the influences of the fractional order, the stiffness constant, the viscosity coefficient, and the axial-speed fluctuation amplitude on steady-state responses. It is noticeable that the amplitudes and existing intervals of steady-state responses predicted by Kirchhoff’s fractional material model are much larger than those predicted by Mote’s fractional material model.