Random Vibrations - Stochastic Structural Dynamics - Lecture Slides, Slides of Stochastic Processes

Some basics concept of Stochastic Structural Dynamics are Moment of Input, Monte Carlo Simulation Approach, Multi-Dimensional Random Variables, Probabilistic Model.Main pouints of this lecture are: Random Vibrations, Random Processes, Joint Description, Cross Covariance Function, Joint Stationarity, Strong Sense Stationarity, Covariance Matrix, Phase Spectrum, Complex Coherency Function, Gaussian Random Process

Typology: Slides

2012/2013

Uploaded on 04/24/2013

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Random processes-4
Random vibrations of sdof systems-1
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Random processes-4Random vibrations of sdof systems-

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Recall•Gaussian random process•Poisson process•Simple Random walk•Wiener process•Brownian motion•Random pulse process•Gaussian white noise processMean square derivative

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  ^ ^

^ 

^      ^

^ ^

^ ^

^ ^

^

1 1 1

2 2 1 1 2

1

1 2

2

1 2 1

2 1

2

Description of

,^
,^ ;^ ,

n i^

i i V V

V VV^

V^

V^

VV

V t P V^ t^

v P V^ t^

v^ V^

t^ v P^ V^

t^ v p^ v t m^ t^

vp^ v t dv C^ t^

t^

v^ m^

t^ v^

m^ t^

p^ v^ v^ t^ t^

dv dv

  

^    ^

^
^
^
^
^
^
^
^
^
 ^
^
^
^
 ^

      

 

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5

^ ^

^ 

^ ^

^ 

^

^ ^

^ ^

^ ^

^

1 1

2 2 1

1 1 2

1

1 2

2

1 2 1

2 1

2

Joint description of

( ) and

( )

, ; , ,^

,^ ;^ ,

n^

m i^ i^

j^ j

i^

j

UV UV^

U^

V^

UV

U t^

V t

P U^ t^

u^ V^

t^ v P^ U

t^ u

V^ s^

v

p^ u v t sC^ t^ t

u^ m^

t^ v^

m^ t^

p^ u^

v^ t^ t^

du dv

^

 ^    ^

^

^

 ^

^

 ^

^

^ ^

^

^ 

^

^

^ ^

 ^

^

 ^

 ^

^

 ^

 ^

^

^

 ^

 ^

  ^  

   

 ^

 Cross covariance function

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^

^

^

^

^

^ ^

^

^ ^

^ ^

^  ^ ^

^ 

^ ^

 ^ 

^

^  

^ ^

(^1 2) ^ 

1 2

1 2

1 2

1 2

Covariance matrix

,^

,

,^

,^

,

UU^

UV VU^

VV

UU^

UV VU^

VV

UV UV^

C^ t^ t VU

C

t^

t

C t^

t^

C^

t^ t^

C^ t

t

C^

C

C^

C^

C

C^

U^ t V

t^

V^ t^

U^ t

C^

C

^

 ^

 ^

^

^

 ^

^

^

^

^

^

^

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^ ^

^ ^

^  ^ ^

^ ^

^ 

^ ^

^ ^

^ 

^    ^ ^

^  ^ ^

exp   1

exp 2

exp amplitude of cross PSD functionphase spectrumRe^

co-spectrum Im^

quadrature spectrum

UV^

UV UV^

UV UV^

UV UV UV^

UV UV^

UV S^

R^

i^ d

R^

S^

i^ d

S^
S^

i

S p^

S

q^

S

^

^  

^   

     

 

     

^
  ^
^
^
^
^
^

 

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^ ^

^   ^ ^ 

^ ^

^ ^

^  

^ ^

^   ^ ^ 

^   

Complex coherency functioncoh coh^

coh^

exp Coherencycoh 0 coh^

1 coh^

0 lack of linear dependency between two processesTwo processes are linearly relatedcoh

UV UV

UU^

VV UV^

UV UV UV

UU^

VV UV UV UV

S S S

iS S S ^

^  

   ^

^   ^  

  ^

 

 ^

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   ^

   ^ ^

^

^  ^ 

0

Let^
and^ ( ) be defined as
( )^ ( ) ( ) ( )=stationary Gaussian random process with zero mean.( )=zero mean Gaussian white noise independent of
with^
Determine coh

UV

U^ t^
V t
U t^
S t
V t^
S t^ W
t
S t W t^
S t
W^ t W
t^
S

^

^   

 ^ 
^ 
Exercise

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 

^ ^

^ 

 ^

 ^    ^ ^

^  

 ^ 

^

^ ^

 ^ ^

^ ^

 ^ 0   2

( )^ ( )^2 ( )^ deterministic envelope function( )=zero mean stationary Gaussian random process( )^

exp^

exp^

;^

0

0

SS

X^

S X^ t^

e t S t e t S t e t^ A

t^

t

X^ t^

e t S t X^ t^ X

t^

e t S t e t

S t

e t e t

R

t^ e^

t

^ 

^

^

   ^

^

^ 

^ 

^

^

 ^ 

^

^ 

Example

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^ ^

^ ^

^  1 2 1

2 Let^ ( ) be a random process with continuous state andcontinuous parameter (time ).Let^

be n time instants.

This defines

random variables ,^

,^ ,^

.

( ) is said to posse

n n X t

t

t^ t^

t n X^ t^

X^ t^

X^ tX t ^

Markov Property

  ^ ^

^ ^

^ ^

^ 

^ ^

(^1)  

1

2

2

1 1

1

1 1 2 ss Markov property if |^

,^

,^ ,

| for any

and any choice of

.

n^ n

n

n^

n^

n

n^ n

n

n

n

P^ X^

t^ x

X^ t

x

X^

t^

x^

X^ t^

x

P^ X^

t^ x

X^ t

x n^

t^ t^

t

^

^

^

^

^

^

^

^

^

^

^

^

^

 ^ ^

 

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^

^

^

^

^

^

^  ^

^ 

^ ^

^

 1 1

2 2

1 1

1 1

1 1

2 2

1 1

1 1

1 1 2 2 1

1

2 2 1 1

1 1

3 3 2

2 1

1

3

,^ |^

,^ ;^

,^ ;^

,^ ,^

,^ |^

,^ |^

,^ ;^

,^ ;^

,^ ,^

,^ |^

Description of a Markov process,

,^ ;^ ,

,^ |^ ,

,^ ;^ ,

;^ ,

X^ n^

n^ n^

n^ n^

n^

X^ n^

n^ n^

n

X^ n^

n^ n^

n^ n^

n^

X^ n^

n^ n^

n

P^ x^

t^ x^

t^ x^

t^

x^ t^

P^ x^

t^ x^

t

p^ x^

t^ x^

t^ x^

t^

x^ t^

p^ x^

t^ x^

t

p x^ t p x^ t

x^ t^

p x^ t

x^ t^

p x^ t

p x^ t

x^ t^

x^ t^

p x

^ ^

^ ^

^ 

^ ^

^ ^

^ 

 ^

^

  

^ ^

^ ^

^

^ ^

^ ^

^

^

^

^ ^

3 2

2 1 1 2

2 1 1

1 1

3 3 2 2

2 2 1 1

1 1

1 1

1 1

1 1

1 1

,^ |^ ,^ ;^2

,^

,^ |^ ,

,^ |^

,^

,^ |^ ,

,^ ;^

,^ ;^

;^ ,^

,^ |^

,^

n

n^ n^ n

n

t^ x^

t^ x^ t

p x

t^ x

t^ p x

t

p x^ t

x^ t

p x

t^ x

t^ p x

t

p x^ t

x^

t^

x^ t^

p x^ t

x^

t^ p x

t

^ ^

^   ^ ^

^  

 ^

^ 

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Chapman - Kolmogorov - Smoluchowski Equation

t^ t ^1

t^ t ^2 t^  ^

^ ^

^ ^

^

 ^

^ ^

^ ^

^

^ 

^ ^

^

^ ^

2 2 1

1

2 2 1

1 1

1

2 2

1 1 2 2

1 1

1 1

1 1

2 2 1

1

2 2

1 1

1 1

2 2

1 1 ,^ ;^ ,^

,^ |^ ,^
,^ ;^ , ;
,^ |^ , ;
,^
,^ |^ ,^
,^ |^ ,^
,^ |^ , ;
,^
,^ |^ ,
,^ |^ ,^
,^ |^ ,

p x^ t^

x^ t^

p x^ t^

x^ t^ p x

t

p x^ t^

x^ x^ t

dx p x^ t^

x^ x^ t

p x^

x^ t^ p x

t^ dx

p x^ t^

x^ t^

p x^ t^

x^ x^ t

p x^

x^ t^ dx

p x^ t^

x^ p x

x^ t^

dx  

 

^

    

  

 

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^

^

^

^ ^

2 2

1 1

2 2

1 1

,^ |^

,^

,^ |^

,^

,^ |^

p x^

t^ x^

t^

p x^

t^ x^

p x^

x^ t^

dx

^

^ 

Consistency condition for the process to be Markov

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