Linear vibrations of continuum with fractional derivatives
© Dönmez Demir et al.; licensee Springer 2013
Received: 18 December 2012
Accepted: 12 April 2013
Published: 25 April 2013
In this paper, linear vibrations of axially moving systems which are modelled by a fractional derivative are considered. The approximate analytical solution is obtained by applying the method of multiple scales. Including stability analysis, the effects of variation in different parameters belonging to the application problems on the system are calculated numerically and depicted by graphs. It is determined that the external excitation force acting on the system has an effect on the stiffness of the system. Moreover, the general algorithm developed can be applied to many problems for linear vibrations of continuum.
Fractional derivatives are useful for describing the occurrence of vibrations in engineering practice. The studies involving fractional calculus and its applications to mechanical problems appear widely in different studies . The advances in fractional calculus focus on modern examples in differential and integral equations, physics, signal processing, fluid mechanics, viscoelasticity, mathematical biology and electrochemistry .
The general solution procedure including all the problems instead of separately solving each problem is quite advantageous. Many different linear or nonlinear models addressing vibrations of continuum appear in the literature. Some of these works are as follows: Pakdemirli  developed a general operator technique to analyse the vibrations of a continuous system with an arbitrary number of coupled differential equations. Özhan and Pakdemirli [4–6] suggested the general solution procedure to investigate a more general class of continuous systems such as gyroscopic and viscoelastic systems. Ghayesh et al.  considered a general solution procedure for the vibrations of systems with cubic nonlinearities subjected to nonlinear and time-dependent boundary conditions. Hence, a general solution is adapted to solve the dynamic problems constituting continuum.
In recent years, there has been a growing interest in the area of fractional variational calculus and its applications [8, 9]. Fractional calculus, which is used successfully in various fields such as mathematics, science and engineering, is one of the generalisations of classical calculus. The merits of using a fractional differential operator lie in the fact that few parameters are needed to accurately describe the constitutive law of damping materials . Bagley and Calico  modelled the mechanical properties of damping materials by fractional order time derivatives. The mechanical scientific community recognised the significance of fractional calculus for modelling viscoelastic material behaviour thanks to Bagley et al. . Also, they studied longitudinal vibrations of rods and flexural vibrations of beams based on viscoelastic fractional derivative models . For solving dynamic problems with a fractional derivative, the analysis of free damped vibrations of various mechanical systems, whose behaviour is described by linear viscoelastic models with fractional derivatives, were studied by Rossikhin and Shitikova . Mainardi  considered the problems in continuum mechanics related to mathematical modelling of viscoelastic bodies. Cooke et al.  investigated the response of a viscoelastic beam with a fractional derivative. Skaar et al.  used a fractional standard linear solid model. French and Rogers  presented a small group of structural dynamics problems for which fractional calculus was adopted.
The general solution allows one to investigate the effects on a dynamic analysis of continuum whose damping term is modelled by a fractional derivative. An engineering problem which is a special application of the general model developed in this study was formerly considered in . In our previous study , the analysis of primary and parametric resonance for the external excitation term having ε-order was performed. As the forced term is obtained in one-order, sum or difference type of resonance also appears in the present model. The method of multiple scales is used in the analysis. Thus, the amplitude and phase modulation equations are produced in terms of operators. In addition, the variations of the curves with respect to the dimensionless parameters are presented. Finally, the effects of fractional damping on the linear vibrations of continuum are investigated in detail.
2 Equation of motion
where represents the displacement, x and t are the spatial and time variables. ε is a small dimensionless parameter, F is the external excitation force, and are the internal and the external excitation frequencies, respectively. defines the fractional derivative of order α. The dot denotes differentiation with respect to time t; , and are self-adjoint operators involving only the spatial variable x, and are linear operators of boundary conditions. Here, the associated boundary conditions are linear, homogeneous and free from the time.
3 Method of multiple scales
Then, five cases occur as follows.
4 Case studies
In this section, we assume that one dominant mode of vibrations exists. Depending on the numerical values of natural frequency, five different cases occur.
4.1 away from and 0, away from
where is constant.
4.2 close to and away from
4.3 close to 0 and away from
4.4 away from and close to
4.5 Sum and difference type of resonance
5.1 The longitudinal vibrations of a tensioned rod
In this problem, three different cases arise at ε-order as follows.
5.1.1 away from and 0, away from
5.1.2 close to 0 and away from
5.1.3 close to and away from
5.2 The dynamic analysis of an axially loaded viscoelastic beam resting on foundation
where represents the particular solution from a non-homogeneous part. For the solution at ε-order, two different cases arise as follows.
5.2.1 away from and 0, away from
5.2.2 away from 0 and , close to
6 Conclusion and discussions
In this study, the general model subject to internal and external excitation is developed. The general model proposed for continuum is linear and one-dimensional. The effect of the damping term which is obtained from viscoelastic material properties is modelled with a fractional derivative. The dynamic analysis of the general model is examined by the method of multiple time scales. The approximate solutions are derived in terms of operators. The external force term is considered at order one. This consideration leads to sum and difference type of resonance in addition to primary and parametric resonance cases. The application of the general solution to two specific engineering problems is presented. The solvability boundaries are approximately obtained and numerically illustrated. It is shown that the order of the fractional derivative has an effect on natural frequencies and stability boundaries. It is shown that the stable region becomes smaller with increasing fractional order. And also, the coefficient of a fractional damping term has similar effects to fractional order.
Dedicated to Professor Hari M Srivastava.
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