random vibrations: response of a linear system

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Random vibrations, stochastic vibration, dynamics,

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  • ,H

    x

    y H xsystem

    input output

    x y

    Response of a linear system

    sinx x t

    siny H x t

  • sinx x t

    siny H x t

    sinx x t

    siny H x t

    ( ) sin ( )x x t ( ) ( ) sin ( ) ( )y x H t

    2

    0

    1 ( )( ) lim

    2xx

    xS

    22

    0

    ( ) ( )1( ) lim

    2yy

    x HS

    2

    ( ) ( ) ( )yy xxS H S

    Response spectrum of a linear system

  • ( )F t

    ( )u t

    k c

    ( )F t

    ( )u tm t

    t

    ;2

    e

    k c

    m k m

    A single-mass-spring system under random loading

  • 1 2

    2 22

    1( )

    1 2e e

    H H

    k

    2

    2arctan with 0

    1

    e

    e

    (0.1)

    frequency

    ; natural frequency

    damping factor ; 2

    e e k m

    c k m

    Dynamic single degree of freedom system

    k c

    ( )F t

    ( )u tm

  • FFS

    ( )H

    uuS

    1

    2k

    0

    0

    2

    u

    Analysis of the single-mass-spring system

  • FFS

    2H

    uuS

    2

    1

    2k

    e

    0

    2area

    4

    e S

    k

    0S

    21 k

    2

    0S k

    Response spectrum for a single-mass-spring system under

    stationary load with white-noise spectrum

  • wind

    10 s

  • respons

    10 s Te = 0.5 s

  • FFS

    0S ( )FF eS

    white-noise approximation

    real spectrum

    e

    Approximation of the load spectrum by a white-noise

    spectrum with intensity

    0 ( )FF eS S

  • S Du u u Fk

    2 2 22 2 2( )( )

    ( )21

    FF eFFuu

    ee

    SSS

    k k

    Response to arbitrary load

    Split the response into

    a quasi static part and a white noise part:

    22

    2 2

    ( )1

    4

    F e FF eu

    F

    S

    k

  • 3 6 2

    3 2

    0

    9 5

    70 10 N ; 8 10 Nm ; 20,000 kg

    50 10 N s ; 3 m ; 0.01

    50 years 1.5 10 s ; 10 Nm

    F

    p

    EI m

    S l

    T M

    ( )F t

    T t

    F

    e

    ( )FFS

    0S

    ( ) ( )F t m u t

    l HE 200 A

    Portal frame with data

  • Spring stiffness k :

    6

    3

    247.1 10 N/m

    EIk

    l

    Natural frequency:

    3

    2419 rad/se

    k EI

    m m l

    Mean response

    10.0098 m 9.8 mmu F

    k

    Variance of the response:

    2 6 20

    21.47 10 m

    4

    eu

    S

    k

    Standard deviation:

    0.0012 m 1.2 mm u

    ( ) ( )F t m u t

    l HE 200 A

  • ( )F t( )u t

    = full-plastic hinge A

    Considered limit state of the frame

  • 1(0)

    ( ) at

    u

    u t t t

    ( )u t

    0 T t

    ( )u t

    10 t T t

    The event failure in [0,T]

  • ( )u t

    t

    1a

    2a 3a

    ia

    na

    T

    ( )aF

    Peak values of the narrow-band process

  • Yield occurs if: 1

    4A p pM F l M

    Corresponding value of u: 4

    0.0188 m 18.8 mmp p

    p

    F Mu

    k kl

    Reliability index: 18.8 9.8

    7.51.2

    p u

    u

    u

    Exceedance probabtilities:

    APT:

    single peak:

    50 year:

    ( )u t

    t

    1a

    2a 3a

    ia

    na

    T

    example portal frame

    14{ ( ) } 1 ( ) ( ) 3 10p N NP u t u

    213{ } exp 6.1 10

    2i pP a u

    3{ ( ) in [0, ]} { } 2.7 10p i pP u t u T n P a u

  • Fatigue strength of steel

    2 double amplitude [N/mm ]s

    400

    100

    200

    50

    20 5 6 710 10 10 N

    reinforced beam

    welded beam

    s

    st

    average

    exceedance probability

  • Fatigue

    105 106 Nf

    S

    bfS

    CN

    S = stress range

    Nf = number of cucles till fracture

    C, b = material constants

    2

  • Fatigue Miner Rule

    )2

    b1(}22{

    C

    Tf)]T(D[E

    ds2

    sexp

    s

    C

    )s2(Tf)]T(D[E

    )s(N

    ds)s(fn

    )s(N

    )s(n)]T(D[E

    )1Diffailure(N

    n...

    N

    n

    N

    nD

    bs

    e

    2

    2

    2s

    b

    e

    i

    i

    2

    2

    1

    1

  • 54

    3

    2

    1

    00 1 2 3 4 x

    ( )x

    0.50 1.772

    0.75 1.225

    1.00 1.000

    1.25 0.906

    1.50 0.886

    1.75 0.919

    2.00 1.000

    x x

    The Gamma function

    for integers: (x) = (x-1)!

  • / 2 / 2k c c k

    v1

    2

    105 tons

    300 tons

    m

    m

    Schematisation of the coal-dust mill

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