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

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.

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“…Using references, 51–53 such term can be substituted by a linear damping force and a restoring force as follows …”

Section: Nanobeam Model Based On Nonlocal Continuum Approachmentioning

confidence: 99%

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

Jha

Dasgupta

2019

Proceedings of the Institution of Mechanical Engineers, Part C:

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“…Substituting 5and 13into (9), and after some mathematical procedure, the equation governing the evolution of ( ) can be obtaineḋ…”

Section: Equivalent Systemmentioning

confidence: 99%

“…Hu et al [8] investigated stationary response of a strongly nonlinear oscillator with fractional derivative damping under bounded noise excitation. Yang et al [9] estimated stationary response of a nonlinear system with Caputo-type fractional derivative under Gaussian white noise. These three works are all achieved on the basis of stochastic averaging method [10].…”

Section: Introductionmentioning

confidence: 99%

Stochastic Response of Dynamical Systems with Fractional Derivative Term under Wide-Band Excitation

Zhao

2016

Mathematical Problems in Engineering

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Transient solution of a fractional stochastic dynamical system under wide-band noise excitation is investigated. Generalized Harmonic Balance technique is firstly used to approximate restoring force of the given system as an amplitude-dependent form. In this way, stochastic averaging method then can be applied to transform the system into an Ito differential equation. Furthermore, the fractional derivative in the integral-differential form can be equivalent to a combination of periodic functions after the averaging procedure. As the following, Galerkin method therein is utilized to obtain the transient probability density functions by solving associated Fokker-Planck-Kolmogorov (FPK) equation. As an example, the Rayleigh oscillator is studied to illustrate the efficiency and accuracy of the proposed approaches. Numerical results show that exact stationary solution and transient solution derived from Galerkin method are in good agreement with those from Monte Carlo Simulation.

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“…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]. Failla and Pirrotta presented a numerical method to calculate the response of a system under stochastic excitation based on the discretization of fractional derivative [24].…”

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

Cited by 48 publications

References 36 publications

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

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

Stochastic Response of Dynamical Systems with Fractional Derivative Term under Wide-Band Excitation

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

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