Stationary response of Duffing oscillator with hardening stiffness and fractional derivative

Chen, Lincong; Wang, Weihua; Li, Zhongshen; Zhu, Wei

doi:10.1016/j.ijnonlinmec.2012.08.001

Cited by 80 publications

(47 citation statements)

References 35 publications

(35 reference statements)

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“…The above says that the equivalent oscillator (59), as well as the fractional oscillator (58), never oscillates at ω → ∞ for 1 < α < 2 because its damping is infinitely large in that case. Due to…”

Section: Remarkmentioning

confidence: 98%

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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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Section: Remarkmentioning

confidence: 98%

“…According to the Newton's second law, the inertia force in the system of the fractional oscillator (58) corresponds to the first item on the left side of its equivalent system (59). That is,…”

Section: Theorem 2 (Equivalent Mass I)mentioning

confidence: 99%

Three Classes of Fractional Oscillators

2018

Symmetry

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“…This means that the memory property of economic control can magnify the amplitude of the economy system. This indicates that the memory property of economic control actually plays a negative role in making the economy system under control, and we can compute the effect of memory property by Equation (31). So the policy maker can make effective economic policy to make the economy under control.…”

Section: The Effect Of the Fractional-ordermentioning

confidence: 99%

Study on the Business Cycle Model with Fractional-Order Time Delay under Random Excitation

Lin

et al. 2017

Entropy

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Time delay of economic policy and memory property in a real economy system is omnipresent and inevitable. In this paper, a business cycle model with fractional-order time delay which describes the delay and memory property of economic control is investigated. Stochastic averaging method is applied to obtain the approximate analytical solution. Numerical simulations are done to verify the method. The effects of the fractional order, time delay, economic control and random excitation on the amplitude of the economy system are investigated. The results show that time delay, fractional order and intensity of random excitation can all magnify the amplitude and increase the volatility of the economy system.

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“…Huang and Jin [16] employed the stochastic averaging and generalized harmonic functions procedure for a SDOF strongly nonlinear oscillator under Gaussian white noise excitations; they found that the order of fractional derivative has a significant effect on the response of system. Response of a strong nonlinear oscillator [17], its reliability function [18], and its stochastic/asymptotic stability [19,20] have been studied by Chen et al They could separate fractional derivative into the equivalent quasilinear dissipative force and quasilinear restoring force [21]. Yang et al studied the stationary and stochastic response of nonlinear system with fractional derivative under white Gaussian noise input [22,23].…”

Section: Introductionmentioning

confidence: 99%

Dynamic Analysis of a Plate on the Generalized Foundation with Fractional Damping Subjected to Random Excitation

Hosseinkhani

Younesian

Farhangdoust

2018

Mathematical Problems in Engineering

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Stochastic response of a plate on the generalized foundation driven by random excitation is solved in this paper. Governing differential equation is obtained by employing the Galerkin method. The generalized harmonic function technique is applied to the governing equation of motion. Using the stochastic averaging method (SAM), the system is approximated by the time homogeneous diffusive Markov process. Corresponding approximate stationary probability function is achieved by solving associated FokkerPlank-Kolmogorov (FPK). An analytical solution is presented for the stationary probability of the amplitude and velocity. Validity of the stationary probability is verified by Monte-Carlo simulation. Parametric study is carried out to investigate effects of foundation parameters and excitation intensity on the stationary probability function. It is found that the fractional properties act similar to the foundation stiffness and damping and can be employed as a new control parameter for the support design.

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Stationary response of Duffing oscillator with hardening stiffness and fractional derivative

Cited by 80 publications

References 35 publications

Three Classes of Fractional Oscillators

Three Classes of Fractional Oscillators

Study on the Business Cycle Model with Fractional-Order Time Delay under Random Excitation

Dynamic Analysis of a Plate on the Generalized Foundation with Fractional Damping Subjected to Random Excitation

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