scholarly journals Vibration Equation of Fractional Order Describing Viscoelasticity and Viscous Inertia

Open Physics ◽  
2019 ◽  
Vol 17 (1) ◽  
pp. 850-856 ◽  
Author(s):  
Jun-Sheng Duan ◽  
Yun-Yun Xu
Keyword(s):  
Fractional Derivative ◽  
Fractional Order ◽  
Fractional Derivatives ◽  
Harmonic Excitation ◽  
Steady State Response ◽  
Vibration System ◽  
Derivative Operator ◽  
Vibration Equation ◽  
State Response ◽  
Frequency Relation

Abstract The steady state response of a fractional order vibration system subject to harmonic excitation was studied by using the fractional derivative operator ${}_{-\infty} D_t^\beta,$where the order β is a real number satisfying 0 ≤ β ≤ 2. We derived that the fractional derivative contributes to the viscoelasticity if 0 < β < 1, while it contributes to the viscous inertia if 1 < β < 2. Thus the fractional derivative can represent the “spring-pot” element and also the “inerterpot” element proposed in the present article. The viscosity contribution coefficient, elasticity contribution coefficient, inertia contribution coefficient, amplitude-frequency relation, phase-frequency relation, and influence of the order are discussed in detail. The results show that fractional derivatives are applicable for characterizing the viscoelasticity and viscous inertia of materials.

Steady-State Response to Periodic Excitation in Fractional Vibration System

2016 ◽  
Vol 32 (1) ◽  
pp. 25-33 ◽  
Author(s):  
C. Huang ◽  
J.-S. Duan
Keyword(s):  
Steady State ◽  
Fractional Derivative ◽  
Fractional Power ◽  
Steady State Response ◽  
Periodic Excitation ◽  
Vibration System ◽  
Principle Of Superposition ◽  
State Response ◽  
Harmonic Excitations ◽  
Frequency Relation

AbstractThe steady-state response to periodic excitation in the linear fractional vibration system was considered by using the fractional derivative operator . First we investigated the response to the harmonic excitation in the form of complex exponential function. The amplitude-frequency relation and phase-frequency relation were derived. The effect of the fractional derivative term on the stiffness and damping was discussed. For the case of periodic excitation, we decompose the periodic excitation into a superposition of harmonic excitations by using the Fourier series, and then utilize the results for harmonic excitations and the principle of superposition, where our adopted tactics avoid appearing a fractional power of negative numbers to overcome the difficulty in fractional case. Finally we demonstrate the proposed method by three numerical examples.

scholarly journals Response of a fractional nonlinear system to harmonic excitation by the averaging method

Open Physics ◽  
2015 ◽  
Vol 13 (1) ◽  
Author(s):  
Jun-Sheng Duan ◽  
Can Huang ◽  
Li-Li Liu
Keyword(s):  
Nonlinear System ◽  
Fractional Derivative ◽  
Nonlinear Vibration ◽  
Averaging Method ◽  
Harmonic Excitation ◽  
Vibration System ◽  
Derivative Operator ◽  
Nonlinear Vibration System

AbstractIn this work, we consider a fractional nonlinear vibration system of Duffing type with harmonic excitation by using the fractional derivative operator

Steady periodic response for a vibration system with distributed order derivatives to periodic excitation

2017 ◽  
Vol 24 (14) ◽  
pp. 3124-3131 ◽  
Author(s):  
Jun-Sheng Duan ◽  
Dumitru Baleanu
Keyword(s):  
Fourier Series ◽  
Steady State ◽  
Fractional Derivative ◽  
Harmonic Excitation ◽  
Periodic Excitation ◽  
Vibration System ◽  
Principle Of Superposition ◽  
Derivative Operator ◽  
Numerical Examples ◽  
Distributed Order

Steady-state periodic responses for a vibration system with distributed order derivatives are investigated, where the fractional derivative operator [Formula: see text] is utilized. The response to complex harmonic excitation is derived and the amplitude–frequency and phase–frequency relations are obtained. For a periodic excitation, we decompose it into the Fourier series, and then make use of the principle of superposition and the results of harmonic excitations to obtain the response. Finally, we examine three numerical examples by using the proposed method.

scholarly journals Fractional-Order Colour Image Processing

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 457
Author(s):  
Manuel Henriques ◽  
Duarte Valério ◽  
Paulo Gordo ◽  
Rui Melicio
Keyword(s):  
Image Processing ◽  
Edge Detection ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Fractional Derivatives ◽  
Grey Scale ◽  
Colour Image ◽  
Colour Image Processing ◽  
Image Processing Algorithms ◽  
Processing Algorithms

Many image processing algorithms make use of derivatives. In such cases, fractional derivatives allow an extra degree of freedom, which can be used to obtain better results in applications such as edge detection. Published literature concentrates on grey-scale images; in this paper, algorithms of six fractional detectors for colour images are implemented, and their performance is illustrated. The algorithms are: Canny, Sobel, Roberts, Laplacian of Gaussian, CRONE, and fractional derivative.

scholarly journals Necessary optimality conditions of a reaction-diffusion SIR model with ABC fractional derivatives

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Moulay Rchid Sidi Ammi ◽  
Mostafa Tahiri ◽  
Delfim F. M. Torres
Keyword(s):  
Optimal Control ◽  
Fractional Derivative ◽  
Optimality Conditions ◽  
Fractional Derivatives ◽  
Necessary Optimality Conditions ◽  
Reaction Diffusion ◽  
Derivative Operator ◽  
Sir Epidemic ◽  
Non Local

<p style='text-indent:20px;'>The main aim of the present work is to study and analyze a reaction-diffusion fractional version of the SIR epidemic mathematical model by means of the non-local and non-singular ABC fractional derivative operator with complete memory effects. Existence and uniqueness of solution for the proposed fractional model is proved. Existence of an optimal control is also established. Then, necessary optimality conditions are derived. As a consequence, a characterization of the optimal control is given. Lastly, numerical results are given with the aim to show the effectiveness of the proposed control strategy, which provides significant results using the AB fractional derivative operator in the Caputo sense, comparing it with the classical integer one. The results show the importance of choosing very well the fractional characterization of the order of the operators.</p>

Study on the Method of Modified Random Decrement for Transfer Function

2003 ◽  
Author(s):  
Zhenzhong Zhang ◽  
Shijian Zhu
Keyword(s):  
Transfer Function ◽  
Steady State Response ◽  
Vibration System ◽  
State Response ◽  
Modified Method ◽  
Conventional Process ◽  
Random Decrement ◽  
Initial Excitation ◽  
Function Calculation ◽  
Random Decrement Method

This paper presents a modified random decrement method to obtain the steady-state response of vibration system. The expressions are deduced. Two numerical simulations and verification are given. The efficiencies of the modified method and the conventional process are compared. It is shown that the method is more efficient than the conventional process to diminish the influence of the initial excitation on the transfer function calculation of the vibration system.

scholarly journals FRACTIONAL DERIVATIVE AS FRACTIONAL POWER OF DERIVATIVE

2007 ◽  
Vol 18 (03) ◽  
pp. 281-299 ◽  
Author(s):  
VASILY E. TARASOV
Keyword(s):  
Fourier Series ◽  
Fractional Derivative ◽  
Taylor Series ◽  
Fractional Derivatives ◽  
Fractional Power ◽  
Weyl Quantization ◽  
Fractional Powers ◽  
Derivative Operator ◽  
Quantization Procedure ◽  
Definition Of

Definitions of fractional derivatives as fractional powers of derivative operators are suggested. The Taylor series and Fourier series are used to define fractional power of selfadjoint derivative operator. The Fourier integrals and Weyl quantization procedure are applied to derive the definition of fractional derivative operator. Fractional generalization of concept of stability is considered.

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