A general methodology is presented for time-dependent reliability and random vibrations of nonlinear vibratory systems with random parameters excited by non-Gaussian loads. The approach is based on polynomial chaos expansion (PCE), Karhunen–Loeve (KL) expansion, and quasi Monte Carlo (QMC). The latter is used to estimate multidimensional integrals efficiently. The input random processes are first characterized using their first four moments (mean, standard deviation, skewness, and kurtosis coefficients) and a correlation structure in order to generate sample realizations (trajectories). Characterization means the development of a stochastic metamodel. The input random variables and processes are expressed in terms of independent standard normal variables in N dimensions. The N-dimensional input space is space filled with M points. The system differential equations of motion (EOM) are time integrated for each of the M points, and QMC estimates the four moments and correlation structure of the output efficiently. The proposed PCE–KL–QMC approach is then used to characterize the output process. Finally, classical MC simulation estimates the time-dependent probability of failure using the developed stochastic metamodel of the output process. The proposed methodology is demonstrated with a Duffing oscillator example under non-Gaussian load.

References

References
1.
Soong
,
T. T.
, and
Grigoriu
,
M.
,
1993
,
Random Vibration of Mechanical and Structural Systems
,
Prentice Hall
,
Englewood Cliffs, NJ
.
2.
Roberts
,
J. B.
, and
Spanos
,
P. D.
,
1999
,
Random Vibration and Statistical Linearization
,
Dover Publications
,
Mineola, NY
.
3.
Li
,
C.-C.
, and
Kiureghian
,
A. D.
,
1993
, “
Optimal Discretization of Random Fields
,”
J. Eng. Mech.
,
119
(
6
), pp.
1136
1154
.
4.
Zhang
,
J.
, and
Ellingwood
,
B.
,
1994
, “
Orthogonal Series Expansions of Random Fields in Reliability Analysis
,”
J. Eng. Mech.
,
120
(
12
), pp.
2660
2677
.
5.
Sudret
,
B.
, and
Der Kiureghian
,
A.
,
2000
, “
Stochastic Finite Element Methods and Reliability—A State of the Art Report
,” University of California, Berkeley, CA, Report No.
UCB/SEMM-2000/08
.http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.453.7433&rep=rep1&type=pdf
6.
Spanos
,
P. D.
,
Kougioumtzoglou
,
I. A.
, and
Soize
,
C.
,
2011
, “
On the Determination of the Power Spectrum of Randomly Excited Oscillators Via Stochastic Averaging: An Alternative Approach
,”
Probab. Eng. Mech.
,
26
(
1
), pp.
10
15
.
7.
Spanos
,
P. D.
, and
Kougioumtzoglou
,
I. A.
,
2012
, “
Harmonic Wavelets Based Statistical Linearization for Response Evolutionary Power Spectrum Determination
,”
Probab. Eng. Mech.
,
27
(
1
), pp.
57
68
.
8.
Shields
,
M. D.
,
Deodatis
,
G.
, and
Bocchini
,
P.
,
2011
, “
A Simple and Efficient Methodology to Approximate a General Non-Gaussian Stationary Stochastic Process by a Translation Process
,”
Probab. Eng. Mech.
,
26
(
4
), pp.
511
519
.
9.
Joo
,
H. K.
, and
Sapsis
,
T.
,
2016
, “
A Moment-Equation-Copula-Closure Method for Nonlinear Vibrational Systems Subjected to Correlated Noise
,”
Probab. Eng. Mech.
,
46
, pp.
120
132
.
10.
Mourelatos
,
Z. P.
,
Majcher
,
M.
, and
Geroulas
,
V.
,
2016
, “
Time-Dependent Reliability Analysis of Vibratory Systems With Random Parameters
,”
ASME J. Vib. Acoust.
,
138
(
3
), p.
031007
.
11.
Tsianika
,
V.
,
Geroulas
,
V.
,
Mourelatos
, and
Baseski
,
I.
,
2017
, “
A Methodology for Fatigue Life Estimation of Linear Vibratory Systems Under Non-Gaussian Loads
,”
SAE
Paper No. 2017-01-0197.
12.
Melchers
,
R. E.
,
1999
,
Structural Reliability Analysis and Prediction
,
2nd ed.
,
Wiley
,
Chichester, UK
.
13.
Hu
,
Z.
,
Li
,
H.
,
Du
,
X.
, and
Chandrashekhara
,
K.
,
2013
, “
Simulation-Based Time-Dependent Reliability Analysis for Composite Hydrokinetic Turbine Blades
,”
Struct. Multidiscip. Optim.
,
47
(
5
), pp.
765
781
.
14.
Hu
,
Z.
, and
Du
,
X.
,
2013
, “
Time-Dependent Reliability Analysis With Joint Upcrossing Rates
,”
Struct. Multidiscip. Optim.
,
48
(
5
), pp.
893
907
.
15.
Wang
,
Z.
, and
Wang
,
P.
,
2012
, “
A Nested Extreme Response Surface Approach for Time-Dependent Reliability-Based Design Optimization
,”
ASME J. Mech. Des.
,
134
(
12
), p.
121007
.
16.
Andrieu-Renaud
,
C.
,
Sudret
,
B.
, and
Lemaire
,
M.
,
2004
, “
The PHI2 Method: A Way to Compute Time-Variant Reliability
,”
Reliab. Eng. Saf. Syst.
,
84
(
1
), pp.
75
86
.
17.
Madsen
,
P. H.
, and
Krenk
,
S.
,
1984
, “
An Integral Equation Method for the First Passage Problem in Random Vibration
,”
ASME J. Appl. Mech.
,
51
(
3
), pp.
674
679
.
18.
Rice
,
S. O.
,
1944
, “
Mathematical Analysis of Random Noise
,”
Bell Syst. Tech. J.
,
23
(
3
), pp.
282
332
.
19.
Rackwitz
,
R.
,
1998
, “
Computational Techniques in Stationary and Non-Stationary Load Combination—A Review and Some Extensions
,”
J. Struct. Eng.
,
25
(
1
), pp.
1
20
.https://www.researchgate.net/publication/279895900_Computational_techniques_in_stationary_and_non-stationary_load_combination_-_A_review_and_some_extensions
20.
Hu
,
Z.
, and
Du
,
X.
,
2012
, “
Reliability Analysis for Hydrokinetic Turbine Blades
,”
Renewable Energy
,
48
, pp.
251
262
.
21.
Mourelatos
,
Z. P.
,
Majcher
,
M.
,
Pandey
,
V.
, and
Baseski
,
I.
,
2015
, “
Time-Dependent Reliability Analysis Using the Total Probability Theorem
,”
ASME J. Mech. Des.
,
137
(
3
), p.
031405
.
22.
Shinozuka
,
M.
, and
Jan
,
C.
,
1972
, “
Digital Simulation of Random Processes and Its Applications
,”
J. Sound Vib.
,
25
(
1
), pp.
111
128
.
23.
Yamazaki
,
Y.
, and
Shinozuka
,
M.
,
1988
, “
Digital Generation of Non-Gaussian Stochastic Fields
,”
J. Eng. Mech.
,
114
(
7
), pp.
1183
1197
.
24.
Sakamoto
,
S.
, and
Ghanem
,
R.
,
2002
, “
Simulation of Multi-Dimensional Non-Gaussian Non-Stationary Random Fields
,”
Probab. Eng. Mech.
,
17
(
2
), pp.
167
176
.
25.
Xiu
,
D.
, and
Karniadakis
,
G.
,
2002
, “
The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations
,”
SIAM J. Sci. Comput.
,
24
(
2
), pp.
619
644
.
26.
Dick
,
J.
,
Kuo
,
F. Y.
, and
Sloan
,
I. H.
,
2013
, “
High Dimensional Integration—The Quasi Monte Carlo Way
,”
Acta Numerica
,
22
, pp.
133
288
.
27.
Ye
,
Q.
,
Li
,
W.
, and
Sudjianto
,
A.
,
2000
, “
Algorithmic Construction of Optimal Symmetric Latin Hypercube Designs
,”
J. Stat. Plann. Inference
,
90
(
1
), pp.
145
159
.
28.
Newland
,
D. E.
,
1993
,
An Introduction to Random Vibrations, Spectral and Wavelet Analysis
,
3rd ed.
,
Dover Publications
,
Mineola, NY
.
You do not currently have access to this content.