# An Analysis of the Dynamical Behaviour of Systems with Fractional Damping for Mechanical Engineering Applications

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Background: A 1 DOF System with Fractional Damping

#### 2.1. Poles of the System

#### 2.2. Dynamic Response

#### 2.2.1. Response to Initial Conditions

#### 2.2.2. Response to Impulse

#### 2.2.3. Response to Step

## 3. Application: Bearing Support

^{3}is considered. The natural frequencies, mode shapes and response of the shaft when changing the parameters of the bearings are studied, the two bearings being identical (${k}_{1}={k}_{N-1}=k$, ${c}_{1}={c}_{N-1}=c$ and ${\alpha}_{1}={\alpha}_{N-1}=\alpha $).

#### 3.1. Natural Frequencies and Mode Shapes

^{$\alpha $}/m respectively. When the shaft is discretised with 60 beam elements in its length the results shown in Figure 11 are obtained. It can be observed that, as deduced for the 1 DOF system, changing the order of the derivative affects the stiffness of the supports and, thus, the vibration frequencies.

^{$\alpha $}/m in the computation of the mode shapes presented in Figure 12. The phase of the mode shape in each point of the shaft is represented with the aim of bringing to light the complex nature of the modes: if the mode were normal, the phase would change abruptly from $-\pi $ to $\pi $ (or vice versa) in the nodal points, but it is not what occurs.

#### 3.2. Dynamic Response

**Case 1**: a viscous case ($\alpha =$ 1) with low stiffness (${k}_{\mathrm{ref}}={10}^{4}$ N/m) and low damping (${c}_{\mathrm{ref}}=50$ N s/m) that is used for reference, in which the rigid body movement prevails.**Case 2**: a viscous case ($\alpha =$ 1) with low stiffness ($k={k}_{\mathrm{ref}}$) and supercritical damping ($c=10{c}_{\mathrm{ref}}$), in which the shaft returns to the equilibrium position without oscillating.**Case 3**: a fractionally damped case ($\alpha =$ 0.6) with low stiffness ($k={k}_{\mathrm{ref}}$) and high damping ($c=100{c}_{\mathrm{ref}}$) so that the system returns to its original position oscillating around a variable equilibrium position.**Case 4**: a fractionally damped case ($\alpha =$ 0.6) with high stiffness ($k=10{k}_{\mathrm{ref}}$) and low damping ($c={c}_{\mathrm{ref}}$), in which the movement is a combination of the rigid body motion and the first modes of the system.

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Adolfsson, K.; Enelund, M.; Olsson, P. On the fractional order model of viscoelasticity. Mech. Time Depend. Mater.
**2005**, 9, 15–34. [Google Scholar] [CrossRef] - Podlubny, I. Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Spplications; Mathematics in Science and Engineering; Academic Press: Cambridge, MA, USA, 1999. [Google Scholar]
- Torvik, P.J.; Bagley, R.L. On the Appearance of the Fractional Derivative in the Behavior of Real Materials. J. Appl. Mech.
**1984**, 51, 294–298. [Google Scholar] [CrossRef] - Bagley, R.L.; Torvik, P.J. A theoretical basis for the application of fractional calculus to viscoelasticity. J. Rheol.
**1983**, 27, 201–210. [Google Scholar] [CrossRef] - Di Paola, M.; Pirrotta, A.; Valenza, A. Visco-elastic behavior through fractional calculus: An easier method for best fitting experimental results. Mech. Mater.
**2011**, 43, 799–806. [Google Scholar] [CrossRef] [Green Version] - Nutting, P.G. A new general law of deformation. J. Frankl. Inst.
**1921**, 191, 679–685. [Google Scholar] [CrossRef] - Gemant, A. A method of analyzing experimental results obtained from elasto-viscous bodies. Physics
**1936**, 7, 311–317. [Google Scholar] [CrossRef] - Pinnola, F.P.; Zavarise, G.; Prete, A.D.; Franchi, R. On the appearance of fractional operators in non-linear stress–strain relation of metals. Int. J. Non-Linear Mech.
**2018**, 105, 1–8. [Google Scholar] [CrossRef] [Green Version] - Makris, N. Three-dimensional constitutive viscoelastic laws with fractional order time derivatives. J. Rheol.
**1997**, 41, 1007–1020. [Google Scholar] [CrossRef] - Alotta, G.; Barrera, O.; Cocks, A.C.; Di Paola, M. On the behavior of a three-dimensional fractional viscoelastic constitutive model. Meccanica
**2017**, 52, 2127–2142. [Google Scholar] [CrossRef] [Green Version] - Alotta, G.; Barrera, O.; Cocks, A.; Paola, M.D. The finite element implementation of 3D fractional viscoelastic constitutive models. Finite Elem. Anal. Des.
**2018**, 146, 28–41. [Google Scholar] [CrossRef] - Heymans, N.; Podlubny, I. Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives. Rheol. Acta
**2006**, 45, 765–771. [Google Scholar] [CrossRef] [Green Version] - Di Paola, M.; Pinnola, F.P.; Zingales, M. A discrete mechanical model of fractional hereditary materials. Meccanica
**2013**, 48, 1573–1586. [Google Scholar] [CrossRef] [Green Version] - Schiessel, H.; Metzler, R.; Blumen, A.; Nonnenmacher, T.F. Generalized viscoelastic models: their fractional equations with solutions. J. Phys. Math. Gen.
**1995**, 28, 6567. [Google Scholar] [CrossRef] - Naber, M. Linear fractionally damped oscillator. Int. J. Differ. Equations
**2010**, 2010. [Google Scholar] [CrossRef] [Green Version] - Achar, B.N.N.; Hanneken, J.W.; Enck, T.; Clarke, T. Dynamics of the fractional oscillator. Phys. Stat. Mech. Appl.
**2001**, 297, 361–367. [Google Scholar] [CrossRef] - Achar, B.N.N.; Hanneken, J.W.; Clarke, T. Response characteristics of a fractional oscillator. Phys. Stat. Mech. Appl.
**2002**, 309, 275–288. [Google Scholar] [CrossRef] - Achar, B.N.N.; Hanneken, J.W.; Clarke, T. Damping characteristics of a fractional oscillator. Phys. Stat. Mech. Appl.
**2004**, 339, 311–319. [Google Scholar] [CrossRef] - Shokooh, A.; Suárez, L. A comparison of numerical methods applied to a fractional model of damping materials. J. Vib. Control.
**1999**, 5, 331–354. [Google Scholar] [CrossRef] - Suarez, L.E.; Shokooh, A. An eigenvector expansion method for the solution of motion containing fractional derivatives. J. Appl. Mech.
**1997**, 64, 629–635. [Google Scholar] [CrossRef] - Fenander, A. Modal synthesis when modeling damping by use of fractional derivatives. AIAA J.
**1996**, 34, 1051–1058. [Google Scholar] [CrossRef] - Cortés, F.; Elejabarrieta, M.J. Finite element formulations for transient dynamic analysis in structural systems with viscoelastic treatments containing fractional derivative models. Int. J. Numer. Methods Eng.
**2007**, 69, 2173–2195. [Google Scholar] [CrossRef] - Cortés, F.; Elejabarrieta, M.J. Homogenised finite element for transient dynamic analysis of unconstrained layer damping beams involving fractional derivative models. Comput. Mech.
**2007**, 40, 313–324. [Google Scholar] [CrossRef] - Mendiguren, J.; Cortés, F.; Galdos, L. A generalised fractional derivative model to represent elastoplastic behaviour of metals. Int. J. Mech. Sci.
**2012**, 65, 12–17. [Google Scholar] [CrossRef] - Oldham, K.; Spanier, J. The Fractional Calculus Theory and Applications of Differentiation and Integration to Arbitrary Order; Elsevier: Amsterdam, The Netherlands, 1974. [Google Scholar]
- Hilfer, R. Applications of Fractional Calculus in Physics; World Scientific: Singapore, 2000. [Google Scholar]
- Miller, K.S.; Ross, B. An Introduction to the Fractional Calculus and Fractional Differential Equations; Wiley-Interscience: Hoboken, NJ, USA, 1993. [Google Scholar]
- Bagley, R.L.; Calico, R.A. Fractional order state equations for the control of viscoelasticallydamped structures. J. Guid. Control. Dyn.
**1991**, 14, 304–311. [Google Scholar] [CrossRef] - Kilbas, A.A.A.; Srivastava, H.M.; Trujillo, J.J. Theory and Applications of Fractional Differential Equations; Elsevier Science Limited: Amsterdam, The Netherlands, 2006; Volume 204. [Google Scholar]
- Yuan, L.; Agrawal, O.P. A numerical scheme for dynamic systems containing fractional derivatives. Trans. Am. Soc. Mech. Eng. J. Vib. Acoust.
**2002**, 124, 321–324. [Google Scholar] [CrossRef] [Green Version] - Pinnola, F.P. Statistical correlation of fractional oscillator response by complex spectral moments and state variable expansion. Commun. Nonlinear Sci. Numer. Simul.
**2016**, 39, 343–359. [Google Scholar] [CrossRef] - Matsubara, M.; Rahnejat, H.; Gohar, R. Computational modelling of precision spindles supported by ball bearings. Int. J. Mach. Tools Manuf.
**1988**, 28, 429–442. [Google Scholar] [CrossRef] - Cortés, F.; Elejabarrieta, M.J. Computational methods for complex eigenproblems in finite element analysis of structural systems with viscoelastic damping treatments. Comput. Methods Appl. Mech. Eng.
**2006**, 195, 6448–6462. [Google Scholar] [CrossRef] - Cortés, F.; Elejabarrieta, M.J. An approximate numerical method for the complex eigenproblem in systems characterised by a structural damping matrix. J. Sound Vib.
**2006**, 296, 166–182. [Google Scholar] [CrossRef] - Cortés, F.; Jesús Elejabarrieta, M. Finite element analysis of the seismic response of damped structural systems including fractional derivative models. J. Vib. Acoust.
**2014**, 136, 050901. [Google Scholar] [CrossRef]

**Figure 1.**Distribution of poles for different values of $\alpha $. The solutions that comply with (5) are highlighted in red.

**Figure 2.**Evolution of damped frequency ${\omega}_{\mathrm{d}}$ with the order of the derivative $\alpha $. The mass, stiffness and damping parameters are unitary. The damped frequency ranges from ${\omega}_{\mathrm{d}}=\sqrt{\frac{k+c}{m}}=\sqrt{2}$ to ${\omega}_{\mathrm{d}}={\omega}_{0}\sqrt{1-{\xi}^{2}}=0.866$.

**Figure 3.**Evolution of the damped frequency ${\omega}_{\mathrm{d}}$ with the damping parameter c for different values of the order of the derivative $\alpha $.

**Figure 4.**Time response of a fractionally damped system for different values of the order of the derivative.

**Figure 5.**Response of a fractionally damped system in the frequency domain: (

**a**) from t = 0 to t = 20, (

**b**) for t > 20.

**Figure 6.**Response of a single-degree-of-freedom (1 DOF) fractionally damped system with unitary mass and stiffness parameters, the order of the derivative being $\alpha $ = 0.9 and the damping parameter $c=$2 when subjected to initial conditions $\left\{{x}_{0},{v}_{0}\right\}=\left\{1,0\right\}$.

**Figure 8.**Response to impulse of a system with critical damping for different values of the order of the derivative $\alpha $.

**Figure 10.**Shaft supported by a bearing in each end, where N stands for the number of degrees of freedom of the finite element model of the shaft. The bearing support affects the $(N-1)$th degree of freedom, as the Nth is related to the rotation of the end.

**Figure 11.**Evolution of the frequency of the first two modes with the order of the derivative $\alpha $.

**Figure 12.**First five mode shapes of the supported shaft: real part and phase. The vibration frequencies of the mode shapes are the following: ${\omega}_{1}=$ 116.20 rad/s, ${\omega}_{2}=$ 320.24 rad/s, ${\omega}_{3}=$ 1201.0 rad/s, ${\omega}_{4}=$ 3144.6 rad/s and ${\omega}_{5}=$ 6194.3 rad/s.

**Figure 13.**Displacement of the right end of the shaft: (

**a**) case 1, (

**b**) case 2, (

**c**) case 3, (

**d**) case 4.

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Zarraga, O.; Sarría, I.; García-Barruetabeña, J.; Cortés, F.
An Analysis of the Dynamical Behaviour of Systems with Fractional Damping for Mechanical Engineering Applications. *Symmetry* **2019**, *11*, 1499.
https://doi.org/10.3390/sym11121499

**AMA Style**

Zarraga O, Sarría I, García-Barruetabeña J, Cortés F.
An Analysis of the Dynamical Behaviour of Systems with Fractional Damping for Mechanical Engineering Applications. *Symmetry*. 2019; 11(12):1499.
https://doi.org/10.3390/sym11121499

**Chicago/Turabian Style**

Zarraga, Ondiz, Imanol Sarría, Jon García-Barruetabeña, and Fernando Cortés.
2019. "An Analysis of the Dynamical Behaviour of Systems with Fractional Damping for Mechanical Engineering Applications" *Symmetry* 11, no. 12: 1499.
https://doi.org/10.3390/sym11121499