scholarly journals Анализ добротности вынужденных колебаний дробного линейного осциллятора

2020 ◽  
Vol 90 (7) ◽  
pp. 1059
Author(s):  
Р.И. Паровик
Keyword(s):  
Quality Factor ◽  
Frequency Characteristic ◽  
Fractional Derivatives ◽  
Harmonic Balance Method ◽  
Forced Oscillations ◽  
Oscillatory Process ◽  
Fractional Oscillator ◽  
Analytical Formulas ◽  
Derivatives Of ◽  
Fractional Orders

Using the harmonic balance method, analytical formulas are obtained for calculating the amplitude-frequency and phase-frequency characteristics, as well as the quality factor of the forced oscillations of a linear fractional oscillator. It was established that the characteristics under study depend on the dissipative properties of the medium - memory effects, which are described by derivatives of fractional orders. It is shown that fractional orders affect the attenuation of the oscillatory process and are associated with its quality factor. The calculated curves of the characteristics of the forced oscillations of a linear linear fractional oscillator showed that fractional orders can be considered as control parameters of the oscillatory process in a dissipative medium. Key words: quality factor, amplitude-frequency characteristic, phase-frequency characteristic, fractional derivatives, memory.

scholarly journals Frequency characteristics of the fractional oscillator Van der Pol

2019 ◽  
Vol 127 ◽  
pp. 02010
Author(s):  
Roman Parovik
Keyword(s):  
Quality Factor ◽  
Harmonic Balance ◽  
Fractional Derivatives ◽  
Harmonic Balance Method ◽  
Oscillatory System ◽  
Frequency Characteristics ◽  
Van Der Pol ◽  
Fractional Oscillator ◽  
Analytical Formulas ◽  
Frequency Phase

Into this paper, the amplitude-frequency and phase-frequency characteristics of the Van der Polar fractional oscillator are studied in order to establish their relationship with the orders of fractional derivatives included in the model equation. Using the harmonic balance method, analytical formulas were obtained for the amplitude-frequency, phase-frequency characteristics, as well as the quality factor – the energy characteristic of the oscillatory system. It was shown that the quality factor depends on the orders of fractional derivatives, and change in their values can lead to both an increase and a decrease in the quality factor.

scholarly journals Mathematical Model of Fractional Duffing Oscillator with Variable Memory

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 2063
Author(s):  
Valentine Kim ◽  
Roman Parovik
Keyword(s):  
Mathematical Model ◽  
Fractional Derivative ◽  
Quality Factor ◽  
Numerical Scheme ◽  
Duffing Oscillator ◽  
Harmonic Balance Method ◽  
Forced Oscillations ◽  
Model Parameters ◽  
Liouville Type ◽  
Fractional Oscillator

The article investigates a mathematical model of the Duffing oscillator with a variable fractional order derivative of the Riemann–Liouville type. The study of the model is carried out using a numerical scheme based on the approximation of the fractional derivative of the Riemann–Liouville type by a discrete analog—the fractional derivative of Grunwald–Letnikov. The adequacy of the numerical scheme is verified using specific examples. Using a numerical algorithm, oscillograms and phase trajectories are constructed depending on the values of the model parameters. Chaotic regimes of the Duffing fractional oscillator are investigated using the Wolf–Bennetin algorithm. The forced oscillations of the Duffing fractional oscillator are investigated using the harmonic balance method. Analytical formulas for the amplitude-frequency, phase-frequency characteristics, and also the quality factor are obtained. It is shown that the fractional Duffing oscillator possesses different modes: regular, chaotic, multi-periodic. The relationship between the order of the fractional derivative and the quality factor of the oscillatory system is established.

scholarly journals Амплитудно-частотные и фазово-частотные характеристики вынужденных колебаний нелинейного дробного осциллятора

2019 ◽  
Vol 45 (13) ◽  
pp. 25
Author(s):  
Р.И. Паровик
Keyword(s):  
Computer Simulation ◽  
Quality Factor ◽  
Fractional Derivatives ◽  
Model Equation ◽  
Oscillatory System ◽  
Forced Oscillations ◽  
Control Parameters ◽  
System A ◽  
Fractional Oscillator

In the work, using the amplitude-frequency (AFC) and phase-frequency characteristics (PFC) of forced oscillations of a non-linear fractional oscillator, their connection with the orders of fractional derivatives, which are included in its model equation, is substantiated. It is shown, using computer simulation, that the orders of fractional derivatives are related to the quality factor of an oscillatory system. A decrease in the higher order (“fractional” inertia) leads to a decrease in the quality factor, and a decrease in the lower order (“fractional” friction) leads to an increase in the quality factor. Therefore, we come to two mechanisms for controlling the Q of the oscillatory system, where the orders of fractional derivatives play the role of control parameters.

scholarly journals STUDY OF THE SINGULAR POINTS OF THE FRACTIONAL OSCILLATOR VAN DER POL-DUFFING

2019 ◽  
pp. 47-54
Author(s):  
Е.Р. Новикова
Keyword(s):  
Asymptotic Stability ◽  
Differential Equations ◽  
Fractional Order ◽  
Fractional Derivatives ◽  
Oscillatory System ◽  
Memory Effects ◽  
Oscillatory Process ◽  
Van Der Pol ◽  
Phase Trajectories ◽  
Fractional Oscillator

В работе проводится исследование на асимптотическую устойчивость точек покоя дробного осциллятора Ван дер ПоляДуффинга. Дробный осциллятор Ван дер Поля Дуффинга представляет собой колебательную систему двух дифференциальных уравнений с производными дробных порядков в смысле ГерасимоваКапуто. Порядки дробных производных характеризуют свойства среды (эффекты памяти), в которой происходит колебательный процесс и могут быть одинаковыми (соизмеримыми) или разными (несоизмеримыми). С помощью теорем для соизмеримой и несоизмеримой систем на конкретных примерах исследуется асимптотическая устойчивость точек покоя дробного осциллятора Ван дер ПоляДуффинга. Результаты исследований были подтверждены с помощью построения соответствующих осциллограмм и фазовых траекторий A study is conducted on the asymptotic stability of the rest points of the fractional oscillator Van der PolDuffing. The fractional van der PolDuffing oscillator is an oscillatory system of two differential equations with fractional order derivatives in the sense of GerasimovCaputo. The orders of fractional derivatives characterize the properties of the medium (memory effects) in which the oscillatory process takes place and can be the same (commensurate) or different (incommensurable). Using theorems for commensurable and incommensurable systems, the asymptotic stability of the rest points of the fractional van der PolDuffing oscillator is investigated with concrete examples. The results of the studies were confirmed by constructing the appropriate waveforms and phase trajectories.

The properties of Bessel-type fractional derivatives and integrals

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.

scholarly journals Time-Domain Analysis of Fractional Electrical Circuit Containing Two Ladder Elements

Electronics ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 475
Author(s):  
Ewa Piotrowska ◽  
Krzysztof Rogowski
Keyword(s):  
Fractional Derivative ◽  
Fractional Derivatives ◽  
Electric Circuit ◽  
Theoretical Models ◽  
Time Domain Analysis ◽  
Electrical Circuit ◽  
State Variable ◽  
State Space System ◽  
Derivatives Of ◽  
Different Order

The paper is devoted to the theoretical and experimental analysis of an electric circuit consisting of two elements that are described by fractional derivatives of different orders. These elements are designed and performed as RC ladders with properly selected values of resistances and capacitances. Different orders of differentiation lead to the state-space system model, in which each state variable has a different order of fractional derivative. Solutions for such models are presented for three cases of derivative operators: Classical (first-order differentiation), Caputo definition, and Conformable Fractional Derivative (CFD). Using theoretical models, the step responses of the fractional electrical circuit were computed and compared with the measurements of a real electrical system.

scholarly journals Fractional Diffusion Equations for Lattice and Continuum: Grünwald-Letnikov Differences and Derivatives Approach

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Vasily E. Tarasov
Keyword(s):  
Fractional Derivatives ◽  
Three Dimensional ◽  
Lattice Models ◽  
Fractional Diffusion ◽  
Lattice Diffusion ◽  
Diffusion Equations ◽  
Difference Operators ◽  
Fractional Diffusion Equations ◽  
Nonlocal Media ◽  
Derivatives Of

Fractional diffusion equations for three-dimensional lattice models based on fractional-order differences of the Grünwald-Letnikov type are suggested. These lattice fractional diffusion equations contain difference operators that describe long-range jumps from one lattice site to another. In continuum limit, the suggested lattice diffusion equations with noninteger order differences give the diffusion equations with the Grünwald-Letnikov fractional derivatives for continuum. We propose a consistent derivation of the fractional diffusion equation with the fractional derivatives of Grünwald-Letnikov type. The suggested lattice diffusion equations can be considered as a new microstructural basis of space-fractional diffusion in nonlocal media.

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