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Research Papers

Stationary Response of Multidegree-of-Freedom Strongly Nonlinear Systems to Fractional Gaussian Noise

[+] Author and Article Information
Qiang Feng Lü, Mao Lin Deng

Department of Mechanics,
Zhejiang University,
Hangzhou 310027, China;
State Key Laboratory of Fluid Power and
Mechatronic Systems,
Zhejiang University,
Hangzhou 310027, China;
Key Laboratory of Soft Machines and
Smart Devices of Zhejiang Province,
Zhejiang University,
Hangzhou 310027, China

Wei Qiu Zhu

Department of Mechanics,
Zhejiang University,
Hangzhou 310027, China;
State Key Laboratory of Fluid Power and
Mechatronic Systems,
Zhejiang University,
Hangzhou 310027, China;
Key Laboratory of Soft Machines and
Smart Devices of Zhejiang Province,
Zhejiang University,
Hangzhou 310027, China
e-mail: wqzhu@zju.edu.cn

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received April 22, 2017; final manuscript received July 25, 2017; published online August 18, 2017. Editor: Yonggang Huang.

J. Appl. Mech 84(10), 101001 (Aug 18, 2017) (14 pages) Paper No: JAM-17-1213; doi: 10.1115/1.4037409 History: Received April 22, 2017; Revised July 25, 2017

The stationary response of multidegree-of-freedom (MDOF) strongly nonlinear system to fractional Gaussian noise (FGN) with Hurst index 1/2 < H < 1 is studied. First, the system is modeled as FGN-excited and -dissipated Hamiltonian system. Based on the integrability and resonance of the associated Hamiltonian system, the system is divided into five classes: partially integrable and resonant, partially integrable and nonresonant, completely integrable and resonant, completely integrable and nonresonant, and nonintegrable. Then, the averaged fractional stochastic differential equations (SDEs) for five classes of quasi-Hamiltonian systems with lower dimension and involving only slowly varying processes are derived. Finally, the approximate stationary probability densities and other statistics of two example systems are obtained by numerical simulation of the averaged fractional SDEs to illustrate the application and compared with those from original systems to show the advantages of the proposed procedure.

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References

Figures

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Fig. 1

Samples of processes I1,I2,H3,Φ, Θ2,Q3,Q4,P4 of system (43)

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Fig. 2

stationary probability densities p(I1), p(H3) of system (43) under FGN or GWN excitation. Solid line is obtained from the method in Ref. [20]. Other curves are obtained from the method proposed in this paper.

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Fig. 3

Contour of stationary probability densities p(I1,I2), p(I1,H3) of system (43): (a), (c) simulated from fractional SDEs (47); (b), (d) simulated from original system (43)

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Fig. 4

Stationary probability densities p(I1),p(I2),p(H3) of system (43)

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Fig. 5

Mean-values E[I1],E[I2],E[H3] of system (43) as a function of Hurst index H

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Fig. 6

Mean-square values E[I12],E[I22],E[H32] of system (43) as a function of Hurst index H

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Fig. 7

Contour of stationary probability densities p(q1,q2),p(q1,q3) of system (43): (a), (c) simulated from fractional SDEs (47); (b), (d) simulated from original system (43)

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Fig. 8

Stationary probability densities p(q1), p(q2), p(q3), p(q4) of system (43)

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Fig. 9

Stationary probability density of phase difference p(ϕ) of system (43)

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Fig. 10

Mean-square values E[Q12],E[Q22],E[Q32],E[Q42] of system (43) as a function of Hurst index H

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Fig. 11

The autocorrelation function R(τ) of FGN WH(t) with Hurst index 0.5, 0.6, 0.8, and 0.98

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Fig. 12

Samples of FGN with different Hurst index 0.5, 0.6, 0.8, and 0.98

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Fig. 13

Autocorrelation function RQ3(τ) of response Q3(t) of system (43) under FGN excitation with different Hurst index 0.5, 0.9, and 0.98. Part of (a) (i.e., 30<τ<100) are zoomed in (b).

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Fig. 14

Sample functions of processes I1,I2,H3,Φ of system (43) under FGN excitation with different Hurst index 0.5, 0.8, 0.98, and 1

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Fig. 15

The mechanical model of system (58)

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Fig. 16

The Kaimal spectrum and the spectral density of FGN with parameters: v*2=0.0264, z=10 m, v¯(z)=40 m/s, and Hurst=0.973

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Fig. 17

Stationary probability densities p(H) of system (58)

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Fig. 18

Stationary probability densities p(q1), p(q2) of system (58)

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Fig. 19

Sample of X(t) generated from Eqs. (A5) and (A6), respectively, where α(X,t)=−γX+D/ω2X, β(X,t)=D/ωX with parameter γ=0.03,D=0.1,ω=1

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