A new linearization model with density response based on information closure scheme is proposed for the prediction of dynamic response of a stochastic nonlinear system. Firstly, both probability density function and maximum entropy of a nonlinear stochastic system are estimated under the available information about the moment response of the system. With the estimated entropy and property of entropy stability, a robust stability boundary of the nonlinear stochastic system is predicted. Next, for the prediction of response statistics, a statistical linearization model is constructed with the estimated density function through a priori information of moments from statistical data. For the accurate prediction of the system response, the excitation intensity of the linearization model is adjusted such that the response of maximum entropy is invariant in the linearization model. Finally, the performance of the present linearization model is compared and supported by employing two examples with exact solutions, Monte Carlo simulations, and Gaussian linearization method.

Issue Section:
Technical Papers
Keywords:
Gaussian processes, Monte Carlo methods, maximum entropy methods, stochastic systems, linearisation techniques
Topics:
Density, Entropy, Stability, Stochastic systems
1.
Risken, H., 1989, The Fokker-Planck Equation: Methods of Solution and Applications, 2nd ed. Springer-Verlag, Berlin.
2.
Lin, Y. K., and Cai, G. Q., 1995, Probabilistic Structural Dynamics: Advanced Theory and Applications, McGraw-Hill, New York.
3.
Chang
,
R. J.
,
1993
, “
Extension in Techniques for Stochastic Dynamic Systems
,”
Control and Dynamic Systems
,
55
, pp.
429
–
470
.
4.
Bergman, L. A., Spencer, B. F., Wojtkiewicz, S. F., and Johnson, E. A., 1996, “Robust Numerical Solution of the Fokker-Planck Equation for Second Order Dynamical Systems under Parametric and External White Noise Excitations,” Nonlinear Dynamics and Stochastic Mechanics, Langford, W., Kliemann, W., and Sri Namachchivaya, N., eds., Vol. 9, pp. 23–37, American Mathematical Society, Providence.
5.
Socha
,
L.
, and
Soong
,
T. T.
,
1991
, “
Linearization in Analysis of Nonlinear Stochastic Systems
,”
Appl. Mech. Rev.
,
44
, pp.
399
–
422
.
6.
Leithead
,
W. E.
,
1990
, “
A Systematic Approach to Linear Approximation of Nonlinear Stochastic Systems. Part 1: Asymptotic Expansions
,”
Int. J. Control
,
51
, pp.
71
–
91
.
7.
Chang
,
R. J.
,
1990
, “
Model Based Discrete Linear State Estimator for Nonlinearizable Systems with State-Dependent Noise
,”
ASME J. Dyn. Syst., Meas., Control
,
112
, pp.
774
–
781
.
8.
Chang
,
R. J.
,
1992
, “
Non-Gaussian Linearization Method for Stochastic Parametrically and Externally Excited Nonlinear Systems
,”
ASME J. Dyn. Syst., Meas., Control
,
114
, pp.
20
–
26
.
9.
Beaman
,
J. J.
, and
Hedrick
,
J. K.
,
1981
, “
Improved Statistical Linearization for Analysis and Control of Nonlinear Stochastic Systems. Part 1: An Extended Statistical Linearization Technique
,”
ASME J. Dyn. Syst., Meas., Control
,
103
, pp.
14
–
21
.
10.
Bru¨ckner
,
A.
, and
Lin
,
Y. K.
,
1987
, “
Generalization of Equivalent Linearization Method for Nonlinear Random Vibration Problems
,”
Int. J. Non-Linear Mech.
,
22
, pp.
227
–
235
.
11.
Wojtkiewicz
,
S. F.
,
Spencer
, Jr.,
B. F.
, and
Bergman
,
L. A.
,
1996
, “
On the Cumulant-Neglect Closure Method in Stochastic Dynamics
,”
Int. J. Non-Linear Mech.
,
31
, pp.
657
–
684
.
12.
Haken, H., 1988, Information and Self-Organization, Springer-Verlag, Berlin.
13.
Chang
,
R. J.
,
1991
, “
Maximum Entropy Approach for Stationary Response of Nonlinear Stochastic Oscillators
,”
ASME J. Appl. Mech.
,
58
, pp.
266
–
271
.
14.
Sobczyk
,
K.
, and
Trebicki
,
J.
,
1990
, “
Maximum Entropy Principle in Stochastic Dynamics
,”
Probab. Eng. Mech.
,
5
, pp.
102
–
110
.
15.
Trebicki
,
J.
, and
Sobczyk
,
K.
,
1996
, “
Maximum Entropy Principle and Non-Stationary Distributions of Stochastic Systems
,”
Probab. Eng. Mech.
,
11
, pp.
169
–
178
.
16.
Jumarie
,
G.
,
1990
, “
Solution of the Multivariate Fokker-Planck Equation by Using a Maximum Path Entropy Principle
,”
J. Math. Phys.
,
31
, pp.
2389
–
2392
.
17.
Phillis
,
Y. A.
,
1982
, “
Entropy Stability of Continuous Dynamic Systems
,”
Int. J. Control
,
35
, pp.
323
–
340
.
18.
Chang
,
R. J.
, and
Lin
,
S. J.
,
2002
, “
Information Closure Method for Dynamic Analysis of Nonlinear Stochastic Systems
,”
ASME J. Dyn. Syst., Meas., Control
,
124
, pp.
353
–
363
.
19.
Jaynes
,
E. T.
,
1957
, “
Information Theory and Statistical Mechanics
,”
Phys. Rev.
,
106
, pp.
620
–
630
.
20.
Young
,
G. E.
, and
Chang
,
R. J.
,
1987
, “
Prediction of the Response of Nonlinear Oscillators under Stochastic Parametric and External Excitations
,”
Int. J. Non-Linear Mech.
,
28
, pp.
151
–
160
.
21.
Dimentberg
,
M. F.
,
1982
, “
An Exact Solution to a Certain Nonlinear Random Vibration Problem
,”
Int. J. Non-Linear Mech.
,
17
, pp.
231
–
236
.
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