Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 1393954, 9 pages

https://doi.org/10.1155/2017/1393954

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

^{1}School of Arts, Anhui Polytechnic University, Wuhu 241000, China^{2}School 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.

#### 1. Introduction

Axially moving structures are one of common elements in many mechanical systems, which is extensively used in engineering fields, such as paper sheets, magnetic tapes, power transmission belts, chains, fiber textiles, aerial tramways, pipes conveying fluids, and thread lines. In many cases, due to initial, parametric, and external excitations, the generation of unwanted transverse vibrations may limit their applications. Therefore, the dynamic behaviors of such devices have been widely investigated by numerous scholars for the past decades and are still of interesting today [1–10]. Moreover, according to the demand of the actual engineering problems, some researchers have applied their theoretical results to design and optimize the axially moving structures [11, 12].

In general, small imperfections induced by either geometrical or dynamic sources may bring about the occurrence of an unsteady axial speed or tension. For example, when an axially moving belt is installed on rotating pulleys, the torsional vibration of the pulleys would lead to a small fluctuation in the axial moving velocity, and consequently this system may exhibit more complicated dynamics behaviors like the parametric resonances. Thus, the parametric resonances of the axially moving structures caused by the pulsatile transport speed or tension have received extensive attention. Fung et al. [13] applied the Galerkin method and the numerical integration technique to investigate the parametric vibration of a viscoelastic string with the nonuniform transport speed. Pellicano et al. [14] used the approximate analytical and the experimental methods to study the primary and the parametric resonances of a power transmission belt with the fluctuation of the tension. Chen et al. [15] employed the averaging method to investigate the stability problems of an axially accelerating tensioned beam under the condition of the subharmonic and combination resonances. Pakdemirli and Öz [16] adopted a perturbation technique to discuss stable regions of a simply supported axially moving beams subjected to sum- and difference-type combination resonances. Ghayesh [17] utilized the method of multiple scales to study the stability characteristics for principal and combination parametric resonances of an axially moving string with the partial elastic support.

With the development of engineering technique, the various complicated materials are utilized to fabricate the axially moving structures. In order to better understand energy dissipation mechanism of such materials, some viscoelastic constitutive relations like the Kelvin-Voigt form have been used to describe mechanical behaviors of the materials, which is widely applied in axially moving continuums such as strings [18, 19], beams [20, 21], belts [22, 23], and plates [24, 25]. In addition, some structures made of more complex viscoelastic materials are explored. Marynowski and Kapitaniak [26] presented a mathematical model of an axially moving viscoelastic beam with the three-parameter Zener element and investigated both regular and chaos motions using the Galerkin method. Chen et al. [27] analytically studied nonlinear parametric responses of an axially moving string composed of the complicated viscoelastic material based on the Boltzmann superposition principle. Wang and Chen [28] developed the differential quadrature scheme to determine the stable boundary of an axially moving viscoelastic beam with the standard linear solid in the case of the principal resonance. Ding and Chen [29] employed the material time derivative to characterize the viscoelastic property of an axially moving viscoelastic beam. They investigated the stability in principle resonance of the system by using analytical and numerical methods.

On the other hand, the axially moving structures made by rubber-like materials are widely applied in some engineering fields. To exactly describe the viscoelastic features, some scholars have adopted the fractional derivative theory to model the structures. For example, Chen et al. [30] established the fractional dynamic model of an axially moving string and analyzed the transient responses using the Galerkin and numerical methods. Yang et al. [31, 32] employed the multiscale method to investigate the nonlinear free and parametric vibrations of an axially moving viscoelastic string with a fractional order damping.

The above-mentioned studies are concentrated on axially moving strings based on Mote’s model. Besides, the nonlinear integro-partial-differential equation called Kirchhoff’s model is also used for describing the transverse motion of axially moving strings [33–35]. Nevertheless, the applications of Kirchhoff’s model in axially moving structures obeying the fractional differentiation law are rather limited, and the difference of fractional parametric resonances between the Mote’s model and the Kirchhoff’s model remains unclear. Therefore, the present paper further explores the nonlinear parametric vibration of an axially moving string with the fractional viscoelastic damping based on the literature [32].

#### 2. Equations of Motion

Consider a uniform, axially moving viscoelastic string made by rubber-like materials, with linear density , the length , and the length cross-sectional area , traveling at time-dependent axially speed between two fixed supports at both ends, shown in Figure 1. The symbol represents the transverse displacement, is the coordinate along the axial direction, and is the time. On the basis of the Euler beam theory and the Newton second law [34], the Kirchhoff’s mathematical model governing transverse motion can be obtained as where represents the initial tension and denotes the axial disturbed stress. According to the Lagrangian strain [35], the disturbed strain accounted for the geometric nonlinearity can be expressed as In the work, the viscoelastic damping of the string is supposed as obeying the fractional derivative Kelvin-Voigt model [32]. Thus, the stress-strain relation can be given as where and represent the stiffness constant and viscosity coefficient of the string, respectively, and is a constant which is used to describe the viscosity characteristic. Considering the fractional derivative operator defined by the Riemann-Liouville form [36, 37], we write the expression in (3) as follows:where is the Gamma function. The fractional derivative operator is applied to investigate an intermediate viscoelastic characteristic between the elastic string () and Kelvin-Voigt viscoelastic string (). For calculating the fractional differentiation operator, we introduce the following property aswhere denotes complex number .