111

Stochastic Structural Dynamics

Lecture-38

Dr C S ManoharDepartment of Civil Engineering

Professor of Structural EngineeringIndian Institute of ScienceBangalore 560 012 India

manohar@civil.iisc.ernet.in

Problem solving session-2

2

1 2 3

Problem 17Consider the vector random variable given by

= . It is given that is normal with mean vector and correlation matrix given by

1 4 1 6= 2 and = 1 9 0 .

3 6 0 19We

t

t

Y

Y Y Y Y YR

R YY

21 2 3

now form the random process .

Find the mean, autocorrelations and cross correlationsof ( ) and .

X t Y Y t Y t

X t X t

3

2

2 2

1 2 1 2

22 2

2 21 1 2 1 1 2

2 22 2

1 2

Define 1

11 2 1 2 3

3

2 2 6

4 1 6 1 4 61 1 9 0 1 1 9

6 0 19 6 19

,

t t

t

t t t

t t

XX

N t t t X t N t Y

X t N t Y t t t t

X t B Ct t

X t X t N t YY N t

t tt t t t t t

t t

R t t

2 2 2 21 2 1 2 1 2 1 2

2 2 4

4 9 6 6 19

4 2 21 19

t t t t t t t t

X t t t t

4

2

2 2 2 21 2 1 2 1 2 1 2 1 2

2 2 4

22 2 4 2

2 4 2 4 2 3

2 3 4

1 2 3

, 4 9 6 6 19

4 2 21 19

4 2 21 19 1 2 3

4 2 21 19 (1 4 9 4 6 12 )4 6 11 12 19

XX

X

X t t t

R t t t t t t t t t t

X t t t t

t t t t t t

t t t t t t t tt t t t

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

50

100

150

200

250

time

varia

nce

x(t)

5

2 2 2 21 2 1 2 1 2 1 2 1 2

21 2 1 2 1 2 1 2

2

21 2 1 2 1 3 1 2 3 2

22 1 1 2

2

1 2 1 2 1 21 2

, 4 9 6 6 19

, 1 9 12 38

Check

2

1 12 9 38

, 9 76

XX

XX

XX

R t t t t t t t t t t

X t X t R t t t t t tt

X t X t Y Y t Y t Y Y t

t t t t ok

X t X t R t t t tt t

6

Problem 18

Consider the random process exp

where deterministic constant, 1, is a random variable with pdf and

characteristic function , and =a random variable that is indepen

X t a j t

a jp

dent of anddistributed uniformly in , . Show that

is proportional to , and

is proportional to XX

XX

R

S p

7

*

2

2

2

2 2

exp

exp exp exp 0

exp exp

exp

exp

XX

XX

X t a j t

a j t a j t j

X t X t

a j t j t

a j

R a

S a j d a p

8

2 2

Problem 19: A random process is given by

2 - - 2where ( ) is a zero mean stationary random processwith PSD function

Determine the PSD function of ( ).

XX

Y t

Y t X t X t X tX t

CS

Y t

9

2 - - 2

2 - - 2 0

2 - - 2

2 2

2 4 2

2 2

6 4 4

2 2

XX XX XX

XX XX XX

XX XX XX

YY XX XX XX

XX XX

Y t X t X t X t

Y t X t X t X t

Y t X t X t X t

Y t Y t R R R

R R R

R R R

R R R R

R R

10

exp

Consider exp

exp

exp exp exp UU

S R i d

R a i d

R u i u a d

i a R u i u d i a S

11

2 2

exp

exp exp

6 4 4

2 2

6 4 exp exp

exp 2 exp 2

6 8cos 2cos 2

6 8cos 2cos 2

UU

YY XX XX XX

XX XX

YY XX XX

XX

XX

YY

R i d S

R a i d i a S

R R R R

R R

S S S i i

S i i

S

CS

12

-100 -80 -60 -40 -20 0 20 40 60 80 10010

-20

10-15

10-10

10-5

100

105

w rad/s

psd

Sxx

Syy

13

Problem 20Consider two independent random processes and which have zero mean and are stationary. Define

where is a deterministic constant.Determine PSD function of ( ).

X t Y t

Z t X t Y tZ t

0

XX

ZZ XX YY

ZZ YY

Z t X t Y t

Z t X t Y t X t Y t

Z t Z t X t Y t X t Y t

X t X t Y t Y t Y t

R R R

S S S d

14

0

Problem 21A simply supported beam of span carries a distributed load ( ). The load is modeled as a segment of stationary

random process as ( ) 1 such that

0 and . Determine the

Lf x

f x F x

x x x

followingBending moment at midspanJoint pdf of reactions at the two supports.

15

A B

f x

0

00 0

0 0

0

0 0 0

0

1 1 1

2

2 2

L

B

L L

B

L

L

B

R L xf x dx

R xf x dx xF x dxL L

F L F x x dxL

F L F F LR x x dxL

16

0 0

0

2 20 0

1 2 1 2 1 20 0

20

1 2 0 1 2 1 20 0

2 2 220 0 0

00

2 20 0 0

2

2

3

~2 3

L

B

L L

B

L L

L

B

F L FR x x dxL

F L FR x x x x dx dxL

F x x I x x dx dxL

F F LIx I dxL

F L F LIR N

17

0

00 0

0 0

0

2 2 20 0 0 0

2 20 0 0

1 1 1

2

&2 2 3

~2 3

L

A

L L

A

L

A A

B

R L L x f x dx

R L x f x dx L x F x dxL L

F L F L x x dxL

F L F L F LIR R

F L F LIR N

18

0 0

2 20

1 2 1 2 1 220 0

2 20

1 2 0 2 1 1 220 0

2 2 2 20 0 0

020

2 2 2 20 0 0 00

2 2 2 20 0 0 0 0

2 2

6

3 62~

2 6 3

A B

L L

L L

L

A

B

F L F LR R

F L x x x x dx dxL

F L x x I x x dx dxL

F F I LI x L x dxL

F LI F LIF LR

NR F L F LI F LI

19

2

00

2

00 0

2

00 0

0

Similarly one can study

( )2 2 2

1 ( )2 2

1 1 ( )2 2

Exercise: complete the characterization of

L

A

LL

LL

R LL LM M x f x dx

L LM L x f x dx x f x dxL

LL x F x dx x f x dx

M

20

0

2 21 2 1 2

Problem 22A cantilever beam carries a randomly distributed loadas shown below. The load ( ) is modeled as

1 ; 0 &

1 1exp2 2

Determine the bending moment at a section m

q x

q x q f x f x

f x f x x x

x

easured from the free end.

x

q x

21

0

00

2 22

0 0 00

22

0

2 20 1 2 1 2 1 2

0 0

2 2 2 20 1 2 1 2 1 2

0 0

1

2 2

2

1 1exp2 2

x

x

x

x x

x x

M x x q d

x q f d

x xM x x q d q x q

xM x q

q x x f f d d

q x x d d

22

2 2 2 2 20 1 2 1 2 1 2

0 0

2 2 2 2 20 1 2 1 2

0 0

2 2 2 20 2 1 2 1 2

0 0

2 2 2 20 1 1 2 1 2

0 0

2 20 1 2

1 1exp2 2

1 1exp2 2

1 1exp2 2

1 1exp2 2

1 1exp2 2

x x

M

x x

x x

x x

x q x x d d

q x d d

q x d d

q x d d

q

2 21 2 1 2

0 0

x x

d d

23

22 2 2 2 2 2 2

0 0

22 2 20

2

0

2 1 exp22

1 exp2 2

with

1 exp22

M

x

x xx q x x xq

q x

xx dx

24

Problem 23A cantilever beam of span carries a series of concentrated loads. The point of application of theseloads are distributed as Poisson points on 0 to . The magnitude of the loads are modeled

L

L as a sequence

of iid-s with a common Rayleigh distribution withparamter . Determine the characteristic funciton of thereaction .R

25

1w 2w 3w nw

R

M

1

2

2 2

exp ; 0,1, 2,!

exp ; 02

N L

nn

n

w

R w

aLP N L n aL n

nw wp w w

26

1 1

1 1

1

0

2 2

exp exp

0 exp

exp exp!

exp exp 1!

: 1 exp 1 erf2

N L N L

n R in n

k

ik n

kk

wk

kk

w wk

w

R w i R i w

P N L i w N L k P N L k

aLaL i aL

k

aLi aL aL i

k

ii i

Note (prove)2

27

1 2 2 1 2 1

Problem 24Verify if

, exp - - sin -

can be a valid autocovariance function of a zero meanrandom process. It is given that , , 0.

R t t t t t t

1 2 2 1

1 2 1 1 2 2

Required characteristics, ,

, 0

, , ,

No, since the function is not positive definite;notice , 0

R t t R t t

R t t

R t t R t t R t t

R t t

28

2

Problem 25Consider to be a stationary random process with

zero mean. Define ( ) - where and ( ) are determinisitc. Determine autocovariance of( ). It is given that

exp 1XX

X t

Y t X t a t X ta t

Y t

R

29

( ) -

( ) - 0

( )

- -

-

-

- -

-

Simplify using

XX XX XX

XX

n m

n m

Y t X t a t X t

Y t X t a t X t

Y t Y t

X t a t X t X t a t X t

X t X t X t a t X t

a t X t X t

a t X t a t X t

R a t R a t R

a t a t R

d X t d Y tdt dt

1n m

m XYn m

d Rd

30

2

2

2

2

2

2

2

exp 1

exp sgn 1

exp sgn

exp sgn sgn sgn

exp sgn

Use

sgn &

and derive

XX

XX

XX

R

d Rd

dU tU U t

dtd Rd

31

Problem 26An undamped sdof system is set into free vibrationby imparting random initial displacement and velocity.Characterize the system response. Determine the conditionsunder which the response can become stationary.

32

2

2

2

2

0; 0 ; 0

cos sin

cos sin

cos sin

cos sin cos sin

, cos cos cos sin

sin cos sin sin

xx

x x x u x v

x t A t B tvx t u t t

vx t u t t

x t x t

v vu t t u t t

uvR t u t t t t

vuvt t t t

33

2

2

2

22 2

2

2

2

, cos cos cos sin

sin cos sin sin

Take 0 &

, cos cos sin sin

, cosConditions for existence of stochastic steady state

xx

xx

xx xx

uvR t u t t t t

vuvt t t t

vuv u

R t t t t t

R t R

2

2 22

are

0, 0, 0 &v

u v uv u

34

Problem 27An input ( ) 0 for 0 and ( ) exp(-2 ) for 0 to a linear system produces the output

1 exp 2 exp 4 . The system is now 2

excited by a Gaussian white noise excitation with unit stren

f t t f t t t

y t t t

gth. Determine the PSD of the steady state response.

35

2

2

( ) exp(-2 ) 1

21 exp 2 exp 42

1 1 12 2 4

1 1 11 2 12 2 4 11 2 4 4

21

16YY

f t t U t

Fi

y t t t U t

Yi i

ii iHi i

i

S H

36

exp ; 0 0

expexp

10 0

1 exp exp

1

x x t x

tx t A t

x A

x t t t

Hi

Notice

37

Problem 28Consider a random process

sinwhere is determinstic, is a random variable distributed uniformly in 0 to 2 , and ( ) is azero mean stationary Gaussian random process.It may be a

X t P t Y tP

Y t

ssumed that ( ) and are independent.Determine the joint pdf of and .

Are and uncorrelated? independent?

Y tX t X t

X t X t

38

2

2

sin

0

sin sin

sin sin

cos2

is wide sense stationary

YY

YY

X t F t Y t

X t

X t X t F t Y t F t Y t

F t t R

F R

X t

39

sincos

2 2

2 2

2 2

2 2

sin

cos

, | ,

1 1exp sin cos2 2

, , |

1 1 1exp sin cos2 2 2

12

y x F tXX YYy x F t

XX XX

X t F t Y t

X t F t Y t

p x x p y y

x F t x F t

p x x p x x p d

x F t x F t d

2 22 2

1 1exp sin cos2 2

x F x F d

40

2 22 2

2 2 22 2 2

2 2 2 2 202 2 2

,

1 1 1exp sin cos2 2 2

1 1 1exp exp sin cos2 2 2

1 1exp ; ,2 2

sin

1|2

XX

X

p x x

x F x F d

Fx x F x x d

Fx x F I x x x x

X t F t Y t

p x

2

2

1exp sin2

|

|

X X

XX

x F t

p x p x p d

p x p x p d

41

It can be verified that,

and are uncorrelated because they are stationaryrandom processes.

and are not independent

XXX Xp x x p x p x

X t X t

X t X t

Remark

42

2 2

2 22 2

Problem 29

Let ( ) be a random process with

and . Show that

P for some in

1 1 ; 02

Note: This is the generalization of Chebychev's

X

X X

X

b

X X X Xa

X t X t t

X t t t

X t t t a t b

a b t t dt

inequalityfor random processes.

43

22

2 2

2

2 2 2

2

We have for a random process ( )1P sup E sup Prove this

Also,

2

2

12

1sup

a t b a t b

t

ab

tt

a

a t b

Y t

Y t Y t

dY t Y a Y u Y u dudu

dY b Y u Y u dudu

dY t Y a Y b Y u Y u dudu

Y t

[Hints]

2 2

2

t

a

dY a Y b Y u Y u dudu

44

12 2 2

2 2 2

12 2

2

We have

E E E

1sup2

E E

Substitute in the above to getthe required result.

a t b

t

a

X

UV U V

E Y t E Y a Y b

d Y u Y u dudu

Y t X t t

45

2

2

Problem 30Let ( ) be a stationary random process with zero meanand autocovariance function given by

1 exp22

How many times can we differentiate this process?

Determine 0.75 i

XX

X t

R

P X t

f it is given that the

process is Gaussian and =1.

46

2

2

2 2

2 2

1 exp22

exp2

exp exist 1, 2,2

is differentiable at =0 forall orders.

is differentiable to any order (in the mean square

XX

XX

nn

XX

R

S

d n

R

X t n

sense)

47

2

2

2

2 2

2 2

2 2 2

22

2 2

2

2 2

1 exp22

1 exp22

1 1 exp22

1 exp22

1 1 10 12 2

1exp 2 exp2

0

XX

XX

XX

XX

X

R

d Rd

d Rd

R

p x x x

P X t

0.75

2.75 exp 0.97x dx

48

Problem 31Let ( ) be a Poisson random process with arrival rate .

Define 1 .Determine the mean and covariance of ( ).Note: is known as semi random telegraph signal.

N t

N t

X tX t

X t

1

1

t

X t

49

2 4

3 5

1 .

1 if ( ) 0 or ( ) is even1 if ( ) is odd

1 0 or ( ) is even

exp 1 exp cosh2! 4!

1 ( ) is odd

exp exp s3! 5!

N tX t

N t N tX t

N t

P X t P N t N t

t tt t t

P X t P N t

t tt t t

inh

1 1 1 1

exp cosh sinh 2exp 2

t

X t P X t P X t

t t t t

50

even odd

1 if there are even number of occurrences in to .

1 if there are odd number of occurrences in to .

1 exp 1 exp exp 2! !

, exp 2

n n

n n

XX XX

X t X tt t

X t X tt t

X t X t

n nR t t R

51

Problem 32A random walk is performed on a two-dimensional plane with auniform step size of . At every step the direction is a randomvariable. -s can be taken to be an iid sequence with a common

i

i

pdf that is uniformly distributed in 0 to 2 . Find the distribution of the

-coordinate after steps.x n

X

x

y

52

1

1

12

00

0

0

00

cos

exp exp cos

exp cos

1exp cos exp cos2

1 exp2

1 cos

N

ii

N

X jj

n

ji

j j j

nX

nX

n

X

i X i

i

i i d J

J

p x J i x d

J xd

53

0

00

2 2 4 4

0 2 2 2

2 2 4 4

0 2 2 2 2

2 2 2 2

2

2

2

1 cos

12 2 4

Consider the limit such that .

12 2 4

1 exp2 4

1 exp

nX

nX

nn

nn

n

X

J

p x J xd

J

n n c

c cJn n

c cn

xp xcc

; x

54

th

Problem 33Let the time interval 0 to be divided into a sequence of equal intervals of length . Consider a sequence

1of Bernoulli trials with P success = . Define2

1 if success on trial 1 if f

TT

n

nX t

th 1ailure on trial

Find mean and autocorrelation of ( ).

Furtheremore, let be a random variable distributeduniformly in 0 to and independent of ( ).Define - . Determine the me

n T t nTn

X t

eT X t

Y t X t e

an andautocorrelaiton of ( ).Y t

55

th

th

22 2

1 21 2

1 if success on trial 1

1 if failure on trial1 11 1 02 2

1 11 1 12 2

1 if 1 ,0 otherwise

nX t n T t nT

n

X t

X t

n T t t nTX t X t

56

1 2 1 1

1 2

1 2 1 2

1 2

1 21 2

E =E E E 0

E E

E E

E 0 if

If

1 if -E

0 otherwise

Y t X t

Y t Y t X t

Y t Y t X t X t

X t X t

X t X t t t T

t t T

e T t tX t X t

57

1 2 1 2

1 2

E 1 -

1

Y t Y t P e T t t

t tT

T T

1

2 1t t

1 2E Y t Y t

58

Problem 34Given a postive function ( ) find a stochastic process whose PSD is ( ). Existence theorem

SS

Determine a LTI system with H and

pass a zero mean stationary Gaussian white noise withunit strength through this system. The output processwould be a zero mean stationary process with PSD= .

i S

S

59

22Determine and define .

Clearly, 0 & 1; also,

has the properties of a pdf of a random variable.

Let ( ) cos where and are random variables

with

Sa S d f

a

f f d f f

f

X t a t

Alternative

2

1 2 1 20

22

~ , ~ 0,2 ,& .

( ) 0 Prove it; start with finding mean conditioned on

( ) ( ) cos cos ,

cos OK2XX XX

f U

X t

X t X t a t a t p d d

aR f d S a f

60

0 10 20 30 40 50 60 70 80-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

t s

x(t)

61

0 10 20 30 40 50 60 70 80-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

t s

x(t)

0 10 20 30 40 50 60 70 80-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

t s

x(t)

PSD is an ensemble propertyThese two time histories

represent samplesfrom two different processes having thesame mean and PSD function.

Remarks

62

RemarkThe following three processes share the same form of PSDThe sample realizations are dramatically different.

1 ; ( ) : Poisson process with rate

cos ; ~ ; ~ 0,2 ;

Steady st

N tX t N t

SX t t U

S d

0ate response ; 0X X t X X

63

1

1

t

X t

0 10 20 30 40 50 60 70 80-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

t s

x(t)

0 10 20 30 40 50 60 70 80-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

t s

x(t)

64

Discussion on properties of processes withIndependent increments