Pseudorandom Number Generator - Stochastic Structural Dynamics - Lecture Slides, Slides of Stochastic Processes

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Monte Carlo simulation approach-

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2

Scalar/vectorGaussian/non-GaussianCompletely specified/partially specifiedTransformation methodAcc

    

R Pseudorandom number generatorSimulation of random variablesM

ethods ecall

ept-Reject method

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4

 









































1

1

1

1

2

1

cos

sin

cos

sin

cos

sin

cos

sin

cos

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

m XX

n

n

n

X

t

X

t

a

t

b

t

a

t

b

t

a

t

b

t

a

t

b

t

R































 

















 



































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5









































1

1

1

Consider the psd function

1

cos

2

1

cos

2

XX

n

n

n

n

XX

n

n

n

n

XX

n

n

n

n

S

S

R

S

R

S





  







  













 



 





































 

Fourier representation of a Gaussian random process(continued)

 

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0

5

10

15

20

25

30

35

40

45

50

0

frequency rad/s

psd

1

n





n







Area

2

n

n

n

S













 





By discretizing the psd function as shown we can simulatesamples of

( ) using the Fourier representation

cos

sin

;

n

n

n

n

n

n

X t

X

t

a

t

b

t

n

















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



 









2

2

1

max

0

0

max

Simulate samples of a zero mean stationary Gaussianrandom process with following properties:

exp

;

2

2

2;

6

cos

sin

5 ;

120;

;

0.2513 rad/s

N

n

n

n

n

i

n

I

S I X

t

a

t

b

t

T

s N

n



































 

 



























Example

120

30.1593 rad/s.

0.0419 s

t







 

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time s

x(t)

a samplemeanstd devtarget stdevtarget mean

1

0

0

0

s

a

m

p

l

e

s

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x

PDF

simulationtarget

1

0

0

0

s

a

m

p

l

e

s

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2

2

4

2

2

2

2

2

2

2

2

2

Let

rad/s;

g

b

g

b

g

g

g

g

g

b

g

b

g

g

g

yy

g

g

g

g

g

mu

c

u

x

k

u

x

u

u

u

x

x

y

u

S

I

I

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141414

10

10

10

0

10

1

Frequency rad/s

PSD (m/s 2

2

/(rad/s)

TargetSimulation

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161616

Accl m/s

(^2)

Probability

SimulationNormal PDF





2

2

2

exp

cos

;

1;

1;

1

1

exp

;

2

2

X

R

x

p

x

x





 







 

















 







 

 

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0

1

2

3

4

5

6

7

8

9

10

0

1

2

Frequency (rad/s)

PSD

Target and Simulated PSD

TargetSimulation

5

0

0

s

a

m

p

l

e

s

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x(t)

Samples of the random process X(t)

x(t)

x(t)

x(t)

x(t)

Time (s)

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Let

( ) be a random process whose first order pdf and

the ACF functions are available. No further information abou

X t

Simulation of partially specified non - Gaussianrandom processes : Nataf's transformation

 

 

 

 

 

 

 

 

 

2

t

the process is available.

need not be stationary.

How to simulate samples of

( )?

Define

so that

0 &

Introduce a new random process Z(t) through the transformation

X

X

Y

X

t

X t

X

t

m

t

Y

t

t

Y

t

Y

t

Z t

P

Y

 t



























 





 

Here

PDF of N 0,

random variable.

Z

is a zero mean Gaussian random process with an unknown

covariance function.

t



  

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