







































Study with the several resources on Docsity
Earn points by helping other students or get them with a premium plan
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
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
1 / 47
This page cannot be seen from the preview
Don't miss anything!








































Random processes-4Random vibrations of sdof systems-
Docsity.com
22
Docsity.com
4
^ ^
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
^ ^
Docsity.com
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
Docsity.com
(^7) Docsity.com
8
^
^
^
^
^
^ ^
^
^ ^
^ ^
^ ^ ^
^
^ ^
^
^
^
^ ^
(^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
^
^
^
^
^
^
^
^
^
^
^
^
Docsity.com
10
^ ^
^ ^
^ ^ ^
^ ^
^
^ ^
^ ^
^
^ ^ ^
^ ^ ^
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^
i^ d
R^
i^ d
i
S p^
q^
^
^
^
Docsity.com
11
^
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
i S S S ^
^
^
^ ^
^
Docsity.com
13
^
^ ^
^
^ ^
0
UV
^
^
Docsity.com
14
^ ^
^
^
^ ^ ^
^
^
^
^ ^
^ ^
^ ^
^ 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
Docsity.com
16
^ ^
^ ^
^ 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^ t X 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
^
^
^
^
^
^
^
^
^
^
^
^
^
^ ^
Docsity.com
17
^
^
^
^
^
^
^ ^
^
^ ^
^
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
X^ n^
n^ n^
n^ n^
n^
X^ n^
n^ n^
n
X^ n^
n^ n^
n^ n^
n^
X^ n^
n^ n^
n
^ ^
^ ^
^
^ ^
^ ^
^
^ ^
^ ^
^
^ ^
^ ^
^
^
^
^ ^
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
n
n^ n^ n
n
^ ^
^ ^ ^
^
^
Docsity.com
19
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
^
Docsity.com
20
2 2
1 1
2 2
1 1
Docsity.com