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This article is part of the series Recent Trends on Boundary Value Problems and Related Topics.

Open Access Research

Application of fractional calculus in the dynamics of beams

D Dönmez Demir1*, N Bildik1 and BG Sinir2

Author affiliations

1 Department of Mathematics, Faculty of Art & Science, Celal Bayar University, Manisa, 45047, Turkey

2 Department of Civil Engineering, Faculty of Engineering, Celal Bayar University, Manisa, 45140, Turkey

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Citation and License

Boundary Value Problems 2012, 2012:135  doi:10.1186/1687-2770-2012-135

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/135


Received:29 August 2012
Accepted:19 October 2012
Published:15 November 2012

© 2012 Dönmez Demir et al.; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper deals with a viscoelastic beam obeying a fractional differentiation constitutive law. The governing equation is derived from the viscoelastic material model. The equation of motion is solved by using the method of multiple scales. Additionally, principal parametric resonances are investigated in detail. The stability boundaries are also analytically determined from the solvability condition. It is concluded that the order and the coefficient of the fractional derivative have significant effect on the natural frequency and the amplitude of vibrations.

Keywords:
perturbation method; fractional derivative; method of multiple scales; linear vibrations

1 Introduction

Many researchers have demonstrated the potential of viscoelastic materials to improve the dynamics of fractionally damped structures. Fractional derivatives are practically used in the field of engineering for describing viscoelastic features in structural dynamics [1]. Namely, linear or non-linear vibrations of axially moving beams have been studied extensively by many researchers [2]. Fractional derivatives are used in the simplest viscoelastic models for some standard linear solid. It can be seen that the vibrations of the continuum are modeled in the form of a partial differential equation system [3]. These damping models involve ordinary integer differential operators that are relatively easy to manipulate [4]. On the other hand, fractional derivatives have more advantages in comparison with classical integer-order models [5].

The partial differential equations of fractional order are increasingly used to model problems in the continuum and other areas of application. The field of fractional calculus is of importance in various disciplines such as science, engineering, and pure and applied mathematics [6]. The numerical solution for the time fractional partial differential equations subject to the initial-boundary value is introduced by Podlubny [5]. The finite difference method for a fractional partial differential equation is presented by Zhang [7]. Galucio et al. developed a finite element formulation of the fractional derivative viscoelastic model [4]. Chen et al. studied the transient responses of an axially accelerating viscoelastic string constituted by the fractional differentiation law [8]. Applications of the method of multiple scales to partial differential systems arising in non-linear vibrations of continuous systems were considered by Boyacı and Pakdemirli [9]. The method of multiple scales is one of the most common perturbation methods used to investigate approximate analytical solutions of dynamical systems. The dynamic response of the continuum is analyzed by using this method.

In this paper, longitudinal vibrations of the beam with external harmonic force are studied. The model developed is used to show the applicability of the fractional damped model and to find an approximate solution of the problem. The Riemann-Liouville fractional operator is emphasized among several definitions of a fractional operator [10,11]. On the other hand, the approximate solution of the beam modeled by a fractional derivative is obtained and an application of the fractional damped model is also given. Additionally, the effects of a fractional damping term on a dynamical system are investigated. Finally, it is seen that the fractional derivative also has an effect on damping as a result of the previous studies in the literature.

2 The equation of motion

The problem of giving the longitudinal vibration of a harmonic external forced beam is given by

(1)

(2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M3">View MathML</a> is the transverse displacements of the beam and ε is a small dimensionless parameter; m denotes the mass and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M4">View MathML</a> is the damping coefficient; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M5">View MathML</a> is the external excitation amplitude, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M6">View MathML</a> is the external excitation frequencies, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M7">View MathML</a> denotes the fractional derivative of order α. Here, also, the dot denotes partial differentiation with respect to time <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M8">View MathML</a>, and prime denotes the derivative with respect to spatial <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M9">View MathML</a>. On the other hand, it is assumed that the tension T is characterized as a small periodic perturbation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M10">View MathML</a> on the steady-state tension <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M11">View MathML</a>, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M12">View MathML</a>

(3)

where Ω is the frequency of a beam [12]. Introducing the dimensionless parameters as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M13">View MathML</a>

(4)

we have the new dimensionless parameters

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M14">View MathML</a>

(5)

where ρ is density, A is the cross-sectional area, and L is the length of the beam. Thus, the equation in the non-dimensional form is presented as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M15">View MathML</a>

(6)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M16">View MathML</a> equals εη. For simply supported beams, non-dimensional boundary conditions are

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M17">View MathML</a>

(7)

3 The method of multiple scales

In this section, an approximate solution will be searched by using the method of multiple scales. This method is known as the direct-perturbation method which can be applied directly to the partial differential equation. In higher-order schemes and for finite mode truncations, the method yields better approximations to the real problem [13]. Let us consider the expansion

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M18">View MathML</a>

(8)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M19">View MathML</a> is the usual fast-time scale and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M20">View MathML</a> is the slow-time scales. Now, the time derivatives are given by

(9)

(10)

(11)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M24">View MathML</a>. Here, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M25">View MathML</a> can be used for calculating the fractional derivative of the exponential function, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M26">View MathML</a> are the Riemann-Liouville fractional derivatives [1]. Substituting Eqs. (8)-(11) into Eqs. (6) and (7) and separating into terms at each order of ε, we have the following:

(12)

(13)

(14)

(15)

At order one, the solution is obtained as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M31">View MathML</a>

(16)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M32">View MathML</a> represents the natural frequency, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M33">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M34">View MathML</a> are complex amplitudes and their conjugates, respectively. Now, substituting (16) into Eq. (12), we obtain the boundary value problems

(17)

(18)

and

(19)

(20)

Thus, the solutions of Eqs. (17) and (19) are

(21)

(22)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M41">View MathML</a> is a particular solution for Eq. (19). Here, the particular solution of Eq. (19) changes with respect to the selection of the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M42">View MathML</a>. Let us substitute (16) into Eq. (14) for the solution of order ε, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M43">View MathML</a>

(23)

Thus, different cases arise depending on the numerical value of variation frequency. These cases will be treated in the following sections.

4 Case studies

In this section, we assume that one dominant mode of vibrations exists. As a result of the previous studies in the literature, it is seen that the results are the same in the finite mode analysis and in the infinite mode analysis [3,14]. Therefore, we consider one dominant mode of vibration in this study.

4.1 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M44">View MathML</a> close to 0, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M45">View MathML</a> away from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M32">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M47">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M48">View MathML</a>)

For this case, we consider the case of the nearness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M49">View MathML</a> to zero is expressed as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M50">View MathML</a>

(24)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M51">View MathML</a> is a detuning parameter. Then, Eq. (23) becomes

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M52">View MathML</a>

(25)

where cc and NST denote complex conjugates and non-secular terms, respectively. Thus, the solution of Eq. (25) is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M53">View MathML</a>

(26)

where the first term is related to the secular terms and the second term is related to the non-secular terms. Now, substituting Eq. (26) into Eq. (25), we obtain the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M54">View MathML</a>

(27)

with the boundary conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M55">View MathML</a>

(28)

Using the solvability condition [15], we then find

(29)

Thus, by the normalization given as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M57">View MathML</a>, then Eq. (29) turns into

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M58">View MathML</a>

(30)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M59">View MathML</a>

(31)

Then, the amplitude solution for the first order of the problem is as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M60">View MathML</a>

(32)

and the displacement is also obtained:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M61">View MathML</a>

(33)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M62">View MathML</a> is a constant (determined by enforcing initial conditions). Additionally, the supplementary natural frequency from the fractional derivative is also given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M63">View MathML</a>

(34)

As seen in Figure 1, the fractional derivative α-order has an effect on the displacement-time curves. In Figure 2 and Figure 3, the effect of the variation of the coefficient λ is observed for the different functions on displacement-time curves.

thumbnailFigure 1. Displacement-time curves for different values of the order of the fractional derivative for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M64">View MathML</a>(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M65">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M66">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M67">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M68">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M69">View MathML</a>).

thumbnailFigure 2. Displacement-time graph for different values ofλfor<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M64">View MathML</a>(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M71">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M66">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M67">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M74">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M75">View MathML</a>).

thumbnailFigure 3. Displacement-time graph for various values ofλfor<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M76">View MathML</a>(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M77">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M66">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M67">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M80">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M75">View MathML</a>).

4.2 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M44">View MathML</a> close to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M83">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M45">View MathML</a> away from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M32">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M86">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M48">View MathML</a>)

If we consider the parametric resonance, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M88">View MathML</a>

(35)

Hence, the solvability condition requires that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M89">View MathML</a>

(36)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M90">View MathML</a> is given by (31). To perform the stability analysis, one introduces the transformation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M91">View MathML</a>

(37)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M92">View MathML</a> can be written as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M93">View MathML</a>

(38)

Substituting Eq. (38) into Eq. (37) and also obtaining the result placed into Eq. (36) (and separating into real and imaginary parts), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M94">View MathML</a>

(39)

For a non-trivial solution (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M95">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M96">View MathML</a>), the determinant of the coefficient matrix must be

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M97">View MathML</a>

(40)

Here, λ also must be zero for the steady-state condition. Thus, the stability boundaries are determined as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M98">View MathML</a>

(41)

Inserting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M99">View MathML</a> into Eq. (29), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M100">View MathML</a>

(42)

for the external excitation frequency. Thus, the two different values of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M44">View MathML</a> denote the stability boundaries for small ε. Additionally, it is seen that the stability boundaries depend not only on natural frequency but also on α.

The variation of an unstable region for different values of λ is observed in Figure 4. Since the rigidity of the system is increased by decreasing the value of λ, the unstable region reduces expeditiously for smaller values of λ.

thumbnailFigure 4. Stability boundaries for different values ofλfor<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M64">View MathML</a>(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M103">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M104">View MathML</a>).

The variation of an unstable region with some different values of α for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M105">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M106">View MathML</a> is shown in Figure 5. Here, it is expected that the critical value of a becomes zero for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M107">View MathML</a>. This situation is clearly observed in Figure 5. On the other hand, the unstable region diminishes while α is increasing. Finally, the effect of the variation of α on the critical value of a is presented in Figure 6. Figure 7 shows that critical value a changes nonlinearly with the order of fractional derivative.

thumbnailFigure 5. Stability boundaries for different values ofαfor<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M64">View MathML</a>(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M109">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M104">View MathML</a>).

thumbnailFigure 6. Critical value ofaversus the value ofηfor various fractional orders (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M111">View MathML</a>).

thumbnailFigure 7. Critical value ofaversus the value ofαfor various damping coefficients (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M111">View MathML</a>).

4.3 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M44">View MathML</a> away from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M83">View MathML</a> and 0, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M45">View MathML</a> away from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M32">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M117">View MathML</a>, 0, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M118">View MathML</a>)

This case corresponds to the absence of any resonances. Then, Eq. (23) turns into

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M119">View MathML</a>

(43)

where cc is a complex conjugate and NST denotes non-secular terms. Substituting Eq. (26) into Eq. (43), we obtain the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M120">View MathML</a>

(44)

with the boundary conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M121">View MathML</a>

(45)

Using the solvability condition [15], we find

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M122">View MathML</a>

(46)

By the normalization, then Eq. (46) becomes

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M123">View MathML</a>

(47)

Thus, the displacement is obtained as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M124">View MathML</a>

(48)

On the other hand, the amplitude is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M125">View MathML</a>

(49)

The displacement-time variation for different values of α is seen in Figure 8. Also, it is shown that the damping increases while the value of coefficient λ diminishes in Figure 9.

thumbnailFigure 8. Displacement-time graph for the different fractional order for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M64">View MathML</a>(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M66">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M68">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M129">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M75">View MathML</a>).

thumbnailFigure 9. Displacement-time graph for different values ofλfor<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M64">View MathML</a>(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M66">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M67">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M103">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M75">View MathML</a>).

4.4 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M44">View MathML</a> away from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M83">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M45">View MathML</a> close to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M32">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M140">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M141">View MathML</a>)

This case deals with the primary resonance <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M142">View MathML</a> when the frequency of the transverse loading is approximately equal to the natural frequency. Then, the steady-state solutions of amplitude-phase modulation equations and their stability can be discussed. Using the polar form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M143">View MathML</a>

(50)

and substituting Eq. (50) into the equation below,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M144">View MathML</a>

(51)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M145">View MathML</a>

(52)

we then obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M146">View MathML</a>

(53)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M147">View MathML</a>. Separating the equation into real and imaginary parts and also substituting the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M148">View MathML</a>

(54)

into Eq. (53), we find

(55)

(56)

By the same mathematical manipulation, the stability boundaries are calculated as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M151">View MathML</a>

(57)

4.5 Sum type of resonance (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M152">View MathML</a>)

In this case, we consider the sum or difference of internal and external forced frequency since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M153">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M154">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M48">View MathML</a>. Likewise, Eq. (23) is arranged once again; it is found that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M156">View MathML</a>

(58)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M157">View MathML</a>

(59)

Substituting Eq. (50) into Eq. (58) and also separating the equation into real and imaginary parts, we get

(60)

(61)

Inserting Eq. (54) into Eqs. (60) and (61), then we have

(62)

(63)

Therefore, the stability boundaries are obtained as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/135/mathml/M162">View MathML</a>

(64)

5 Conclusion

In this study, the effects of the damping term modeled with a fractional derivative on the dynamic analysis of a beam having viscoelastic properties subject to the harmonic external force are investigated. The parametric or primary resonances in simple supported beams, the governing equation of which involves a fractional derivative, are also analyzed. It is concluded that the value of the natural frequency of the beam modeled with a fractional damper is greater than that of the beam modeled with a classical damper. The fractional derivative has no effect on the static behavior, but it has a significant impact on the dynamic behavior. Furthermore, it is seen that the unstable region in the resonance case diminishes when the order of the fractional derivative increases.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors read and approved the final manuscript.

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