Responses of Duffing oscillator with fractional damping and random phase

Xu, Yong; Li, Yongge; Liu, Di; Jia, Wantao; Huang, Hui

doi:10.1007/s11071-013-1002-9

Cited by 68 publications

(21 citation statements)

References 23 publications

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“…Based on what have discussed in Section 2, the equivalent stochastic system associated with system (34) is governed bÿ (35) in which…”

Section: Examplementioning

confidence: 99%

“…Di Paola et al [32] numerically investigated the stochastic response of a linear viscoelastic system under stationary and non-stationary random excitations by discretizing the fractional derivative operator and increasing the system dimension, in which the key idea has been utilized by Failla and Pirrotta [33] to estimate the stochastic response of fractionally damped Duffing oscillators subjected to a stochastic input. Xu and Li [34,35] put forward a new approach combining the L-P method and the multiple-scale method to obtain the response of stochastic oscillator with fractional derivative. Liu [36] investigated the principal resonance responses of SDOF systems with small fractional derivative damping subjected to narrow-band random parametric excitation by using multiple scale method.…”

Section: Introductionmentioning

confidence: 99%

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Stochastic response of a class of self-excited systems with Caputo-type fractional derivative driven by Gaussian white noise

Yang

et al. 2015

Chaos, Solitons & Fractals

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“…Based on what have discussed in Section 2, the equivalent stochastic system associated with system (34) is governed bÿ (35) in which…”

Section: Examplementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Stochastic response of a class of self-excited systems with Caputo-type fractional derivative driven by Gaussian white noise

Yang

et al. 2015

Chaos, Solitons & Fractals

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“…one says that f 1 (t) = f 2 (t), (45) in the sense of Fourier transform (Gelfand and Vilenkin [67], Papoulis [76]), implying…”

Section: Equivalence Of Functions In the Sense Of Fourier Transformmentioning

confidence: 99%

“…The above implies that a null function as a difference between f 1 (t) and f 2 (t) is allowed for (45). An example relating to oscillation theory is the unit step function.…”

Section: Equivalence Of Functions In the Sense Of Fourier Transformmentioning

confidence: 99%

Three Classes of Fractional Oscillators

2018

Symmetry

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This article addresses three classes of fractional oscillators named Class I, II and III. It is known that the solutions to fractional oscillators of Class I type are represented by the Mittag-Leffler functions. However, closed form solutions to fractional oscillators in Classes II and III are unknown. In this article, we present a theory of equivalent systems with respect to three classes of fractional oscillators. In methodology, we first transform fractional oscillators with constant coefficients to be linear 2-order oscillators with variable coefficients (variable mass and damping). Then, we derive the closed form solutions to three classes of fractional oscillators using elementary functions. The present theory of equivalent oscillators consists of the main highlights as follows. (1) Proposing three equivalent 2-order oscillation equations corresponding to three classes of fractional oscillators; (2) Presenting the closed form expressions of equivalent mass, equivalent damping, equivalent natural frequencies, equivalent damping ratio for each class of fractional oscillators; (3) Putting forward the closed form formulas of responses (free, impulse, unit step, frequency, sinusoidal) to each class of fractional oscillators; (4) Revealing the power laws of equivalent mass and equivalent damping for each class of fractional oscillators in terms of oscillation frequency; (5) Giving analytic expressions of the logarithmic decrements of three classes of fractional oscillators; (6) Representing the closed form representations of some of the generalized Mittag-Leffler functions with elementary functions. The present results suggest a novel theory of fractional oscillators. This may facilitate the application of the theory of fractional oscillators to practice.

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“…Gorenflo and Abdel-Rehim [12], Jumarie [13], Ishteva et al [14], and Malinowska and Torres [15], respectively studied the definitions and numerical methods of fractional-order calculus for Grünwald-Letnikov, Riemann-Liouville, and Caputo. Xu et al [16] combined Lindstedt-Poincare method and multiscale method to study fractional-order Duffing oscillator to harmonic excitation with random phase and analyzed the stochastic jump and bifurcation in the oscillator. Li et al [17][18][19] have done a lot of researches in the mathematical theory of fractional-order calculus and established some efficient numerical algorithms.…”

Section: Introductionmentioning

confidence: 99%

Subharmonic Resonance of Van Der Pol Oscillator with Fractional-Order Derivative

Shen

Wei

Sui

et al. 2014

Mathematical Problems in Engineering

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The subharmonic resonance of van der Pol (VDP) oscillator with fractional-order derivative is studied by the averaging method. At first, the first-order approximate solutions are obtained by the averaging method. Then the definitions of equivalent linear damping coefficient (ELDC) and equivalent linear stiffness coefficient (ELSC) for subharmonic resonance are established, and the effects of the fractional-order parameters on the ELDC, the ELSC, and the dynamical characteristics of system are also analysed. Moreover, the amplitude-frequency equation and phase-frequency equation of steady-state solution for subharmonic resonance are established. The corresponding stability condition is presented based on Lyapunov theory, and the existence condition for subharmonic resonance (ECSR) is also obtained. At last, the comparisons of the fractional-order and the traditional integer-order VDP oscillator are fulfilled by the numerical simulation. The effects of the parameters in fractional-order derivative on the steady-state amplitude, the amplitude-frequency curves, and the system stability are also studied.

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Responses of Duffing oscillator with fractional damping and random phase

Cited by 68 publications

References 23 publications

Stochastic response of a class of self-excited systems with Caputo-type fractional derivative driven by Gaussian white noise

Stochastic response of a class of self-excited systems with Caputo-type fractional derivative driven by Gaussian white noise

Three Classes of Fractional Oscillators

Subharmonic Resonance of Van Der Pol Oscillator with Fractional-Order Derivative

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