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 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 Анализ вынужденных колебаний дробного осциллятора

2019 ◽  
Vol 45 (1) ◽  
pp. 34 ◽  
Author(s):  
А.В. Псху ◽  
С.Ш. Рехвиашвили
Keyword(s):  
Differential Equation ◽  
Fractional Derivative ◽  
Frequency Dependence ◽  
Classical Model ◽  
Forced Oscillations ◽  
Viscous Damping ◽  
Q Factor ◽  
Fractional Oscillator ◽  
The Relationship ◽  
Good Agreement

AbstractA model of forced oscillations of an oscillator based on the fractional integro-differential formalism has been considered. It is shown that this model is in good agreement with the classical model of forced oscillations of an oscillator with viscous damping. The parameters of the frequency dependence of stationary oscillations of a fractional oscillator are calculated, and the relationship between the order of fractional derivative in the differential equation of oscillations and the Q factor of the system is determined.

scholarly journals The application of fractional derivatives through Riemannliouville approach to xanthan gum viscoelasticity

2018 ◽  
Vol 7 (4) ◽  
pp. 2633
Author(s):  
Ira Sumiati ◽  
Endang Rusyaman ◽  
Betty Subartini ◽  
Sukono . ◽  
Ruly Budiono
Keyword(s):  
Mathematical Model ◽  
Fractional Derivative ◽  
Xanthan Gum ◽  
Fractional Derivatives ◽  
Complex Modulus ◽  
Concentrate Solution ◽  
Loss Modulus ◽  
Model Parameters ◽  
Good Ability ◽  
The Mathematical Model

Fractional derivatives are derivative with non-integer order, one of which is used for mathematical modeling of viscoelasticity. In this research, the fractional derivative was used to obtain a mathematical model of viscoelasticity. The method used was a fractional derivative through the Riemann-Liouville approach. The mathematical model of viscoelasticity obtained was a complex modulus consisting of storage and loss modulus. This model was applied to xanthan gum concentrate solution 0.5%, 1.0%, 2.0%, 3.0%, and 4.0% with simplified model parameters. The results obtained that the storage and loss modulus increased with increasing concentration of the solution. In addition, the modulus storage was always greater than the modulus loss for all concentrations of the solution. This suggests that the elastic properties of the xanthan gum solution are more dominant than their viscosity properties for all concentrations. Therefore, the viscoelasticity model using Riemann-Liouville fractional derivatives has a good ability to investigate the viscoelasticity behavior of all xanthan gum concentrations.  

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.

Dynamic Stability of Hovercraft in Heave

1970 ◽  
Vol 37 (4) ◽  
pp. 895-900 ◽  
Author(s):  
H. J. Davies ◽  
G. A. Poland
Keyword(s):  
Mathematical Model ◽  
Dynamic Behavior ◽  
Dynamic Stability ◽  
Equilibrium Position ◽  
Forced Oscillation ◽  
Dynamic Equilibrium ◽  
Forced Oscillations ◽  
Two Dimensional ◽  
Oscillation Characteristics ◽  
Nonlinear Nature

The regimes of flow governing the dynamic behavior of a two-dimensional mathematical model of an edge-jet Hovercraft in heaving motion are described and the equations associated with such regimes derived. Both the free and forced-oscillation characteristics are studied. The nonlinear nature of the system manifests itself, in the case of the forced oscillations, as a shift in the dynamic equilibrium position resulting in a loss of mean hoverheight.

scholarly journals The identification of a device based on a Peltier element by the least squares method

2020 ◽  
pp. 17-27
Author(s):  
Vladimir Grinkevich ◽  
Keyword(s):  
Mathematical Model ◽  
Least Squares ◽  
Least Squares Method ◽  
Control Object ◽  
Object Identification ◽  
Model Parameters ◽  
Peltier Element ◽  
Transport Delay ◽  
Non Linear ◽  
The Mathematical Model

The evaluation of the mathematical model parameters of a non-linear object with a transport delay is considered in this paper. A temperature controlled stage based on a Peltier element is an identification object in the paper. Several input signal implementations are applied to the input of the identification object. The least squares method is applied for the calculation of the non-linear differential equitation parameters which describe the identification object. The least squares method is used due to its simplicity and the possibility of identification non-linear objects. The parameters values obtained in the process of identification are provided. The plots of temperature changes in the temperature control system with a controller designed based on the mathematical model of the control object obtained as a result of identification are shown. It is found that the mathematical model obtained in the process of identification may be applied to design controllers for non-linear systems, in particular for a temperature stage based on a Peltier element, and for self-tuning controllers. However, the least square method proposed in the paper cannot estimate the transport delay time. Therefore it is required to evaluate the time delay by temperature transient processes. Dynamic object identification is applied when it is required to obtain a mathematical model structure and evaluate the parameters by an input and output control object signal. Also, identification is applied for auto tuning of controllers. A mathematical model of a control object is required to design the controller which is used to provide the required accuracy and stability of control systems. Peltier elements are applied to design low-power and small- size temperature stage . Hot benches based on a Peltier element can provide the desired temperature above and below ambient temperature.

scholarly journals A mathematical model of common-cold epidemics on Tristan da Cunha

1971 ◽  
Vol 69 (3) ◽  
pp. 423-433 ◽  
Author(s):  
B. J. Hammond ◽  
D. A. J. Tyrrell
Keyword(s):  
Mathematical Model ◽  
Common Cold ◽  
Average Duration ◽  
Model Parameters ◽  
Epidemic Threshold ◽  
Tristan Da Cunha ◽  
Simulation Techniques ◽  
Time Courses ◽  
The Mathematical Model ◽  
Computer Simulation Techniques

SUMMARYRecords of seven common-cold outbreaks on the island of Tristan da Cunha are compared with the corresponding time courses given by the mathematical model of Kermack & McKendrick (1927) and with an alternative model that directly involves a constant average duration of individual infection. Using computer simulation techniques the latter model is shown to be preferred and is then closely matched to the field data to obtain values for the model parameters. Consideration is then given to the intensity of epidemics predicted by the model and to the distribution of the actual epidemics relative to the theoretical epidemic threshold.

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