To the question on the correctness of fractional derivative models in dynamic problems of viscoelastic bodies

Rossikhin, Yury A.; Shitikova, Marina V.; Krusser, Anastasiya I.

doi:10.1016/j.mechrescom.2016.09.002

Cited by 17 publications

(5 citation statements)

References 5 publications

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“…Similar conclusions have been presented by other authors, e.g Bagley [34] and Pritz [35]. Another disadvantage is that this model cannot describe the behavior of some types of viscoelastic auxetic materials possessing negative Poisson's ratios [36]. The fractional Zener solid model does not have the aforementioned disadvantages [36] and is therefore being used increasingly in modeling the dynamics of solids.…”

Section: Introductionsupporting

confidence: 73%

“…Another disadvantage is that this model cannot describe the behavior of some types of viscoelastic auxetic materials possessing negative Poisson's ratios [36]. The fractional Zener solid model does not have the aforementioned disadvantages [36] and is therefore being used increasingly in modeling the dynamics of solids. The fractional Zener solid model has been employed in beam dynamics.…”

Section: Introductionmentioning

confidence: 99%

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Transient vibrations of a fractional Zener viscoelastic cantilever beam with a tip mass

Freundlich

2021

Meccanica

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The presented work concerns the kinematically excited transient vibrations of a cantilever beam with a mass element fixed to its free end. The Euler–Bernoulli beam theory and the fractional Zener model of the beam material are assumed. A fractional Caputo derivative is used to formulate a viscoelastic material law. A characteristic equation, modal frequencies, eigenfunction and orthogonality conditions are achieved for the beam considered. The equations of motion of the system are solved numerically. A numerical solution of a multi-term fractional differential equation is obtained by means of a conversion to a mixed system of ordinary and fractional differential equations, each of the order of $$0 < \gamma \le 1$$ 0 < γ ≤ 1 . The transient time histories of the beam vibrations during the passage through resonance are calculated. A comparison between the beam responses obtained with a fractional and an integer viscoelastic material model is presented. The calculations performed reveal that use of the fractional damping affects on the time histories of the system. The calculated beam responses show that for some values of the order of the fractional derivative $$\gamma$$ γ , the amplitudes occurring in the area of the second resonance are greater than those obtained in the area of the first resonance, which does not occur in the case of the integer order of the fractional derivative. Moreover, an evaluation is made of the difference between the results obtained for the calculations using the fractional Zener model and the fractional Kelvin model. It is shown that for some physical beam parameters, the calculation results obtained using both models are virtually the same for both models, which means that the the simpler, fractional Kelvin–Voigt material can be used instead of the fractional Zener material model. This simplifies the solution and decreases the time needed to make the numerical calculations.

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Section: Introductionsupporting

confidence: 73%

Section: Introductionmentioning

confidence: 99%

Transient vibrations of a fractional Zener viscoelastic cantilever beam with a tip mass

Freundlich

2021

Meccanica

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“…Based on the MPA algorithm, F i is added to the iterative process for the IDE, the improved mutation strategy as shown in Equations ( 6) and (7).…”

Section: Multiple Mutation Strategies Fusionmentioning

confidence: 99%

“…In recent years, fractional calculus has been widely adopted to describe batteries [1,2], thermal systems [3], diffusion phenomenon [4,5], dynamic systems [6,7], power systems [8,9], and financial system [10], and it has been verified that the fractional-order model is more accurate than the integer-order model. Energy storage links, electromagnetic coupling, and friction have been shown to have fractional-order properties [11].…”

Section: Introductionmentioning

confidence: 99%

Fractional‐order system identification based on an improved differential evolution algorithm

Liang

Chen

et al. 2021

Asian Journal of Control

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With the continuous increasing of system modeling accuracy and control performance requirements, the fractional‐order models are applied to describe the dynamic characteristics of the real systems more accurately over traditional integer‐order models. Traditional identification algorithms for fractional‐order system require stringent information on the derivatives of the objective function and initial values of the parameters, which is difficult in practical applications. The differential evolution (DE) algorithm has the advantage of simple implementation and superior performance for solving multi‐dimensional complex optimization problems which is suitable for the identification of fractional‐order systems. Aiming at improving the solution accuracy and convergence speed, an improved DE (IDE) algorithm for fractional‐order system identification is proposed. The movement strategy of the marine predator algorithm (MPA) and the mutation strategy of randomly selecting one from the optimal individual population as the basis vector are both adopted in this proposed IDE algorithm. The parameter information of the successfully mutated individual is archived during the iteration process. The scaling factor and crossover probability factor are adaptively adjusted according to the archived information to enhance the best solution accuracy obtained in the search process. By testing 13 sets of unimodal and multimodal functions with high‐performance optimization algorithms, the accuracy performance of the proposed IDE solution is verified. The IDE algorithm is applied to identify the parameters of the fractional‐order system of permanent magnet synchronous motor with simulation and experiment. The identification results clearly show the effectiveness of the proposed methodology.

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“…With the accelerated progress in studies of auxetics made in last few years, there are new potential opportunities for researchers. Auxetic materials are still in developing stage and require intensive research [8,9].…”

Section: Introductionmentioning

confidence: 99%

Mathematical Model to Study the Impact Response of a Viscoelastic Auxetic Plate

Shitikova

Rossikhin

Ajeneza

2019

KEM

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In the present paper, a mathematical model has been constructed in order to describe the impact response of a linear Kirchhoff-Love plate made of viscoelastic auxetic material possessing fractional viscosity. Auxetic’s Poisson’s ratio is a time-dependent value changing from negative to positive magnitudes with time. In the case of a linear plate, the solution out of the contact domain is found through the Green function, and within the contact zone via the modified Hertz contact theory. Integral equations for the contact force and local indentation have been obtained.

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To the question on the correctness of fractional derivative models in dynamic problems of viscoelastic bodies

Cited by 17 publications

References 5 publications

Transient vibrations of a fractional Zener viscoelastic cantilever beam with a tip mass

Transient vibrations of a fractional Zener viscoelastic cantilever beam with a tip mass

Fractional‐order system identification based on an improved differential evolution algorithm

Mathematical Model to Study the Impact Response of a Viscoelastic Auxetic Plate

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