The nonstationary random response of a class of lightly damped nonlinear oscillators subjected to Gaussian white noise is considered. An approximate analytical method for determining the response envelope statistics is presented. Within the framework of stochastic averaging, the procedure relies on the Markovian modeling of the response envelope process through the definition of an equivalent linear system with response-dependent parameters. An approximate solution of the associated Fokker-Planck equation is derived by resorting to a Galerkin scheme. Specifically, the nonstationary probability density function of the response envelope is expressed as the sum of a time-dependent Rayleigh distribution and of a series expansion in terms of a set of properly selected basis functions with time-dependent coefficients. These functions are the eigenfunctions of the boundary-value problem associated with the Fokker-Planck equation governing the evolution of the probability density function of the response envelope of a linear oscillator. The selected basis functions possess some notable properties that yield substantial computational advantages. Applications to the Van der Pol and Duffing oscillators are presented. Appropriate comparisons to the data obtained by digital simulation show that the method, being nonperturbative in nature, yields reliable results even for large values of the nonlinearity parameter.

1.
Crandall
,
S. H.
, and
Mark
,
W. D.
, 1963,
Random Vibration in Mechanical Systems
,
Academic Press
,
San Diego, CA
.
2.
Lin
,
Y. K.
, 1967,
Probabilistic Theory of Structural Dynamics
,
McGraw-Hill
,
New York
.
3.
Clough
,
R. W.
, and
Penzien
,
J.
, 1993,
Dynamics of Structures
,
2nd ed.
,
McGraw-Hill
,
New York
.
4.
Caughey
,
T. K.
, 1971, “
Nonlinear Theory of Random Vibrations
,”
Advances in Applied Mechanics
, Vol.
11
,
C. S.
Yih
, ed.
Academic Press
,
San Diego, CA
, pp.
209
235
.
5.
Caughey
,
T. K.
, and
Dienes
,
J. K.
, 1961, “
Analysis of a Nonlinear First-Order System With a White Noise Input
,”
J. Appl. Phys.
0021-8979,
32
(
11
), pp.
2476
2479
.
6.
Gardiner
,
C. W.
, 1983,
Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences
,
Springer-Velarg
,
Berlin
.
7.
Andronov
,
A.
,
Pontryagin
,
L.
, and
Witt
,
A.
, 1933, “
On the Statistical Investigation of Dynamical Systems
,”
Zh. Eksp. Teor. Fiz.
0044-4510,
3
, pp.
165
180
(in Russian).
8.
Kramers
,
H. A.
, 1940, “
Brownian Motion in a Field of Force and Diffusion Model of Chemical Reactions
,”
Physica (Utrecht)
0031-8914,
7
, pp.
284
304
.
9.
Caughey
,
T. K.
, and
Ma
,
F.
, 1982, “
The Exact Steady-State Solution of a Class of Non-Linear Stochastic Systems
,”
Int. J. Non-Linear Mech.
0020-7462,
17
(
3
), pp.
137
142
.
10.
Dimentberg
,
M. F.
, 1982, “
An Exact Solution to a Certain Non-Linear Random Vibration Problem
,”
Int. J. Non-Linear Mech.
0020-7462,
17
(
4
), pp.
231
236
.
11.
Roberts
,
J. B.
, and
Spanos
,
P. D.
, 1986, “
Stochastic Averaging: an Approximate Method of Solving Random Vibration Problems
,”
Int. J. Non-Linear Mech.
0020-7462,
21
, pp.
111
134
.
12.
Stratonovich
,
R. L.
, 1986,
Topics in the Theory of Random Noise
, Vols.
1
and
2
,
Gordon & Breach
,
New York
.
13.
Khasminskii
,
R. Z.
, 1966, “
A Limit Theorem for Solutions of Differential Equations With Random Right Hand Sides
,”
Theor. Probab. Appl.
0040-585X,
11
, pp.
390
405
.
14.
Bogoliubov
,
N.
, and
Mitropolski
,
A.
, 1961,
Asymptotic Methods in the Theory of Non-linear Oscillations
,
Gordon & Breach
,
New York
.
15.
Iwan
,
W. D.
,
Spanos
,
P.-T.
, 1978, “
Response Envelope Statistics for Nonlinear Oscillators with Random Excitation
,”
ASME J. Appl. Mech.
0021-8936,
100
(
1
), pp.
170
174
.
16.
Spanos
,
P.-T. D.
, 1976, “
Linearization Techniques for Non-Linear Dynamical Systems
,” Report No. EERL 70-04, Earthquake Engineering Research Laboratory,
California Institute of Technology
, Pasadena, CA.
17.
Spanos
,
P. D.
,
Cacciola
,
P.
, and
Muscolino
,
G.
, 2004, “
Stochastic Averaging of Preisach Hysteretic Systems
,”
J. Eng. Mech. Div., Am. Soc. Civ. Eng.
0044-7951,
130
(
11
), pp.
1257
1267
.
18.
Bouc
,
R.
, 1994, “
The Power Spectral Density of Response for a Strongly Non-Linear Random Oscillator
,”
J. Sound Vib.
0022-460X,
175
(
3
), pp.
317
331
.
19.
Spanos
,
P.-T. D.
, 1978, “
Stochastic Analysis of Oscillators with Non-Linear Damping
,”
Int. J. Non-Linear Mech.
0020-7462,
13
, pp.
249
259
.
20.
Spanos
,
P.-T. D.
, 1981, “
A Method for Analysis of Non-Linear Vibrations Caused by Modulated Random Excitation
,”
Int. J. Non-Linear Mech.
0020-7462,
16
, pp.
1
11
.
21.
Spanos
,
P.-T. D.
, and
Iwan
,
W. D.
, 1978, “
Computational Aspects of Random Vibration Analysis
,”
J. Eng. Mech. Div., Am. Soc. Civ. Eng.
0044-7951,
104
(
EM6
), pp.
1043
1415
.
22.
Spanos
,
P.-T. D.
, 1978, “
Non-Stationary Random Vibration of a Linear Structure
,”
Int. J. Solids Struct.
0020-7683,
14
, pp.
861
867
.
23.
Spencer
, Jr.
B. F.
, and
Bergman
,
L. A.
, 1993, “
Numerical Solutions of the Fokker-Planck Equations for Nonlinear Stochastic Systems
,”
Nonlinear Dyn.
0924-090X,
4
, pp.
357
372
.
24.
Wojtkiewicz
,
S. F.
,
Johnson
,
E. A.
,
Bergman
,
L. A.
,
Grigoriu
,
M.
, and
Spencer
, Jr.,
B. F.
, 1999, “
Response of Stochastic Dynamical Systems Driven by Additive Gaussian and Poisson White Noises: Solution of a Forward Generalized Kolmogorov Equation by a Spectral Finite Difference Method
,”
Comput. Methods Appl. Mech. Eng.
0045-7825,
168
(
1–4
), pp.
73
89
.
25.
Masud
,
A.
, and
Bergman
,
L. A.
, 2005, “
Application of Multi-Scale Finite Element Methods to the Solution of the Fokker-Planck Equation
,”
Comput. Methods Appl. Mech. Eng.
0045-7825
194
, pp.
1513
1526
.
26.
Zhu
,
W.
, and
Lin
,
Y. K.
, 1991, “
Stochastic Averaging of Energy Envelopes
,”
J. Eng. Mech.
0733-9399,
117
, pp.
1890
1905
.
27.
Krenk
,
S.
, and
Roberts
,
J. B.
, 1999, “
Local Similarity in Non-Linear Random Vibration
,”
ASME J. Appl. Mech.
0021-8936,
66
, pp.
225
235
.
You do not currently have access to this content.