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1. Introduction
The model of axially moving systems that play essential roles is always observed in a wide range of engineering devices, such as power transmission belts, magnetic tapes, paper sheets, chains, pipes conveying fluids, aerial tramways, and fiber textiles. Therefore, research on the dynamic behaviors of such systems has been conducted in the past decades and is still of interest today [1–3]. Especially, the problems of axially moving beams and micro/nano scaled beams axially loaded have widely been tackled in the analysis of some aspects, such as free vibration [4], stability analysis [5, 6], discretization approaches [7], modeling techniques [8], different solution methods [9, 10], and nonlinear dynamics [5–20]. In the architectural design and construction industry, the materials that exhibit excellent damping characteristics have been widely utilized to fabricate structures to enhance their performance from the viewpoint of vibration control. A growing body of research activities has therefore been appearing to explore the nonlinear vibrations of axially moving viscoelastic beams [4–18]. To better understand the damping mechanism, some classical constitutive models such as Kelvin–Voigt [5–19] and three-parameter Zener model [20] are adopted to effectively describe the dynamical responses of the viscoelastic materials. Although the classical models contain combinations of elastic and viscous elements, they do not have sufficient parameters to handle the different shapes of the hysteresis loops reflecting the nature of viscoelastic materials and structures. Consequently, the fractional calculus [21] has been introduced in constitutive relations to obtain a satisfactory solution for the real viscoelastic responses of the materials over a large range of frequency [22, 23]. Although there remain some mathematical issues unsolved, the fractional calculus-based modern viscoelasticity problems are becoming the focus of attention [24–32].
This research is devoted to investigating the dynamic behavior of an axially moving viscoelastic beam under a transverse harmonic excitation. It is assumed that the material of the beam obeys the Kelvin–Voigt model based on the fractional Caputo definition. Remarkably speaking, by one-term Garlerkin’s technique, the governing equation of the beam is discretized into a nonlinear Duffing type equation that has the nonlinear fractional operator [26–29]. The first order averaging method is utilized to derive the modulation equations governing the steady state amplitudes and phases of the system. Then, the stability of the solution is studied by the Routh–Hurwitz criterion [33]. Finally, the results of representative calculations are described and briefly discussed from the vibration control point of view.
2. Equation of Motion
The present study considers a uniform axially moving viscoelastic beam, shown in Figure 1, with density
[figure(s) omitted; refer to PDF]
The viscoelasticity of the beam material obeys the fractional Kelvin–Voigt model [27, 28, 30] and its constitutive relationship is given as follows:
In which
In equation (2),
Substitution of equations (2), (3) and (5) into equation (1), yields the governing equation of transverse motion of the axially viscoelastic beam
In the present investigation, the boundary conditions are taken to be simply supported, i.e.,
Using the following dimensionless scheme,
Equation (6) becomes
With boundary conditions
3. Steady State Responses and Stability Analysis
In this study, one term Galerkin’s method is applied to discretize equation (9), and its solution is assumed to be as follows:
For simplicity in the following analysis, equation (12) is rewritten as follows:
With
Remarkably, equations (12) or (13) is a nonlinear Duffing type equation having a nonlinear fractional operator
In what follows, the primary resonance of the fractional system described by (13) will be investigated by the linear averaging method. We mainly focus on examining the effects of the damping parameters and the fractional order on the steady state response of the beam. The stability of the stationary solutions is examined by the Routh–Hurwitz criterion [33].
To obtain the primary resonance of the fractional oscillator (equation (13)), we introduce a small parameter
Differentiating the second equation in equation (17) with respect to
Substituting equations (17) and (19) into equation (14) leads to the following equation:
Based on the averaging method, the right-hand side of equation (20) can be replaced by its averages over one period when
Since we consider the steady state only, the fractional order derivative of
Subsequently, the average in equation (21) can also be integrated over one period, when the fractional derivative of
A direct calculation yields
The small parameter
Then, the variation of the amplitude
In what follows, the stability of the solutions of the system in the neighborhood of the equilibrium state is approximately analyzed by exploring the eigenvalues of the Jacobian matrix of equation (27) evaluated at the fixed points of interest. In order to do that, the small disturbances of the response amplitude and phase
The eigenfunction of the linearized equation (29) is as follows:
From the Routh–Hurwitz criterion [33], the steady-state response is asymptotically stable if and only if the real parts of the eigenvalues are negative. It can be obtained by the following equation:
4. Verifications and Numerical Simulations
In the following, a parametric investigation in primary resonance conditions has been conducted to reveal the influences of the control variables on the steady state responses of the beam from the vibration control point of view. The approximate results have been achieved and illustrated by frequency- and forcing amplitude-response curves, in which the solid lines stand for the stable results and the dotted ones correspond to the unstable results. The distinction between the stable solutions and unstable ones is determined by the inequalities (equation (34)) using the Routh–Hurwitz criterion [33]. In calculations, the system parameters in equation (12) have been taken as follows [14]:
4.1. Effects of Fractional Order on System Response
In Figures 2 and 3, the forcing frequency
[figure(s) omitted; refer to PDF]
The influence of the fractional order
In addition, the amplitude-response curves are also depicted in Figure 4 to verify the above conclusions, where the response amplitudes are plotted as the function of forcing amplitude. Similar conclusions can easily be drawn.
4.2. Effects of Viscoelastic Coefficient on the Dynamic Behaviors
The influence of the viscoelastic coefficient
4.3. Verifications
To verify the previous discussions, the analytical results have been compared with the numerical solutions achieved from the direct numerical integration technique. The primary resonance of the system is shown in Figure 5, in which the control parameter-set adopted is chosen as
[figure(s) omitted; refer to PDF]
5. Conclusions
By the averaging technique, in the present investigations, the primary resonant response of the axially moving viscoelastic beam system has been analytically investigated from the viewpoint of vibration control. The influences of different system parameters, such as fractional order and the viscoelastic coefficient, on the dynamic responses of the viscoelastic beam are graphically illustrated. From the investigation, the following conclusions have been made:
(1) It has shown that the hysteresis or jump phenomenon occurs owing to the existence of multiple solutions on response curves, while the hysteresis area contracts with the increase of the fractional order or viscoelastic coefficient.
(2) A critical value of these two parameters exists, beyond which the hysteresis region can be eliminated. The increase in them reduces the vibration amplitudes.
(3) Both the fractional order and viscoelastic coefficient are powerful factors in stabilizing the system.
Disclosure
A preprint of this paper [34] has previously been presented on Engineering Archive (doi: 10.31224/osf.io/2wsv6).
Acknowledgments
The research is supported by the National Natural Science Foundation of China (Nos. 51808212, 51708205, and 11502160), the Project Funds for Talents Introduction of Taishan University (No. Y2014-01-18), and the Natural Science Foundation of Shandong Province (Nos. ZR2019MA017, ZR2021MA086).
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Abstract
The nonlinear vibrations of an axially moving viscoelastic beam under the transverse harmonic excitation are examined. The governing equation of motion of the viscoelastic beam is discretized into a Duffing system with nonlinear fractional derivative using Galerkin’s method. The viscoelasticity of the moving beam is described by the fractional Kelvin–Voigt model based on the Caputo definition. The primary resonance is analytically investigated by the averaging method. With the aid of response curves, a parametric study is conducted to display the influences of the fractional order and the viscosity coefficient on steady-state responses. The validations of this study are given through comparisons between the analytical solutions and numerical ones, where the stability of the solutions is determined by the Routh–Hurwitz criterion. It is found that suppression of undesirable responses can be achieved via changing the viscosity of the system.
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Details

1 College of Civil Engineering, Hunan University of Technology, Zhuzhou, Hunan 412007, China
2 School of Information Science and Technology, Taishan University, Taian, Shandong, China
3 School of Mathematics and Statics, Taishan University, Taian, Shandong, China