Primary resonance of Duffing oscillator with two kinds of fractional-order derivatives

Shen, Yongjun; Yang, Shaopu; Xing, Haijun; Ma, Hui

doi:10.1016/j.ijnonlinmec.2012.06.012

Cited by 95 publications

(51 citation statements)

References 31 publications

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“…Duffing oscillator is one of the most common and typical models in nonlinear dynamical systems. Among these nonlinear systems the well-known Duffing oscillator is quite suitable to model the large deformation structure in many physics and engineering fields [7][8][9][10]. Moreover, time delay is an unavoidable and very common problem when those systems were controlled.…”

Section: Introductionmentioning

confidence: 99%

Heteroclinic Bifurcation Behaviors of a Duffing Oscillator with Delayed Feedback

Wen

Chen

Guo

2018

Shock and Vibration

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The heteroclinic bifurcation and chaos of a Duffing oscillator with forcing excitation under both delayed displacement feedback and delayed velocity feedback are studied by Melnikov method. The Melnikov function is analytically established to detect the necessary conditions for generating chaos. Through the analysis of the analytical necessary conditions, we find that the influences of the delayed displacement feedback and delayed velocity feedback are separable. Then the influences of the displacement and velocity feedback parameters on heteroclinic bifurcation and threshold value of chaotic motion are investigated individually. In order to verify the correctness of the analytical conditions, the Duffing oscillator is also investigated by numerical iterative method. The bifurcation curves and the largest Lyapunov exponents are provided and compared. From the analysis of the numerical simulation results, it could be found that two types of period-doubling bifurcations occur in the Duffing oscillator, so that there are two paths leading to the chaos in this oscillator. The typical dynamical responses, including time histories, phase portraits, and Poincare maps, are all carried out to verify the conclusions. The results reveal some new phenomena, which is useful to design or control this kind of system.

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Section: Introductionmentioning

confidence: 99%

Heteroclinic Bifurcation Behaviors of a Duffing Oscillator with Delayed Feedback

Wen

Chen

Guo

2018

Shock and Vibration

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“…The first one is to simply add fractional-order derivative term into the original integerorder system, so as to establish a fractional-order system. For example, Shen et al [12][13][14][15][16] studied several linear and nonlinear fractional-order oscillators by the averaging method or incremental harmonic balance method and found that the fractional-order derivatives had both damping and stiffness effects on the dynamical response in those oscillators. Chen et al [17,18] studied the response of some nonlinear fractional-order oscillator under Gaussian white noise excitation.…”

Section: Introductionmentioning

confidence: 99%

Primary Resonance of van der Pol Oscillator under Fractional-Order Delayed Feedback and Forced Excitation

Chen

Tang

et al. 2017

Shock and Vibration

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The primary resonance of van der Pol oscillator under fractional-order delayed negative feedback and forced excitation is studied. Firstly, the approximate analytical solution is obtained based on the averaging method, and it could be found that the fractionalorder delayed feedback has not only the property of delayed velocity feedback but also that of delayed displacement feedback. Moreover, the amplitude-frequency equation for the steady-state solution is established, and its stability conditions are also obtained. Then, the results of the approximate analytical solution and numerical integration are compared and analyzed. The agreement between the two methods is very high, so that the correctness and accuracy of the approximate analytical solution are verified. Finally, the effects of all the parameters in the fractional-order delayed feedback on the amplitude-frequency curves are analyzed. It could be concluded that fractional-order delayed feedback has important influences on the dynamical behavior of van der Pol oscillator, which is very significant to the optimization and control of a similar system.

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“…[7][8][9][10][11] In recent years, the field of the fractional-order derivative has attracted interest in several areas including physics, chemistry, engineering, and even finance and social sciences. [12][13][14][15][16][17] It has received an increasing interest due to the fact that fractional-order derivative could reflect a lot of natural phenomena more reasonably.…”

Section: Introductionmentioning

confidence: 99%

“…Recently, Shen et al [14][15][16]26 obtained approximately analytical solutions of some fractionalorder dynamical systems based on averaging method, and they verified that fractional-order derivative should be considered as a damping and stiffness factor simultaneously in dynamical system. Optimal control theory for the integer-order isolation system had acquired more mature development, and the implementation methods for optimal control had been provided in some literatures.…”

Section: Introductionmentioning

confidence: 99%

“…It is difficult to deal with fractionalorder derivative when studying fractional-order system by optimal control theory. Based on the heuristic level in Shen et al, [14][15][16]26 the fractional-order derivative is studied in section ''Processing of fractional-order derivative.'' The optimal control aim is to minimize the displacement and velocity amplitude with different excitation frequency.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Optimal design for fractional-order active isolation system

Huh

Shen

Yang

2015

Advances in Mechanical Engineering

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The optimal control of fractional-order active isolation system is researched based on the optimal control theory, and the effect of fractional-order derivative on passive isolation system is also analyzed. The mechanical model is established where viscoelastic features of isolation materials are described by fractional-order derivative. The viscoelastic property of the fractional-order derivative in dynamical system is studied and the fractional-order derivative could be divided into linear stiffness and linear damping. It is found that both the fractional coefficient and the fractional order could affect not only the resonance amplitude through the equivalent linear damping coefficient but also the resonance frequency by the equivalent linear stiffness. Based on optimal control theory, the feedback gain of fractional-order active isolation system under harmonic excitation is obtained, which is changed with the excitation frequency. The statistical responses of the displacement and velocity for passive and active vibration isolation systems subjected to random excitation are also presented, which further verifies the excellent performance of fractional-order derivative in vibration control engineering.

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Primary resonance of Duffing oscillator with two kinds of fractional-order derivatives

Cited by 95 publications

References 31 publications

Heteroclinic Bifurcation Behaviors of a Duffing Oscillator with Delayed Feedback

Heteroclinic Bifurcation Behaviors of a Duffing Oscillator with Delayed Feedback

Primary Resonance of van der Pol Oscillator under Fractional-Order Delayed Feedback and Forced Excitation

Optimal design for fractional-order active isolation system

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