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Stochastic Structural Dynamics
Lecture-8
Dr C S ManoharDepartment of Civil Engineering
Professor of Structural EngineeringIndian Institute of ScienceBangalore 560 012 India
Random processes-3
22
exp
1 exp2
XX XX
XX XX
S R i d
R S i d
Recall
33
2
1 2 1 22
1 2 12
21 1
22112
Let ( ) be a random process and consider its 1st and 2nd order pdf-s.
1 1; exp ;22
1, ; ,2 1
1exp2 1
XX
XX
XX
X t
x m tp x t x
tt
p x x t tr
x m x
r
Gaussian random process
22 2 1 1 2 2
1221 22
1 2
1 1 2 2 1 1 2 2 12 1 2
2
,; ; ; ; ,X X X X XX
m x m x mr
x xm m t m m t t t r r t t
44
1
1
1
1 2 1 2
1122
Continuing further, consider time instants and
associated random variables .
Let the jpdf of be given by
, , , ; , , ,
1 1exp ; 1,22
ni i
ni i
ni i
XX X n n
tin
i
n t
X t
X t
p x x x t t t
x S x x i nS
S
1 2
1 2
: & is positive definite.
is said to be a Gaussian random process if the above form of
pdf is true for any and for any cho
j i X i j X j
t
tX X X n
n
X t m t X t m t
Note S S S
m t m t m t
x x x x
X t
n
Definition
1ice of .ni it
55
1 2
1 2 1 2
1 2 1 2 1 2 1 2
(a) A Gaussian random process is completely specified through its mean and covariance , .
(b) ( ) is stationary & ,
, ; , , ;
( ) is 2nd order
X XX
X X XX XX
XX XX
m t C t t
X t m t m C t t C t t
p x x t t p x x t t
X t
Remarks
SSS is SSS.
(c) A stationary Gaussian random process with zero mean iscompletely described by its autocovariance function or itspdf function.
(d) Linear transformation of Gaussian random processes p
X t
reserve theGaussian nature. Gaussian distributed loads on linear systems produceGaussian distributed responses.
66
01
Let be a zero mean, stationary, Gaussian random process defined as
cos sin ;
Here ~ 0, , ~ 0, ,
0 ,
n n n n nn
n n n n
n k n
X t
X t a t b t n
a N b N
a a n k b b
Fourier representation of a Gaussian random process
Assumptions
1
0 ,
0 , 1,2, ,
cos sin 0
k
n k
n n n nn
n k
a b n k
X t a t b t
77
1 1
1 1
2
1
cos sin cos sin
cos sin cos sin
cos
n n n n n n n nn n
n n n n n n n nn m
XX n nn
X t X t a t b t a t b t
a t b t a t b t
R
is a WSS random process.
is Gaussian.
is a SSS process.
X t
X t
X t
88
1
1
1
2
Consider the psd function
1 cos2
1 cos2
Compare this with
cos
XX n n nn
XX n n nn
XX n n nn
XX n
S S
R S
R S
R
Fourier representation of a Gaussian random process (continued)
1
2By choosing , we see that the two ACF-s2
coincide.
nn
n nn
S
99
0 5 10 15 20 25 30 35 40 45 500
0.005
0.01
0.015
frequency rad/s
psd
1n n
2Area 2n n nS
01
By discretizing the psd function as shown we can simulatesamples of ( ) using the Fourier representation
cos sin ; n n n n nn
X t
X t a t b t n
10
Ensemble ofrealizations of randomprocesses can be digitally simulated
10
11
1
2 22
2
22 2
22 2
2 22 2
Let be an iid sequence of random variables
withP
Psuch that 1.
P P
P P
Var
i iX
X x p
X x qp q
X X x x X x x
x p q
X X x x X x x
x p q
X X X
x p q x p q
x p q x p q
Simple random walk
2 22 2
1
4
p q
x p q p q pq x
12
1
1 1
2
2
Let be the time axis and let us divide theinterval (0, ) into n subintervals each of width such that
.Define
Var 4
4
n
ii
n n
ii i
tt
tn t t
S t X
S t X p q x
n p q xxt p qt
S t t pq x
xt pqt
13
0 00 0
00
is known as a simple random walk.( ) is a discrete state, discrete parameter random process.
Consider the limit of 0 as 0
lim lim
and
lim Var lim
x xt t
xt
S tS t
x t
xS t p qt
S t
Remarks
2
00
4 0
In the limit of 0 as 0, ( ) becomesa deterministic function.This is not an interesting limit from probabilistic point of view.
xt
xt pqt
x t S t
14
2
2
Consider the following limit of the simple random walk0 as 0
with
1 1; 1 ; 12 2
Var
This is an interesting limit!
x t
t tx t p q
S t t
S t t
Wiener Process
15
The resulting process is known as the Wiener process.This is a process with continuous state and continuous parameter.The process is a Gaussian process (central limit theorem).The process i
Remarks
s nonstationary.If 0, the process is known as a Brownian motion process.
1616
Random events and Poisson process
1 1 2 2 1 1 2 2
Let ( ) be the number of events occuring randomly in the interval 0, .If there exists probability functions
, , , ; ,
then we say that ( ) is a counting procN NN
N t t
P n t P N t n P n t n t P N t n N t n
N t
ess (discrete state, continuous parameter random process).
Inter-arrival time
time t1t 2t2s1s
1717
1 1 2 2 1 1
1 1 2 2
( ) is said to be a Poisson process with stationary increments if the following conditions are satisfied
(a)
That is, |
where , & , are
N t
P N t N s n N t N s m P N t N s n
s t s t
Independent arrivals :
1 1 2 2 mutually exclusive and & .
( ) :
( ) 1 ( ) 1 ; 0.
(c) :
( ) 1 & ( ) 1
s t s t
b
P N t dt N t P N t dt h N t h dt
P N t dt N t dt P N t dt N t
Stationary arrival rule
Negligible probability for simultaneous arrivals
0.
Under these conditions it can be shown that
( ) exp ; 0,1,2, , .!
ktP N t k t k
k
1s 2s1t 2t
1818
, , 0, 1, 1,
0, 1
1,
, , 1 1,
, ,, 1, 1, ,
, , 1,
, exp exp 1
N N N N N
N
N
N N N
N NN N N N
N N N
N n N
P n t dt P n t P dt P n t P dt
P dt dt
P dt dt
P n t dt P n t dt P n t dt
P n t dt P n tP n t P n t P n t P n t
dtd P n t P n t P n tdt
P n t A t t P n
Proof
0
,
This equation can be used to recursively evaluate , by varying as 0,1,2,
t
N
d
P n t nn
1919
00
0
0
1
Thus with =0, we have
0, exp exp 1,
Clearly, 1, 1 0
0, exp
We have 0,0 0 0 1 counting begins after 0
1 0, expConsider now 1.
1,
t
N N
N
N
N
N
N
n
P t A t t P d
P P N
P t A t
P P N t
A P t tn
P t A
0
10
1
1
exp exp 0,
exp exp exp
exp exp
We have 1,0 0 1 0
0= 1, exp
t
N
t
N
N
t t P d
A t t d
A t t t
P P N
A P t t t
2020
0 0
Repeating this process for 2,3, we get
, exp ; 0,1,2, ,!
If the stationary arrival rule is relaxed, the above model can bemodified to read as
1, exp!
n
N
nt t
N
n
tP n t t n
n
P n t d dn
Remark
; 0,1,2, ,n
21
1
Here we construct a random process by viewing it as a superposition of pulsesarriving randomly in time.
counting process
a random pulse that commences at time .
N t
k kk
k k k
X(t) W t,τ
N t
W t,τ τ
Random pulses
1
Consider the subclass
iid sequence of rvs; independent of ;indicate the intensity of the -th event.a deterministic pulse arriving at time ; 0 .
N t
k kk
k k
k k k k
X(t) Y w t,τ
Y τ k kw t,τ τ w t,τ t τ
22
1 2
1
0
min2
1 2 1 20
2 2 2
0
By imposing the condition , we can write the above equation as
;
It can be shown that
,
,&
N T
k kk
t
X Y
t ,t
XX
t
X
t T
X(t) Y w t,τ T t
m t m w t,τ λ τ d
C t ,t E Y w t ,τ w t ,τ λ τ dτ
σ t E Y w t,τ λ τ dτ
23
1 2
Consider a random phenomenon E, which occurs as a Poisson processwith constant arrival rate . Let be the times at which the event E occurs. Let be the random variable representin
k
i
t ,t , ,tZ
Example
max
g the intensity measure of E occuring at the time instant .
Let , 1, 2, be an iid sequence with common PDF .
Let be the maximum value of observed
over the time interval 0 .
i
i Z
i
t
Z i P z
Z t Z
,t
24
max
max
max
max0
0
0
0
Consider
exp!
exp 1
If 1 exp
exp exp
This is the
kZ
Zk
kk
Zk
Z
Z
Z
P Z z | N t k P z
P z P Z z | N t k P N t k
tP z t
k
t P z
P z z z
P z t z z
PDF of a Gumbel RV. The above model has been used to model the maximum earthquake ground acceleration in the time interval 0 to t.
2525
Let be a random process. In formulating problems of mechanicswe need to differentiate random processes. For example, if ( ) is displacement,we wo
X t
X t
Differentiation and integration of random processes
0
i 1
i+1
uld be interested in velocity and acceleration.Recall: for deterministic functions
lim
By selecting a sequence of -s, of the form ,
such that ,we obtain a sequence of numbe
ni
i
z t z tdzdt
rs
and we seek to determine lim .ii iii
z t z ty y
2626
When is a random process, the sequence
- , =1,2,
is a sequence of random variables.
ii
i
X t
X t X tY i
What is meant by convergence of a sequenceof random variables?
2727
There are several valid modes of convergence of random variables.Consequently, the associated calculus also would be built basedon a chosen mode of convergence of random variables.
A sequenceDefinition
1 2
2
of random variables , , , , is said toconverge to the random variable in the mean square sense if
lim 0.
This is denoted by l.i.m. .
The calculus based on this definition of converg
n
in
in
X X XX
X X
X X
ence of rvs iscalled the mean square calculus.This leads to the definition of mean square derivative and mean square integral.
2828
2 21 2 1 0
1 2 1 2
0
1 2 1 2 1 2
0 2
1 21 2
2
21 2
1 2 1 21 2
Consider
l.i.m.
lim
, , ,lim
,,
Similarly, it can be shown that
,,
& more gen
XX XX XX
XXXX
XXXX
X t X tX t X t X t
X t X t X t X t
R t t R t t R t tt
R t tR t t
t
R t tX t X t R t t
t t
1 2
1 2
1 2
erally,
,n mn mXX
n m n mt t t t
R t td X d Xdt dt t t
2929
1 2
2
(a) When we say that a random variable existsin the mean square sense?
Answer: when .
(b) Thus for to exist in the mean square sense, itsvariance must be finite. This means,
lim
X
t t t
X
X t
Remarks
21 2
1 2
,.
(c) If ( ) and ( ) are jointly stationary, show that
1
XX
n m n mm XY
n m n m
R t tt t
X t Y t
d X t d Y t d Rdt dt d
30
Show that for a zero mean, stationaryrandom process, the process and its
derivative are uncorrelated.
1 exp2
1 cos 2
XX XX
XX XX XX
XX
X t
X t
R S i d
S d S S
dR
Example :
1 sin2
0 0
XXXX
XX
RS d
d
X t X t R
31
1 2 1 2
1 2
1 22 1
2
1 2
1 21 2
22
1 21 2
1 2
: Given , min ,
determine , .
, if
=0 if ,
,
XX
XX
XX
XX
XX
R t t t t
R t t
R t tt t
tt t
R t tU t t
t
R t tt t
t t
Example
32
1 2 1 2 1 2
1 2
1 21 1 2
2
1 1 2
1 21 1 2
22
1 21 2
1 2
: Given , min ,
determine , .
, if
= + if ,
,
XX
XX
XX
XX
XX
R t t t t t t
R t t
R t tt t t
tt t t
R t tt U t t
t
R t tt t
t t
Example
33
Area under psd (=variance) .The process is physically unrealizable.Analogous to a concentrated load in mechanicsand an impulse in dynamics.
R I
S I
White noise
34
Two random processes
