A new nonlinear duffing system with sequential fractional derivatives

Bezziou, Mohamed; Jebril, Iqbal H.; Dahmani, Zoubir

doi:10.1016/j.chaos.2021.111247

Cited by 18 publications

(8 citation statements)

References 15 publications

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“…Given that the direct method (Lyapunov first method) has been successfully used in the study of similar problems, this work will also use it in the stability analysis of fractional Duffing systems with a nonlinear time delay term [12]. The problem of motion stability is also important for the study of fractional Duffing system with nonlinear time delay term in this work [15,16].…”

Section: Introductionmentioning

confidence: 99%

Dynamics analysis for a class of fractional Duffing systems with nonlinear time delay terms

Yang

Jiao

2023

Math Methods in App Sciences

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The dynamic characteristics of the fractional Duffing system with nonlinear time delay term are studied according to the average method and the harmonic balance method. First, the steady‐state solutions of the fractional Duffing system with a nonlinear time delay term are obtained by applying the average method and the harmonic balance method. The analytical expressions of the amplitude–frequency characteristic curves under the average method and the harmonic balance method are also established. Second, the stability criterion of the system is obtained according to the indirect method of studying the motion stability problem, and the concept of the stability condition parameter is proposed in this process. Third, the transient solution of the fractional Duffing system with a nonlinear time delay term is obtained utilizing the average method, and the approximate analytical solution of the fractional Duffing system with nonlinear time delay term is obtained. Finally, the amplitude–frequency characteristic curves of the different fractional differential terms, coefficients of different fractional differential terms, magnitude of different time delay quantities, and feedback strength of different delay are analyzed by numerical simulation. Furthermore, the amplitude–frequency characteristic curves obtained by the average method and the harmonic balance method under different order of fractional differential term are compared and analyzed by numerical simulation. The time sequence and phase diagrams of the system under different order of fractional differential term and different time delay quantities are analyzed by numerical simulation. In addition, the correctness of the analytical analysis is verified by comparing the analytical results with the numerical simulation results under the average method and the harmonic balance method, respectively, by numerical simulation.

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Section: Introductionmentioning

confidence: 99%

Dynamics analysis for a class of fractional Duffing systems with nonlinear time delay terms

Yang

Jiao

2023

Math Methods in App Sciences

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“…Fractional calculus is essentially a non-integer-order calculus, in which the order of differentiation or integration can be real or complex numbers [8,9,10]. The basic operation of fractional calculus is a fractional-order differentiation, a D α t , which denotes the fractional-order differential operator [11,12,13], i.e.…”

Section: Introductionmentioning

confidence: 99%

Design Fractional-order PID Controllers for Single-Joint Robot Arm Model

Batiha¹,

Njadat²,

Batyha³

et al. 2022

ijasca

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he major goal of the this work is to present an optimal design of the Fractional-order Proportional-Derivative-Integral (FoPID) controller for the single-joint arm dynamics. For meeting this aim, the Particle Swarm Optimization (PSO) algorithm will be implement to tune the parameters of such controller. Six FoPID-controllers will be generated in accordance with two kinds of approaches (Continued Fraction Expansion (CFE) and Outstaloup’s approaches) for Laplacian operators, coupled with three fitness functions (IAE, ITAE, ITSE). These controllers will be competed to each other to determine which one can provide to the closed-loop system of the single-joint robot arm model a good rise time, short settling time, and an excellent overshoot. Keywords: Fractional-order model; Oustaloup approximation, Continued

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“…The introduction of fractional order theory greatly improves this defect, and it is found that fractional order damping not only presents damping characteristics but also stiffness characteristics in the simulation of dynamics problems. At present, a variety of dynamical systems have been studied by various scholars, such as Duffing system 14–16 . According to the vibration theory, the response of the system can be divided into free vibration and forced vibration.…”

Section: Introductionmentioning

confidence: 99%

“…At present, a variety of dynamical systems have been studied by various scholars, such as Duffing system. [14][15][16] According to the vibration theory, the response of the system can be divided into free vibration and forced vibration. The free vibration is caused by the initial conditions of the equation, and the forced vibration is caused by the external excitation.…”

Section: Introductionmentioning

confidence: 99%

Dynamical analysis of fractional oscillator system with cosine excitation utilizing the average method

Xie

Wang

Chen

et al. 2022

Math Methods in App Sciences

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In this paper, an analytical and numerical algorithm for solving displacement response of fractional oscillator system with cosine excitation is established. The analytical method is to solve steady‐state response and transient response solutions of the system by means of average method and then the total displacement response solution is the sum of steady‐state solution and transient solution. The numerical method is to use the Grünwald–Letnikov definition of fractional derivatives to discretize the fractional differential term in the system and then to reduce the order of the original system. Then, the approximate response solution of the system under the general periodic excitation is obtained by using the Fourier series expansion method and the superposition principle of linear system. Finally, the effectiveness and feasibility of the analytical technique are verified by numerical simulation, and the influence of fractional order, linear damping coefficient and fractional derivative coefficient on the amplitude of the steady‐state response and the total displacement response of the system is analyzed.

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A new nonlinear duffing system with sequential fractional derivatives

Cited by 18 publications

References 15 publications

Dynamics analysis for a class of fractional Duffing systems with nonlinear time delay terms

Dynamics analysis for a class of fractional Duffing systems with nonlinear time delay terms

Design Fractional-order PID Controllers for Single-Joint Robot Arm Model

Dynamical analysis of fractional oscillator system with cosine excitation utilizing the average method

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