Fractional Derivative Damping Research Articles

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

Numerical investigation of rotor-bearing systems with fractional derivative material damping models

Numerical investigation of rotor-bearing systems with fractional derivative material damping models

Bifurcations analysis of a multiple attractors energy harvesting system with fractional derivative damping under random excitation

The multiple attractors energy harvester, which can convert mechanical energy into electrical energy, is a typical birhythmic system. This paper investigates the stochastic bifurcation of the system with fractional derivative damping under different noise excitations. Variable transformation and conversion mechanism for decoupling the electromechanical equations are utilized to obtain the approximate equivalent system. Probability density functions of the system are generated via the stochastic averaging method. Numerical results are presented to verify the effectiveness of the method. In the deterministic case, the fractional order can effectively control the birhythmic properties. In the stochastic case, the effects of fractional derivative damping and different noises on the stochastic P-bifurcations of the system are discussed separately. It concludes that appropriately changing noise parameters and fractional derivative parameters can increase the harvested energy from vibrations. This study highlights the theoretical contribution in energy harvesting and hopes to provides ideas for optimizing the energy harvesting performance.

Stationary Response of a Kind of Nonlinear Stochastic Systems with Variable Mass and Fractional Derivative Damping

Viscoelasticity and variable mass are common phenomena in Micro-Electro-Mechanical Systems (MEMS), and could be described by a fractional derivative damping and a stochastic process, respectively. To study the dynamic influence cased by the viscoelasticity and variable mass, stationary response of a kind of nonlinear stochastic systems with stochastic variable-mass and fractional derivative, damping is investigated in this paper. Firstly, an approximately equivalent system of the studied nonlinear stochastic system is presented according to the Taylor expansion technique. Then, based on stochastic averaging of energy envelope, the corresponding Fokker–Plank–Kolmogorov (FPK) equation is deduced, which gives an approximated analytical solution of stationary response. Finally, a nonlinear oscillator with variable mass and fractional derivative damping is proposed in numerical simulations. The approximated analytical solution is compared with Monte Carlo numerical solution, which could verify the effectiveness of the obtained results.

Stochastic analysis of a nonlinear energy harvester with fractional derivative damping

Stochastic analysis of a nonlinear energy harvester with fractional derivative damping

The effect of fractional damping and time-delayed feedback on the stochastic resonance of asymmetric SD oscillator

This paper proposes the stiffness nonlinearities and asymmetric SD (smooth and discontinuous) oscillator under time-delayed feedback control with the fractional derivative damping. With the effect of displacement and velocity feedback, the oscillator can be adjusted to the desired vibration state, and then the stochastic resonance (SR) is achieved. This article discusses the contribution of various system parameters and time-delayed feedback to SR, especially which induced by fractional order. It should be noted that this paper provides effective guidance for fault diagnosis and weak signal detection, energy harvesting, vibration isolation and vibration reduction.

Stochastic bifurcation analysis of a friction-damped system with impact and fractional derivative damping

Stochastic bifurcation analysis of a friction-damped system with impact and fractional derivative damping

A method for the experimental identification of equivalent viscoelastic models from vibration of thin plates

Aim of the present study is the identification of equivalent viscoelastic models for layered thin walled structures, obtained from vibration measurement only, able to fit the experimental data on a relatively wide frequency range by means of a minimum number of parameters. A novel approach is proposed, based on a definition of an equivalent modal damping ratio applied to the circle-fit technique, to overcome the difficulties related to the identification of modal parameters when adopting non-conventional viscoelastic models. When the structural internal dissipative effects are dominant, this procedure identifies the parameters of an equivalent Young’s modulus in the frequency domain, representing the viscoelastic properties of a homogenized structure as a scalar function with frequency-dependent real and imaginary parts. The proposed procedure is applied to the analysis of Aluminum plates coated by damping pads and of plates made by Quiet Aluminum. To fit the experimentally found equivalent modal damping ratios, several viscoelastic models are adopted and compared (viscous, hysteretic, generalized Maxwell, fractional derivative damping, and in particular the Fractional Kelvin-Voigt model), assessing the accuracy of the identified parameters by comparison of numerically simulated with experimentally measured frequency response functions.

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.

The generalized differential transform method for solution of a free vibration linear differential equation with fractional derivative damping

Journal of Applied Mathematics and Computational Mechanics, Prace Naukowe Instytutu Matematyki i Informatyki, Politechnika Częstochowska, Scientific Research of the Institute of Mathematics and Computer Science, Czestochowa University of Technology

Stochastic analysis of monostable vibration energy harvesters with fractional derivative damping under Gaussian white noise excitation

To the best of authors’ knowledge, the dynamical behaviors of vibration energy harvesters with fractional derivative damping have not been discussed by researchers with the help of the stochastic averaging method. As the fractional-order models are more accurate than the classical integer-order models, so it is necessary to investigate the dynamical behaviors of fractional vibration energy harvesters. This paper aims to investigate the stochastic response of monostable vibration energy harvesters with fractional derivative damping under Gaussian white noise excitation. First, we can get the equivalent stochastic system with the help of variable transformation. Then, the approximately analytical solutions of the equivalent stochastic system can be obtained by the stochastic averaging method. Third, the numerical results are considered as the benchmark to prove the effectiveness of the proposed method. The results indicate that the proposed method has a satisfactory level of accuracy. We also discuss the effect of system parameters on the mean square voltage.

Stochastic stability of Duffing oscillator with fractional derivative damping under combined harmonic and Poisson white noise parametric excitations

The stochastic stability of a Duffing oscillator with fractional derivative damping under combined harmonic and Poisson white noise parametric excitations is investigated. A stochastic averaging method and the Khasminskii’s procedure are applied to evaluate the largest Lyapunov exponent. Then the asymptotic Lyapunov stability with probability one of the original system is determined approximately by using the largest Lyapunov exponent. Finally, the analytical results are verified by those from the Monte Carlo simulation of the original system.

Response of non-linear oscillator driven by fractional derivative term under Gaussian white noise

This paper aimed to investigate the stationary response of non-linear system with fractional derivative damping term under Gaussian white noise excitation. The corresponding Fokker–Plank–Kolmogorov (FPK) equation can be deduced by utilizing the stochastic averaging method and Stratonovich–Khasminskii theorem in the first place. And then we can solve the FPK equation to obtain the stationary probability densities (SPDs) of amplitude, which in fact can be used to describe the response of system. Furthermore, the analytical results coincide with the Monte Carlo results. Finally, one found that reducing fractional derivative order is able to enhance the response of system and increasing fractional coefficient can weaken the response of system. So the fractional derivative damping term has a great effect on the response of Duffing-Van der Pol oscillator. In addition, the response can also be influenced by other system parameters.

Circuit implementation of a piezoelectric buckled beam and its response under fractional components considerations

In this paper, an analog testing circuit and determinist averaging method for a vibration energy harvesting system with fractional derivative and nonlinear damping under a sinusoidal vibration source is proposed in order to predict the system response and its stability. The objective of this paper is to show that there is a possibility to make a pre-experimental design of the structure by using analog circuit and discussing the performance of a system with fractional derivative. Bifurcation diagram, poincare maps and power spectral density are provided to deeply characterize the dynamic of the system. These results are corroborated by using 0–1 test. By using the Melnikov method, we find the necessary condition for which homoclinic bifurcation occurs. Understanding and predicting this bifurcation is very judicious in the energy harvesting field because it may lead to different types of motion in the perturbed system. The appearance of chaotic vibrations increases the frequency’s bandwidth of the harvester thereby, allowing to harvest more energy. The pre-experimental investigation is carried out through appropriate software electronic circuit (Multisim®). The corresponding electronic circuit is designed exhibiting transient to chaos in accord with numerical simulations. The impact of fractional derivatives is presented upon the power generated by the system. In addition, by combining the harmonic force and a random excitation, the stochastic resonance appears, giving rise to large amplitude of vibration and consequently, enhancing the performance of the system. The results obtained in this work show the interest of using the electronic circuit to make the experiment analysis of the physical structure and also, the effects of the use of piezoelectric material exhibiting fractional properties in this research field.

Stochastic P-bifurcation analysis of a fractional smooth and discontinuous oscillator via the generalized cell mapping method

The smooth and discontinuous oscillator with fractional derivative damping under combined harmonic and random excitations is investigated in this paper. The short memory principle is introduced so that the evolution process of the oscillator with fractional derivative damping can be described by the Markov chain. Then the stochastic generalized cell mapping method is used to obtain the steady-state probability density functions of the response. The stochastic response and bifurcation of the oscillator with fractional derivative damping are discussed in detail. We found that both the smoothness parameter, the noise intensity, the amplitude and frequency of the harmonic force can induce the occurrence of stochastic P-bifurcation in the system. Monte Carlo simulation verifies the effectiveness of the method we adopt in the paper.

Nonlinear analysis and analog simulation of a piezoelectric buckled beam with fractional derivative

In this article, an analog circuit for implementing fractional-order derivative and a harmonic balance method for a vibration energy harvesting system under pure sinusoidal vibration source is proposed in order to predict the system response. The objective of this paper is to discuss the performance of the system with fractional derivative and nonlinear damping ( $\mu_{b}$ ). Bifurcation diagram, phase portrait and power spectral density (PSD) are provided to deeply characterize the dynamics of the system. These results are corroborated by the 0-1 test. The appearance of the chaotic vibrations reduces the instantaneous voltage. The pre-experimental investigation is carried out through appropriate software electronic circuit (Multisim). The corresponding electronic circuit is designed, exhibiting periodic and chaotic behavior, in accord with numerical simulations. The impact of fractional derivative and nonlinear damping is presented with detail on the output voltage and power of the system. The agreement between numerical and analytical results justifies the efficiency of the analytical technique used. In addition, by combining the harmonic excitation with the random force, the stochastic resonance phenomenon occurs and improves the harvested energy. It emerges from these results that the order of fractional derivative μ and nonlinear damping $\mu_{b}$ play an important role in the response of the system.

Response of strongly nonlinear vibro-impact system with fractional derivative damping under Gaussian white noise excitation

This paper aims to investigate the strongly nonlinear vibro-impact dynamic system with fractional derivative damping under Gaussian white noise excitation, and this system considers both nonlinear factors and non-smooth factors. With the help of non-smooth transformation, the original system is rewritten as a smooth system, which is an easier handled form. For the altered function, we obtain the approximate stationary solutions analytically by generalized stochastic averaging method. To prove the validity of the approximate analytical methods, an efficient scheme with high accuracy is adopt to simulate the fractional derivative, then the fourth-order Runge–Kutta approach is used to obtain the numerically response statistics. Meanwhile the analytical solutions are verified by the numerical simulation solutions. At last, we research the responses of this system. In this context, we can consider the influences to this system caused by the fractional order, the restitution coefficient and the noise intensity through changing the values of corresponding parameters. The observation and investigation state that fractional derivative term, impact conditions and stochastic excitation can influence the responses of the fractional vibro-impact dynamic system.

Hard-Spring Bistability and Effect of System Parameters in a Two-Degree-of-Freedom Vibration System with Damping Modeled by a Fractional Derivative

The harmonic balance coupled with the continuation algorithm is a well-known technique to follow the periodic response of dynamical system when a control parameter is varied. However, deriving the algebraic system containing the Fourier coefficients can be a highly cumbersome procedure, therefore this paper introduces polynomial homotopy continuation technique to investigate the steady state bifurcation of a two-degree-of-freedom system including quadratic and cubic nonlinearities subjected to external and parametric excitation forces under a nonlinear absorber. The fractional derivative damping is considered to examine the effects of different fractional order, linear and nonlinear damping coefficients on the steady response. By means of polynomial homotopy continuation, all the possible steady state solutions are derived analytically, i.e. without numerical integration. Coexisting periodic solutions, saddle-node bifurcation and various effects of fractional damping on the steady state response are found and investigated. It is shown that the fractional derivative order and damping coefficient change the bifurcating curves qualitatively and eliminate the saddle-node bifurcation during resonance. Moreover, the system response depicts bigger and bigger region of hard-spring bistability with increasing fractional derivative order, but the region of hard-spring bistability of steady response becomes gradually small and then disappears when we increase the linear and nonlinear damping coefficients. In addition, the analytical results are verified by comparison with the numerical integration ones, it can be found that the present approximate resonance responses are in good agreement with numerical ones.

Stochastic responses of Van der Pol vibro-impact system with fractional derivative damping excited by Gaussian white noise.

This paper focuses on the study of the stochastic Van der Pol vibro-impact system with fractional derivative damping under Gaussian white noise excitation. The equations of the original system are simplified by non-smooth transformation. For the simplified equation, the stochastic averaging approach is applied to solve it. Then, the fractional derivative damping term is facilitated by a numerical scheme, therewith the fourth-order Runge-Kutta method is used to obtain the numerical results. And the numerical simulation results fit the analytical solutions. Therefore, the proposed analytical means to study this system are proved to be feasible. In this context, the effects on the response stationary probability density functions (PDFs) caused by noise excitation, restitution condition, and fractional derivative damping are considered, in addition the stochastic P-bifurcation is also explored in this paper through varying the value of the coefficient of fractional derivative damping and the restitution coefficient. These system parameters not only influence the response PDFs of this system but also can cause the stochastic P-bifurcation.

A method to stochastic dynamical systems with strong nonlinearity and fractional damping

In this paper, a new technique is proposed to deal with strongly nonlinear stochastic systems with fractional derivative damping and random harmonic excitation. Combining the advantages of Linstedt–Poincare (L–P) method and multiple scales method, introducing a different frequency expansion form and a time transformation, a series of perturbation equations is obtained according to the powers of parameter. Then, we eliminate the secular producing terms to solve the perturbation equations to derive the second-order approximate solution. Furthermore, the steady-state frequency–amplitude function in deterministic case is analyzed, and the first-order and second-order steady-state moments of the amplitude are also discussed in the presence of random harmonic excitation. In order to explore the effectiveness of the proposed approximate method, two classical examples were proposed to verify the theoretical results through numerical simulations. Especially, the method can be used to investigate some types of extremely strong odd nonlinear terms via the discussions of each example.