t = X t h t n d { ( ) } ( ) ( ) = = - - PowerPoint PPT Presentation

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t = X t h t n d { ( ) } ( ) ( ) = = - - PowerPoint PPT Presentation

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Outline Outline Single Degree of Freedom System Single Degree of Freedom System Nonstationary Nonstationary & Transient Responses & Transient Responses Examples Examples Double Fourier Transforms

SLIDE 1

1

G. Ahmadi

ME 529 - Stochastics

G. Ahmadi

ME 529 - Stochastics

Outline Outline

Single Degree of Freedom System

Single Degree of Freedom System

Nonstationary

Nonstationary & Transient Responses & Transient Responses

Examples

Examples

Double Fourier Transforms

Double Fourier Transforms

Fourier Transforms of Stochastic

Fourier Transforms of Stochastic Processes Processes

Response of Non

Response of Non-

stationary System

stationary System

G. Ahmadi

ME 529 - Stochastics

Single Degree of Freedom Single Degree of Freedom Initial Conditions Initial Conditions

( ) ( ) ( ) ( )

t n t X t X t X

+ +

2 ω ξ ω & & &

( ) ( )

= = X X &

( ) { }

= t n E

( ) ( )

τ δ π τ

S R 2 =

n(t n(t) ) -

White Noise

White Noise

G. Ahmadi

ME 529 - Stochastics

Impulse Response Function Impulse Response Function System Response System Response

( )

t sin e t h

d t d

ω

ξω −

= 1

( )2

1 2

1 ξ ω ω − =

( ) ( ) ( )

∫

− =

d n t h t X τ τ τ

( ) { }

= t X E

Response Mean Response Mean

SLIDE 2

2

G. Ahmadi

ME 529 - Stochastics

Mean Mean-

Square Response

Square Response

For white excitation, integrating over delta function For white excitation, integrating over delta function

( )

{ }

( ) ( ) ( )

∫ ∫

− − − =

t t nn

d d R t h t h t X E

2 1 2 1 2 1 2

τ τ τ τ τ τ

( )

{ }

( )

∫

h S t X E

2 2

2 τ τ π ( )

∫

−

t d d

d sin e S t

2 2 2

2 τ τ ω ω π σ

τ ξω

( ) ( )

[ ]

⎭ ⎬ ⎫ ⎩ ⎨ ⎧ + + − =

−

t sin t sin e S t

d d

t d

ξ ω ω ω ξω ω ω ξω π σ

ξω

2 2 1 1 2 2

2 2 2 2 2 3 2

G. Ahmadi

ME 529 - Stochastics

For For ξ ξ = 0 = 0

Approximate Solution for Small Damping, Approximate Solution for Small Damping, Caughey Caughey and and Stumpf Stumpf (1963) (1963)

( ) ( ) ( ) ( ) ( )

∫ ∫ ∫

∞

− − − =

t t nn x

d d d S t h t h t

2 1 2 1 2 1 2

cos 1 τ τ ω τ τ ω ω π τ τ σ ( ) ( ) ( )

[ ]

⎭ ⎬ ⎫ ⎩ ⎨ ⎧ + + − ≈

−

t sin t sin e S t

d d

t d

ξ ω ω ω ξω ω ω ξω ω σ

ξω

2 2 1 1 4

2 2 2 2 2 3 2

( ) ( )[ ]

t sin t S t

ω ω ω ω σ 2 2 4

2 2

− =

Mean Mean-

Square Response

Square Response

G. Ahmadi

ME 529 - Stochastics

G. Ahmadi

ME 529 - Stochastics

SLIDE 3

3

G. Ahmadi

ME 529 - Stochastics

Expected Value Expected Value

Linear System Linear System

( ) ( ) ( )

∫

+∞ ∞ −

− = τ τ τ d h t X t Y

( ) ( ) ( ) ( ) ( )

t h t d h t t

X X Y

* η τ τ τ η η = − = ∫

+∞ ∞ −

( ) ( ) ( ) ( ) ( )

2 2 1 2 1 2 1

* , , , t h t t R d h t t R t t R

XX XX XY

= − = ∫

+∞ ∞ −

τ τ τ

( ) ( ) ( ) ( ) ( )

1 2 1 2 1 2 1

* , , , t h t t R d h t t R t t R

XY XY YY

= − = ∫

+∞ ∞ −

τ τ τ

( ) ( ) ( ) ( )

1 2 2 1 2 1

* * , , t h t h t t R t t R

XX YY

=

Autocorrelation Autocorrelation

G. Ahmadi

ME 529 - Stochastics

Linear System Response Linear System Response

( ) ( )

( )

∫ ∫

+∞ ∞ − +∞ ∞ − − −

= Γ

2 1 2 1 2 1

2 2 1 1

, , dt dt e t t R

t t i ω ω

ω ω

( ) ( ) ( )

( )

∫ ∫

∞ + ∞ − ∞ + ∞ − −

Γ =

2 1 2 1 2 2 1

2 2 1 1

, 2 1 , ω ω ω ω π

ω ω

d d e t t R

t t i

( ) ( )

1 2 * 2 1

, , ω ω ω ω Γ = Γ

( ) ( )

2 1 * 2 1

, , ω ω ω ω Γ = − − Γ

( ) ( ) ( ) ( )

2 * 1 2 1 2 1

, , ω ω ω ω ω ω H H

XX YY

Γ = Γ

( ) ( ) ( )

1 2 1 2 1

2 , ω ω δ ω π ω ω − = Γ S

X(t X(t) stationary ) stationary

Inverse Inverse

Properties Properties

G. Ahmadi

ME 529 - Stochastics

Exists Exists iff iff Inverse Inverse

∴ ∴ FT of stationary processes does not exist FT of stationary processes does not exist