There has been no significant progress in developing new techniques for obtaining exact stationary probability density functions (PDFs) of nonlinear stochastic systems since the development of the method of generalized probability potential in 1990s. In this paper, a general technique is proposed for constructing approximate stationary PDF solutions of single degree of freedom (SDOF) nonlinear systems under external and parametric Gaussian white noise excitations. This technique consists of two novel components. The first one is the introduction of new trial solutions for the reduced Fokker–Planck–Kolmogorov (FPK) equation. The second one is the iterative method of weighted residuals to determine the unknown parameters in the trial solution. Numerical results of four challenging examples show that the proposed technique will converge to the exact solutions if they exist, or a highly accurate solution with a relatively low computational effort. Furthermore, the proposed technique can be extended to multi degree of freedom (MDOF) systems.

References

References
1.
Yong
,
Y.
, and
Lin
,
Y. K.
,
1987
, “
Exact Stationary-Response Solution for Second Order Nonlinear Systems Under Parametric and External White-Noise Excitations
,”
ASME J. Appl. Mech.
,
54
(
2
), pp.
414
418
.
2.
Lin
,
Y. K.
, and
Cai
,
G. Q.
,
1988
, “
Exact Stationary Response Solution for Second Order Nonlinear Systems Under Parametric and External White Noise Excitations: Part II
,”
ASME J. Appl. Mech.
,
55
(
3
), pp.
702
705
.
3.
Huang
,
Z. L.
, and
Zhu
,
W. Q.
,
2000
, “
Exact Stationary Solutions of Stochastically and Harmonically Excited and Dissipated Integrable Hamiltonian Systems
,”
J. Sound Vib.
,
230
(
3
), pp.
709
720
.
4.
Zhu
,
W. Q.
, and
Huang
,
Z. L.
,
2001
, “
Exact Stationary Solutions of Stochastically Excited and Dissipated Partially Integrable Hamiltonian Systems
,”
Int. J. Non-Linear Mech.
,
36
(
1
), pp.
39
48
.
5.
Zhu
,
W. Q.
,
2006
, “
Nonlinear Stochastic Dynamics and Control in Hamiltonian Formulation
,”
ASME Appl. Mech. Rev.
,
59
(
4
), pp.
230
248
.
6.
Zhu
,
W. Q.
,
1988
, “
Stochastic Averaging Methods in Random Vibration
,”
ASME Appl. Mech. Rev.
,
41
(
5
), pp.
189
199
.
7.
Wen
,
Y. K.
,
1989
, “
Methods of Random Vibration for Inelastic Structures
,”
ASME Appl. Mech. Rev.
,
42
(
2
), pp.
39
52
.
8.
Socha
,
L.
, and
Soong
,
T. T.
,
1991
, “
Linearization in Analysis of Nonlinear Stochastic Systems
,”
ASME Appl. Mech. Rev.
,
44
(
10
), pp.
399
422
.
9.
Zhu
,
W. Q.
,
1996
, “
Recent Developments and Applications of the Stochastic Averaging Method in Random Vibration
,”
ASME Appl. Mech. Rev.
,
49
(
10S
), pp.
S72
S80
.
10.
Proppe
,
C.
,
Pradlwarter
,
H. J.
, and
Schuëller
,
G. I.
,
2003
, “
Equivalent Linearization and Monte Carlo Simulation in Stochastic Dynamics
,”
Probab. Eng. Mech.
,
18
(
1
), pp.
1
15
.
11.
Socha
,
L.
,
2005
, “
Linearization in Analysis of Nonlinear Stochastic Systems: Recent Results—Part I: Theory
,”
ASME Appl. Mech. Rev.
,
58
(
3
), pp.
178
205
.
12.
Socha
,
L.
,
2005
, “
Linearization in Analysis of Nonlinear Stochastic Systems, Recent Results—Part II: Applications
,”
ASME Appl. Mech. Rev.
,
58
(
5
), pp.
303
315
.
13.
Schuëller
,
G. I.
,
2006
, “
Developments in Stochastic Structural Mechanics
,”
Arch. Appl. Mech.
,
75
(
10
), pp.
755
773
.
14.
Brückner
,
A.
, and
Lin
,
Y. K.
,
1987
, “
Generalization of the Equivalent Linearization Method for Non-Linear Random Vibration Problems
,”
Int. J. Non-Linear Mech.
,
22
(
3
), pp.
227
235
.
15.
Saha
,
N.
, and
Roy
,
D.
,
2007
, “
The Girsanov Linearization Method for Stochastically Driven Nonlinear Oscillators
,”
ASME J. Appl. Mech.
,
74
(
5
), pp.
885
897
.
16.
Wang
,
R.
,
Kusumoto
,
S.
, and
Zhang
,
Z.
,
1996
, “
A New Equivalent Non-Linearization Technique
,”
Probab. Eng. Mech.
,
11
(
3
), pp.
129
137
.
17.
Zhu
,
W. Q.
, and
Lei
,
Y.
,
1997
, “
Equivalent Nonlinear System Method for Stochastically Excited and Dissipated Integrable Hamiltonian Systems
,”
ASME J. Appl. Mech.
,
64
(
1
), pp.
209
216
.
18.
Cavaleri
,
L.
, and
Di Paola
,
M.
,
2000
, “
Statistic Moments of the Total Energy of Potential Systems and Application to Equivalent Non-Linearization
,”
Int. J. Non-Linear Mech.
,
35
(
4
), pp.
573
587
.
19.
Zhu
,
W. Q.
, and
Deng
,
M. L.
,
2004
, “
Equivalent Non-Linear System Method for Stochastically Excited and Dissipated Integrable Hamiltonian Systems-Resonant Case
,”
J. Sound Vib.
,
274
(
3–5
), pp.
1110
1122
.
20.
Iyengar
,
R. N.
, and
Dash
,
P. K.
,
1978
, “
Study of the Random Vibration of Nonlinear Systems by the Gaussian Closure Technique
,”
ASME J. Appl. Mech.
,
45
(
2
), pp.
393
399
.
21.
Crandall
,
S. H.
,
1980
, “
Non-Gaussian Closure for Random Vibration of Non-Linear Oscillators
,”
Int. J. Non-Linear Mech.
,
15
(
4
), pp.
303
313
.
22.
Ibrahim
,
R. A.
,
Soundararajan
,
A.
, and
Heo
,
H.
,
1985
, “
Stochastic Response of Nonlinear Dynamic Systems Based on a Non-Gaussian Closure
,”
ASME J. Appl. Mech.
,
52
(
4
), pp.
965
970
.
23.
Roberts
,
J. B.
, and
Spanos
,
P. D.
,
1986
, “
Stochastic Averaging: An Approximate Method of Solving Random Vibration Problems
,”
Int. J. Non-Linear Mech.
,
21
(
2
), pp.
111
134
.
24.
Horse
,
J. R. R.
, and
Spanos
,
P. D.
,
1992
, “
A Generalization to Stochastic Averaging in Random Vibration
,”
Int. J. Non-Linear Mech.
,
27
(
1
), pp.
85
101
.
25.
Zhu
,
W. Q.
, and
Yang
,
Y. Q.
,
1997
, “
Stochastic Averaging of Quasi-Nonintegrable-Hamiltonian Systems
,”
ASME J. Appl. Mech.
,
64
(
1
), pp.
157
164
.
26.
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
.
27.
Li
,
W.
, and
Ibrahim
,
R. A.
,
1990
, “
Monte Carlo Simulation of Coupled Nonlinear Oscillators Under Random Excitations
,”
ASME J. Appl. Mech.
,
57
(
4
), pp.
1097
1099
.
28.
Pradlwarter
,
H. J.
, and
Schuëller
,
G. I.
,
1997
, “
On Advanced Monte Carlo Simulation Procedures in Stochastic Structural Dynamics
,”
Int. J. Non-Linear Mech.
,
32
(
4
), pp.
735
744
.
29.
Yu
,
J. S.
,
Cai
,
G. Q.
, and
Lin
,
Y. K.
,
1997
, “
A New Path Integration Procedure Based on Gauss–Legendre Scheme
,”
Int. J. Non-Linear Mech.
,
32
(
4
), pp.
759
768
.
30.
Naess
,
A.
, and
Moe
,
V.
,
2000
, “
Efficient Path Integration Methods for Nonlinear Dynamic Systems
,”
Probab. Eng. Mech.
,
15
(
2
), pp.
221
231
.
31.
Kougioumtzoglou
,
I. A.
,
Di Matteo
,
A.
,
Spanos
,
P. D.
,
Pirrotta
,
A.
, and
Di Paola
,
M.
,
2015
, “
An Efficient Wiener Path Integral Technique Formulation for Stochastic Response Determination of Nonlinear MDOF Systems
,”
ASME J. Appl. Mech.
,
82
(
10
), p.
101005
.
32.
Sun
,
J. Q.
, and
Hsu
,
C. S.
,
1990
, “
The Generalized Cell Mapping Method in Nonlinear Random Vibration Based Upon Short-Time Gaussian Approximation
,”
ASME J. Appl. Mech.
,
57
(
4
), pp.
1018
1025
.
33.
Chen
,
L. C.
, and
Sun
,
J. Q.
,
2016
, “
The Closed-Form Solution of the Reduced Fokker–Planck–Kolmogorov Equation for Nonlinear Systems
,”
Commun. Nonlinear Sci. Numer. Simul.
,
41
, pp.
1
10
.
34.
Er
,
G. K.
,
2000
, “
Exponential Closure Method for Some Randomly Excited Non-Linear Systems
,”
Int. J. Non-Linear Mech.
,
35
(
1
), pp.
69
78
.
35.
Er
,
G. K.
,
2014
, “
Probabilistic Solutions of Some Multi-Degree-of-Freedom Nonlinear Stochastic Dynamical Systems Excited by Filtered Gaussian White Noise
,”
Comput. Phys. Commun.
,
185
(
4
), pp.
1217
1222
.
36.
Paola
,
M. D.
, and
Sofi
,
A.
,
2002
, “
Approximate Solution of the Fokker–Planck–Kolmogorov Equation
,”
Probab. Eng. Mech.
,
17
(
4
), pp.
369
384
.
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