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Research Papers

# Responses of Linear and Nonlinear Oscillators to Fractional Gaussian Noise With Hurst Index Between 1/2 and 1

[+] Author and Article Information
Mao Lin Deng

Department of Mechanics,
State Key Laboratory of Fluid Power
Transmission and Control,
Zhejiang University,
Hangzhou 310027, China

Wei Qiu Zhu

Department of Mechanics,
State Key Laboratory of Fluid Power
Transmission and Control,
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 2, 2015; final manuscript received July 2, 2015; published online July 22, 2015. Editor: Yonggang Huang.

J. Appl. Mech 82(10), 101008 (Jul 22, 2015) Paper No: JAM-15-1180; doi: 10.1115/1.4031009 History: Received April 02, 2015

## Abstract

The responses of linear and nonlinear oscillators to fractional Gaussian noise (fGn) are studied. First, some preliminary concepts and properties of fractional Brownian motion (fBm) and fGn with Hurst index $1/2 are introduced. Then, the exact sample solution, correlation function, spectral density, and mean-square value of the response of linear oscillator to fGn are obtained. Based on the sample solution, it is proved that the long-range correlation index of displacement response of linear oscillator is the same as that of excitation fGn, i.e., $2-2H$, while the velocity response has no such long-range correlation. An interesting discovery is that the ratio of kinetic energy to total energy decreases as increasing Hurst index $H$. Finally, for the responses of one and two degrees-of-freedom (DOF) nonlinear oscillators to fGn, the equivalent linearization method is applied to obtain the sample functions, correlation functions and mean-square values of the responses. Plenty of digital simulation results are obtained to support these solutions. It is shown that the approximate solution is effective for weakly nonlinear oscillators and it is feasible to apply the equivalent linearization to study multi-DOF weakly nonlinear oscillators.

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## Figures

Fig. 1

Mean-square displacements E[X2] and mean-square velocity E[V2] of oscillator (27). The system parameters are k=2, D=0.1, γ=2ξk. (a) Analytical solution for mean-square E[X2] from Eq. (29); (b) analytical solution for mean-square E[V2] from Eq. (31).

Fig. 2

Ratio of kinetic energy to total energy rK (a) and ratio of potential energy to total energy rP (b) with different damping ratio ξ. The system parameters are the same as those in Fig. 1.

Fig. 3

Plot of ratio rK and rP against damping ratio ξ with different Hurst index H. The system parameters are the same as those in Fig. 2.

Fig. 5

Mean-square of displacements E[X2] and velocity E[V2] of oscillator (44). The system parameters are k=2, D=0.1, λ=0.75k. Analytical solutions come from Eq. (51).

Fig. 4

Autocorrelation function RX(τ) of response X(t) of oscillator (27). The system parameters are k=2, D=0.1, γ=0.5. Part of data in (a) (i.e., 0.1< τ < 600) are zoomed in and show in (b). The analytical results come from Eq. (39).

Fig. 6

The influence of nonlinear parameter λ on mean-square displacement E[X2]. H=0.75. The other system parameters are the same as Fig. 5.

Fig. 7

Two sample functions calculated from Eq. (35)X(t)|k→ke and simulated from Eq. (27). The system parameters are k=2, D=0.1, γ=k, λ=3k, H=0.75, initial state [X(0),V(0)]T=[0,0]T.

Fig. 8

Autocorrelation function RX(τ)|k→ke of response X(t) of nonlinear oscillator (44). Where RX(τ)|k→ke means RX(τ) is determined in Eq. (39) and k is replaced by ke in Eq. (50). k=2, D=0.1, γ=k, and H=0.75.

Fig. 9

Mean-square of displacements E[X12] and E[X22] of oscillator (52). The system parameters are k1=k2=2, ks=0.5, D=0.1, λ1=0.75k1, λ2=0.5k2. The analytical solutions come from Eq. (55).

Fig. 10

The influence of nonlinear parameter λ1 on mean-square displacement E[X12]. λ2=0, H=0.75. The other system parameters are the same as Fig. 9.

Fig. 11

Two sample functions calculated from Eq. (21)X1(t)|C→Ce,K→Ke and simulated from Eq. (52). k1=k2=2, ks=0.5, D=0.1, λ1=3k1, λ2=2k2, γ=k1, H=0.75, initial state [X1(0),V1(0),X2(0),V2(0)]T=0.

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