We study the reliability of stochastically excited and controlled single first-integral systems and present an approximate analytical solution for the first-passage rate (FPR). By introducing the stochastic averaging method (SAM), we reduce the dimension of the original system to an averaged one-dimensional controlled Ito^ differential equation. We then modify the classic Laplace integral method (LIM) and apply it to deal with the arduous integrals in the expression of reliability function. The procedure of acquiring the analytical solution for the reliability is illuminated in detail as well. In addition, we provide two controlled single first-integral nonlinear vibration systems, namely, the classical bistable model and the two coupled nonlinear oscillators, as examples. By comparing the results obtained from the modified Laplace integral method (MLIM) to Monte Carlo simulations (MCS), we verify the effectiveness and exactness of the proposed procedure. We identified two properties in the obtained analytical solution: One is that the solutions are independent of the initial system state. The other is that they are only effective in the high passage threshold range. Finally, a reasonable explanation has been given to explain these two properties.

References

References
1.
Hänggi
,
P.
,
Talkner
,
P.
, and
Borkovec
,
M.
,
1990
, “
Reaction Rate Theory: Fifty Years After Kramers
,”
Rev. Mod. Phys.
,
62
(
2
), pp.
251
341
.
2.
Connors
,
K. A.
,
1990
,
Chemical Kinetics: The Study of Reaction Rates in Solution
,
Wiley
,
New York
.
3.
Reimann
,
P.
,
Schmid
,
G. J.
, and
Hänggi
,
P.
,
1999
, “
Universal Equivalence of Mean First Passage Time and Kramers Rate
,”
Phys. Rev. E
,
60
(
1
), pp.
R1
R4
.
4.
Houston
,
P. L.
,
2001
,
Chemical Kinetics and Reaction Dynamics
,
McGraw-Hill
,
Boston, MA
.
5.
Ebeling
,
W.
,
Schimansky-Geier
,
W. L.
, and
Romanovsky
,
Y. M.
,
2002
,
Stochastic Dynamics of Reacting Biomolecules
,
Word Scientific
,
Singapore
.
6.
Noh
,
J. D.
, and
Rieger
,
H.
,
2004
, “
Random Walks on Complex Networks
,”
Phys. Rev. Lett.
,
92
(
11
), p.
118701
.
7.
Drebenkov
,
D. S.
,
2016
, “
Universal Formula for the Mean First Passage Time in Planar Domains
,”
Phys. Rev. Lett.
,
117
(
26
), p.
260201
.
8.
Holcman
,
D.
, and
Schuss
,
Z.
,
2017
, “
100 Years After Smoluchowski: Stochastic Processes in Cell Biology
,”
J. Phys. A
,
50
(
9
), p.
093002
.
9.
Song
,
Y.
,
Liu
,
Z. G.
,
Duan
,
F. C.
,
Lu
,
X. B.
, and
Wang
,
H. R.
,
2018
, “
Study on Wind-Induced Vibration Behavior of Railway Catenary in Spatial Stochastic Wind Field Based on Nonlinear Finite Element Procedure
,”
ASME J. Vib. Acoust.
,
140
(
1
), p.
011010
.
10.
Pontryagin
,
L. S.
,
Boltyanski
,
V. G.
,
Gamkrelidze
,
R. V.
, and
Mischenko
,
E. F.
,
1962
,
Mathematical Theory of Optimal Processes
,
Wiley
,
New York
.
11.
Bellman
,
R.
,
1957
,
Dynamic Programming
,
Princeton University Press
,
Princeton, NJ
.
12.
Zhu
,
W. Q.
,
2006
, “
Nonlinear Stochastic Dynamics and Control in Hamiltonian Formulation
,”
ASME Appl. Mech. Rev.
,
59
(
4
), pp.
230
248
.
13.
Zhu
,
W. Q.
,
Huang
,
Z. L.
, and
Deng
,
M. L.
,
2002
, “
Feedback Minimization of First-Passage Failure of Quasi Non-Integrable Hamiltonian Systems
,”
Int. J. Non-Linear Mech.
,
37
(
6
), pp.
1057
1071
.
14.
Zhu
,
W. Q.
, and
Deng
,
M. L.
,
2004
, “
Optimal Bounded Control for Minimizing the Response of Quasi Non-Integrable Hamiltonian Systems
,”
Nonlinear Dyn.
,
35
(
1
), pp.
81
100
.
15.
Pichler
,
L.
, and
Pradlwarter
,
H. J.
,
2009
, “
Evolution of Probability Densities in the Phase Space for Reliability Analysis of Non-Linear Structures
,”
Struct. Saf.
,
31
(
4
), pp.
316
324
.
16.
Hsu
,
C. S.
,
1981
, “
A Generalized Theory of Cell-to-Cell Mapping for Nonlinear Dynamical Systems
,”
ASME J. Appl. Mech.
,
48
(
3
), pp.
634
642
.
17.
Sun
,
J. Q.
, and
Hsu
,
C. S.
,
1988
, “
First-Passage Time Probability of Non-Linear Stochastic Systems by Generalized Cell Mapping Method
,”
J. Sound Vib.
,
124
(
2
), pp.
233
248
.
18.
Sun
,
J. Q.
, and
Hsu
,
C. S.
,
1989
, “
The Generalized Cell Mapping Method in Nonlinear Random Vibration Based Upon Short-Time Gaussian Approximation
,”
ASME J. Appl. Mech.
,
57
(
4
), pp.
1018
1025
.
19.
Sun
,
J. Q.
,
1995
, “
Random Vibration Analysis of a Nonlinear-System With Dry Friction Damping by the Short-Time Gaussian Cell Mapping Method
,”
J. Sound Vib.
,
180
(
5
), pp.
785
795
.
20.
Han
,
Q.
,
Xu
,
W.
,
Yue
,
X. L.
, and
Zhang
,
Y.
,
2015
, “
First-Passage Time Statistics in a Bistable System Subject to Poisson White Noise by the Generalized Cell Mapping Method
,”
Commun. Nonlinear Sci. Numer. Simul.
,
23
(
1–3
), pp.
220
228
.
21.
Wiener
,
N.
,
1930
, “
Generalized Harmonic Analysis
,”
Acta Math.
,
55
, pp.
117
258
.
22.
Feynman
,
R. P.
,
1948
, “
Space-Time Approach to Non-Relativistic Quantum Mechanics
,”
Rev. Mod. Phys.
,
20
(
2
), pp.
367
387
.
23.
Naess
,
A.
,
Iourtchenko
,
D.
, and
Batsevych
,
O.
,
2011
, “
Reliability of Systems With Randomly Varying Parameters by the Path Integration Method
,”
Probab. Eng. Mech.
,
26
(
1
), pp.
5
9
.
24.
Kougioumtzoglou
,
I. A.
, and
Spanos
,
P. D.
,
2013
, “
Response and First-Passage Statistics of Nonlinear Oscillators Via a Numerical Path Integral Approach
,”
J. Eng. Mech.
,
139
(
9
), pp.
1207
1217
.
25.
Kougioumtzoglou
,
I. A.
, and
Spanos
,
P. D.
,
2014
, “
Stochastic Response Analysis of the Softening Duffing Oscillator and Ship Capsizing Probability Determination Via a Numerical Path Integral Approach
,”
Probab. Eng. Mech.
,
35
, pp.
67
74
.
26.
Stratonovich
,
R. L.
,
1963
,
Topics in the Theory of Random Noise
,
Gordon Breach
,
New York
.
27.
Khasminskii
,
R. Z.
,
1966
, “
A Limit Theorem for the Solutions of Differential Equations With Random Right-Hand Sides
,”
Theory Probab. Appl.
,
11
, pp.
390
405
.
28.
Roberts
,
J. B.
, and
Spanos
,
P. D.
,
1986
, “
Stochastic Averaging: An Approximate Method of Solving Random Vibration Problems
,”
Int. J. Non-Linear Mech.
,
21
(
2
), pp.
111
134
.
29.
Lin
,
Y. K.
, and
Cai
,
G. Q.
,
1995
,
Probabilistic Structural Dynamics: Advanced Theory and Applications
,
McGraw-Hill
,
New York
.
30.
Zhu
,
W. Q.
,
1988
, “
Stochastic Averaging Methods in Random Vibration
,”
ASME Appl. Mech. Rev.
,
41
(
5
), pp.
189
199
.
31.
Zhu
,
W. Q.
,
1996
, “
Recent Developments and Applications of the Stochastic Averaging Method in Random Vibration
,”
ASME Appl. Mech. Rev.
,
49
(10S), pp.
72
80
.
32.
Gan
,
C. B.
, and
Zhu
,
W. Q.
,
2001
, “
First-Passage Failure of Quasi-Non-Integrable-Hamiltonian System
,”
Int. J. Non-Linear Mech.
,
36
(
2
), pp.
209
220
.
33.
Zhu
,
W. Q.
,
Deng
,
M. L.
, and
Huang
,
Z. L.
,
2001
, “
First-Passage Failure of Quasi-Integrable Hamiltonian Systems
,”
ASME J. Appl. Mech.
,
69
(
3
), pp.
274
282
.
34.
Zhu
,
W. Q.
,
Huang
,
Z. L.
, and
Deng
,
M. L.
,
2003
, “
First-Passage Failure and Its Feedback Minimization of Quasi-Partially Integrable Hamiltonian Systems
,”
Int. J. Non-Linear Mech.
,
38
(
8
), pp.
1133
1148
.
35.
Wang
,
S. L.
,
Jin
,
X. L.
,
Wang
,
Y.
, and
Huang
,
Z. L.
,
2014
, “
Reliability Evaluation and Control for Wideband Noise-Excited Viscoelastic Systems
,”
Mech. Res. Commun.
,
62
, pp.
57
65
.
36.
Li
,
W.
,
Chen
,
L. C.
,
Trisovic
,
N.
,
Cvetkovic
,
A.
, and
Zhao
,
J. F.
,
2015
, “
First Passage of Stochastic Fractional Derivative Systems With Power-Form Restoring Force
,”
Int. J. Non-Linear Mech.
,
71
, pp.
83
88
.
37.
Sun
,
J. J.
,
Xu
,
W.
, and
Lin
,
Z. F.
,
2018
, “
Research on the Reliability of Friction System Under Combined Additive and Multiplicative Random Excitations
,”
Commun. Nonlinear Sci. Numer. Simul.
,
54
, pp.
1
12
.
38.
Hammersley
,
J. M.
, and
Handscomb
,
D. C.
,
1964
,
Monte-Carlo Methods
,
Methuen
,
London
.
39.
Rubinstein
,
R. Y.
,
1981
,
Simulations and the Monte-Carlo Method
,
Wiley
,
New York
.
40.
Pradlwarter
,
H. J.
,
Schuёller
,
G. I.
,
Koutsourelakis
,
P. S.
, and
Charmpis
,
D. C.
,
2007
, “
Application of Line Sampling Simulation Method to Reliability Benchmark Problems
,”
Struct. Saf.
,
29
(
3
), pp.
208
221
.
41.
Pradlwarter
,
H. J.
, and
Schuёller
,
G. I.
,
1997
, “
On Advanced MCS Procedures in Stochastic Structural Dynamics
,”
J. Nonlinear Mech.
,
32
(
4
), pp.
735
744
.
42.
Pradlwarter
,
H. J.
, and
Schuёller
,
G. I.
,
2004
, “
Excursion Probability of Non-Linear Systems
,”
Int. J. Non-Linear Mech.
,
39
(
9
), pp.
1447
1452
.
43.
Au
,
S. K.
, and
Beck
,
J. L.
,
2001
, “
Estimates of Small Failure Probabilities in High Dimensions by Subset Simulation
,”
Probab. Eng. Mech.
,
16
(
4
), pp.
263
277
.
44.
Koutsourelakis
,
P. S.
,
Pradlwarter
,
H. J.
, and
Schuёller
,
G. I.
,
2004
, “
Reliability of Structures in High Dimensions—Part I: Algorithms and Applications
,”
Probab. Eng. Mech.
,
19
(
4
), pp.
409
417
.
45.
Vanvinckenroye
,
H.
, and
Denoël
,
V.
,
2017
, “
Average First-Passage Time of a Quasi-Hamiltonian Mathieu Oscillator With Parametric and Forcing Excitations
,”
J. Sound Vib.
,
406
, pp.
328
345
.
46.
Vanvinckenroye
,
H.
, and
Denoël
,
V.
,
2018
, “
Second-Order Moment of the First Passage Time of a Quasi-Hamiltonian Oscillator With Stochastic Parametric and Forcing Excitations
,”
J. Sound Vib.
,
427
, pp.
178
187
.
47.
Asmussen
,
S.
,
2000
,
Ruin Probabilities
,
World Scientific
,
Singapore
.
48.
Cramer
,
H.
, and
Leadbetter
,
M. R.
,
1967
,
Stationary and Related Stochastic Processes
,
Wiley
,
New York
.
49.
Deng
,
M. L.
,
Fu
,
Y.
, and
Huang
,
Z. L.
,
2014
, “
Asymptotic Analytical Solutions of First-Passage Rate to Quasi-Nonintegrable Hamiltonian Systems
,”
ASME J. Appl. Mech.
,
81
(
8
), p.
081012
.
50.
Wong
,
E.
, and
Zakai
,
M.
,
1965
, “
On the Relation Between Ordinary and Stochastic Differential Equations
,”
Int. J. Eng. Sci.
,
3
(
2
), pp.
213
229
.
51.
Tabor
,
M.
,
1989
,
Chaos and Integrability in Nonlinear Dynamics
,
Wiley
,
New York
.
52.
Khasminskii
,
R. Z.
,
1968
, “
On the Averaging Principle for Stochastic Differential Ito^ Equations
,”
Kibernetika
,
4
(
3
), pp.
260
279
.
53.
Zhu
,
W. Q.
, and
Yang
,
Y. Q.
,
1997
, “
Stochastic Averaging of Quasi Non-Integrable-Hamiltonian Systems
,”
ASME J. Appl. Mech.
,
64
(
1
), pp.
157
164
.
54.
Miller
,
P. D.
,
2006
,
Applied Asymptotic Analysis
,
American Mathematical Society
,
Providence, RI
.
55.
vandenBrink
,
A. M.
, and
Dekker
,
H.
,
1997
, “
Reaction Rate Theory: Weak- to Strong-Friction Turnover in Kramers' Fokker-Planck Model
,”
Physica A
,
237
(
3–4
), pp.
515
553
.
56.
Cramer
,
H.
, and
Leadbetter
,
M. R.
,
2004
,
Stationary and Related Stochastic Processes: Sample Function Properties and Their Applications
,
Dover Publications
,
New York
.
You do not currently have access to this content.