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![Page 1: Stochastic Structural Dynamics and Some Recent Developments Y. K. Lin Center for Applied Stochastics Research Florida Atlantic University Boca Raton, FL.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649e555503460f94b4d0b0/html5/thumbnails/1.jpg)
Stochastic Structural Dynamics and Some Recent Developments
Y. K. Lin
Center for Applied Stochastics ResearchFlorida Atlantic UniversityBoca Raton, FL 33431
U.S.A.
![Page 2: Stochastic Structural Dynamics and Some Recent Developments Y. K. Lin Center for Applied Stochastics Research Florida Atlantic University Boca Raton, FL.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649e555503460f94b4d0b0/html5/thumbnails/2.jpg)
)(cos)()(sin)]([)(cos)](sin[ ttPttPttdt
dc 211
t
Pt
P[1+t
c
l
![Page 3: Stochastic Structural Dynamics and Some Recent Developments Y. K. Lin Center for Applied Stochastics Research Florida Atlantic University Boca Raton, FL.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649e555503460f94b4d0b0/html5/thumbnails/3.jpg)
,1)(If t
excitation additive)(
excitation tivemultiplica)(
)()()](1[)(
2
1
21
t
t
ttttdt
d
P
c
![Page 4: Stochastic Structural Dynamics and Some Recent Developments Y. K. Lin Center for Applied Stochastics Research Florida Atlantic University Boca Raton, FL.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649e555503460f94b4d0b0/html5/thumbnails/4.jpg)
More generally
);(),(),( ttgtfX kjkjj XX
mj ...,,2,1
nk ...,,2,1
![Page 5: Stochastic Structural Dynamics and Some Recent Developments Y. K. Lin Center for Applied Stochastics Research Florida Atlantic University Boca Raton, FL.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649e555503460f94b4d0b0/html5/thumbnails/5.jpg)
Historical Development
(1) Physicists – Brownian Motion
Einstein (1905)
W(t) = Gaussian white noise
Ornstein – Uhlenbeck (1930)
Wang – Uhlenbeck (1945)
W(t) = vector of Gaussian white noises
GaussianInput
Linear System
GaussianOutput
)(tWY
)(tWkYYc
)(][][][ tY WkYcYm
→ →
![Page 6: Stochastic Structural Dynamics and Some Recent Developments Y. K. Lin Center for Applied Stochastics Research Florida Atlantic University Boca Raton, FL.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649e555503460f94b4d0b0/html5/thumbnails/6.jpg)
(2) Electrical Engineers–GeneralizedHarmonic Analysis for Communication Systems
Weiner (1930)Khintchine (1934)Rice (1944)
vector of weakly stationary random processes
Objective – obtain correlation functions (or spectral densities) of the response from those of excitations.
)(tF
)(][][][ tFYkYcYm
![Page 7: Stochastic Structural Dynamics and Some Recent Developments Y. K. Lin Center for Applied Stochastics Research Florida Atlantic University Boca Raton, FL.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649e555503460f94b4d0b0/html5/thumbnails/7.jpg)
(3) Mechanical and Aerospace Engineers
Turbulence
Flight vehicles excited by turbulence, jet noise, rocket noise
Rayleigh (1919)Pontryagin, Andronov, Vitt (1933)Taylor (1935)C. C. Lin (1944)CrandallCaugheyBolotin…..
(4) Civil Engineers – Winds, earthquakes, road roughness
Housner (1941)…..
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Solution Forms
(1) Input Statistical Properties
→ Output Statistical Properties
Possible if (a) system is linear, and (b) inputs are additive.
(2) Input Probability Distribution
→ Output Probability Distribution
Possible if (a) system is linear, and
(b) inputs are additive and Gaussian.
![Page 9: Stochastic Structural Dynamics and Some Recent Developments Y. K. Lin Center for Applied Stochastics Research Florida Atlantic University Boca Raton, FL.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649e555503460f94b4d0b0/html5/thumbnails/9.jpg)
Some exact solutions are obtainable, when
(1) System is nonlinear
(2) Some inputs are multiplicative
Use mathematical theory of diffusive Markov processes.
![Page 10: Stochastic Structural Dynamics and Some Recent Developments Y. K. Lin Center for Applied Stochastics Research Florida Atlantic University Boca Raton, FL.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649e555503460f94b4d0b0/html5/thumbnails/10.jpg)
Markov Random Process
1111 xtXxtXxtX nnnn )(,,)()(Prob
),,(
)()(Prob
11
11
nnnn
nnnn
txtxF
xtXxtX
11 ttt nn
One-Step Memory
Generalization to Multi-Dimensional Markov Process X(t)
Transition probability distribution
),,( 00 ttF xx
Transition probability density
),,( 00 ttq xx
![Page 11: Stochastic Structural Dynamics and Some Recent Developments Y. K. Lin Center for Applied Stochastics Research Florida Atlantic University Boca Raton, FL.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649e555503460f94b4d0b0/html5/thumbnails/11.jpg)
Fokker-Planck-Kolmogorov (FPK) Equation for Markov Random Process
Reduced FPK Equation for Stationary Markov Process
)(),,( 00 xxx pttq
0)(2
1)(
2
pbxx
pax jk
kjj
j
jx The jth component of x
drift coefficients ja
jkb diffusion coefficients
0)(2
1)(
2 2
qbxx
qax
qt jk
kjj
j
![Page 12: Stochastic Structural Dynamics and Some Recent Developments Y. K. Lin Center for Applied Stochastics Research Florida Atlantic University Boca Raton, FL.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649e555503460f94b4d0b0/html5/thumbnails/12.jpg)
Exact Probability Solutions for Multi-Dimensional Nonlinear Systems (Restricted to Gaussian white noise excitations)
Early Works (additive excitation only)
(1) Nonlinear stiffness, linear damping(Pontryagin, et. al 1933).
(2) Additional requirement for MDF systems – equipartition of kinetic energy.
(3) Nonlinear damping (Caughey) – replacing constant damping coefficient by a function of total potential.
Adding Multiplicative Excitations
(1) First success (Dimentberg 1982)(2) Detailed balance (Yong-Lin 1987)(3) Generalized stationary potential (Lin-Cai 1988).(4) Removing the restriction of equipartition energy (Cai-Lin-Zhu).
![Page 13: Stochastic Structural Dynamics and Some Recent Developments Y. K. Lin Center for Applied Stochastics Research Florida Atlantic University Boca Raton, FL.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649e555503460f94b4d0b0/html5/thumbnails/13.jpg)
A Single-Degree-of-Freedom System
)(),()(),( tWYYgYuYYhY jj
),( YYh = damping term
)(Yu = stiffness term
)(tW j = Gaussian white noises
)()]()([ jkkj KtWtWE 2
jkK = a constant (cross-spectral density of )kj WW and
![Page 14: Stochastic Structural Dynamics and Some Recent Developments Y. K. Lin Center for Applied Stochastics Research Florida Atlantic University Boca Raton, FL.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649e555503460f94b4d0b0/html5/thumbnails/14.jpg)
Fokker-Planck-Kolmogorov (F-P-K) Equation for )(andfor),(Densityy ProbabilitStationaryJoint tYY(t)xxp
21
px
ggKxuxxh
xx
px j
kjk2
12121
2 )(),(
022
2
)( pggx
K kjjk
)( ofvariablestate tYx 2
)( ofvariablestate tYx 1
)(and)(ofdensityspectraljoint tWtWK kjjk
termcorrectionZakaiWong
jkjk gx
gK2
)on dependswhenoccursit( 2xg j
![Page 15: Stochastic Structural Dynamics and Some Recent Developments Y. K. Lin Center for Applied Stochastics Research Florida Atlantic University Boca Raton, FL.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649e555503460f94b4d0b0/html5/thumbnails/15.jpg)
Additive Excitation Only );(tW
Y
UYmYm j
jjjjj
nj ,2,1
jjjj YXYX 1,Let
Under the condition
jK
m
jj
jj allfor,
),,(
2
1exp 1
2
1nnjj
n
jxxUxmCp
Equi-partition of kinetic energy
![Page 16: Stochastic Structural Dynamics and Some Recent Developments Y. K. Lin Center for Applied Stochastics Research Florida Atlantic University Boca Raton, FL.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649e555503460f94b4d0b0/html5/thumbnails/16.jpg)
Method of Generalized Stationary Potential (Lin-Cai)
)()(
(2)(1)
)(
)(
kkj
jjkjk
jjj
bbxb
aaxa
Reduced F-P Equation
0][][ )2()()1(
sjj
sj
jkk
sjj
pax
pbx
pax
Sufficient Conditions
0][ (2)
sjj
pax
0][ )()1( s
jjk
ksj pb
xpa
(a)
(b)
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Single DOF
02
2
2121
212
)(
])(),([
pggx
K
px
ggKxuxxh
xx
px
kjjk
jkjk
)(),()(),( tWXXgXuXXhX jj 211212 21 XX
or)(2)]()([where 2121 ttKtWtWE jkkj
jtWYYgYuYYhY jj oversum,)(),()(),(
Reduced FPK equation
![Page 18: Stochastic Structural Dynamics and Some Recent Developments Y. K. Lin Center for Applied Stochastics Research Florida Atlantic University Boca Raton, FL.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649e555503460f94b4d0b0/html5/thumbnails/18.jpg)
Split the Wong-Zakai correction term, if exists, into two parts
2x
pggK kjjk
0
The F-P-K equation can be re-arranged as follows:
2
111
2 x
pxuxu
x
px
)()(
2x
pxxhxxh )],(),([ 2121
(B)
(A)
)(),( 1212
xuxxhx
ggK j
kjk
![Page 19: Stochastic Structural Dynamics and Some Recent Developments Y. K. Lin Center for Applied Stochastics Research Florida Atlantic University Boca Raton, FL.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649e555503460f94b4d0b0/html5/thumbnails/19.jpg)
Replacing the F-P-K equation by the sufficient conditions
02
111
2
x
pxuxu
x
px )(( (A)
02
2121
x
pggKpxxhxxh kjjk),(),( (B)
Solving for (A)
)](exp[),( Cxxp 21
where
dvvuvuxx
)]()([ 1
0
222
1
Restriction [from (B)]:
2
2121
xggK
xxhxxh
kjjk ),(),(
)('
(generalized stationary potential)
![Page 20: Stochastic Structural Dynamics and Some Recent Developments Y. K. Lin Center for Applied Stochastics Research Florida Atlantic University Boca Raton, FL.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649e555503460f94b4d0b0/html5/thumbnails/20.jpg)
Detailed Balance - a special case - (Haken)
jj xxtt ~,:reversalTime
)]~()([2
1
)]~()([2
1
Then
odd,1
even,1
)oversum(no,~
or,
velocity)(e.g.~:variableodd
nt)displaceme(e.g.~:variableeven
)2(
(1)j
xaxaaa
xaxaaa
ε
jxx
xx
xx
jjjRjj
jjjIj
j
jjj
jj
jj
![Page 21: Stochastic Structural Dynamics and Some Recent Developments Y. K. Lin Center for Applied Stochastics Research Florida Atlantic University Boca Raton, FL.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649e555503460f94b4d0b0/html5/thumbnails/21.jpg)
EXAMPLE 1: System in detailed balance
21
202
1222
1
0 3322
2/1332221
201122
332211
321
2202
1221
31202
],)2(
)(1exp[)2(),(
then,If
and,spectra with
noiceswhiteGaussiantindependen)(),(),(
),()](1[)]()([
xx
dvKK
vhKKCxxp
KK
KKK
tWtWtW
YY
tWYtWYtWhY
![Page 22: Stochastic Structural Dynamics and Some Recent Developments Y. K. Lin Center for Applied Stochastics Research Florida Atlantic University Boca Raton, FL.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649e555503460f94b4d0b0/html5/thumbnails/22.jpg)
EXAMPLE 2: System not in detailed balance
1
0
2111
22
22
211
21
21122
211
21
212
])(
2
1
2
1[exp),(
then),)(/(,/If
).()(,0
,noiseswhiteGaussiantindependen)(),(
)()()()()(
x
dvvu
xKxvK
KCxxp
KKK
YuYu
tWtW
tWtWYYYuYYYYY
![Page 23: Stochastic Structural Dynamics and Some Recent Developments Y. K. Lin Center for Applied Stochastics Research Florida Atlantic University Boca Raton, FL.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649e555503460f94b4d0b0/html5/thumbnails/23.jpg)
A General Approximation Scheme
),()],(),([x
errorResidual
)()('
).,(fornrestrictiothesatisfynotdoes),(where
)(),(),(
2212121
12
2
2121
21212
21
xxpxxhxxH
xux
ggKggKxh
xxhxxH
tWXXgXXHX
XX
s
jiijjiij
ii
![Page 24: Stochastic Structural Dynamics and Some Recent Developments Y. K. Lin Center for Applied Stochastics Research Florida Atlantic University Boca Raton, FL.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649e555503460f94b4d0b0/html5/thumbnails/24.jpg)
Method of Weighted Residual
averageensemble
),()],(),([
partbynIntegratio
),(
),(functionweightingaChoose
`
E
xxMx
xxhxxHE
dxdxxxM
xxM
M
M
0
0
212
212
2121
21
![Page 25: Stochastic Structural Dynamics and Some Recent Developments Y. K. Lin Center for Applied Stochastics Research Florida Atlantic University Boca Raton, FL.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649e555503460f94b4d0b0/html5/thumbnails/25.jpg)
Dissipation Energy Balancing
)(forequationanis(4)
).(involves),(:Note
(5)])(exp[),(
(4)0),()],(),([
BalancingEnergynDissipatio
unchanged.forceveconservatikeep(2)
unchanged.sidehandrightthekeep(1)
)(),(),(
SystemngSubstituti
)(),(),(
SystemOriginal
21
21
212121212
22
xxh
dCxxp
dxdxxxpxxhxxHx
xM
u(X)
tWXXgXXhX
tWXXgXXHX
s
s
ii
ii
![Page 26: Stochastic Structural Dynamics and Some Recent Developments Y. K. Lin Center for Applied Stochastics Research Florida Atlantic University Boca Raton, FL.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649e555503460f94b4d0b0/html5/thumbnails/26.jpg)
Reduction of Dimensionality
(1) Averaging techniques (generalization of Krylov-Bogoliubov-Mitropolsky techniques for deterministic systems)
(a) Stochastic averaging – linear or weakly nonlinear stiffness terms.
(b) Quasi-conservative averaging – strongly nonlinear stiffness term.
(c) Second-order averaging.
(2) Slaving principle (Haken)
Master – slow motion
Slave – fast motion
![Page 27: Stochastic Structural Dynamics and Some Recent Developments Y. K. Lin Center for Applied Stochastics Research Florida Atlantic University Boca Raton, FL.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649e555503460f94b4d0b0/html5/thumbnails/27.jpg)
Stochastic Averaging (Stratonovich, 1963; Khasminskii, 1966)
An Example – A column excited by horizontal and vertical earthquakes
m
Consider one dominant mode:
)()]([ tYtYY 21200 12
![Page 28: Stochastic Structural Dynamics and Some Recent Developments Y. K. Lin Center for Applied Stochastics Research Florida Atlantic University Boca Raton, FL.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649e555503460f94b4d0b0/html5/thumbnails/28.jpg)
Transformation
,sin)(
)()(,cos)(
0
0
tAY
tttAY
)(cos1
)(coscossin2
)(sin1
)(cossinsin2
20
12
00
20
102
0
tA
t
ttAAA
2/121 order)(),(,order:Assumption tt
vectorMarkova)(
)(
t
tA
![Page 29: Stochastic Structural Dynamics and Some Recent Developments Y. K. Lin Center for Applied Stochastics Research Florida Atlantic University Boca Raton, FL.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649e555503460f94b4d0b0/html5/thumbnails/29.jpg)
processMarkovscarlaraitself
,averagedtheoftindependenAveraged φ(t)A(t)
)()()(
)()(
/
tdBA
dtAdA
21
02220
2011
20
02220
011200
24
22
8
3
For systems with strongly nonlinear stiffness:
A is replaced by total energy U (or more generally Hamiltonian)
![Page 30: Stochastic Structural Dynamics and Some Recent Developments Y. K. Lin Center for Applied Stochastics Research Florida Atlantic University Boca Raton, FL.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649e555503460f94b4d0b0/html5/thumbnails/30.jpg)
Quasi-Conservative Averaging
For undamped free vibration
Integrating
Combining (a) and (b)
)(])(22,[)(22,()(22
)](),(),([
2/1
2/1
tWYUYgYuYfYu
tWYYgYYfY
jj
jj
Eq.(a) is now replaced by Eqs. (b) and (c)
odd )(where),(),(),()( 2/1 utWYYgYYfYuY jj
)(2)]()([ jkkj KtWtWE
0)( YuY
amplitude)(2
1 2 YUY
a
YU
dYTT
04/1 )([2
44 = quasi-period
(a)
(b)
(c)
![Page 31: Stochastic Structural Dynamics and Some Recent Developments Y. K. Lin Center for Applied Stochastics Research Florida Atlantic University Boca Raton, FL.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649e555503460f94b4d0b0/html5/thumbnails/31.jpg)
System Failures
(i) First-Passage Failure
(ii) Fatigue Failure
(iii) Motion Instability
![Page 32: Stochastic Structural Dynamics and Some Recent Developments Y. K. Lin Center for Applied Stochastics Research Florida Atlantic University Boca Raton, FL.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649e555503460f94b4d0b0/html5/thumbnails/32.jpg)
Stochastic Stability Concepts
(1) Lyapunov Stability with Probability One (Sample Stability):
00
21
)(provided
)(Prob sup0
0
xX
Xx
t
tUtt
(2) Stability in Probability
(3) Stability in nL
00
21
)(provided
)(Prob
xX
X
t
t
00 )(provided
(t)E
xX
X
t
εn
![Page 33: Stochastic Structural Dynamics and Some Recent Developments Y. K. Lin Center for Applied Stochastics Research Florida Atlantic University Boca Raton, FL.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649e555503460f94b4d0b0/html5/thumbnails/33.jpg)
For Linear System
Stability in Probability Sample Stability
Stability in Sample Stability
1L
![Page 34: Stochastic Structural Dynamics and Some Recent Developments Y. K. Lin Center for Applied Stochastics Research Florida Atlantic University Boca Raton, FL.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649e555503460f94b4d0b0/html5/thumbnails/34.jpg)
Column under Fluctuating Axial Load
P0 - P1cost
W
21
0
220
20
2
2
104
4
1
0
0
m
P
PPm
xtx
txtW
WtPPWEIWcWm
cr
cos2x
yieldto,sin)(),(Let
]cos[
0
![Page 35: Stochastic Structural Dynamics and Some Recent Developments Y. K. Lin Center for Applied Stochastics Research Florida Atlantic University Boca Raton, FL.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649e555503460f94b4d0b0/html5/thumbnails/35.jpg)
Mathiew-Hill Equation
Strutt diagram
0cos2 200 xtxx
![Page 36: Stochastic Structural Dynamics and Some Recent Developments Y. K. Lin Center for Applied Stochastics Research Florida Atlantic University Boca Raton, FL.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649e555503460f94b4d0b0/html5/thumbnails/36.jpg)
The Column Problem
)(cos2sin
)(cossinsin2
sin)(
)(,cos)(
processrandomstationaryband-wide)(
0)(12
200
02
0
0
0
200
tdt
d
tAAdt
dA
tAX
tttAX
t
XtXX
![Page 37: Stochastic Structural Dynamics and Some Recent Developments Y. K. Lin Center for Applied Stochastics Research Florida Atlantic University Boca Raton, FL.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649e555503460f94b4d0b0/html5/thumbnails/37.jpg)
Averaged A(t) – independent of the averaged , itself a scalar Markov process
F1(t)
)()(
)(
/
tdBA
dtAdA
212
01120
011200
24
28
3
Condition for stability in probability
)( 011 24
)(t
![Page 38: Stochastic Structural Dynamics and Some Recent Developments Y. K. Lin Center for Applied Stochastics Research Florida Atlantic University Boca Raton, FL.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649e555503460f94b4d0b0/html5/thumbnails/38.jpg)
EXAMPLE – BRIDGE IN TURBULENT WIND
TWO TORSIONAL AND TWO VERTICAL MODES
lengthn correlatio turbulenceL
lengthspan bridge
![Page 39: Stochastic Structural Dynamics and Some Recent Developments Y. K. Lin Center for Applied Stochastics Research Florida Atlantic University Boca Raton, FL.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649e555503460f94b4d0b0/html5/thumbnails/39.jpg)
)1.0 ratio (advance
tcoefficien thrust andfrequency lag of in terms boundariesStability
![Page 40: Stochastic Structural Dynamics and Some Recent Developments Y. K. Lin Center for Applied Stochastics Research Florida Atlantic University Boca Raton, FL.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649e555503460f94b4d0b0/html5/thumbnails/40.jpg)
FIRST PASSAGE FAILURE
. within conservednot is )( of measureyprobabilit The *
. with timedecreasing is functions sample ofnumber totalThe*
space safe
. reachesit once removed
bemust function sample a :boundary absorbing*
.first time for the reaches)( when fails System
:failure passage-First
states failure ofunion
f
f
f
f
f
f
Bt
B
B
B
Bt
B
X
X
![Page 41: Stochastic Structural Dynamics and Some Recent Developments Y. K. Lin Center for Applied Stochastics Research Florida Atlantic University Boca Raton, FL.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649e555503460f94b4d0b0/html5/thumbnails/41.jpg)
T = random time when the first-passage failure occurs
![Page 42: Stochastic Structural Dynamics and Some Recent Developments Y. K. Lin Center for Applied Stochastics Research Florida Atlantic University Boca Raton, FL.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649e555503460f94b4d0b0/html5/thumbnails/42.jpg)
Reliability
ff
ff
fttf
jkj
kjj kjk
jjj
n
f
BtBtR
B),t;R(t,B
B),;tR(t,B
ba
Rxx
tbRx
taRt
tR
xxx
t
tTtBtR
000
000
000
00
2
000
000
020100
0
00
iffinite,,;,(
if,0
conditionsBoundary
if,1
:condition Initial
motion. of equations from obtained becan , tscoefficien
0),(2
1,
satisfies)(
,...,,stateinitial
timeinitial
][Prob),;,(
:functiony Reliabilit
0
xx
xx
xx
xx
x
x
![Page 43: Stochastic Structural Dynamics and Some Recent Developments Y. K. Lin Center for Applied Stochastics Research Florida Atlantic University Boca Raton, FL.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649e555503460f94b4d0b0/html5/thumbnails/43.jpg)
Statistical Moments of First-Passage Time T
Condition: aj, bjk independent of t0
ff
ff
nnkjj k
jknjj
fn
fnn
fn
f
BB
BB
nxx
bx
B
dBpTEB
Tn
BtpT
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Average toppling time vs. base excitation level. Solid line: horizontal excitation only; dotted line: combined horizontal and vertical excitations.
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sexcitation verticaland horizontal combined :line Dotted
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![Page 47: Stochastic Structural Dynamics and Some Recent Developments Y. K. Lin Center for Applied Stochastics Research Florida Atlantic University Boca Raton, FL.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649e555503460f94b4d0b0/html5/thumbnails/47.jpg)
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![Page 48: Stochastic Structural Dynamics and Some Recent Developments Y. K. Lin Center for Applied Stochastics Research Florida Atlantic University Boca Raton, FL.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649e555503460f94b4d0b0/html5/thumbnails/48.jpg)
Concluding Remarks
(1) The present review is focused on analytical solutions. The important Monte Carlo simulation techniques are not covered, such as the works by Shinozuka, Schuëller, and Pradlwarter, etc.
(2) Recent works by Arnold and his associates on dynamical systems are not covered.
(3) Numerical solutions, such as those given by Naess and Johnson, Bergman and Spencer, Kloeden et al, etc. also are not covered.