Vibration Systems with Fractional-Order and Distributed-Order Derivatives Characterizing Viscoinertia

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.

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A comparison study of steady-state vibrations with single fractional-order and distributed-order derivatives

Open Physics ◽

10.1515/phys-2017-0095 ◽

2017 ◽

Vol 15 (1) ◽

pp. 809-818 ◽

Cited By ~ 2

Author(s):

Jun-Sheng Duan ◽

Cui-Ping Cheng ◽

Lian Chen

Keyword(s):

Steady State ◽

Weight Function ◽

Fractional Order ◽

Fractional Derivatives ◽

Order Derivative ◽

Similar Response ◽

Driving Frequency ◽

Fractional Order Derivative ◽

The One ◽

Distributed Order

AbstractWe conduct a detailed study and comparison for the one-degree-of-freedom steady-state vibrations under harmonic driving with a single fractional-order derivative and a distributed-order derivative. For each of the two vibration systems, we consider the stiffness contribution factor and damping contribution factor of the term of fractional derivatives, the amplitude and the phase difference for the response. The effects of driving frequency on these response quantities are discussed. Also the influences of the orderαof the fractional derivative and the parameterγparameterizing the weight function in the distributed-order derivative are analyzed. Two cases display similar response behaviors, but the stiffness contribution factor and damping contribution factor of the distributed-order derivative are almost monotonic change with the parameterγ, not exactly like the case of single fractional-order derivative for the orderα. The case of the distributed-order derivative provides us more options for the weight function and parameters.

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Vibration Equation of Fractional Order Describing Viscoelasticity and Viscous Inertia

Open Physics ◽

10.1515/phys-2019-0088 ◽

2019 ◽

Vol 17 (1) ◽

pp. 850-856 ◽

Cited By ~ 2

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.

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The properties of Bessel-type fractional derivatives and integrals

JOURNAL OF ADVANCES IN PHYSICS ◽

10.24297/jap.v11i8.203 ◽

2016 ◽

pp. 3973-3982

Author(s):

V. R. Lakshmi Gorty

Keyword(s):

Fractional Order ◽

Real Line ◽

Computer Algebra System ◽

Fractional Derivatives ◽

Graphical Representation ◽

Fractional Integrals ◽

The Real ◽

Fractional Derivatives And Integrals ◽

Half Axis ◽

Derivatives Of

The fractional integrals of Bessel-type Fractional Integrals from left-sided and right-sided integrals of fractional order is established on finite and infinite interval of the real-line, half axis and real axis. The Bessel-type fractional derivatives are also established. The properties of Fractional derivatives and integrals are studied. The fractional derivatives of Bessel-type of fractional order on finite of the real-line are studied by graphical representation. Results are direct output of the computer algebra system coded from MATLAB R2011b.

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Fractional-Order Colour Image Processing

Mathematics ◽

10.3390/math9050457 ◽

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.

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Variational Problems with Time Delay and Higher-Order Distributed-Order Fractional Derivatives with Arbitrary Kernels

Mathematics ◽

10.3390/math9141665 ◽

2021 ◽

Vol 9 (14) ◽

pp. 1665

Author(s):

Fátima Cruz ◽

Ricardo Almeida ◽

Natália Martins

Keyword(s):

Time Delay ◽

Fractional Derivatives ◽

Variational Problems ◽

Higher Order ◽

Sufficient Optimality Conditions ◽

Fractional Operator ◽

Different Types ◽

Generalized Fractional Derivatives ◽

Necessary And Sufficient ◽

Distributed Order

In this work, we study variational problems with time delay and higher-order distributed-order fractional derivatives dealing with a new fractional operator. This fractional derivative combines two known operators: distributed-order derivatives and derivatives with respect to another function. The main results of this paper are necessary and sufficient optimality conditions for different types of variational problems. Since we are dealing with generalized fractional derivatives, from this work, some well-known results can be obtained as particular cases.

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Mkhitar Djrbashian and his contribution to Fractional Calculus

Fractional Calculus and Applied Analysis ◽

10.1515/fca-2020-0089 ◽

2020 ◽

Vol 23 (6) ◽

pp. 1797-1809

Author(s):

Sergei Rogosin ◽

Maryna Dubatovskaya

Keyword(s):

Cauchy Problem ◽

Fractional Calculus ◽

Fractional Order ◽

Approximation Theory ◽

Fractional Derivatives ◽

Complex Analysis ◽

Integral Transforms ◽

Modern Theory ◽

Survey Paper ◽

The Cauchy Problem

Abstract This survey paper is devoted to the description of the results by M.M. Djrbashian related to the modern theory of Fractional Calculus. M.M. Djrbashian (1918-1994) is a well-known expert in complex analysis, harmonic analysis and approximation theory. Anyway, his contributions to fractional calculus, to boundary value problems for fractional order operators, to the investigation of properties of the Queen function of Fractional Calculus (the Mittag-Leffler function), to integral transforms’ theory has to be understood on a better level. Unfortunately, most of his works are not enough popular as in that time were published in Russian. The aim of this survey is to fill in the gap in the clear recognition of M.M. Djrbashian’s results in these areas. For same purpose, we decided also to translate in English one of his basic papers [21] of 1968 (joint with A.B. Nersesian, “Fractional derivatives and the Cauchy problem for differential equations of fractional order”), and were invited by the “FCAA” editors to publish its re-edited version in this same issue of the journal.

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Fractional-Order PII1/2DD1/2 Control: Theoretical Aspects and Application to a Mechatronic Axis

Applied Sciences ◽

10.3390/app11083631 ◽

2021 ◽

Vol 11 (8) ◽

pp. 3631

Author(s):

Luca Bruzzone ◽

Mario Baggetta ◽

Pietro Fanghella

Keyword(s):

Fractional Order ◽

Position Control ◽

Tracking Error ◽

Experimental Tests ◽

Maximum Torque ◽

Control Effort ◽

Tuning Method ◽

Alternative Approach ◽

Remarkable Reduction ◽

Distributed Order

Fractional Calculus is usually applied to control systems by means of the well-known PIlDm scheme, which adopts integral and derivative components of non-integer orders λ and µ. An alternative approach is to add equally distributed fractional-order terms to the PID scheme instead of replacing the integer-order terms (Distributed Order PID, DOPID). This work analyzes the properties of the DOPID scheme with five terms, that is the PII1/2DD1/2 (the half-integral and the half-derivative components are added to the classical PID). The frequency domain responses of the PID, PIlDm and PII1/2DD1/2 controllers are compared, then stability features of the PII1/2DD1/2 controller are discussed. A Bode plot-based tuning method for the PII1/2DD1/2 controller is proposed and then applied to the position control of a mechatronic axis. The closed-loop behaviours of PID and PII1/2DD1/2 are compared by simulation and by experimental tests. The results show that the PII1/2DD1/2 scheme with the proposed tuning criterium allows remarkable reduction in the position error with respect to the PID, with a similar control effort and maximum torque. For the considered mechatronic axis and trapezoidal speed law, the reduction in maximum tracking error is −71% and the reduction in mean tracking error is −77%, in correspondence to a limited increase in maximum torque (+5%) and in control effort (+4%).

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Study on Self-Excited Vibration Mechanism of Wheel-Rail Lateral Contact System

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.427-429.257 ◽

2013 ◽

Vol 427-429 ◽

pp. 257-261

Author(s):

Li Xia Sun ◽

Jian Wei Yao ◽

Fu Guo Hou ◽

Xin Zhao

Keyword(s):

Feedback Mechanism ◽

Point Of View ◽

Contact System ◽

Vibration System ◽

Lateral Contact ◽

System A ◽

Vibration Model ◽

Excited Vibration ◽

Vibration Mechanism ◽

The Stability

In order to investigate self-excited vibration mechanism of wheel-rail lateral contact system, a two DOF elasticity position wheelset lateral vibration model is established which considers the dry friction; the mechanism of the wheelset lateral self-excited vibration is investigated from the energy point of view. It shows that: the bifurcation diagram of this wheel-rail lateral contact system has a supercritical Hopf bifurcation. The energy of self-excited vibration derives from a part of traction energy; the creep rate in the wheel-rail system act as a feedback mechanism in the wheelset lateral self-excited vibration system. The stability of the wheelset self-excited vibration system depends mainly on the total energy removed from and imported into the system.

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Feedback control for two-degree-of-freedom vibration system with fractional-order derivative damping

Journal of Low Frequency Noise Vibration and Active Control ◽

10.1177/1461348417725958 ◽

2017 ◽

Vol 37 (3) ◽

pp. 554-564

Author(s):

Canchang Liu ◽

Chicheng Ma ◽

Jilei Zhou ◽

Lu Liu ◽

Shuchang Yue ◽

...

Keyword(s):

Feedback Control ◽

Nonlinear Vibration ◽

Fractional Order ◽

Suspension System ◽

Degree Of Freedom ◽

Vehicle Suspension ◽

Vibration System ◽

Fractional Order Derivative ◽

Vehicle Suspension System ◽

Two Degree Of Freedom

A two-degree-of-freedom nonlinear vibration system of a quarter vehicle suspension system is studied by using the feedback control method considered the fractional-order derivative damping. The nonlinear dynamic model of two-degree-of-freedom vehicle suspension system is built and linear velocity and displacement controllers are used to control the nonlinear vibration of the vehicle suspension system. A case of the 1:1 internal resonance is considered. The amplitude–frequency response is obtained with the multiscale method. The asymptotic stability conditions of the nonlinear system can be gotten by using the Routh–Hurwitz criterion and the ranges of control parameters are gained in the condition of stable solutions to the system. The simulation results show that the feedback control can effectively reduce the amplitude of primary resonance, weaken or even eliminate the nonlinear vibration characteristics of the suspension system. Fractional orders have an impact on control performance, which should be considered in the control problem. The study will provide a theoretical basis and reference for the optimal design of the vehicle suspension system.

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Using Fractional Bernoulli Wavelets for Solving Fractional Diffusion Wave Equations with Initial and Boundary Conditions

Fractal and Fractional ◽

10.3390/fractalfract5040212 ◽

2021 ◽

Vol 5 (4) ◽

pp. 212

Author(s):

Monireh Nosrati Sahlan ◽

Hojjat Afshari ◽

Jehad Alzabut ◽

Ghada Alobaidi

Keyword(s):

Fractional Order ◽

Fractional Derivatives ◽

Wave Equations ◽

Bernoulli Polynomials ◽

Fractional Diffusion ◽

Computational Time ◽

Operational Matrices ◽

Algebraic Equations ◽

Diffusion Wave ◽

Klein Gordon

In this paper, fractional-order Bernoulli wavelets based on the Bernoulli polynomials are constructed and applied to evaluate the numerical solution of the general form of Caputo fractional order diffusion wave equations. The operational matrices of ordinary and fractional derivatives for Bernoulli wavelets are set via fractional Riemann–Liouville integral operator. Then, these wavelets and their operational matrices are utilized to reduce the nonlinear fractional problem to a set of algebraic equations. For solving the obtained system of equations, Galerkin and collocation spectral methods are employed. To demonstrate the validity and applicability of the presented method, we offer five significant examples, including generalized Cattaneo diffusion wave and Klein–Gordon equations. The implementation of algorithms exposes high accuracy of the presented numerical method. The advantage of having compact support and orthogonality of these family of wavelets trigger having sparse operational matrices, which reduces the computational time and CPU requirements.

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