introduction to random vibrations
DESCRIPTION
Random vibrations basicsTRANSCRIPT
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Random sine function
( ) sinx t c t
stochastic phase angle, for example uniform on [0,2 ]
deterministic amplitude
deterministic circle frequency
c
f()
2
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Random sine function
x and deterministic, uniform on (0,2)
2 2
0 0
1{ ( )} ( ) ( ) sin( ) 0
2x t x t f d x t d
22 2 2 2
2 2
0 0
1{ ( )} ( ) ( ) sin ( )
2 2
xx t x t f d x t d
( ) sinx t x t
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1.0 1.0
Two important integrals
0.5
( )f y
2y y
( )f y
2
22 2
0 0
sin 0 sin 0.5 2y dy y dy
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1( ) sin( ) ( )k k k k
N
k
x t x t x t
sum of sine functions with random phase angle:
{ ( )} { ( )} 0kx t x t 2 2 21 { ( )} { ( )}
2k kx t x t x
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0
2
0
10
1
1,2,3, 10 ; 4 ; 2 ; 5
exp ; 2 ; 0.1
( ) sin ; 0 50 ; random phase angle
k k k
k k
k k k
k
k T k T
x a b a b
x t x t t
t
( )x t
0 5 10 15 20 25 30 35 40 45 50
10
8
6
4
2
0
2
4
6
8
10
Process with 11 sine functions
k=1,11, xk = 1.0
1= 4.0, 2 = 4.2. 10 = 6.0 rad/s
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1
( ) cos sin sin cosk k k k k k
N
k
x t x t x t
0
2( ) sink k
T
A x t t dtT
0
2( ) cosk k
T
B x t t dtT
2 2 2k k kx A B
2 2 21
1{ ( )}
2k k
N
k
x t A B
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T T
x
t
2kx
k
2kx
k2
T
4
T
base time T
2
T
base time 2T
k
2( ) 2k k kS x 2 2
lim kxxx
S
Definition of the variance spectrum
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1
( ) cos sin sin cosk k k k k k
N
k
x t x t x t
0
2( ) sink k
T
A x t t dtT
0
2( ) cosk k
T
B x t t dtT
2 2 2k k kx A B
2 2 21
1{ ( )}
2k k
N
k
x t A B
2
10
2 2( ) lim withkxx k k k
xS
T
2 2
0
1( ) ( ) ( )
2k xx k xxx x S S d
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2 2
2 2
1 1( ) ( ) cos sin withi tx
T T
T T
S x t e dt x t t i t dt T
*( ) ( )xx x xS S ST
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0
( )ii xxm S d
The standard deviation
0( )x m
central frequency:
0 1 0m m
2
0 2 0m m
width of the spectrum
2
1
0 2
1m
qm m
moments of the spectrum
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a) narrow-band process
b) wide-band process
c) process with two distinctive frequencies
0 02T
16%x
x
x
t
t
xxS
xxS
0
0
21
xxS
0
t
0.5q
0.6q
q
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Fig. A.2: Stochastic process as a series of stochastic variables.
( )x t
t t
( )x t
( )x t t
( )x t n t
t t n t t
Alternative Approach (Annex)
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random variable f(x) x x
two randomvariabels f(x,y) x x y y covxy
n random variables f(x) x x covxy
random process f(x(t)) x(t) x(t) covxy(t1,t2)
stationary random proc f(x) x x Rxx(t)
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Description of continuous processes
Gaussian process:
Mean value for every point in time
Covariance for every two points in time
General process:
Multidimensional probability distribution for every
set X(t1), X(t2), X(t3), .
Stationary Gaussian process:
Mean value X
Autocovariance function RXX()
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Spectrum
0
2( ) ( )cosxx xxS R d
0
( ) ( )cos( )xx xxR S d
2
0
(0) ( )x xx xxR S d
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2 1 2 1
2 1 0 2 1 0 1 0
2 1 0 2 1 0 1 0
2 1 2 1
2 1 2 2 1 2
2 1 2
( ) ( ) ( ) ( )
1( ) ( ) cos ( ) ( ) ( )
21
( ) ( )sin ( ) ( ) ( )2
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( )
nn
n
R a R S a S
R R S S S
R R S S Si
dR R S i S
d
R R R S S S
R R g t dt S
1
2 1 2 1
( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
Hereby is the Fourier transform of ( ) defined by:
( ) ( )
complex gonjugate of ( ) real function
i t
G S
R R g t dt S G S
G g t
G g t e dt
G Gg t
Table A.1: Properties of Fourier transforms.
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( ) 1S
1
( ) 2 ( )S
2area
0 0( ) ( ) ( )S
1area
0 0
( )S
1area
1
( )S
1area
0 1 2
( )R
1area
( ) ( )R
( )R
( ) 1R
1
( )R
0( ) cosR
1
0
( )
sin cos
R
( )R
( )R
1
( )
sin1
R
1( ) ( ) ( ) iR S R e dt
1
2
3
4
5
( )R e
( )R 2 2
1 2( )S
1
0
( )
cos
R
e
( )R
1
2 22 2
0 0
1( )S
0 0
00 0( ) cos sin4
R e
0 0
4
0
22 2 2 2 2
0 0
( )4
S
2
0 1
6
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( ) sinx t c t
sine function
X= 0
2( ) ( )1
cov ( ) cos2
x t x t xxR c
2 21 1(0) ; 22 2
x xx xR c c
Fig. A.4: Auto-covariancefunction of with uniform on .
( )xxR
212c
c and deterministic, uniform on (0,2)
SXX() = c2 (-k)
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( ) sinx t c t
1 2 1 2( ) ( ) 1 ( ) 2 ( )cov {[ ( ) ] [ ( ) ]}x t x t x t x tE x t x t
1 2
2
( ) ( ) 1 2 1 2cov { ( ) ( )} { sin sin }x t x t E x t x t E c t t
1 2
2
( ) ( ) 1 2
2
0
cov sin sin2
x t x t
ct t d
1 2
2 2
( ) ( ) 1 2 1 2
2 2
1 2 1 2
2
1 2
2 2
0 0
2 2
0 0
1 1cov cos cos 2
2 2 2 2
cos sin 24 8
1cos
2
x t x t
c ct t d t t d
c ct t t t
c t t
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1
2
sin
k
k
k k k k
N
k k k
k
S
a S
y a t
random
generator
Sk
Generation of a random process
To Remember (1):
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Construction of the variance spectrum
x
t
Sx () = (1/) 0 T x(t) exp (it) dt
Sxx() = ( /T) Sx Sx*
2
0
(0) ( )x xx xxR S d
Sk
To Remember (2):