Abstract
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.
Keywords:
linear vibrations; dynamic analysis of continuum; fractional derivative; perturbation method1 Introduction
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 [1]. The advances in fractional calculus focus on modern examples in differential and integral equations, physics, signal processing, fluid mechanics, viscoelasticity, mathematical biology and electrochemistry [2].
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 [3] developed a general operator technique to analyse the vibrations of a continuous system with an arbitrary number of coupled differential equations. Özhan and Pakdemirli [46] suggested the general solution procedure to investigate a more general class of continuous systems such as gyroscopic and viscoelastic systems. Ghayesh et al.[7] considered a general solution procedure for the vibrations of systems with cubic nonlinearities subjected to nonlinear and timedependent 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 [10]. Bagley and Calico [11] 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.[12]. Also, they studied longitudinal vibrations of rods and flexural vibrations of beams based on viscoelastic fractional derivative models [13]. 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 [14]. Mainardi [15] considered the problems in continuum mechanics related to mathematical modelling of viscoelastic bodies. Cooke et al.[16] investigated the response of a viscoelastic beam with a fractional derivative. Skaar et al.[17] used a fractional standard linear solid model. French and Rogers [18] 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 [19]. In our previous study [20], the analysis of primary and parametric resonance for the external excitation term having εorder was performed. As the forced term is obtained in oneorder, 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
Let us consider a nonhomogeneous and dimensionless model as follows:
where
3 Method of multiple scales
The method is directly applied to the partial differential equation (1). Thus, we can write
where
where
In order to calculate the fractional derivative of the exponential function, we may use RiemannLiouville derivatives [22]
If we take
is obtained such that
The solution at
where cc denotes complex conjugates. On the other hand, the functions
At εorder, the solution is
where
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
Ω
1
away from
2
ω
n
and 0,
Ω
2
away from
ω
n
This case corresponds to absence of any resonances. Then Eq. (18) turns into
where NST denotes nonsecular terms. If Eq. (17) is substituted into Eq. (19), then
Thus, the solvability condition [23] requires
where
Finally, the amplitude is obtained as
and in the same sense, the displacement is calculated as
where
4.2
Ω
1
close to
2
ω
n
and
Ω
2
away from
ω
n
Principal parametric resonance occurs in this case. Thus, the internal excitation frequency is considered as
where
where
For the stability analysis, one introduces the transformation
where
Substituting (29) and (30) into Eq. (27), and separating real and imaginary parts, the representation of the system of equations with the coefficient matrix is given as
For a nontrivial solution (
For the steady state condition, λ must be zero. Therefore, the stability boundaries are
Inserting
4.3
Ω
1
close to 0 and
Ω
2
away from
ω
n
The nearness of
Arranging Eq. (18), one obtains
Solving Eq. (36),
and
is calculated.
4.4
Ω
1
away from
2
ω
n
and
Ω
2
close to
ω
n
In this case, we consider the primary resonance
where
Substituting the polar form
into Eq. (39) and separating the equation into real and imaginary parts, we obtain
where
By the same mathematical manipulation, the stability boundaries are found as
4.5 Sum and difference type of resonance
Let us consider sum or difference of internal and external excitation frequency as
where
Substituting Eq. (41) into Eq. (46) and separating into real and imaginary parts, we get
For steadystate solutions, the equations must be rearranged according to the condition (44). Then the stability boundaries are obtained as
5 Applications
5.1 The longitudinal vibrations of a tensioned rod
We will investigate longitudinal vibrations of an axial loaded rod with linear fractional damping for application. This problem is quite important in engineering applications. Also, the rods are used as a structural element in many civil and mechanical engineering problems. The governing equation motion of a tensioned rod with fractional damping is introduced as
where
where
the new dimensionless parameters are
where L is the length of the rod. Thus, the dimensionless equation is presented as
For the simply supported beam, the boundary conditions are
Considering Eq. (55), the operators corresponding to the general model are
and the other terms are
Thus, the spatial function
Finally, the solution of eigenvalueeigenfunction problem (62) is
In this problem, three different cases arise at εorder as follows.
5.1.1
Ω
1
away from
2
ω
n
and 0,
Ω
2
away from
ω
n
By the general solution (23), we may write
Thus, the displacement is obtained as
In Figure 1, it is observed that the damping and the natural frequency changed for different modes. The damping decreases and the natural frequency increases as the number of modes enlarges.
Figure 1. Displacementtime graph for different mode values (
5.1.2
Ω
1
close to 0 and
Ω
2
away from
ω
n
This case corresponds to
Thus, the amplitude and displacement are calculated respectively as follows:
and
Furthermore, the supplementary term of the natural frequency from a fractional derivative is
The variation of a supplementary term from a fractional derivative according to αorder is shown in Figure 2. The effects of the order of a fractional derivative on the displacementtime curves
are seen readily in Figure 3. The damping accelerates acutely in the classic damping approach, namely
Figure 2. Supplementary term versusαfor various mode values at
Figure 3. Displacementtime curves for the different values ofα(
5.1.3
Ω
1
close to
2
ω
n
and
Ω
2
away from
ω
n
In this part, we get the principal parametric resonance such that
Figure 4 shows the effects of αorder on the critical value N and the variation of an unstable region with some different values of α. It is observed that the critical value N becomes zero for
Figure 4. Stability boundaries for various fractional order for second mode (
5.2 The dynamic analysis of an axially loaded viscoelastic beam resting on foundation
The fractional viscoelastic beam with axial load is resting on linear elastic foundation.
This type of foundation is known as Winkler foundation. In the linear Winkler foundation
model,
where E represents the modulus of elasticity, I is the moment of inertia and
and the new dimensionless parameters are
where
Thus, the operators corresponding to a general model are
and the other terms are
Then Eqs. (13) and (15) reduce to
and
Finally, the solutions of Eqs. (82) and (84) are
and
where
5.2.1
Ω
1
away from
2
ω
n
and 0,
Ω
2
away from
ω
n
Using the general solution (23), we get
The amplitude and displacement are obtained
and
Thus, the supplementary term of the natural frequency due to a fractional derivative is
5.2.2
Ω
1
away from 0 and
2
ω
n
,
Ω
2
close to
ω
n
In this case, using Eq. (45), the stability boundaries which correspond to the primary
resonance
where the coefficient is
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 onedimensional. 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.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors read and approved the final manuscript.
Acknowledgements
Dedicated to Professor Hari M Srivastava.
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