A new non-Gaussian linearization method is developed for extending the analysis of Gaussian white-noise excited nonlinear oscillator to incorporate sinusoidal excitation. The non-Gaussian linearization method is developed through introducing a modulated correction factor on the linearization coefficient which is obtained by Gaussian linearization. The time average of cyclostationary response of variance and noise spectrum is analyzed through the correction factor. The validity of the present non-Gaussian approach in predicting the statistical response is supported by utilizing Monte Carlo simulations. The present non-Gaussian analysis, without imposing restrictive analytical conditions, can be obtained by solving nonlinear algebraic equations. The non-Gaussian solution effectively predicts accurate sinusoidal and noise response when the nonlinear system is subjected to both sinusoidal and white-noise excitations.

References

References
1.
To
,
C. W. S.
,
2000
,
Nonlinear Random Vibration, Analytical Techniques and Applications
,
Swets & Zeitlinger
,
Lisse, The Netherlands
.
2.
Kovacic
,
I.
, and
Brennan
,
M. J.
,
2011
,
The Duffing Equation: Nonlinear Oscillators and Their Behaviour
,
Wiley
,
New York
.
3.
Jung
,
P.
,
1993
, “
Periodically Driven Stochastic Systems
,”
Phys. Rep.
,
234
(
4–5
), pp.
175
295
.
4.
Ellermann
,
K.
,
2005
, “
On the Determination of Nonlinear Response Distributions for Oscillators With Combined Harmonic and Random Excitation
,”
Nonlinear Dyn.
,
42
(
3
), pp.
305
318
.
5.
Budgor
,
A. B.
,
1977
, “
Studies in Nonlinear Stochastic Processes—III: Approximate Solutions of Nonlinear Stochastic Differential Equations Excited by Gaussian Noise and Harmonic Disturbances
,”
J. Stat. Phys.
,
17
(
1
), pp.
21
44
.
6.
Bulsara
,
A. R.
,
Lindenberg
,
K.
, and
Shuler
,
K. E.
,
1982
, “
Spectral Analysis of a Nonlinear Oscillator Driven by Random and Periodic Forces—I: Linearized Theory
,”
J. Stat. Phys.
,
27
(
4
), pp.
787
808
.
7.
Iyengar
,
R. N.
,
1986
, “
A Nonlinear System Under Combined Periodic and Random Excitation
,”
J. Stat. Phys.
,
44
(
5–6
), pp.
907
920
.
8.
Nayfeh
,
A. H.
, and
Serhan
,
S. J.
,
1990
, “
Response Statistics of Non-Linear Systems to Combined Deterministic and Random Excitations
,”
Int. J. Non-Linear Mech.
,
25
(
5
), pp.
493
505
.
9.
Wagner
,
U. V.
,
2002
, “
On Double Crater-Like Probability Density Functions of a Duffing Oscillator Subjected to Harmonic and Stochastic Excitations
,”
Nonlinear Dyn.
,
28
(3–4), pp.
343
355
.
10.
Rong
,
H.
,
Meng
,
G.
,
Wang
,
X.
,
Xu
,
W.
, and
Fang
,
T.
,
2004
, “
Response Statistic of Non-Linear Oscillator to Combined Deterministic and Random Excitation
,”
Int. J. Non-Linear Mech.
,
39
(6), pp.
871
878
.
11.
Huang
,
Z. L.
,
Zhu
,
W. Q.
, and
Suzuki
,
Y.
,
2000
, “
Stochastic Averaging of Non-Linear Oscillators Under Combined Harmonic and White-Noise Excitations
,”
J. Sound Vib.
,
238
(
2
), pp.
233
256
.
12.
Wu
,
Y. J.
, and
Zhu
,
W. Q.
,
2008
, “
Stochastic Averaging of Strongly Nonlinear Oscillators Under Combined Harmonic and Wide-Band Noise Excitations
,”
ASME J. Vib. Acoust.
,
130
(
5
), p.
051004
.
13.
Huang
,
Z. L.
, and
Zhu
,
W. Q.
,
2004
, “
Stochastic Averaging of Quasi-Integrable Hamiltonian Systems Under Combined Harmonic and White Noise Excitations
,”
Int. J. Non-Linear Mech.
,
39
(
9
), pp.
1421
1434
.
14.
Anh
,
N. D.
, and
Hieu
,
N. N.
,
2012
, “
The Duffing Oscillator Under Combined Periodic and Random Excitations
,”
Probab. Eng. Mech.
,
30
, pp.
27
36
.
15.
Zhu
,
H. T.
, and
Guo
,
S. S.
,
2015
, “
Periodic Response of a Duffing Oscillator Under Combined Harmonic and Random Excitations
,”
ASME J. Vib. Acoust.
,
137
(4), p. 041015.
16.
Yu
,
J. S.
, and
Lin
,
Y. K.
,
2003
, “
Numerical Path Integration of a Non-Homogeneous Markov Process
,”
Int. J. Non-Linear Mech.
,
39
(
9
), pp.
1493
1500
.
17.
Xu
,
W.
,
He
,
Q.
,
Fang
,
T.
, and
Rong
,
H.
,
2004
, “
Stochastic Bifurcation in Duffing System Subjected to Harmonic Excitation and in Presence of Random Noise
,”
Int. J. Non-Linear Mech.
,
39
(
9
), pp.
1473
1479
.
18.
Wagner
,
U. V.
, and
Wedig
,
W. V.
,
2000
, “
On the Calculation of Stationary Solutions of Multi-Dimensional Fokker-Planck Equations by Orthogonal Functions
,”
Nonlinear Dyn.
,
21
(
3
), pp.
289
306
.
19.
Roberts
,
J. B.
, and
Spanos
,
P. D.
,
1999
,
Random Vibration and Statistical Linearization
,
Dover Publications
,
Mineola, NY
.
20.
Crandall
,
S. H.
,
2006
, “
A Half-Century of Stochastic Equivalent Linearization
,”
Struct. Control Health Monit.
,
13
(
1
), pp.
27
40
.
21.
Socha
,
L.
,
2008
,
Linearization Methods for Stochastic Dynamic Systems
,
Springer
,
Berlin
.
22.
Chang
,
R. J.
,
2014
, “
Two-Stage Optimal Stochastic Linearization in Analyzing of Non-Linear Stochastic Dynamic Systems
,”
Int. J. Non-Linear Mech.
,
58
, pp.
295
304
.
23.
Chang
,
R. J.
,
2015
, “
Cyclostationary Gaussian Linerarization for Analyzing Nonlinear System Response Under Sinusoidal Signal and White Noise Excitation
,”
Int. J. Mech. Aerosp. Ind. Mechatronic Manuf. Eng.
,
90
(
5
), pp.
664
671
.https://www.researchgate.net/publication/278022347_Cyclostationary_Gaussian_Linearization_for_Analyzing_Nonlinear_System_Under_Sinusoidal_Signal_and_White_Noise_Excitation
24.
Assaf
,
S. A.
, and
Zirkle
,
L. D.
,
1976
, “
Approximate Analysis of Non-Linear Stochastic Systems
,”
Int. J. Control
,
23
(
4
), pp.
477
492
.
25.
Beaman
,
J. J.
, and
Hedrick
,
J. K.
,
1981
, “
Improved Statistical Linearization for Analysis and Control of Nonlinear Stochastic Systems—Part I: An Extended Statistical Linearization Technique
,”
ASME J. Dyn. Syst. Meas. Control
,
103
(
1
), pp.
14
21
.
26.
Crandall
,
S. H.
,
1985
, “
Non-Gaussian Closure Techniques for Stationary Random Vibration
,”
Int. J. Non-Linear Mech.
,
20
(
1
), pp.
1
8
.
27.
Lee
,
J.
,
1995
, “
Improving the Equivalent Linerarization Technique for Stochastic Duffing Oscillators
,”
J. Sound Vib.
,
186
(
5
), pp.
846
855
.
28.
Falsone
,
G.
,
2005
, “
An Extension of the Kazakov Relationship for Non-Gaussian Random Variables and Its Use in the Non-Liner Stochastic Dynamics
,”
Prob. Eng. Mech.
,
20
(
1
), pp.
45
56
.
29.
Ricciardi
,
G.
,
2007
, “
A Non-Gaussian Stochastic Linearization Method
,”
Int. J. Non-Linear Mech.
,
22
(1), pp.
1
11
.
30.
Chang
,
R. J.
,
1992
, “
Non-Gaussian Linearization Method for Stochastic Parametrically and Externally Excited Nonlinear Systems
,”
ASME J. Dyn. Syst. Meas. Control
,
114
(
1
), pp.
20
26
.
31.
Er
,
G. K.
,
1998
, “
Multi-Gaussian Closure Method for Randomly Excited Non-Linear Systems
,”
Int. J. Non-Linear Mech.
,
33
(
2
), pp.
201
214
.
32.
Pradlwarter
,
H. J.
,
2001
, “
Non-Linear Stochastic Response Distributions by Local Statistical Linearization
,”
Int. J. Non-Linear Mech.
,
36
(
7
), pp.
1135
1151
.
33.
Sobczyk
,
K.
, and
Trebicki
,
J.
,
1990
, “
Maximum Entropy Principle in Stochastic Dynamics
,”
Probab. Eng. Mech.
,
5
(
3
), pp.
102
110
.
34.
Chang
,
R. J.
,
1991
, “
Maximum Entropy Approach for Stationary Response of Nonlinear Stochastic Oscillators
,”
ASME J. Appl. Mech.
,
58
(
1
), pp.
266
271
.
35.
Trebicki
,
J.
, and
Sobczyk
,
K.
,
1996
, “
Maximum Entropy Principle and Non-Stationary Distributions of Stochastic Systems
,”
Probab. Eng. Mech.
,
11
(
3
), pp.
169
178
.
36.
Er
,
G. K.
,
2000
, “
Exponential Closure Method for Some Randomly Excited Non-Linear Systems
,”
Int. J. Non-Linear Mech.
,
35
(
1
), pp.
69
78
.
37.
Ricciardi
,
G.
, and
Elishakoff
,
I.
,
2002
, “
A Novel Local Stochastic Linearization Method Via Two Extremum Entropy Principles
,”
Int. J. Non-Linear Mech.
,
37
(4–5), pp.
785
800
.
38.
Paola
,
M. D.
, and
Sofi
,
A.
,
2002
, “
Approximate Solution of the Fokker–Planck–Kolmogorov Equation
,”
Probab. Eng. Mech.
,
17
(
4
), pp.
369
384
.
39.
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
(
3
), pp.
353
363
.
40.
Chang
,
R. J.
, and
Lin
,
S. J.
,
2004
, “
Statistical Linearization Model for the Response Prediction of Nonlinear Stochastic Systems Through Information Closure Method
,”
ASME J. Vib. Acoust.
,
126
(
3
), pp.
438
448
.
41.
Crandall
,
S. H.
,
2004
, “
On Using Non-Gaussian Distributions to Perform Statistical Linearization
,”
Int. J. Non-Linear Mech.
,
39
(
9
), pp.
1395
1406
.
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