Sliding Mode Control for Two-Degree-Of-Freedom Fractional Zener Oscillator

Fractional-order derivatives provide a powerful tool for the characterization of mechanical properties of viscoelastic materials. A fractional oscillator refers to mechanical model of viscoelastically damped structures, of which the viscoelastic damping is described by constitutive equations involving fractional-order derivatives. This paper proposes active control of vibration in a two-degree-of-freedom fractional Zener oscillator utilizing sliding mode technique. Firstly, with a state transformation, the fractional differential equations of motion are equivalently transformed into a relatively simple form. Meanwhile, a virtual fractional oscillator is generated, which is further used to analyze the original oscillator. Then, the stored energy in the two fractional derivative terms is derived based on the diffusive model of fractional integrator. Thus, the total mechanical energy in the virtual oscillator is determined as the sum of the kinetic energy, the potential energy and the fractional energy. Furthermore, sliding mode control of vibration in the fractional Zener oscillator is designed, of which the Lyapunov function is chosen as the total mechanical energy. Finally, numerical simulations are conducted to validate the effectiveness of the proposed controllers.

Approximate Solutions of Nonlinear Pendulum with Fractional Damping

Because of many real problems are better characterized using fractional-order models, fractional calculus has recently become an intensively developing area of calculus not only among mathematicians but also among physicists and engineers as well. Fractional oscillator and fractional damped structure have attracted the attention of researchers in the field of mechanical and civil engineering [1-6]. This study is dedicated mainly a pendulum with fractional viscous damping. The mathematic model of pendulum is a cubic nonlinear equation governing the oscillations of systems having a single degree of freedom, via Riemann-Liouville fractional derivative. The method of multiple scales is performed to solve the equation by assigning the nonlinear and damping terms to the ε-order. Finally, the effects of the coefficient of a fractional damping term on the approximate solution are observed.

Solving Coupled Fractional Differential Equations Using Differential Transform Method.

This paper presents the solution of coupled equations which are of fractional order using differential transform method. In this paper we extend the scope of differential transform method to system of fractional differential equations so that we get the analytical solutions. The coupled fractional differential equations of a physical system, namely, coupled fractional oscillator with some applications is given via differential transform method. Here we introduce the solution of coupled oscillation of equal fractional order which can be enhanced to unequal fractional order.

A Practical Fractional-Order Sinusoidal Oscillator Design and Implementation

In this study, a new sinusoidal fractional oscillator circuit in which the fractional low-pass filter and fractional differentiator form a closed loop has been introduced. The single series [Formula: see text] pair and Valsa method have been used to imitate the fractional-order capacitor. Two different sinusoidal signals with different phases are available at output ports. An integer version of the proposed oscillator is not possible, only the fractional version is valid, that is a noticeable feature. The introduced circuit has been simulated by employing SPICE software. The proposed fractional oscillator is also verified by implementing circuit with AD817ANs and passive components, experimentally.

Излучение дробного осциллятора

The spectral energy density of the oscillator radiation is calculated in the dipole approximation. The oscillator motion is described by an equation with fractional integro-differentiation. The fractional oscillator model can describe various types of radiation, including those with a nonexponential relaxation law. Found the shape of the spectral line of radiation. The obtained result is compared with the classical Lorentzian spectrum and experimental emission spectra of monochromatic and phosphor LEDs. The order of fractional integro-differentiation in the model sets the magnitude of the broadening of the radiation spectrum.

The periodic response of a fractional oscillator with a spring-pot and an inerter-pot

The fractional oscillation system with two Weyl-type fractional derivative terms $_{ - \infty }D_t^\beta x$ (0 < β < 1) and $_{ - \infty }D_t^\alpha x$ (1 < α < 2), which portray a “spring-pot” and an “inerter-pot” and contribute to viscoelasticity and viscous inertia, respectively, was considered. At first, it was proved that the fractional system with constant coefficients under harmonic excitation is equivalent to a second-order differential system with frequency-dependent coefficients by applying the Fourier transform. The effect of the fractional orders β (0 < β < 1) and α (1 < α < 2) on inertia, stiffness and damping was investigated. Then, the harmonic response of the fractional oscillation system and the corresponding amplitude–frequency and phase–frequency characteristics were deduced. Finally, the steady-state response to a general periodic incentive was obtained by utilizing the Fourier series and the principle of superposition, and the numerical examples were exhibited to verify the method. The results show that the Weyl fractional operator is extremely applicable for researching the steady-state problem, and the fractional derivative is capable of describing viscoelasticity and portraying a “spring-pot”, and also describing viscous inertia and serving as an “inerter-pot”.

Ordinary Differential Equation with Left and Right Fractional Derivatives and Modeling of Oscillatory Systems

We consider the principle of least action in the context of fractional calculus. Namely, we derive the fractional Euler–Lagrange equation and the general equation of motion with the composition of the left and right fractional derivatives defined on infinite intervals. In addition, we construct an explicit representation of solutions to a model fractional oscillator equation containing the left and right Gerasimov–Caputo fractional derivatives with origins at plus and minus infinity. We derive a representation for the composition of the left and right derivatives with origins at plus and minus infinity in terms of the Riesz potential, and introduce special functions with which we give solutions to the model fractional oscillator equation with a complex coefficient. This approach can be useful for describing dissipative dynamical systems with the property of heredity.

Mathematical Model of Fractional Duffing Oscillator with Variable Memory

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.