# Hysteretically Symmetrical Evolution of Elastomers-Based Vibration Isolators within α-Fractional Nonlinear Computational Dynamics

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

_{o}denoting the shear modulus, and τ, γ meaning the stress and the shear strain, respectively. It has to be noted that the authors of the paper [6], using the less-square fit procedure for a set of experimental data, showed that the fractional Voigt-Kelvin model fits the data much better than the hysteretic, classical Kelvin and standard linear solid models.

_{i}are the control coefficients related to each damping component. It has to be noted that extensive examples of FDE can be found in the works [11,12] and within theirs bibliography.

## 3. Results

_{0}, k

_{0}denote damping and stiffness coefficients, respectively.

_{0}, k

_{0}were identically valued as follows: c

_{0}= 1, k

_{0}= 0. Taking into account the nonlinear term in parenthesis within the left side hand of Equation (22), the analysis has to be conducted related to displacement magnitude. In this study, random values for magnitude were taken. The results presented correspond to the 3.3, 1.0, and 0.3 valued displacement magnitudes. The monitored parameters, for each case, were similarly adopted than the previous linear case: Timed evolution, spectral composition, hysteretic loops with Poincare maps overlapped, and hysteresis axis according to the pseudo-elastic component provided by the model. The results for the three cases were presented in Figure 10, Figure 11 and Figure 12, respectively (details were mentioned within the figure captions).

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Tenreiro Machado, J.A.; Galhano, A. Fractional Dynamics: A Statistical Perspective. J. Comput. Nonlinear Dyn.
**2008**, 3, 021201. [Google Scholar] [CrossRef] - Babiarz, A.; Legowski, A. Human arm fractional dynamics. In Trends in Advanced Intelligent Control, Optimization and Automation, Advances in Intelligent Systems and Computing 577; Mitkowski, W., Ed.; Springer International Publishing AG: Cham, Switzerland, 2017. [Google Scholar]
- Zhou, Y.; Ionescu, C.; Tenreiro Machado, J.A. Fractional dynamics and its applications. Nonlinear Dyn.
**2015**, 80, 1661–1664. [Google Scholar] [CrossRef] [Green Version] - Chi, C.; Gao, F. Simulating Fractional Derivatives using Matlab. J. Softw.
**2013**, 8, 572–578. [Google Scholar] [CrossRef] - Ozturk, O. A Study on the Damped Free Vibration with Fractional Calculus. IJAMEC
**2016**, 4, 156–159. [Google Scholar] - Koh, C.G.; Kelly, J.M. Application of fractional derivatives to seismic analysis of base-isolated models. Earthq. Eng. Struct. Dyn.
**1990**, 19, 229–241. [Google Scholar] [CrossRef] - Freundlich, J. Vibrations of a simply supported beam with a fractional viscoelastic material model–supports movement excitation. Shock Vib.
**2013**, 20, 1103–1112. [Google Scholar] [CrossRef] - Gómez-Aguilar, J.F.; Yépez-Martínez, H.; Calderón-Ramón, C.; Cruz-Orduña, I.; Escobar-Jiménez, R.F.; Olivares-Peregrino, V.H. Modeling of a Mass-Spring-Damper System by Fractional Derivatives with and without a Singular Kernel. Entropy
**2015**, 17, 6289–6303. [Google Scholar] [CrossRef] - Petráš, I. An Effective Numerical Method and Its Utilization to Solution of Fractional Models Used in Bioengineering Applications. Adv. Differ. Equ.
**2011**, 2011, 652789. [Google Scholar] [CrossRef] - Petráš, I. Fractional Derivatives, Fractional Integrals, and Fractional Differential Equations in Matlab. In Engineering Education and Research Using Matlab; Assi, A., Ed.; InTech: Rijeka, Croatia, 2011. [Google Scholar] [Green Version]
- Rossikhin, Y.A.; Shitikova, M.V. New approach for the analysis of damped vibrations of fractional oscillators. Shock Vib.
**2009**, 16, 365–387. [Google Scholar] [CrossRef] - Rossikhin, Y.A.; Shitikova, M.V.; Shcheglova, T. Forced Vibrations of a Nonlinear Oscillator with Weak Fractional Damping. J. Mech. Mater. Struct.
**2009**, 4, 1619–1636. [Google Scholar] [CrossRef] - Malara, G.; Spanos, P.D. Nonlinear random vibrations of plates endowed with fractional derivative elements. Probab. Eng. Mech.
**2017**. [Google Scholar] [CrossRef] - Di Paola, M.; Failla, G.; Pirrotta, A. Stationary and non-stationary stochastic response of linear fractional viscoelastic systems. Probab. Eng. Mech.
**2012**, 28, 85–90. [Google Scholar] [CrossRef] [Green Version] - Cajić, M.; Karličić, D.; Lazarević, M. Nonlocal Vibration of a Fractional Order Viscoelastic Nanobeam with Attached Nanoparticle. Theor. Appl. Mech.
**2015**, 42, 167–190. [Google Scholar] [CrossRef] - Ma, C. A Novel Computational Technique for Impulsive Fractional Differential Equations. Symmetry
**2019**, 11, 216. [Google Scholar] [CrossRef] - Eltayeb, H.; Bachar, I.; Kılıçman, A. On Conformable Double Laplace Transform and One Dimensional Fractional Coupled Burgers’ Equation. Symmetry
**2019**, 11, 417. [Google Scholar] [CrossRef] - Hilfer, R. Foundations of Fractional Dynamics: A Short Account. In Fractional Dynamics: Recent Advances; Klafter, J., Lim, S., Metzler, R., Eds.; World Scientific: Singapore, 2011; pp. 207–223. [Google Scholar]
- Herrmann, R. Fractional Calculus: An Introduction for Physicists, 2nd ed.; World Scientific Publishing: Singapore, 2014. [Google Scholar]
- Mainardi, F. Fractional Calculus and Waves in Linear Viscoelasticity. An Introduction to Mathematical Models; Imperial College Press: London, UK, 2010. [Google Scholar]
- Ray, S.S. Fractional Calculus with Applications for Nuclear Reactor Dynamics; CRC Press Taylor & Francis Group: Boca Raton, FL, USA; London, UK; New York, NY, USA, 2016. [Google Scholar]
- Anastassiou, G.A.; Argyros, I.K. Intelligent Numerical Methods: Applications to Fractional Calculus; Springer International Publishing: Cham, Switzerland, 2016. [Google Scholar]
- Bratu, P. Viscous nonlinearity for interval energy dissipation. Int. J. Acoust. Vib.
**2000**, 4, 321–326. [Google Scholar] - Marin, M.; Baleanu, D.; Vlase, S. Effect of microtemperatures for micropolar thermoelastic bodies. Struct. Eng. Mech.
**2017**, 61, 381–387. [Google Scholar] [CrossRef] - Iancu, V.; Vasile, O.; Gillich, G.R. Modelling and Characterization of Hybrid Rubber-Based Earthquake Isolation Systems. Mater. Plast.
**2012**, 4, 237–241. [Google Scholar] - Nastac, S. Theoretical and experimental researches regarding the dynamic behavior of the passive vibration isolation of systems. In Research Trends in Mechanics; Dinel, P., Chiroiu, V., Toma, I., Eds.; Romanian Academy Publishing House: Bucharest, Romania, 2008; Volume II, pp. 234–262. [Google Scholar]
- Leopa, A.; Nastac, S.; Debeleac, C. Researches on damage identification in passive vibro-isolation devices. Shock Vib.
**2012**, 19, 803–809. [Google Scholar] [CrossRef] - Nastac, S. On Nonlinear Computational Assessments of Passive Elastomeric Elements for Vibration Isolation. RJAV
**2014**, 2, 130–135. [Google Scholar] - Nastac, S.; Debeleac, C. On Shape and Material Nonlinearities Influences about the Internal Thermal Dissipation for Elastomer-Based Vibration Isolators. PAMM
**2014**, 1, 751–752. [Google Scholar] [CrossRef] - Nastac, S. On Fractional Order Dynamics of Elastomers-based Vibration Insulation Devices. Acta Electrotech.
**2019**, 60, 169–182. [Google Scholar] - Xiao, D.; Fang, F.; Buchan, A.G.; Pain, C.C.; Navon, I.M.; Du, J.; Hu, G. Non-linear model reduction for the Navier–Stokes equations using residual DEIM method. J. Comput. Phys.
**2014**, 263, 1–18. [Google Scholar] [CrossRef]

**Figure 1.**Timed response of α-order fractional damping model with respect to 1 Hz harmonic displacement excitation. Fractional orders were mentioned on graph legend.

**Figure 2.**Spectral composition of α-order fractional damping model with respect to 1 Hz harmonic displacement excitation, in terms of magnitude and phase, respectively.

**Figure 3.**The response of α-fractional damping model with respect to 1 Hz harmonic displacement excitation. Poincare maps, with snapshots at 1s, were doted on graphs (named PM on graph legend).

**Figure 4.**Hysteresis axes for loops in Figure 3, denoting the pseudo-elastic component, virtually emulated by the α-fractional damping model.

**Figure 5.**The results of additional analysis, supposing large number of α-fractional orders, in terms of hysteresis, including Poincare maps (

**a**), with detail on the left hand side (

**b**), timed evolution (

**c**), and correspondent hysteresis axes (

**d**).

**Figure 6.**Schematic diagram related to the equivalence between α-fractional dissipative model and emulated linear Voigt-Kelvin model.

**Figure 7.**Comparative hysteresis of initial α-fractional dissipative and emulated linear Voigt-Kelvin models respectively, for α-order valued to 0.4 (

**a**) and 0.7 (

**b**). The identified values of virtual linear system were mentioned on each graph.

**Figure 8.**The evolution of pseudo-elastic component (

**a**) and pseudo-dissipative component (

**b**), respectively, for the emulated linear Voigt-Kelvin model, in respect to α-order of initial fractional derivative model.

**Figure 9.**The evolution of losses coefficient for the emulated linear Voigt-Kelvin model, in respect to α-order of initial fractional derivative model.

**Figure 10.**The response of α-order fractional damping VdP-based model, with respect to 1 Hz and 3.3 magnitude harmonic displacement excitation, in terms of timed evolution (

**a**), magnitude and phase spectral composition (

**b**), hysteresis loops with Poincare map (

**c**), and pseudo-elastic component (

**d**).

**Figure 11.**The response of α-order fractional damping VdP-based model, with respect to 1 Hz and unitary magnitude harmonic displacement excitation, in terms of timed evolution (

**a**), magnitude and phase spectral composition (

**b**), hysteresis loops with Poincare map (

**c**), and pseudo-elastic component (

**d**).

**Figure 12.**The response of α-order fractional damping VdP-based model, with respect to 1 Hz and 0.3 magnitude harmonic displacement excitation, in terms of timed evolution (

**a**), magnitude and phase spectral composition (

**b**), hysteresis loops with Poincare map (

**c**), and pseudo-elastic component (

**d**).

**Figure 13.**Responses hysteresis provided by the initial α-fractional dissipative and the emulated VdP-based models, respectively. The simulations were performed for displacement magnitude gained to 1.8, and in respect to emulated VdP model according to Equation (23) (

**a**), and respectively, to Equation (24) (

**b**). The α-order was valued to 0.7.

**Figure 14.**Responses hysteresis provided by the initial α-fractional dissipative and the emulated VdP-based models, respectively. The simulations were performed for displacement magnitude unitary gained, and in respect to emulated VdP model according to Equation (23) (

**a**), and respectively, to Equation (24) (

**b**). The α-order was valued to 0.7.

**Figure 15.**Responses hysteresis provided by the initial α-fractional dissipative and the emulated VdP-based models, respectively. The simulations were performed for displacement magnitude gained to 0.8, and in respect to emulated VdP model according to Equation (23) (

**a**), and respectively, to Equation (24) (

**b**). The α-order was valued to 0.7.

**Figure 16.**Hysteretic response of rational-power model, for $\alpha $ ∈ [2.3; 3.7] (

**a**), [1.3; 2.7] (

**b**) and [0.3; 1.7] (

**c**). It was assumed a single time-period of 1 Hz excitation signal, and ratio between rigidity to damping, for linear components, valued to 20 s

^{−1}. Within these diagrams, the symbol α must be considered as the rational power of additional dissipative component (multiplied by 20 from linear damping coefficient) into the computational model, with specific values mentioned on graph legends.

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**MDPI and ACS Style**

Nastac, S.; Debeleac, C.; Vlase, S.
Hysteretically Symmetrical Evolution of Elastomers-Based Vibration Isolators within α-Fractional Nonlinear Computational Dynamics. *Symmetry* **2019**, *11*, 924.
https://doi.org/10.3390/sym11070924

**AMA Style**

Nastac S, Debeleac C, Vlase S.
Hysteretically Symmetrical Evolution of Elastomers-Based Vibration Isolators within α-Fractional Nonlinear Computational Dynamics. *Symmetry*. 2019; 11(7):924.
https://doi.org/10.3390/sym11070924

**Chicago/Turabian Style**

Nastac, Silviu, Carmen Debeleac, and Sorin Vlase.
2019. "Hysteretically Symmetrical Evolution of Elastomers-Based Vibration Isolators within α-Fractional Nonlinear Computational Dynamics" *Symmetry* 11, no. 7: 924.
https://doi.org/10.3390/sym11070924