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.

References

References
1.
Zhu
,
W. Q.
,
2006
, “
Nonlinear Stochastic Dynamics and Control in Hamiltonian Formulation
,”
ASME Appl. Mech. Rev.
,
59
(
4
), pp.
230
248
.
2.
Zeng
,
Y.
, and
Zhu
,
W. Q.
,
2011
, “
Stochastic Averaging of Quasi Non-Integrable Hamiltonian Systems Under Poisson White Noise Excitation
,”
ASME J. Appl. Mech.
,
78
(
2
), p.
021002
.
3.
Liu
,
W. Y.
, and
Zhu
,
W. Q.
,
2015
, “
Lyapunov Function Method for Analyzing Stability of Quasi-Hamiltonian Systems Under Combined Gaussian and Poisson White Noise Excitations
,”
Nonlinear Dyn.
,
81
(
4
), pp.
1879
1893
.
4.
Jia
,
W. T.
,
Zhu
,
W. Q.
,
Xu
,
Y.
, and
Liu
,
W. Y.
,
2013
, “
Stochastic Averaging of Quasi-Integrable and Resonant Hamiltonian Systems Under Combined Gaussian and Poisson White Noise Excitations
,”
ASME J. Appl. Mech.
,
81
(
4
), p.
041009
.
5.
Xu
,
Y.
,
Guo
,
R.
,
Liu
,
D.
,
Zhang
,
H. Q.
, and
Duan
,
J. Q.
,
2014
, “
Stochastic Averaging Principle for Dynamical Systems With Fractional Brownian Motion
,”
AIMS Discrete Continuous Dyn. Syst., Ser. B
,
19
(
4
), pp.
1197
1212
.
6.
Xu
,
Y.
,
Duan
,
J. Q.
, and
Xu
,
W.
,
2011
, “
An Averaging Principle for Stochastic Dynamical Systems With Lévy Noise
,”
Physica D
,
240
(
17
), pp.
1395
1401
.
7.
Calif
,
R.
, and
Schmitt
,
F. G.
,
2012
, “
Modeling of Atmospheric Wind Speed Sequence Using a Lognormal Continuous Stochastic Equation
,”
J. Wind Eng. Ind. Aerodyn.
,
109
, pp.
1
8
.
8.
Guo
,
W. J.
,
Wang
,
Y. X.
,
Xie
,
M. X.
, and
Cui
,
Y. J.
,
2009
, “
Modeling Oil Spill Trajectory in Coastal Waters Based on Fractional Brownian Motion
,”
Mar. Pollut. Bull.
,
58
(
9
), pp.
1339
1346
.
9.
Sliusarenko
,
O. Y.
,
Gonchar
,
V. Y.
,
Chechkin
,
A. V.
,
Sokolov
,
I. M.
, and
Metzler
,
R.
,
2010
, “
Kramers-Like Escape Driven by Fractional Gaussian Noise
,”
Phys. Rev. E.
,
81
(
4
), p.
041119
.
10.
Ai
,
B.
,
He
,
Y.
, and
Zhong
,
W.
,
2010
, “
Transport in Periodic Potentials Induced by Fractional Gaussian Noise
,”
Phys. Rev. E.
,
82
(
6
), p.
061102
.
11.
Kou
,
S. C.
, and
Xie
,
X. S.
,
2004
, “
Generalized Langevin Equation With Fractional Gaussian Noise: Subdiffusion Within a Single Protein Molecule
,”
Phys. Rev. Lett.
,
93
(
18
), p.
180603
.
12.
Diop
,
M. A.
, and
Garrido-Atienza
,
M. J.
,
2014
, “
Retarded Evolution Systems Driven by Fractional Brownian Motion With Hurst Parameter H > 1/2
,”
Nonlinear Anal. Theor.
,
97
, pp.
15
29
.
13.
Deng
,
M. L.
, and
Zhu
,
W. Q.
,
2015
, “
Responses of Linear and Nonlinear Oscillators to Fractional Gaussian Noise With Hurst Index Between 1/2 and 1
,”
ASME J. Appl. Mech.
,
82
(
10
), p.
101008
.
14.
Lin
,
S. J.
,
1995
, “
Stochastic Analysis of Fractional Brownian Motions
,”
Stochastics
,
55
(
1–2
), pp.
121
140
.
15.
Decreusefond
,
L.
,
1999
, “
Stochastic Analysis of the Fractional Brownian Motion
,”
Potential Anal.
,
10
(
2
), pp.
177
214
.
16.
Zähle
,
M.
,
1998
, “
Integration With Respect to Fractal Functions and Stochastic Calculus—I
,”
Probab. Theory Relat. Fields
,
111
(
3
), pp.
333
374
.
17.
Duncan
,
T. E.
,
Hu
,
Y.
, and
Pasik-Duncan
,
B.
,
2000
, “
Stochastic Calculus for Fractional Brownian Motion I
,”
Theory SIAM J. Optim.
,
38
(
2
), pp.
582
612
.
18.
Biagini
,
F.
,
Hu
,
Y. Z.
,
Øksendal
,
B.
, and
Zhang
,
T. S.
,
2008
,
Stochastic Calculus for Fractional Brownian Motion and Applications
,
Springer-Verlag
,
London
.
19.
Zhu
,
W. Q.
,
Huang
,
Z. L.
, and
Yang
,
Y. Q.
,
1997
, “
Stochastic Averaging of Quasi-Integrable Hamiltonian Systems
,”
ASME J. Appl. Mech.
,
64
(
4
), pp.
975
984
.
20.
Zhu
,
W. Q.
,
Huang
,
Z. L.
, and
Suzuki
,
Y.
,
2002
, “
Stochastic Averaging and Lyapunov Exponent of Quasi Partially Integrable Hamiltonian Systems
,”
Int. J. Nonlinear Mech.
,
37
(
3
), pp.
419
437
.
21.
Zhu
,
W. Q.
, and
Yang
,
Y. Q.
,
1997
, “
Stochastic Averaging of Quasi-Nonintegrable-Hamiltonian Systems
,”
ASME J. Appl. Mech.
,
64
(
1
), pp.
157
164
.
22.
Baker
,
C. J.
,
1995
, “
The Development of a Theoretical Model for the Windthrow of Plants
,”
J. Theor. Biol.
,
175
(
3
), pp.
355
372
.
23.
Peltola
,
H.
, and
Kellomäki
,
S.
,
1993
, “
A Mechanistic Model for Calculating Windthrow and Stem Breakage of Scots Pines at Stand Age
,”
Silva Fennica
,
27
(
2
), pp.
99
111
.
24.
Letchford
,
C. W.
,
2001
, “
Wind Loads on Rectangular Signboards and Hoardings
,”
J. Wind Eng. Ind. Aerodyn.
,
89
(
2
), pp.
135
151
.
25.
Simiu
,
E.
, and
Scanlan
,
R. H.
,
1996
,
Wind Effects on Structures: Fundamentals and Application to Design
,
Wiley
,
New York
.
26.
Kaimal
,
J. C.
,
Wyngaard
,
J. C.
,
Izumi
,
Y.
, and
Cote
,
O. R.
,
1972
, “
Spectral Characteristics of Surface-Layer Turbulence
,”
Q. J. R. Meteorol. Soc.
,
98
(
417
), pp.
563
589
.
27.
Davenport
,
A. G.
,
1961
, “
The Spectrum of Horizontal Gustiness Near the Ground in High Winds
,”
Q. J. R. Meteorol. Soc.
,
87
(
372
), pp.
194
211
.
28.
Von Karman
,
T.
,
1948
, “
Progress in the Statistical Theory of Turbulence
,”
Proc. Natl. Acad. Sci. U.S.A.
,
34
(
11
), pp.
530
539
.
29.
Vepa
,
R.
,
2011
, “
Nonlinear Optimal Control of a Wind Turbine Generator
,”
IEEE Trans. Energy Convers.
,
26
(
2
), pp.
468
478
.
30.
van der Male
,
P.
,
van Dalen
,
K. N.
, and
Metrikine
,
A. V.
,
2016
, “
The Effect of the Nonlinear Velocity and History Dependencies of the Aerodynamic Force on the Dynamic Response of a Rotating Wind Turbine Blade
,”
J. Sound. Vib
,
383
, pp.
191
209
.
31.
Mishura
,
Y. S.
,
2008
,
Stochastic Calculus for Fractional Brownian Motion and Related Processes
,
Springer-Verlag
,
Berlin
.
You do not currently have access to this content.