The field of random vibrations of large-scale systems with millions of degrees-of-freedom (DOF) is of significant importance in many engineering disciplines. In this paper, we propose a method to calculate the time-dependent reliability of linear vibratory systems with random parameters excited by nonstationary Gaussian processes. The approach combines principles of random vibrations, the total probability theorem, and recent advances in time-dependent reliability using an integral equation involving the upcrossing and joint upcrossing rates. A space-filling design, such as optimal symmetric Latin hypercube (OSLH) sampling, is first used to sample the input parameter space. For each design point, the corresponding conditional time-dependent probability of failure is calculated efficiently using random vibrations principles to obtain the statistics of the output process and an efficient numerical estimation of the upcrossing and joint upcrossing rates. A time-dependent metamodel is then created between the input parameters and the output conditional probabilities allowing us to estimate the conditional probabilities for any set of input parameters. The total probability theorem is finally applied to calculate the time-dependent probability of failure. The proposed method is demonstrated using a vibratory beam example.

References

1.
Singh
,
A.
,
Mourelatos
,
Z. P.
, and
Li
,
J.
,
2010
, “
Design for Lifecycle Cost Using Time-Dependent Reliability
,”
ASME J. Mech. Des.
,
132
(
9
), p.
091008
.
2.
Singh
,
A.
,
Mourelatos
,
Z. P.
, and
Li
,
J.
,
2010
, “
Design for Lifecycle Cost and Preventive Maintenance Using Time-Dependent Reliability
,”
Adv. Mater. Res.
,
118–120
, pp.
10
16
.
3.
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
.
4.
Rice
,
S. O.
,
1954
, “
Mathematical Analysis of Random Noise
,”
Bell Syst. Tech. J.
,
23
(
3
), pp.
282
332
.
5.
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
.
6.
Sudret
,
B.
,
2008
, “
Analytical Derivation of the Outcrossing Rate in Time-Variant Reliability Problems
,”
Struct. Infrastruct. Eng.
,
4
(
5
), pp.
353
362
.
7.
Zhang
,
J.
, and
Du
,
X.
,
2011
, “
Time-Dependent Reliability Analysis for Function Generator Mechanisms
,”
ASME J. Mech. Des.
,
133
(
3
), p.
031005
.
8.
Savage
,
G. J.
, and
Son
,
Y. K.
,
2009
, “
Dependability-Based Design Optimization of Degrading Engineering Systems
,”
ASME J. Mech. Des.
,
131
(
1
), p.
011002
.
9.
Singh
,
A.
, and
Mourelatos
,
Z. P.
,
2010
, “
On the Time-Dependent Reliability of Non-Monotonic, Non-Repairable Systems
,”
SAE Int. J. Mater. Manuf.
,
3
(
1
), pp.
425
444
.
10.
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
.
11.
Li
,
J.
, and
Mourelatos
,
Z. P.
,
2009
, “
Time-Dependent Reliability Estimation for Dynamic Problems Using a Niching Genetic Algorithm
,”
ASME J. Mech. Des.
,
131
(
7
), p.
071009
.
12.
Hu
,
Z.
, and
Du
,
X.
,
2012
, “
Reliability Analysis for Hydrokinetic Turbine Blades
,”
Renewable Energy
,
48
, pp.
251
262
.
13.
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
.
14.
Hu
,
Z.
, and
Du
,
X.
,
2013
, “
Time-Dependent Reliability Analysis With Joint Upcrossing Rates
,”
Struct. Multidiscip. Optim.
,
48
(
5
), pp.
893
907
.
15.
Singh
,
A.
, and
Mourelatos
,
Z. P.
,
2010
, “
Time-Dependent Reliability Estimation for Dynamic Systems Using a Random Process Approach
,”
SAE Int. J. Mater. Manuf.
,
3
(
1
), pp.
339
355
.
16.
Singh
,
A.
,
Mourelatos
,
Z. P.
, and
Nikolaidis
,
E.
,
2011
, “
Time-Dependent Reliability of Random Dynamic Systems Using Time-Series Modeling and Importance Sampling
,”
SAE Int. J. Mater. Manuf.
,
4
(
1
), pp.
929
946
.
17.
Au
,
S. K.
, and
Beck
,
J. L.
,
2003
, “
Subset Simulation and Its Application to Seismic Risk Based on Dynamic Analysis
,”
J. Eng. Mech.
,
129
(
8
), pp.
901
917
.
18.
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
.
19.
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
.
20.
Wehrwein
,
D.
, and
Mourelatos
,
Z. P.
,
2009
, “
Optimization of Engine Torque Management Under Uncertainty for Vehicle Driveline Clunk Using Time-Dependent Metamodels
,”
ASME J. Mech. Des.
,
131
(
5
), p.
051001
.
21.
Nikolaidis
,
E.
,
Mourelatos
,
Z. P.
, and
Pandey
,
V.
,
2011
,
Design Decisions Under Uncertainty With Limited Information
,
CRC Press, Taylor & Francis Group
,
London
.
22.
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
.
23.
Melchers
,
R. E.
,
1999
,
Structural Reliability Analysis and Prediction
, 2nd ed.,
Wiley
,
Chichester, UK
.
24.
Liang
,
J.
,
Mourelatos
,
Z. P.
, and
E.
Nikolaidis
,
2007
, “
A Single-Loop Approach for System Reliability-Based Design Optimization
,”
ASME J. Mech. Des.
,
129
(
12
), pp.
1215
1224
.
25.
Newland
,
D. E.
,
1993
,
An Introduction to Random Vibrations, Spectral and Wavelet Analysis
, 3rd ed.,
Dover Publications
,
Mineola, NY
.
26.
Wirsching
,
P. H.
,
Paez
,
T. L.
, and
Ortiz
,
H.
,
1995
,
Random Vibrations: Theory and Practice
,
Wiley
,
New York
.
27.
Mourelatos
,
Z. P.
,
Angelis
,
D.
, and
Skarakis
,
J.
,
2012
, “
Vibration and Optimization Analysis of Large-Scale Structures Using Reduced-Order Models and Reanalysis Methods
,”
Advances in Vibration Engineering and Structural Dynamics
,
F.
Beltran-Carbajal
, ed.,
InTech
,
Rijeka
,
Croatia
.
28.
Norouzi
,
M.
, and
Nikolaidis
,
E.
,
2012
, “
Efficient Method for Reliability Assessment Under High-Cycle Fatigue
,”
Int. J. Reliab., Qual. Saf. Eng.
,
19
(
5
), p.
1250022
.
29.
Ye
,
Q.
,
Li
,
W.
, and
Sudjianto
,
A.
,
2003
, “
Algorithmic Construction of Optimal Symmetric Latin Hypercube Designs
,”
J. Stat. Plann. Inference
,
90
, pp.
145
159
.
30.
Roberts
,
J. B.
, and
Spanos
,
P. D.
,
1999
,
Random Vibration and Statistical Linearization
,
Dover Publications
,
Mineola, NY
.
31.
Mohamad
,
M. A.
, and
Sapsis
,
T. P.
,
2015
, “
Probabilistic Description of Extreme Events in Intermittently Unstable Dynamical Systems Excited by Correlated Stochastic Processes
,”
SIAM/ASA J. Uncertainty Quantif.
,
3
(
1
), pp.
709
736
.
32.
Li
,
C. C.
, and
Kiureghian
,
A. D.
,
1993
, “
Optimal Discretization of Random Fields
,”
J. Eng. Mech.
,
119
(
6
), pp.
1136
1154
.
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