Stochastic Structural Dynamics and Some Recent Developments Y. K. Lin Center for Applied Stochastics Research Florida Atlantic University Boca Raton, FL.

Stochastic Structural Dynamics and Some Recent Developments

Y. K. Lin

Center for Applied Stochastics ResearchFlorida Atlantic UniversityBoca Raton, FL 33431

U.S.A.

)(cos)()(sin)]([)(cos)](sin[ ttPttPttdt

dc 211

t

Pt

P[1+t

c

l

,1)(If t

excitation additive)(

excitation tivemultiplica)(

)()()](1[)(

2

1

21

t

t

ttttdt

d

P

c

More generally

);(),(),( ttgtfX kjkjj XX

mj ...,,2,1

nk ...,,2,1

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

→ →

(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

(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)…..

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.

Some exact solutions are obtainable, when

(1) System is nonlinear

(2) Some inputs are multiplicative

Use mathematical theory of diffusive Markov processes.

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

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

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).

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

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

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

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)

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

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

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)

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

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

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

A General Approximation Scheme

),()],(),([x

errorResidual

)()('

).,(fornrestrictiothesatisfynotdoes),(where

)(),(),(

2212121

12

2

2121

21212

21

xxpxxhxxH

xux

ggKggKxh

xxhxxH

tWXXgXXHX

XX

s

jiijjiij

ii

Method of Weighted Residual

averageensemble

),()],(),([

partbynIntegratio

),(

),(functionweightingaChoose

`

E

xxMx

xxhxxHE

dxdxxxM

xxM

M

M

0

0

212

212

2121

21

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

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

Stochastic Averaging (Stratonovich, 1963; Khasminskii, 1966)

An Example – A column excited by horizontal and vertical earthquakes

m

Consider one dominant mode:

)()]([ tYtYY 21200 12

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

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)

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)

System Failures

(i) First-Passage Failure

(ii) Fatigue Failure

(iii) Motion Instability

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

For Linear System

Stability in Probability Sample Stability

Stability in Sample Stability

1L

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

Mathiew-Hill Equation

Strutt diagram

0cos2 200 xtxx

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

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

EXAMPLE – BRIDGE IN TURBULENT WIND

TWO TORSIONAL AND TWO VERTICAL MODES

lengthn correlatio turbulenceL

lengthspan bridge

)1.0 ratio (advance

tcoefficien thrust andfrequency lag of in terms boundariesStability

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

T = random time when the first-passage failure occurs

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

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

001n

001n

0

100

2

010

0j

0

00

0

0

if finite,)],([

if 0,)],([

conditionsBoundary

1

)1()(2

1)(a

equation) Pontryagin ed(generaliz satisfies),(

),,(][),(

ofmoment th The

),,(: ofdensity y probabilit The

xx

xx

xx

x

xx

x

An example for first-passage failure

-

0

t coefficienn restitutio

)cos()()1()sin()y1()(sgn

cGcG xMRgMRgI

Average toppling time vs. base excitation level. Solid line: horizontal excitation only; dotted line: combined horizontal and vertical excitations.

time topplingaverage

5.0

ofdensity spectral

ofdensity spectral

)/MRg(

1

2/10

h

v

Gh

Gv

K

K

yK

xK

I

sexcitation verticaland horizontal combined :line Dotted

only excitation horizontal :line Solid

level excitation base vs. time topplingofdeviation Standard

R

sexcitation verticaland horizontal combined :line Dotted

only. excitation horizontal :line Solid scale. size vs. time topplingAverage

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.