# Application of fractional calculus in the dynamics of beams

- D Dönmez Demir
^{1}Email author, - N Bildik
^{1}and - BG Sinir
^{2}

**2012**:135

**DOI: **10.1186/1687-2770-2012-135

© Dönmez Demir et al.; licensee Springer 2012

**Received: **29 August 2012

**Accepted: **19 October 2012

**Published: **15 November 2012

## 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

*ε*is a small dimensionless parameter;

*m*denotes the mass and $\stackrel{\u02c6}{\eta}$ is the damping coefficient; $\stackrel{\u02c6}{F}$ is the external excitation amplitude, $\stackrel{\u02c6}{\mathrm{\Omega}}$ is the external excitation frequencies, and ${D}^{\alpha}$ denotes the fractional derivative of order

*α*. Here, also, the dot denotes partial differentiation with respect to time $\stackrel{\u02c6}{t}$, and prime denotes the derivative with respect to spatial $\stackrel{\u02c6}{x}$. On the other hand, it is assumed that the tension

*T*is characterized as a small periodic perturbation $\epsilon {T}_{1}cos\mathrm{\Omega}$ on the steady-state tension ${T}_{0}$,

*i.e.*,

*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

*εη*. For simply supported beams, non-dimensional boundary conditions are

## 3 The method of multiple scales

*ε*, we have the following:

*ε*, then

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 ${\mathrm{\Omega}}_{1}$ close to 0, ${\mathrm{\Omega}}_{2}$ away from ${\omega}_{n}$ (${\mathrm{\Omega}}_{1}\cong 0$, ${\mathrm{\Omega}}_{2}\ne {\omega}_{n}$)

*cc*and

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

### 4.2 ${\mathrm{\Omega}}_{1}$ close to $2{\omega}_{n}$, ${\mathrm{\Omega}}_{2}$ away from ${\omega}_{n}$ (${\mathrm{\Omega}}_{1}\cong 2{\omega}_{n}$, ${\mathrm{\Omega}}_{2}\ne {\omega}_{n}$)

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

for the external excitation frequency. Thus, the two different values of ${\mathrm{\Omega}}_{1}$ 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*.

*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*.

*α*for $\lambda =5$ and $\eta =5$ is shown in Figure 5. Here, it is expected that the critical value of

*a*becomes zero for $\alpha =0$. 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.

### 4.3 ${\mathrm{\Omega}}_{1}$ away from $2{\omega}_{n}$ and 0, ${\mathrm{\Omega}}_{2}$ away from ${\omega}_{n}$ (${\mathrm{\Omega}}_{1}\ne 2{\omega}_{n}$, 0, ${\mathrm{\Omega}}_{2}\ne {\omega}_{n}$)

*cc*is a complex conjugate and

*NST*denotes non-secular terms. Substituting Eq. (26) into Eq. (43), we obtain the equation

### 4.4 ${\mathrm{\Omega}}_{1}$ away from $2{\omega}_{n}$ and ${\mathrm{\Omega}}_{2}$ close to ${\omega}_{n}$ (${\mathrm{\Omega}}_{1}\ne 2{\omega}_{n}\ne 0$, ${\mathrm{\Omega}}_{2}\cong {\omega}_{n}$)

### 4.5 Sum type of resonance (${\mathrm{\Omega}}_{1}+{\mathrm{\Omega}}_{2}\cong {\omega}_{n}+\epsilon {\sigma}_{n}$)

## 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.

## Declarations

## Authors’ Affiliations

## References

- Rossikhin YA, Shitikova MV: Application of fractional calculus for dynamic problems of solid mechanics: novel trends and recent results.
*Appl. Mech. Rev.*2010., 63: Article ID 010801 - Fung R-F, Huang J-S, Chen Y-C, Yao C-M: Nonlinear dynamic analysis of the viscoelastic string with a harmonically varying transport speed.
*Comput. Struct.*1998, 66(6):777–784. 10.1016/S0045-7949(98)00001-7View Article - Pakdemirli M, Boyacı H: The direct-perturbation methods versus the discretization-perturbation method: linear systems.
*J. Sound Vib.*1997, 199(5):825–832. 10.1006/jsvi.1996.0643View Article - Galucio AC, Deu J-F, Ohayon R: A fractional derivative viscoelastic model for hybrid active-passive damping treatments in time domain - application to sandwich beams.
*J. Intell. Mater. Syst. Struct.*2005, 16: 33–45. 10.1177/1045389X05046685View Article - Podlubny I Mathematics in Science and Engineering 198. In
*Fractional Differential Equations*. Academic Press, San Diego; 1999. - Agrawal OP: Formulation of Euler-Lagrange equations for fractional variational problems.
*J. Math. Anal. Appl.*2002, 272: 368–379. 10.1016/S0022-247X(02)00180-4MathSciNetView Article - Zhang Y: A finite difference method for fractional partial differential equation.
*Appl. Math. Comput.*2009, 215: 524–529. 10.1016/j.amc.2009.05.018MathSciNetView Article - Chen L-Q, Zhao W-J, Zu JW: Transient responses of an axially accelerating viscoelastic string constituted by a fractional differentiation law.
*J. Sound Vib.*2004, 278: 861–871. 10.1016/j.jsv.2003.10.012View Article - Boyacı H, Pakdemirli M: A comparison of different versions of the method of multiple scales for partial differential equations.
*J. Sound Vib.*1997, 204(4):595–607. 10.1006/jsvi.1997.0951View Article - Shimizu N, Zhang W: Fractional calculus approach to dynamic problems of viscoelastic materials.
*JSME Int. J. Ser. C Mech. Syst. Mach. Elem. Manuf.*1999, 42(4):825–837.View Article - Rossikhin YA, Shitikova MV: Application of fractional calculus for analysis of nonlinear damped vibrations of suspension bridges.
*J. Eng. Mech.*1998, 124: 1029–1036. 10.1061/(ASCE)0733-9399(1998)124:9(1029)View Article - Zhang L, Zu JW: Nonlinear vibration of parametrically excited moving belts, part I: dynamic response.
*J. Appl. Mech.*1999, 66(2):396–403. 10.1115/1.2791062View Article - Öz HR, Pakdemirli M, Boyacı H: Non-linear vibrations and stability of an axially moving beam with time dependent velocity.
*Int. J. Non-Linear Mech.*2001, 36: 107–115. 10.1016/S0020-7462(99)00090-6View Article - Pakdemirli M, Boyacı H: Comparison of direct-perturbation methods with discretization-perturbation methods for nonlinear vibrations.
*J. Sound Vib.*1995, 186: 837–845. 10.1006/jsvi.1995.0491View Article - Nayfeh AH:
*Introduction to Perturbation Techniques*. Wiley-Interscience, New York; 1981.

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