Introduction to Random vibrations

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Random vibrations basics

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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

  • 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

  • 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

  • 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

  • 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

  • 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

  • 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

  • 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

  • 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

  • 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

  • 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

  • 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)

  • 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)

  • 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()

  • Spectrum

    0

    2( ) ( )cosxx xxS R d

    0

    ( ) ( )cos( )xx xxR S d

    2

    0

    (0) ( )x xx xxR S d

  • 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.

  • ( ) 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

  • ( ) 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)

  • ( ) 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

  • 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):

  • 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):

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