Oscilacije Elasticnih Tela

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  • 7.

    , , , . . , , , . , , . . . . , , . , , , . . , . , , () , . , . 7.1

    , , , . . , , . . , . , , , . A i B tako da u svakom preseku postoji normalni napon =z . , , (.7.1), .

    dz dmy

    7.1

  • F

    r F r

    zd+= . ( ) , , . , :

    zy

    ) (7.1.1) ,( tzyy = . dm F

    r F r ( )FFF == rr :

    FFdmarrr +=

    Oy :

    sin)sin(22

    FdFdmty

    z += . (7.1.2)

    zd , : dz dz

    zdda z )(sin)(sinsin)sin(

    ==+ . ,

    y , :

    zy= tgsin , dz

    zydz 2

    2)(sin

    = . :

    AFAdzdVdm === , , : , A , , (7.1.1) :

    ==

    2

    2

    22

    2

    2 0 c

    zyc

    ty , (7.1.3)

    . , .

    c

    (7.1.3) .

  • (7.1.1) , . (7.1.3) : ) , (7.1.4) ()( tTzZy = :

    TZdt

    TdZtyTZT

    dzZd

    zy &&==

    ==

    2

    2

    2

    2

    2

    2

    2

    2 , .

    (7.1.3) : 0 (7.1.5) 2 = TZcTZ &&, :

    hTc

    TZZ == 2

    &&, (7.1.6)

    , ,

    h

    z t . (7.1.5) : (7.1.7) .0 ,0 2 == ThcThZZ && , : h

    (7.1.8)

    ).sin()cos( : 0 3)

    ),sinh()cosh( : 0 2)

    , : 0 1)

    2

    2

    kzDkzCZkh

    kzDkzCZkh

    DCzZh

    +==

    +==

    (7.1.4) ) , : , 0( lzz == 0)( 0),( ,0)0( 0),0( ==== lZtlyZty . (7.1.9) (7.1.8) 3) , (7.1.7) :

    h

    (7.1.10) ),( 0 ,0 22222 ckTTZkZ ==+=+ && : ) (7.1.11) sin()cos( kzDkzCZ += )sin()cos( tBtAT += . (7.1.12) (7.1.11) (7.1.9), : (7.1.13) ,0 0)0( == CZ , )0( D , :

  • ,0)sin( =kl (7.1.14) :

    l

    nkn= ),,2,1( = Kn . (7.1.15)

    , (7.1.11) (17.1.2), :

    n

    (7.1.16) )sin( zkDZ nnn =

    ==+=

    lcncktBtAT nnnnnn

    n )sin()cos( , (7.1.17) , (7.1.4), [ )sin()sin()cos( zktBtAy nnnnnn ] += , (7.1.18) : nnnnnn DBDAA == B , . (7.1.3) (7.1.18), . :

    , (7.1.19) [ )sin()sin()cos(1

    zktBtAy nn

    nnnn

    =+= ]

    :

    ) , (7.1.20) cos()sin(1

    nnnn

    n tzkRy =

    = :

    n

    nnnn A

    BBAR =+= arctg , n22 . (7.1.21)

    nn BA , (7.1.19) , nnR , (7.1.20), . ) n- (.7.2) .

    sin( zkR nn

    . )0( 0 =t , : )( ),()0,( ,0

    00 zt

    yzfzytt

    =

    ==

    =, (7.1.22)

    (20) :

    )(sin ,)(sin11

    zzl

    nBzfzl

    nAn

    nnn

    n =

    =

    =

    = . (7.1.23)

    nnA B , :

  • ==

    l

    nmnm

    ldzz

    lmz

    ln

    0 21 0

    sinsin ),...,2,1,( =nm . (7.1.24)

    (7.1.23)

    zl

    msin [ ]l,0 , (7.1.24), :

    =

    = l lnn dzzlnz

    cndzz

    lnzf

    lA

    0 0sin)(2B ,sin)(2

    . (7.1.25) (7.1.19) (7.1.20) . . , n , . 7.2 .

    7.2

    7.1.1 : b, , , . 7.3. .

  • 7.3

    : , , :

    .0)()0,(

    2),(

    501

    20,

    50)()0,( ==

    == zzy

    bzbzb

    bzz

    zfzy & (1)

    (7.1.10) :

    bnc

    bn

    n == . (2) :

    0,2

    sin25

    2sin50

    2sin50

    2

    2/22

    2/

    0==+= n

    b

    b

    b

    n Bn

    nbdz

    bznzb

    bdz

    bznz

    bA

    , (3) :

    =

    =1

    22 cossin2sin1

    252),(

    nt

    bn

    bznn

    nbtzy

    . (4)

    7.1.2 l :

    lzvzzy

    lzyzfzy 5sin)()0,(,2sin)()0,( 00 ==== & . (1)

    , A F . : (7.1.24) :

    ====

    ===

    5,0

    5,5sin5sin2

    2,02,

    sin2sin2

    0

    5

    0

    00

    0

    00

    n

    nFAlvv

    dzlzn

    lzv

    lB

    nny

    dzlzn

    lzy

    lA

    l

    nn

    l

    n

    (2)

    :

  • tA

    Fll

    zFAlvt

    AF

    llzytzy

    5sin5sin5

    2cos2sin),( 00 += , (3) , , . 7.2 ()

    ( ) , . . , :

    l

    ) , , . , ) , ) , , . , , :

    Oz

    z w

    . (7.2.1) lztzww = 0 ),( z z :

    zw

    z = ,

    zwEE zz == , (7.2.2)

    E . (7.2.1) , . , (.7.4). dz dm

    7.4 F

    r F

    r

  • .

    ktwa

    rr2

    2

    = Oz , , :

    FFdma

    rrr += . (7.2.3) :

    )( d22

    FFFFddmtw

    zz == , (7.2.4)

    , , :

    Fdzz

    dzzwAEdz

    zAEdz

    zAdz

    zFFdAdzdm zzz 2

    2 ,

    ==

    === , (7.2.5)

    : , . (7.2.4) :

    A

    Ec

    zwc

    tw ==

    2

    2

    22

    2

    2 , 0 (7.2.6)

    . . .

    c

    (7.2.6) (7.1.3) , . , , ) , 7.2.7) ()( tTzZw = , (7.2.6), :

    ===+=+

    EkckTTZkZ 222222 0 ,0 && , (7.2.8)

    : )sin()cos( ),sin()cos( tBtATkzDkzCZ +=+= . (7.2.9) (7.2.9), (7.2.7).

    k

    ( )

  • . , , , . . 1. . .7.5 , .

    7.5 : ,0),( ,0),0( == tlwtw (7.2.10) , (7.2.7) (7.2.9) : .0)sin(0)( ,00)0( ==== klDlZCZ (7.2.11) 0D , (7.2.11) :

    ),,2,1( 0)sin( === Knl

    nkkl n . (7.2.13)

    , . , :

    ,sincos ,sin

    +

    =

    = ctl

    nBctl

    nATzl

    nDZ nnnnn (7.2.14)

    +

    = =

    zl

    nctl

    nBctl

    nAwn

    nn sinsincos

    1, (7.2.15)

    : nnnnnn DBDAA == B , . 2. . (.7.6) .

    7.6 , , , , , :

  • ,0 ,00

    =

    =

    == lzz zw

    zw (7.2.16)

    , (7.2.7) (7.2.9) : 0)sin(0)( ,00)0( ==== klClZDZ . (7.2.17) )0( C :

    ),,2,1( 0)sin( === Knl

    nkkl n . (7.2.18)

    (7.2.9):

    +

    =

    = ctl

    nBctl

    nATzl

    nCZ nnnnn sincos ,cos , (7.2.19)

    :

    =

    +

    =1

    cossincosn

    nn zlnct

    lnBct

    lnAw . (7.2.20)

    3. . .7.7 .

    7.7 :

    ,0 ,0),0( =

    =

    =lzzwtw (7.2.21)

    : ,0)cos(0)( ,00)0( ==== klDlZCZ (7.2.22) :

    ),,2,1( 2

    )12( 0)cos( ,0 === Knl

    nkklD n , (7.2.23)

    :

    =

    +

    =1 2

    )12(sin2

    )12(sin2

    )12(cosn

    nn zlnct

    lnBct

    lnAw .(7.2.24)

  • 4. . A (.7.8) , , .

    1m

    7.8

    A , (7.2.7) (7.2.8),

    )()()()( 222

    tTlZtTlZtwa

    lzA ==

    =

    =&& , (7.2.25)

    , ,