Fractional Vibrations: Theory and Applications

A special issue of Fractal and Fractional (ISSN 2504-3110). This special issue belongs to the section "Engineering".

Deadline for manuscript submissions: closed (31 December 2021) | Viewed by 5505

Special Issue Editors

Ocean College, Zhejiang University, Hangzhou 310027, Zhejiang, China
Interests: fractal time series; long-range dependent processes; self-similar processes; ocean waves; sea level time series; network traffic
Special Issues, Collections and Topics in MDPI journals
Engineering School (DEIM), University of Tuscia, Largo dell'Università, 01100 Viterbo, Italy
Interests: wavelets; fractals; fractional and stochastic equations; numerical and computational methods; mathematical physics; nonlinear systems; artificial intelligence
Special Issues, Collections and Topics in MDPI journals
Department of Mathematics, Shiraz University of Technology, 71557-13876 Shiraz, Iran
Interests: wavelets theory; fractional calculus; fractional optimal control; stochastic differential equations

Special Issue Information

Dear Colleagues

Fractional vibrations have attracted interest from researchers in various fields, ranging from mechanical engineering to mechanics of materials. The topic of this Special Issue is the Theory and Applications of Fractional Vibrations, and its focus is two-fold: the theory of fractional vibrations, such as the responses and dynamics of fractional vibrators, and applications, such as theoretical explanations of the Rayleigh damping assumption or structure–fluid coupling vibrations.

Specifically, the goal of this Special Issue is to develop the theory and applications of fractional vibrations except conventional nonlinear ones. We are interested in mechanical engineering applications, and recent advances in the theory of fractional vibrations with a mathematics or mechanics approach.

High quality papers, including research articles and reviews, are invited in, among others, the following areas:

  • Theory and applications of fractional vibrations.
  • Fractional multi-degree-of freedom vibrations.
  • Theoretical explanations of the Rayleigh damping assumption.
  • Fractional beams.
  • Fractional wave equations.
  • Fractional Kelvin–Voigt model
  • Fractional vibrations in marine structures.
  • Fractional vibrations in structure–fluid coupling.
  • Fractional vibrations in hydro–elasticity
  • Fractional noise produced by fractional vibration systems.
  • Dynamics of fractional vibrations.

Prof. Dr. Ming Li
Prof. Dr. Carlo Cattani
Dr. Mohammad Hossein Heydari
Guest Editors

Manuscript Submission Information

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Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Fractal and Fractional is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2700 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Published Papers (3 papers)

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Research

23 pages, 1126 KiB  
Article
Heat and Mass Transfer Impact on Differential Type Nanofluid with Carbon Nanotubes: A Study of Fractional Order System
Fractal Fract. 2021, 5(4), 231; https://doi.org/10.3390/fractalfract5040231 - 18 Nov 2021
Cited by 4 | Viewed by 1379
Abstract
This paper is an analysis of flow of MHD CNTs of second grade nano-fluid under the influence of first order chemical reaction, suction, thermal generation and magnetic field. The fluid is flowing through a porous medium. For the study of heat and mass [...] Read more.
This paper is an analysis of flow of MHD CNTs of second grade nano-fluid under the influence of first order chemical reaction, suction, thermal generation and magnetic field. The fluid is flowing through a porous medium. For the study of heat and mass transfer, we applied the newly introduced differential operators to model such flow. The equations for heat, mass and momentum are established in the terms of Caputo (C), Caputo–Fabrizio (CF) and Atangana–Baleanu in Caputo sense (ABC) fractional derivatives. This shows the novelty of this work. The equations for heat, mass and momentum are established in the terms of Caputo (C), Caputo–Fabrizio (CF) and Atangana–Baleanu in Caputo sense (ABC) fractional derivatives. The solutions are evaluated by employing Laplace transform and inversion algorithm. The flow in momentum profile due to variability in the values of parameters are graphically illustrated among C, CF and ABC models. It is concluded that fluid velocity showed decreasing behavior for χ, P, 2, Mo, Pr, and Sc while it showed increasing behavior for Gr, Gm, κ and Ao. Moreover, ABC fractional operator presents larger memory effect than C and CF fractional operators. Full article
(This article belongs to the Special Issue Fractional Vibrations: Theory and Applications)
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13 pages, 646 KiB  
Article
Comparison of Two Different Analytical Forms of Response for Fractional Oscillation Equation
Fractal Fract. 2021, 5(4), 188; https://doi.org/10.3390/fractalfract5040188 - 27 Oct 2021
Cited by 6 | Viewed by 1325
Abstract
The impulse response of the fractional oscillation equation was investigated, where the damping term was characterized by means of the Riemann–Liouville fractional derivative with the order α satisfying 0α2. Two different analytical forms of the response were obtained [...] Read more.
The impulse response of the fractional oscillation equation was investigated, where the damping term was characterized by means of the Riemann–Liouville fractional derivative with the order α satisfying 0α2. Two different analytical forms of the response were obtained by using the two different methods of inverse Laplace transform. The first analytical form is a series composed of positive powers of t, which converges rapidly for a small t. The second form is a sum of a damped harmonic oscillation with negative exponential amplitude and a decayed function in the form of an infinite integral, where the infinite integral converges rapidly for a large t. Furthermore, the Gauss–Laguerre quadrature formula was used for numerical calculation of the infinite integral to generate an analytical approximation to the response. The asymptotic behaviours for a small t and large t were obtained from the two forms of response. The second form provides more details for the response and is applicable for a larger range of t. The results include that of the integer-order cases, α= 0, 1 and 2. Full article
(This article belongs to the Special Issue Fractional Vibrations: Theory and Applications)
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16 pages, 604 KiB  
Article
Vibration Systems with Fractional-Order and Distributed-Order Derivatives Characterizing Viscoinertia
Fractal Fract. 2021, 5(3), 67; https://doi.org/10.3390/fractalfract5030067 - 12 Jul 2021
Cited by 4 | Viewed by 1620
Abstract
We considered forced harmonic vibration systems with the Liouville–Weyl fractional derivative where the order is between 1 and 2 and with a distributed-order derivative where the Liouville–Weyl fractional derivatives are integrated on the interval [1, 2] with respect to the order. Both types [...] Read more.
We considered forced harmonic vibration systems with the Liouville–Weyl fractional derivative where the order is between 1 and 2 and with a distributed-order derivative where the Liouville–Weyl fractional derivatives are integrated on the interval [1, 2] with respect to the order. Both types of derivatives enhance the viscosity and inertia of the system and contribute to damping and mass, respectively. Hence, such types of derivatives characterize the viscoinertia and represent an “inerter-pot” element. For such vibration systems, we derived the equivalent damping and equivalent mass and gave the equivalent integer-order vibration systems. Particularly, for the distributed-order vibration model where the weight function was taken as an exponential function that involved a parameter, we gave detailed analyses for the weight function, the damping contribution, and the mass contribution. Frequency–amplitude curves and frequency-phase curves were plotted for various coefficients and parameters for the comparison of the two types of vibration models. In the distributed-order vibration system, the weight function of the order enables us to simultaneously involve different orders, whilst the fractional-order model has a single order. Thus, the distributed-order vibration model is more general and flexible than the fractional vibration system. Full article
(This article belongs to the Special Issue Fractional Vibrations: Theory and Applications)
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