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Research Papers

Response of Fractional Oscillators With Viscoelastic Term Under Random Excitation

[+] Author and Article Information
Yong Xu

Department of Applied Mathematics,
Northwestern Polytechnical University,
Xi'an 710072, China
e-mail: hsux3@nwpu.edu.cn

Yongge Li

Department of Applied Mathematics,
Northwestern Polytechnical University,
Xi'an 710072, China
e-mail: liyonge@163.com

Di Liu

Department of Applied Mathematics,
Northwestern Polytechnical University,
Xi'an 710072, China
e-mail: di-lau@hotmail.com

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received September 5, 2013; final manuscript received November 19, 2013; published online February 13, 2014. Assoc. Editor: J. A. Tenreiro Machado.

J. Comput. Nonlinear Dynam 9(3), 031015 (Feb 13, 2014) (8 pages) Paper No: CND-13-1213; doi: 10.1115/1.4026068 History: Received September 05, 2013; Revised November 19, 2013

A system with fractional damping and a viscoelastic term subject to narrow-band noise is considered in this paper. Based on the revisit of the Lindstedt–Poincaré (LP) and multiple scales method, we present a new procedure to obtain the second-order approximate analytical solution, and then the frequency–amplitude response equations in the deterministic case and the first- and second-order steady-state moments in the stochastic case are derived theoretically. Numerical simulation is applied to verify the effectiveness of the proposed method, which shows good agreement with the analytical results. Specially, we find that the new method is valid for strongly nonlinear systems. In addition, the influences of fractional order and the viscoelastic parameter on the system are explored, and the results indicate that the steady-state amplitude will increase at a fixed point with the increase of fractional order or viscoelastic parameter. At last, stochastic jump is investigated via the received Fokker–Planck–Kolmogorov (FPK) equation to compute the stationary solution of probability density functions with its shape changing from one peak to two peaks with the increase of noise intensity, and the phenomena of stochastic jump is consistent with the solution of frequency–amplitude response equations.

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Figures

Grahic Jump Location
Fig. 1

Sketch map of a viscoelastic oscillator model

Grahic Jump Location
Fig. 2

Amplitude frequency responses: (a) mean moments and (b) second-order moments. ɛ = 0.1,μ = 5.0,α = 0.8,κ = 3.0,ω0 = 0.8,δ = 0.1,γ = 20.0,h = 0.05. Analytical results: ——, numerical results: ▴♦▪.

Grahic Jump Location
Fig. 3

Amplitude frequency responses: (a) mean moments and (b) second-order moments. ɛ = 0.1,μ = 5.0,β = 0.3,κ = 3.0,ω0 = 0.8,δ = 0.3,γ = 10.0,h = 0.05. Analytical results: ——, numerical results: ▴♦⋆▪.

Grahic Jump Location
Fig. 4

Amplitude frequency responses: (a) mean moments and (b) second-order moments. ɛ = 0.1,μ = 5.0,α = 0.5,β = 0.3,κ = 3.0,ω0 = 0.8,γ = 10.0,h = 0.05. Analytical results: ——, numerical results: ▴♦▪.

Grahic Jump Location
Fig. 5

Stationary probability density figures: ɛ = 0.1,μ = 5.0,α = 0.8,β = 0.9,κ = 3.0,ω0 = 0.8,δ = 0.1,γ = 10.0,Ω = 1.75. (a) h = 0.07, (b) h = 0.12, (c) h = 0.16, and (d) h = 0.20.

Grahic Jump Location
Fig. 6

Time history: ɛ = 0.1,μ = 5.0,α = 0.2,β = 0.3,κ = 3.0,ω0 = 0.8,δ = 0.3,γ = 10.0,Ω = 1.75. (a) h=0.2 and (b) h=0.3.

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