Fractional Damping Research Articles

Asymptotic behavior of a logarithmic-viscoelastic wave equation with internal fractional damping

Asymptotic behavior of a logarithmic-viscoelastic wave equation with internal fractional damping

Polynomial decay of the energy of solutions of coupled wave equations with locally boundary fractional dissipation

In this paper, we investigate a system of coupled wave equations featuring boundary fractional damping applied to a portion of the domain. We first establish the well-posedness of the system, proving the existence and uniqueness of solutions through semi-group theory. While the system does not exhibit exponential stability, we demonstrate its strong stability. Furthermore, leveraging Arendt and Batty’s general criteria and certain geometric conditions, we prove a polynomial rate of energy decay for the solutions.

Resonance and attraction domain analysis of asymmetric duffing systems with fractional damping in two degrees of freedom

This article establishes a two degree of freedom asymmetric Duffing system model with fractional damping through the study of the Jeffcott rotor system with horizontal support, and analyzes the dynamic behavior of its resonance and attraction domain structures. The second-order approximate solution of the system equation is obtained using a multiscale method, and the slow flow modulation equations of both the amplitudes and phases are extracted. The comparison between approximate and numerical solutions has shown their high consistency and the Routh Hurwitz criterion is utilized to study the stability of the system. Finally, we explore the effect of fractional damping on the vibration behavior of the system using amplitude frequency curves, attractive watersheds, and time trajectory plots, including both negative and positive values of the nonlinear stiffness coefficient. The analysis shows that if the coefficient and order of the fractional damping term are reasonably selected, the horizontal and vertical displacements of the system vibration can be effectively controlled. The importance of this work lies in its application in rotor vibration, which has significant implications for the study of the system under investigation.

Response of MDOF nonlinear system with fractional derivative damping and driven by fractional Gaussian noise

The integration of fractional calculus into stochastic dynamics has introduced two innovative concepts, i.e., fractional derivative damping (FDD) and fractional Gaussian noise (fGn). To date, most studies on fractional stochastic dynamics have exclusively focused on either FDD or fGn, rather than concurrently exploring both. It is the first attempt for this paper to investigate a multi-degree-of-freedom (MDOF) nonlinear system with FDD and driven by fGn at the same time. Employing the generalized harmonic balance technique, FDD is equivalently decomposed into a quasi-linear restoring force and a quasi-linear damping force. Consequently, the original random Lagrangian system with FDD is transformed into the stochastically excited and dissipated Hamiltonian system. Utilizing the characteristic of the relatively flat power spectral density of fGn, it is approximated as wideband noise. This enables the application of the stochastic averaging method for quasi-integrable Hamiltonian system under wideband noise. Two detailed examples are carried out to provide the technical procedures and numerical verifications.

Random flutter analysis of a novel binary airfoil with fractional order viscoelastic constitutive relationship

The fractional order model emerges naturally when the integer order model fails to meet practical requirements and finds extensive applications in engineering practice. However, there is still limited research on the random flutter of fractional order viscoelastic binary airfoils. This study aims to explore the random flutter of such airfoils under the influence of Gaussian white noise excitation by utilizing the pth moment Lyapunov exponent. Firstly, a two-degree-of-freedom fractional order stochastic dynamic equation is established for the viscoelastic binary airfoil through the introduction of fractional order damping. The singular perturbation method is then employed to derive the pth moment Lyapunov exponent of the system. An approximate analytical solution of the pth moment Lyapunov exponent is obtained through Fourier series expansion, which exhibits good agreement with numerical results obtained from Monte Carlo simulations. Subsequently, a comprehensive analysis is conducted to examine the impacts of fractional order, airflow velocity, noise intensity, and key structural parameters on the stochastic flutter of the fractional order viscoelastic binary airfoil. The findings indicate that compared with the typical binary airfoil model, the flutter speed of the fractional-order viscoelastic binary airfoil is significantly increased, the stability of the system is highly sensitive to the natural frequency of the system, and the fractional-order factor has a positive effect on the stochastic stability of the binary airfoil before the critical value. The above research results can guide the design optimization of the fractional-order viscoelastic airfoil.

Advanced Analytical and Numerical Studies on Coupled Schrödinger Equations with Fractional Order Damping

This paper embarks on a thorough analytical and numerical exploration of coupled Schrödinger equations under the influence of fractional order damping mechanisms. By integrating fractional damping, which introduces memory effects and non-local dissipative interactions, into the coupled Schrödinger framework, we aim to dissect and understand the nuanced dynamics that govern these complex quantum systems. The research delves into the mathematical underpinnings, stability characteristics, and the dynamical behaviors that emerge from the intricate balance between quantum coupling and fractional damping effects. Through a blend of analytical rigor and sophisticated numerical simulations, this study unveils new insights into the complex interplay among quantum entanglement, dissipation, and non-linear dynamics, offering potential implications for quantum computing, optical systems, and beyond.

Analytical Study of Coupled Schrödinger Equations with Fractional Damping: Decay Solutions and Stability Analysis

Coupled Schrödinger equations with fractional damping represent a complex yet fascinating area of study in quantum mechanics. This research paper delves into the analytical investigation of such systems, focusing on deriving decay solutions and conducting stability analysis. By combining theoretical frameworks with numerical methods, we explore the behavior of these coupled equations under varying parameters and conditions. Through a thorough analysis, we aim to deepen our understanding of the dynamics and stability properties of quantum systems subject to fractional damping, with potential implications for diverse fields ranging from quantum mechanics to condensed matter physics.

Phase control of escapes in the fractional damped Helmholtz oscillator

We analyze the nonlinear Helmholtz oscillator in the presence of fractional damping, a characteristic feature in several physical situations. In our specific scenario, as well as in the non-fractional case, for large enough excitation amplitudes, all initial conditions are escaping from the potential well. To address this, we incorporate the phase control technique into a parametric term, a feature commonly encountered in real-world situations. In the non-fractional case it has been shown that, a phase difference of ϕOPT≈π, is the optimal value to avoid the escapes of the particles from the potential well. Here, our investigation focuses on understanding when particles escape, considering both the phase difference ϕ and the fractional parameter α as control parameters. Our findings unveil the robustness of phase control, as evidenced by the consistent oscillation of the optimal ϕ value around its non-fractional counterpart when varying the fractional parameter. Additionally, our results underscore the pivotal role of the fractional parameter in governing the proportion of bounded particles, even when utilizing the optimal phase.

Non-Linear Resonances and Chaotic Dynamics of a Rotating Gyroscope Under a Fractional Order Derivative Damping

Non-Linear Resonances and Chaotic Dynamics of a Rotating Gyroscope Under a Fractional Order Derivative Damping

Well-Posedness and $$L^2$$-Decay Estimates for the Navier–Stokes Equations with Fractional Dissipation and Damping

Well-Posedness and $$L^2$$-Decay Estimates for the Navier–Stokes Equations with Fractional Dissipation and Damping

Rich phenomenology of the solutions in a fractional Duffing equation

In this paper, we characterize the chaos in the Duffing equation with negative linear stiffness and a fractional damping term given by a Caputo fractional derivative of order α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document} ranging from 0 to 2. We use two different numerical methods to compute the solutions, one of them new. We discriminate between regular and chaotic solutions by means of the attractor in the phase space and the values of the Lyapunov Characteristic Exponents. For this, we have extended a linear approximation method to this equation. The system is very rich with distinct behaviours. In the limits α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document} to 0 or α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document} to 2, the system tends to basically the same undamped system with a behaviour clearly different from the classical Duffing equation.

Fractional damping induces resonant behavior in the Duffing oscillator

The interaction between the fractional order parameter and the damping parameter can play a relevant role for introducing different dynamical behaviors in a physical system. Here, we study the Duffing oscillator with a fractional damping term. Our findings show that for certain values of the fractional order parameter, the damping parameter, and the forcing amplitude high oscillations amplitude can be induced. This phenomenon is due to the appearance of a resonance in the Duffing oscillator only when the damping term is fractional.

Stochastic bifurcation and dynamic reliability analyses of nonlinear MDOF vehicle system with generalized fractional damping via DPIM

Stochastic bifurcation and dynamic reliability analyses of nonlinear MDOF vehicle system with generalized fractional damping via DPIM

Quasi-stability and Upper Semicontinuity for Coupled Wave Equations with Fractional Damping

Quasi-stability and Upper Semicontinuity for Coupled Wave Equations with Fractional Damping

Subharmonic resonance and fixed-range asymptotic stability of the fractional-order SD oscillator

The smooth and discontinuous (SD) oscillator is a typical system with strong nonlinear characteristics, and it is widely used in low-frequency vibration isolation and energy harvesting. A fractional damping model denoted by the Caputo model is introduced into the SD oscillator to adjust the property of the secondary resonance and evaluate the stability of the system. The influence of the fractional damping model on the one-third subharmonic resonance and the fixed-range asymptotic stability is studied. Residue theory and the Laplace transform are used to solve the fractional damping model. The amplitude–frequency response function and the existence conditions are derived by means of the averaging method. Lyapunov theory is used to determine the stable criteria of steady-state solutions. The cell-mapping method is ameliorated and used to calculate the fixed-range asymptotic stability of the one-third subharmonic resonance. The main results are as follows: a gap in the excitation amplitude occurs in the region of the existence condition of the one-third subharmonic resonance when the smooth parameter is smaller than 1. The generation of one-third subharmonic resonance is totally avoided for all frequencies when the excitation amplitudes are within the gap. The width of the gap, as well as the amplitude of the one-third subharmonic resonance, is affected by the parameters of the fractional damping term. The fixed-range asymptotic stability of the one-third subharmonic resonance is weak when the fractional damping parameters are large, which indicates a low resistance of the one-third subharmonic resonance to the external disturbance. The tuning effects of the fractional damping model on the one-third subharmonic resonance and fixed-range asymptotic stability are beneficial for the applications of SD oscillators.

Fractional Mathieu Differential Equations in Dynamic Stability of Piles

Piles are commonly used in Civil and Mechanical Engineering to provide support to superstructures and to transmit loads to deeper ground layers. During earthquakes, pile instability can result in the collapse of the entire structure. In this paper, the pile is modeled as a column surrounded by Winkler soil foundations with fractional damping. Investigation of the axially loaded pile leads to a fractional Mathieu differential equation of motion. The Bolotin method, employing harmonic balance, is proposed to obtain the approximate instability boundaries of the pile in the stability diagrams. A practical example is presented to conduct a parametric study on pile instability concerning the fractional order. The study observes parametric resonance in the first order, representing the principal instability region, which requires special attention to maintain stability. Furthermore, increasing the fractional damping order leads to a higher critical dynamic load and a slight reduction in the critical frequency ratio in each instability region. Higher-order instability regions exhibit greater sensitivity to changes in the fractional order. These results provided insights into the stability of piles under earthquake conditions.

Blow-up dynamics in nonlinear coupled wave equations with fractional damping and external source

<p>This study focused on a coupled nonlinear wave equation system featuring fractional damping and polynomial source terms within a bounded domain. We demonstrated that, under specific conditions, the solutions exhibited blow-up in a finite time-frame.</p>

Analysis of the stationary probability density of a generalized and bistable van der Pol system excited by colored noise

The stochastic P-bifurcation behavior of bi-stability in a generalized van der Pol oscillator with the fractional damping under colored noise and thermal excitation is investigated. Firstly, using the principle of minimal mean square error and lin-earization method, the non-linear stiffness terms can be equivalent to a linear stiffness which is a function of the system amplitude, and the original system is simplified to an equivalent integer order van der Pol system. Secondly, the system amplitude stationary probability density function is obtained by the stochastic averaging, and then based on the singularity theory, the critical parametric con-ditions for the system amplitude stochastic P-bifurcation are found. Finally, the types of the stationary probability density function of the system amplitude are qualitatively analyzed in each area divided by the transition set curves. The con-sistency between the analytical results and the numerical results acquired from Monte-Carlo simulation also testifies the theoretical analysis in this paper and the method used in this paper can directly guide the design of the fractional order controller to adjust the response of the system.

Nonlinear logarithmic wave equations: Blow-up phenomena and the influence of fractional damping, infinite memory, and strong dissipation

Nonlinear logarithmic wave equations: Blow-up phenomena and the influence of fractional damping, infinite memory, and strong dissipation

Global existence and blow up of solution of wave nonlinear equation with boundary fractional damping and logarithmic source terms

Global existence and blow up of solution of wave nonlinear equation with boundary fractional damping and logarithmic source terms