Abstract
In classical stochastic linearization method, the nonlinear force is replaced by an equivalent linear one. Several nonclassical schemes were suggested in recent years, based on potential or complementary energy criteria. Here, these criteria are compared with each other, and the classical stochastic linearization scheme, to determine their efficacy.
References
1.
Booton
, R.
C.
, 1954,
“The Analysis of Nonlinear Control Systems With Random
Inputs
,” IEEE Trans. Circuit Theory
0018-9324, 1
, pp.
9
–18
.2.
Kazakov
, I.
E.
, 1954,
“An Approximate Method for the Statistical Investigation of
Nonlinear Systems
,” Trudi Voenno–Vozdushnoi
Inzhenernoi Akademii imeni Professora N. E. Zhukovskogo
,
394
, pp. 1
–52
, in
Russian.3.
Caughey
, T.
K.
, 1953,
“Response of Nonlinear Systems to Random
Excitation
,” Lecture Notes, California Institute of
Technology.4.
Caughey
, T.
K.
, 1963,
“Equivalent Linearization Techniques
,”
J. Acoust. Soc. Am.
0001-4966, 35
, pp.
1706
–1711
.5.
Falsone
,
G.
, and
Ricciardi
,
G.
,
2003, “Stochastic
Linearization: Classical Approach and New Developments
,”
Recent Research Developments in Structural Dynamics
,
A.
Luongo
, ed., Rome,
Italy, pp. 81
–106
.6.
Socha
,
L.
,
2005, “Linearization in
Analysis of Nonlinear Stochastic Systems: Recent Results—Part I:
Theory
,” Appl. Mech. Rev.
0003-6900, 58
, pp.
178
–204
; Socha
,
L.
,
2005, “Linearization in
Analysis of Nonlinear Stochastic Systems: Recent Results—Part II:
Applications
,” Appl. Mech. Rev.
0003-6900, 58
, pp.
303
–315
.7.
Bernard
,
P.
, and
Wu
, L. M.
, 1998,
“Stochastic Linearization: The Theory
,”
J. Appl. Probab.
0021-9002, 35
, pp.
718
–730
.8.
Elishakoff
,
I.
,
2000, “Stochastic
Linearization Technique: A New Interpretaion and a Selective
Review
,” Shock Vib. Dig.
0583-1024, 32
(3
), pp.
179
–188
.9.
Wang
,
C.
, and
Zhang
, X.
T.
, 1985,
“An Improved Equivalent Linearization Technique in Nonlinear
Random Vibration
,” Proceedings of the International
Conference on Nonlinear Mechanics
, Beijing, China, pp.
959
–964
.10.
Zhang
, X.
T.
, Elishakoff
,
I.
, and
Zhang
, R.
C.
, 1991,
“A Stochastic Linearization Technique Based on Minimum Mean
Square Derivation of Potential Energies
,” Stochastic
Structural Dynamics: New Theorical Developments
, Y. K.
Lin
and
I.
Elishakoff
, eds.,
Springer-Verlag
,
Berlin
, pp.
327
–338
.11.
Elishakoff
,
I.
, and
Zhang
, R.
C.
, 1992,
“Comparison of the New Energy-Based Versions of the
Stochastic Linearization Technique
,” Nonlinear
Stochastic Mechanics
, N.
Bellomo
and
F.
Casciati
, eds.,
Springer
,
Berlin
, pp.
201
–212
.12.
Elishakoff
,
I.
, and
Zhang
, X.
T.
, 1992,
“An Appraisal of Different Stochastic Linearization
Techniques
,” J. Sound Vib.
0022-460X, 153
, pp.
370
–375
.13.
Falsone
,
G.
, and
Elishakoff
,
I.
,
1994, “Modified
Stochastic Linearization Technique for Colored-Noise Excitation of Duffing
Oscillator
,” Int. J. Non-Linear Mech.
0020-7462, 29
, pp.
65
–69
.14.
Fang
,
J.
,
Elishakoff
,
I.
, and
Caimi
,
R.
,
1995, “Nonlinear
Response of a Beam under Stationary Random Excitation by Improved Stochastic
Linearization Method
,” Appl. Math. Model.
0307-904X, 19
, pp.
106
–111
.15.
Muravyov
,
A.
,
Turner
,
T.
,
Robinson
,
J.
, and
Rizzi
,
S.
,
1999, “A New Stochastic
Equivalent Linearization Implementation for Prediction of Geometrically
Nonlinear Vibrations
,” Proceedings of the
AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics Conference
,
St. Louis, AIAA Paper No. 59-1376, Vol. 2
, pp.
1489
–1497
.16.
Zhang
, X.
T.
, 1992,
“Study of Weighted Energy Technque in Analysis of Nonlinear
Random Vibration
,” Nonlinear Vibration and
Chaos
, Tianjin University Press
,
Tianjin
, pp.
139
–144
.17.
Zhang
, R.
C.
, Elishakoff
,
I.
, and
Shinozuka
,
M.
,
1994, “Analysis of
Nonlinear Sliding Structures by Modified Stochastic Linearization
Methods
,” Nonlinear Dyn.
0924-090X, 5
, pp.
299
–312
.18.
Zhang
, X.
T.
, and Zhang
, R.
C.
, 1999,
“Energy-Based Stochastic Equivalent Linearization With
Optimized Power
,” Stochastic Structural
Dynamics
, B. F.
Spencer
and
E. A.
Johnson
, eds.,
Balkema
,
Rotterdam
, pp.
113
–117
.19.
Elishakoff
,
I.
, and
Bert
, C.
W.
, 2000,
“Complementary Energy Criterion in Nonlinear Stochastic
Dynamics
,” Applications of Statistics and
Probability
, Balkema
,
Rotterdam
, pp.
821
–825
.20.
Elishakoff
,
I.
,
Fang
,
J.
, and
Caimi
,
R.
,
1995, “Random Vibration
of a Nonlinearly Deformed Beam by a New Stochastic Linearization
Technique
,” Int. J. Solids Struct.
0020-7683, 32
, pp.
1571
–1584
.21.
Stratonovich
, R.
L.
, 1961,
Selected Questions of the Fluctuations Theory in Radiotechnics (in
Russian)
, Sovietskoe Radio
Publishers
,
Moscow
.22.
Piszczek
,
K.
, and
Niziol
,
J.
,
1984, Random Vibration of
Mechanical Systems
, Ellis Horwood
,
Chichester
.23.
Roberts
, J.
B.
, 1981,
“Response of Nonlinear Mechanical Systems to Random
Excitation. Part 2: Equivalent Linearization and Other
Methods
,” Shock Vib. Dig.
0583-1024, 13
(4
), pp.
15
–28
.24.
Li
,
W.
,
Chang
, C.
W.
, and Tseng
,
S.
,
2006, “The Linearization
Method Based on the Equivalence of Dissipated Structural
Systems
,” J. Sound Vib.
0022-460X, 295
, pp.
797
–809
.25.
Liang
, J.
W.
, and Feeny
,
B.
,
2006, “Balancing Energy
to Estimate Damping Parameters in Forced Oscillators
,”
J. Sound Vib.
0022-460X, 295
, pp.
988
–998
.26.
Casciati
,
F.
,
Faravelli
,
L.
, and
Hasofer
, A.
M.
, 1993,
“A New Philosophy for Stochastic Equivalent
Linearization
,” Probab. Eng. Mech.
0266-8920, 9
, pp.
179
–185
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