Fractional Order Damping Research Articles

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.

Vibration suppression and P-bifurcation of a randomly excited fractional-order damping system controlled by nonlinear energy sink

Vibration suppression and P-bifurcation of a randomly excited fractional-order damping system controlled by nonlinear energy sink

Quantifying the performance enhancement facilitated by fractional-order implementation of classical control strategies for nanopositioning

For most nanopositioning systems, maximizing positioning bandwidth to accurately track periodic and aperiodic reference signals is the primary performance goal. Closed-loop control schemes are employed to overcome the inherent performance limitations such as mechanical resonance, hysteresis and creep. Most reported control schemes are integer-order and combine both damping and tracking actions. In this work, fractional-order controllers from the positive position feedback family namely: the Fractional-Order Integral Resonant Control (FOIRC), the Fractional-Order Positive Position Feedback (FOPPF) controller, the Fractional-Order Positive Velocity and Position Feedback (FOPVPF) controller and the Fractional-Order Positive, Acceleration, Velocity and Position Feedback (FOPAVPF) controller are designed and analysed. Compared with their classical integer-order implementation, the fractional-order damping and tracking controllers furnish additional design (tuning) parameters, facilitating superior closed-loop bandwidth and tracking accuracy. Detailed simulated experiments are performed on recorded frequency-response data to validate the efficacy, stability and robustness of the proposed control schemes. The results show that the fractional-order versions deliver the best overall performance.

Fractional-Order Models of Damping Phenomena in a Flexible Sensing Antenna Used for Haptic Robot Navigation

In this paper, two types of fractional-order damping are proposed for a single flexible link: internal and external friction, related to the material of the link and the environment, respectively. Considering these dampings, the Laplace transform is used to obtain the exact model of a slewing flexible link by means of the Euler–Bernoulli beam theory. The model obtained is used in a sensing antenna with the aim of accurately describing its dynamic behavior, thanks to the incorporation of the mentioned damping models. Therefore, experimental data are used to identify the damping phenomena of this system in the frequency domain. Welch’s method is employed to estimate the experimental frequency responses. To determine the best damping model for the sensing antenna, a cost function with two weighting forms is minimized for different model structures (i.e., with internal and/or external dampings of integer- and/or fractional-order), and their robustness and fitting performance are analyzed.

Resonance Analysis of Horizontal Nonlinear Vibrations of Roll Systems for Cold Rolling Mills under Double-Frequency Excitations

In this paper, the fractional order differential terms are introduced into a horizontal nonlinear dynamics model of a cold mill roller system. The resonance characteristics of the roller system under high-frequency and low-frequency excitation signals are investigated. Firstly, the dynamical equation of the roller system with a fractional order is established by replacing the normal damping term with a fractional damping term. Secondly, the fast-slow variable separation method is introduced to solve the dynamical equation. The amplitude frequency response characteristics of the system are analyzed. The study finds that there are three equilibrium points. The characteristics of the three equilibrium points and the critical forces causing the bifurcation are investigated. Due to the different orders of the fractional derivatives, various new resonant phenomena are found in the systems with single-well and double-well potentials. Additionally, the double resonance occurs while p = 0.3 or 1.0, and single resonance occurs while p = 1.8. Unlike integer order systems, the critical resonance amplitude of high-frequency signals in fractional order systems depends on the damping strength and is influenced by the fractional order damping. This study provides a broader picture of the vibration characteristics of the roll system for rolling mills.

Suboptimal State Tracking Control Applied to a Nonlinear Fractional-Order Slewing Motion Flexible Structure

Abstract The control of slewing motion flexible structures is important to a number of systems found in engineering and physical sciences applications, such as aerospace, automotive, robotics, and atomic force microscopy. In this kind of system, the controller must provide a stable and well-damped behavior for the flexible structure vibrations, with admissible control signal amplitudes. Recently, many works have used fractional-order derivatives to model complex and nonlinear dynamical behavior present in the mentioned systems. In order to perform digital computer-based control of fractional-order dynamical systems, a time discretization of the equations is necessary. In many cases, the Grünwald–Letnikov method is used, resulting in an implicit integration method. In this work, a nonlinear slewing motion flexible structure is modeled considering a fractional-order viscous damping in the flexible beam motion. To obtain an explicit integration method, based on the Grünwald–Letnikov definition, the discretization of the dynamical equations is performed considering the existence of sample and hold circuits. In addition, real-time suboptimal infinite horizon tracking control system strategies, namely, the linear quadratic tracking and the state-dependent Riccati equation tracking controller, are designed and implemented to control the fractional-order slewing motion flexible system. The general behavior and performance of the control systems are tested for parameter uncertainties related to the order of the fractional derivatives.

Improved Fractional-order Damping Method for Voltage-controlled DFIG System Under Weak Grid

When a doubly-fed induction generator (DFIG) is connected to a weak grid, the coupling between the grid and the DFIG itself will increase, which will cause stability problems. It is difficult to maintain the tracking accuracy and ro-bustness of the phase-locked loop (PLL) in the weak grid, and the risk of instability of the current-controlled DFIG (CC-DFIG) system will increase. In this paper, a new type of voltage-controlled DFIG (VC-DFIG) mode is adopted, which is a grid-forming structure that can independently support the voltage and frequency with a certain adaptability in the weak grid. A small-signal impedance model of the VC-DFIG system is also established. The impedance of DFIG inevitably generates coupling with the grid impedance in the weak grid, especially in parallel compensation grids, and results in resonance. On the basis of the VC-DFIG, impedance stability analysis is performed to study the influences of the control structure and short-circuit ratio. Then, a feedforward damping method is proposed to modify the impedance of the VC-DFIG system at resonance frequencies. The proposed fractional order damping is utilized, which can enhance the robustness and rapidity of resonance suppression under parameter fluctuations. Finally, the experimental results are presented to validate the effectiveness of the proposed control strategy.

Fractional damping enhances chaos in the nonlinear Helmholtz oscillator

The main purpose of this paper is to study both the underdamped and the overdamped dynamics of the nonlinear Helmholtz oscillator with a fractional-order damping. For that purpose, we use the Grunwald–Letnikov fractional derivative algorithm in order to get the numerical simulations. Here, we investigate the effect of taking the fractional derivative in the dissipative term in function of the parameter $$\alpha $$ . Our main findings show that the trajectories can remain inside the well or can escape from it depending on $$\alpha $$ which plays the role of a control parameter. Besides, the parameter $$\alpha $$ is also relevant for the creation or destruction of chaotic motions. On the other hand, the study of the escape times of the particles from the well, as a result of variations of the initial conditions and the undergoing force F, is reported by the use of visualization techniques such as basins of attraction and bifurcation diagrams, showing a good agreement with previous results. Finally, the study of the escape times versus the fractional parameter $$\alpha $$ shows an exponential decay which goes to zero when $$\alpha $$ is larger than one. All the results have been carried out for weak damping where chaotic motions can take place in the non-fractional case and also for a stronger damping (overdamped case), where the influence of the fractional term plays a crucial role to enhance chaotic motions. We expect that these results can be of interest in the field of fractional calculus and its applications.

Nonlinear dynamic analysis of spur gear system based on fractional-order calculus

In this paper, nonlinear dynamic model of spur gear pairs with fractional-order damping under the condition of time-varying stiffness, backlash and static transmission error is established. The general formula of fractional-order damping term is derived by using the incremental harmonic balance method (IHBM), and the approximate analytical solution of the system is obtained by use of the iterative formula. The correctness of the results is verified by comparing with the numerical solutions in the existing literature. The effects of mesh stiffness, internal excitation amplitude and fractional order on the dynamic behavior of the system are analyzed. The results show that changing the fractional order can effectively control the resonance position and amplitude in the meshing process. Both the mesh stiffness and internal excitation can control the collision state and the stability.

On the transient response of plates on fractionally damped viscoelastic foundation

This work underlines the importance of the application of fractional-order derivative damping model in the modelling of the viscoelastic foundation, by demonstrating the effect of various orders of the fractional derivative on the dynamic response of plates resting on the viscoelastic foundation, subjected to concentrated step load. The foundation of the plate is modelled as a fractionally-damped Kelvin–Voigt model. Modal superposition method and Triangular strip matrix approach are used to solve the partial fractional differential equations of motion. The influence of (a) fractional-order derivative, (b) foundation stiffness, and (c) foundation damping viscosity parameter on the dynamic response of the plate are investigated. Theoretical results show that with the increase in the order of derivative, the damping of the system increases, which leads to decreased dynamic response. The results obtained from the fractional-order damping model and integer-order damping model are compared. The results are verified with literature and numerical results (ANSYS).

Nonlinear superharmonic resonance analysis of a nonlocal beam on a fractional visco-Pasternak foundation

This paper investigates the dynamic behavior of a geometrically nonlinear nanobeam resting on the fractional visco-Pasternak foundation and subjected to dynamic axial and transverse loads. The fractional-order governing equation of the system is derived and then discretized by using the single-mode Galerkin discretization. Corresponding forced Mathieu-Duffing equation is solved by using the perturbation multiple time scales method for the weak nonlinearity and by the semi-numerical incremental harmonic balance method for the strongly nonlinear case. A comparison of the results from two methods is performed in the validation study for the weakly nonlinear case and a fine agreement is achieved. A parametric study is performed and the advantages and deficiencies of each method are discussed for order two and three superharmonic resonance conditions. The results demonstrate a significant influence of the fractional-order damping of the visco-Pasternak foundation as well as the nonlocal parameter and external excitation load on the frequency response of the system. The proposed methodology can be used in pre-design procedures of novel energy harvesting and sensor devices at small scales exhibiting nonlinear dynamic behavior.

Dynamic response spectra of fractionally damped viscoelastic beams subjected to moving load

The dynamic response spectra of fractionally damped viscoelastic beams subjected to concentrated moving load are presented. This work quantitatively emphasizes the need for fractional-order damping model in the dynamics of viscoelastic structures, by demonstrating the effect of various orders of fractional derivative damping in beams subjected to concentrated moving loads. The fractional-order Kelvin–Voigt model describes the rheological properties of the viscoelastic material. The Riemann–Liouville fractional derivative of order 0<α ≤ 1 is applied. The complex natural frequencies of viscoelastic beams are studied with respect to α. The forced vibration response is investigated for a wide range of speeds of the moving load, encompassing all practical applications. The results obtained from the fractional-order damping model and the integer-order damping model are compared. With an increase in the order of the fractional derivative, the system damping of the system increases and the dynamic amplification factor (DAF) decreases, especially in the dynamic zone of the sweep parameter. Thus, an integer-order mechanical model over-predicts the damping and under-predicts the dynamic deflections and stresses. The fractional-order of derivative significantly influences the natural frequency, damping coefficient, and DAF. The results are verified with the literature. Communicated by Jie Yang.

Mathematical modeling of a fractionally damped nonlinear nanobeam via nonlocal continuum approach

Modeling of fractionally damped nanostructure is extremely important because of its inherent ability to capture the memory and hereditary effect of several viscoelastic materials extensively used in nanotechnology. The nonlinear free vibration characteristics of a simply-supported nanobeam with fractional-order derivative damping via nonlocal continuum theory are studied in this article. Using Newton’s second law, the equation of motion for the nanobeam embedded in a viscoelastic matrix is derived. The Galerkin method is employed to transform the integro-partial differential equation of motion into a Duffing-type nonlinear ordinary differential equation. The fractional-order damping term is replaced by a combination of linear damping and linear stiffness term. The approximate analytical solution obtained via method of averaging is found to be in good agreement with solution obtained through numerical scheme. Detailed study of system parameters reveals that the fractional-order derivative damping has significant influence on the time response and effective natural frequency of the nanobeam.

Bifurcations in a fractional birhythmic biological system with time delay

This paper presents a bifurcation analysis in a stochastic time-delayed birhythmic oscillator containing fractional derivative. Analytical criterion for birhythmic region is obtained by adopting approximate methods and verified by numerical results. The detailed parameter space study reveals that the rhythmic properties of the oscillator can be efficiently controlled by the fractional order damping. The maximum birhythmic region can be detected in parameter space (α, β) by choosing an approximate fractional order and a larger absolute value of fractional coefficient is conducive to a smaller optimal fractional order. In the stochastic case, the smaller fractional order, the smaller the critical value of time delay and its feedback required for the occurrence of birhythmicity. Similarly, a smaller time delay admits to a smaller value of fractional order and absolute value of fractional coefficient. Astonishingly, fractional derivative and time delay have opposite effects on the direction of the transition. Our study sheds new insight lights on the understanding of nontrivial effects of fractional derivative on a birhythmic oscillator.

Optimal indirect stability of a weakly damped elastic abstract system of second order equations coupled by velocities

In this paper, by means of the Riesz basis approach, we study the stability of a weakly damped system of two second order evolution equations coupled through the velocities (see (1.1)). If the fractional order damping becomes viscous and the waves propagate with equal speeds, we prove exponential stability of the system and, otherwise, we establish an optimal polynomial decay rate. Finally, we provide some illustrative examples.

Dynamic analysis of beams on fractional viscoelastic foundation subject to a variable speed moving load

The present paper investigates the dynamic response of beams resting on fractional viscoelastic foundation subjected to a moving load with variable speeds. The Galerkin with finite difference methods are used to deal with the governing equation of motion. The effect of various parameters, like fractional order derivative, foundation stiffness and damping, speed of moving load on the response of the beam are investigated and discussed.

Fractional-order modeling and dynamic analyses of a bending-torsional coupling generator rotor shaft system with multiple faults

ABSTRACT Unexpected vibrations induced by the crack fault and other unbalance factors in rotor system seriously affect the health and reliability of the generator. Here, to explore the vibration performances, a bending-torsional coupling model of the generator rotor shaft system is established, in which electromagnetic malfunction (unbalanced magnetic pull) and mechanical failures (fractional-order damping, crack and contact-rubbing) are considered. Then, the simulation is done by a modified Adams-Bashforth-Moulton algorithm. Based on the simulation, the correctness of the new coupling model is verified by comparing with previous model and experimental data. At the same time, the new coupling model is analyzed to obtain the dynamic evolutions of the generator rotor shaft system with the changes of crack depth ratio, the fractional order of damping, rotational speed ratio and mass eccentricity of rotor. In addition to this, some critical values and ranges are proposed. Finally, these results can efficiently provide a theoretical reference for the design of generator rotor system and be applied to forecasting and diagnosing vibration faults in generator rotor shaft system.

Analysis of a quintic system with fractional damping in the presence of vibrational resonance

In the present paper, the phenomenon of the vibrational resonance in a quantic oscillator that possesses a fractional order damping and is driven by both the low and the high frequency periodic signals is investigated, and the approximate theoretical expression of the response amplitude at the low-frequency is obtained by utilizing the method of direct partition of motions. Based on the definition of the Caputo fractional derivative, an algorithm for simulating the system is introduced, and the new method is shown to have higher precision and better feasibility than the method based on the Grünwald –Letnikov expansion. Due to the order of the fractional derivative, various new resonance phenomena are found for the system with single-well, double-well, and triple-well potential, respectively. Moreover, the value of fractional order can be treated as a bifurcation parameter, through which, it is found that the slowly-varying system can be transmitted from a bistability system to a monostabillity one, or from tristability to bistability, and finally to monostabillity. Unlike the cases of the integer-order system, the critical resonance amplitude of the high-frequency signal in the fractional system does depend on the damping strength and can be significantly affected by the fractional-order damping. The numerical results given by the new method is found to be in good agreement with the analytical predictions.

The resonant behavior in the oscillator with double fractional-order damping under the action of nonlinear multiplicative noise

We study stochastic resonance (SR) in an oscillator with nonlinear noise, fractional-order external damping, and fractional-order intrinsic damping. Using a moment equation, we derive the exact analytical expression of the output amplitude and find that fluctuations in the output amplitude are non-monotonic. Using numerical simulations we verify the accuracy of this analytical result. We find (i) that nonlinear noise plays a key role in system behavior and that the resonance of the output amplitude is diverse when there is nonlinear noise, (ii) that the order of the fractional-order damping strongly impacts resonant intensity and that the impact on resonant intensity of fractional-order external damping is opposite that of fractional-order intrinsic damping, and (iii) that the evolution of the output amplitude versus the frequency of the external periodic force exhibits three behaviors: a resonance with one peak, a resonance with one peak and one valley, and a resonance with one valley.