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lecture notes on the theory of mechanical vibrations

TRANSCRIPT

  • 3102

    1

  • snoitrabiV fo yroehT

    I 51 , . . . . . . . . . . . . . . . . . . . . 51.1 : )y ,x(f = z . . . . . . . 82 )trebmelA'd( . . . . . . . . . . . . . . . . . . 93 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.3 1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311.4 : )x(f = y . . . . . . . . 512.4 : . . . . . . . . . . . . . . . . . . . . . . . . . . 713.4 : . . . . . . . . . . 815 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 021.5 : . . . . . . . . . . . . . . . . . . . . . . 122.5 : )y ,x(f = z . . . . . . . . . . . . . . . . . . . 326 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 egnargaL . . . . . . . . . . . . . . . . . . . . . . 421.7 . . . . . . . . . . . . . . . . . . . . . . . . 522.7 . . . . 723.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 034.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 639 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 931.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1401 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241.01 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.01 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.01 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.01 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545.01 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6411 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 841.11 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 942.11 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 943.11 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 054.11 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

    2hcalaF roiL dna vegeS nevueRc

  • snoitrabiV fo yroehT

    II 351 . . . . . . . . . . . . . . . . . . . . . . . . 352 . . . . . . . . . . . . . . . . . . . 451.2 ecnanoseR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653 . . . . . . . . . . . . . . . . . . . . . . . . 651.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 854 . . . . . . . . . . . . . . . . . . . . . . 061.4 . . . . . . . . . . . . . . . . . . . . . . . . 265 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361.5 . . . . . . . . . . . . . . . . . . . . . . 462.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 864.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 07

    III 371 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 671.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 872.4 . . . . . . . . . . . . . . 973.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281.5 . . . . . . . . . . . . . . 382.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.5 . . . . . . . . . . . . . . . . . . . . . . 486 . . . . . . . . . . . . . . . . . . . . . . . 781.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 782.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 987 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 893.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 994.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 995.7 . . . . . . . . . . . . . . . . . . . . 001

    3hcalaF roiL dna vegeS nevueRc

  • snoitrabiV fo yroehT

    VI 4011 . . . . . . . . . . . . . . . . . . . . . . . . . . 4011.1 1 > . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5012.1 1 < . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6013.1 1 = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9012 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0111.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1113 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2111.3 . . . . . . . . . . . . . . . . . . . 3112.3 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3113.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 5114 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5111.4 esnopser eslupmI . . . . . . . . . . . . . . . . . . . . . 6112.4 . . . . . . . . . . . . . . . . . . . . . . . . 7113.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 911

    V 2211 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2212 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421

    4hcalaF roiL dna vegeS nevueRc

  • snoitrabiV fo yroehT

    I

    1 ,

    P 3R. . ,

    P3 . . / N ) P3 < N( . 1.1, )a( 4 )2y ,1y ,2x ,1x(, )b( 4 L 2L = 2)2y 1y( + 2)2x 1x(

    3 , ) ,1y ,1x( .

    x

    y

    1m

    2m

    2x 1x)a(

    1y

    2y

    x

    y

    1m

    2m

    2x 1x)b(

    1y

    2yL

    1.1:

    . 2.1,

    2 ,1 . N 1=nN}nq{ N 1=nN}nq{ . , 1=nN}nq{

    .

    5hcalaF roiL dna vegeS nevueRc

  • snoitrabiV fo yroehT

    1

    2

    2.1:

    P :

    P , . . . ,1 = i im i.

    P , . . . ,1 = i ir i.

    P , . . . ,1 = i i f i.

    ir = ir(t , Nq , . . . ,1q

    ))1.1( ,

    , 1=nN}nq{.

    q( ir = ir)t ,n

    . 5 . i

    irdtd

    =N1=n

    irnq

    + nqirt)2.1( ,

    tir, i. ,

    . j)t(nis r +i)t(soc r = r.

    r

    tj)t(soc r +i)t(nis r =

    6 hcalaF roiL dna vegeS nevueRc

  • snoitrabiV fo yroehT

    .

    q( ir = ir. )t ,mq ,n

    = nqir irnq(t , Nq , . . . ,1q

    ) mqir , 2.1

    (irnq

    )mq

    ,0 =

    q , n

    = mq

    m =6 n 0m = n 1irmq

    =irmq

    )3.1( .

    ir mq

    irmq

    =

    mq(ir)

    =

    mq

    (N1=n

    irnq

    + nqirt

    )

    =N1=n

    ir2nqmq

    + nqir2tmq

    1.1

    irmq

    =irmq

    (t ,nq , . . . ,1q

    ),

    mqir =

    irmq

    =N1=n

    nq+ nq

    t=d

    td=

    d

    td

    (irmq

    ),

    mq

    (irdtd

    )=

    d

    td

    (irmq

    ))4.1( .

    7hcalaF roiL dna vegeS nevueRc

  • snoitrabiV fo yroehT

    1.1 : )y ,x(f = z

    x

    y

    z

    m

    )y ,x( f = z

    3.1:

    m )y ,x(f = z.

    .

    kz + jy +ix = r

    kz + jy +ix = r = v

    )y ,x(f = z y ,x , , .

    ,y = 2q ,x = 1q

    ,k)2q ,1q(f + j2q +i1q = r

    8 hcalaF roiL dna vegeS nevueRc

  • snoitrabiV fo yroehT

    = rr

    1q+ 1q

    r

    2q2q

    =

    (+i

    f

    1qk

    )+ 1q

    (+ j

    f

    2qk

    )2q

    + j2q +i1q =

    (f

    1q+ 1q

    f

    2q2q).k

    2q ,1q r 2q ,1q.

    2 )trebmelA'd(

    . .

    1. i iu. nS ,

    N , . . . ,1 = n nS iu

    = iuN1=n

    irnq

    )1.2( ,nS

    iu , 1.2.

    illuonreB 3071 3471 rebmelA'd, , rebmelA'd " "

    . i

    ,irim = i f

    1 : , ,

    nq.

    9 hcalaF roiL dna vegeS nevueRc

  • snoitrabiV fo yroehT

    irim

    0 = irim i f

    iu P , . . . ,1 = i

    P1=i

    = iu i fP1=i

    )2.2( .yticoliv lautriv elbatpecca 1=iP}iu{ iu irim

    2.2 . 1.2 2.2

    P1=i

    i fN1=n

    irnq

    = nSP1=i

    irimN1=n

    irnq

    ,nS

    N1=n

    [P1=i

    i firnq

    ]= nS

    N1=n

    [P1=i

    ir irimnq

    ]nS

    N , . . . ,1 = n nS

    P1=i

    i firnq

    =P1=i

    ir irimnq

    .N , . . . ,1 = n

    = nQP1=i

    i firnq

    ,

    N

    SnQ1=n nq n

    .

    = nQP1=i

    i firnq

    =P1=i

    ir irimnq

    .

    3.1

    = nQP1=i

    i firnq

    =P1=i

    i f1rnq

    ,

    01 hcalaF roiL dna vegeS nevueRc

  • Theory of Vibartions

    3

    P T

    T =Pi=1

    miri ri

    2.

    Qn =Pi=1

    miri riqn

    =d

    dt

    (T

    qn

    ) Tqn

    .

    (ri ri

    )qn

    =riqn ri + ri ri

    qn= 2ri ri

    qn.

    1.4 ,1.3

    qm

    (dridt

    )=

    d

    dt

    (riqm

    )riqm

    =riqm

    .

    d

    dt

    (riqm

    )=

    d

    dt

    (riqm

    )=

    riqm

    .

    ddt

    (Tqn

    )

    d

    dt

    (T

    qn

    )=

    d

    dt

    (Pi=1

    1

    2mi(ri ri

    )qn

    )

    =d

    dt

    (Pi=1

    1

    2mi2ri ri

    qn

    )

    =Pi=1

    miri riqn

    +Pi=1

    miri ddt

    (riqm

    )

    =Pi=1

    miri riqn

    +Pi=1

    miri riqn

    .

    cReuven Segev and Lior Falach 11

  • snoitrabiV fo yroehT

    . nqT

    T

    nq=

    nq

    (P1=i

    1

    2ir irim

    )

    =P1=i

    ir irimnq

    .

    d

    td

    (T

    nq

    )T nq

    =P1=i

    ir irimnq

    .

    = nQd

    td

    (T

    nq

    )T nq

    .N , . . . ,1 = n

    egnargaL.

    1.3 1.1

    k)2q ,1q(f + j2q +i1q = r

    r

    1q+i =

    f

    1qk

    r

    2q+ j =

    f

    2q.k

    ,kzF + j yF +ixF = F

    r F = 1Q1q

    zF + xF =f

    1q,

    r F = 2Q2q

    zF + yF =f

    2q.

    + j2q +i1q = r

    (f

    1q+ 1q

    f

    2q2q),k

    21 hcalaF roiL dna vegeS nevueRc

  • snoitrabiV fo yroehT

    = Tm

    2

    ([1q2)

    +(2q2)

    +

    (f

    1q+ 1q

    f

    2q2q]2)

    .

    T

    1qm =

    [+ 1q

    (f

    1q+ 1q

    f

    2q2q)f

    1q

    ]

    d

    td

    (T

    1q

    )m =

    [+ 1q

    (f2

    )1q( 2

    (1q2)

    2 +f2

    1q2q+ 1q1q

    f2

    )2q( 2

    (2q2)

    +f

    1q+ 1q

    f

    2q2q

    )(f

    1q+ 1q

    f

    2q2q()

    f2

    )1q( q 2

    + 1f2

    1q2q2q

    ])

    T

    1qm =

    (f

    1q+ 1q

    f

    2q2q()

    f2

    )1q( q 2

    + 1f2

    1q2q2q

    ).

    , 1q

    m = 1Q

    [+ 1q

    (f2

    )1q( 2

    (1q2)

    2 +f2

    1q2q+ 1q1q

    f2

    )2q( 2

    (2q2)

    +f

    1q+ 1q

    f

    2q2q

    ]).

    4

    i f

    .nocnon,i f + noc,i f = i f

    = U = noc f[U

    x+i

    U

    y+ j

    U

    zk

    ] )z ,y ,x(U = U. ,

    ) Pz , Py , Px , . . . ,2z ,2y ,2x ,1z ,1y ,1x( U = U

    31hcalaF roiL dna vegeS nevueRc

  • Theory of Vibartions

    i

    f i,con = iU = [U

    xii+

    U

    yij +

    U

    zik

    ].

    N x1, . . . , xP , y1, . . . , yP , z1, . . . , zP Uqn ,

    U

    qn=

    pi=1

    [U

    xi

    xiqn

    +U

    yi

    yiqn

    +U

    yi

    yiqn

    ].

    riqn =xiqn i+

    yiqn j +

    ziqn k

    Qn =Pi=1

    f i riqn

    =Pi=1

    (fi,xi+ fi,y j + fi,zk

    )(xiqn

    i+yiqn

    j +ziqn

    k

    )

    = Pi=1

    (U

    xii+

    U

    yij +

    U

    zik

    )(xiqn

    i+yiqn

    j +ziqn

    k

    )

    = Pi=1

    [U

    xi

    xiqn

    +U

    yi

    yiqn

    +U

    yi

    yiqn

    ]= U

    qn

    Qn

    Qn = Uqn

    .

    cReuven Segev and Lior Falach14

  • snoitrabiV fo yroehT

    1.4 : )x(f = y

    1m2m

    k k))1x( f ,1x(

    ))2x( f ,2x(

    1.4: )x(f = y

    2x = 2q ,1x = 1q .

    ()2q(f ,2q

    )()1q(f ,1q

    )

    = U1

    2y + 12x(k

    2+ )1

    1

    2k(

    2)1y 2y( + 2)1x 2x()

    =1

    2k((1q2)

    +()1q(f

    )2)+

    1

    2k((. )2))1q(f )2q(f( + 2)1q 2q

    U = 1Q1q

    ))1q( f ))1q(f )2q(f( +)1q 2q(( k ))1q( f)1q(f + 1q(k =k =

    [. ])1q( f ))2q(f )1q(f2( + 2q 1q2

    U = 2Q2q

    k =[. ])2q( f ))1q(f )2q(f( + 1q 2q

    51 hcalaF roiL dna vegeS nevueRc

  • snoitrabiV fo yroehT

    q = iri(j f +i

    )

    = T21=i

    im2

    (iq( 2)

    2 f + 1).

    T

    iqqim =

    i(2 f + 1

    )d

    td

    (T

    iq

    )im =

    [iq(2 f + 1

    )iq f f2iq +

    ]im =

    [iq(2 f + 1

    )2 +

    (iq2) f f

    ]T

    iq=

    im2

    (iq( 2)

    f f2)

    im =(iq2) f f

    d

    td

    (T

    iq

    )T iq

    im =

    [iq(2 f + 1

    )+(iq2) f f

    ].iQ =

    ) Pr , . . . ,1r(U = U 0 = nqU

    U nocnon,nQnq

    =d

    td

    (T

    nq

    )T nq

    d

    td

    ()U T(

    nq

    ))U T(

    nq.nocnon,nQ =

    naignargaL U T = L d

    td

    (L

    nq

    )L nq

    )1.4( .N , . . . ,1 = n nocnon,nQ =

    gnargaL.: gnargaL naignargaL. gnargaL . gnargaL .

    61hcalaF roiL dna vegeS nevueRc

  • snoitrabiV fo yroehT

    2.4 :

    1m

    2m

    L

    F

    k k

    x

    y

    x

    2.4:

    2.4 1m k. L 2m .

    ,x .

    ,ix = 1r

    ,j soc L i ) nis L +x( = 2r

    ,ix = 1r

    = 2r( soc L +x

    ).j nis L +i

    , i F = 2f ,0 = 1f

    1f = xQ1rx

    2f +2rx

    F = i i F =

    1f = Q1r

    2f +2r

    i F =[j nis L +i soc L

    ] soc LF =

    = T21=i

    ir irim2

    =2m+ 1m

    2+ soc xL2m+ 2x

    2m222L

    71 hcalaF roiL dna vegeS nevueRc

  • Theory of Vibartions

    U = m2gL cos + k2

    (2x2)

    x

    L

    x=T

    x= (m1 +m2)x+m2L cos

    d

    dt

    (L

    x

    )=

    d

    dt

    (T

    x

    )= (m1 +m2)x+m2L

    ( cos 2 sin

    )L

    x= U

    x= 2kx

    (m1 +m2)x+m2L( cos 2 sin

    )+ 2kx = F

    L

    =T

    = m2Lx cos +m2L

    2

    d

    dt

    (L

    )=

    d

    dt

    (T

    )= m2L

    (x cos x sin

    )+m2L

    2

    L

    =T

    U

    = m2Lx sin m2gL sin

    m2Lx cos +m2L2 +m2gL sin = FL cos .

    : 4.3

    m2

    m1

    :4.3

    cReuven Segev and Lior Falach18

  • snoitrabiV fo yroehT

    3.4 1m , , 2m . 4.4.

    2m

    1m

    x

    y

    g1m 1N

    g2m2N

    x

    y

    4.4:

    , 2m,xa2m = )(nis 1N = xF.0 = )(soc 1N g2m 2N = yF

    1m, xa1m = )(nis 1N = xF. ya1m = g1m)(soc 1N = yF

    ,

    = )(naty,x x

    ( ,y = )(nat )x x

    xa2m = )(nis 1Nxa1m = )(nis 1N

    )xa xa()(nat 1m = g1m)(soc 1N

    91hcalaF roiL dna vegeS nevueRc

  • snoitrabiV fo yroehT

    1N , xa ,xa. 2q 1q 1m

    5.4.

    2m

    1m

    1q

    2q

    5.4:

    1m 2m

    q( = 1r,i2q = 2r ,j)(nis 1q i)2q + )(soc 1

    q( = 1r.i2q = 2r ,j)(nis 1q i)2q + )(soc 1

    = T1m2

    ([2q + )(soc 1q

    2)+()(nis 1q

    ]2)+2m2

    (2q2)

    =2m+ 1m

    2

    (2q2)

    +1m2

    ([1q2)

    )(soc 2q1q2 +],

    .)(nis 1qg1m = U

    d

    td

    (L

    1q

    )L 1q

    1m =[)(soc 2q + 1q

    ],0 = )(nis g1m+

    d

    td

    (L

    2q

    )L 2q

    q)(soc 1m =q)2m+ 1m( + 1

    .0 = 2

    5

    ) 1.1( .

    q(ir = ir,) Nq , . . . ,1

    02 hcalaF roiL dna vegeS nevueRc

  • snoitrabiV fo yroehT

    = irN1=n

    irnq

    .nq

    = TP1=i

    r rim2

    =1

    2

    P1=i

    im

    ([N1=n

    irnq

    nq

    )(

    N1=m

    irmq

    mq

    ])

    =1

    2

    N1=n

    N1=m

    P1=i

    im

    (irnqir mq

    ).mqnq

    ]g[

    = nmgP1=i

    im

    (irnqir mq

    ).

    = q[Nq , . . . ,1q

    T]Nq , . . . ,1q[ = q T] , ,

    = T1

    2

    {qT}

    ]g[{q}

    =1

    2

    N1=m,k

    qmkg)1.5( .mqk

    kmg = mkg , q ])q(g[

    .0 > T])q(g[ = ])q(g[

    1.5 :

    = T1

    2x1m

    2+ 1

    1

    2x2m

    22

    12 hcalaF roiL dna vegeS nevueRc

  • snoitrabiV fo yroehT

    1x

    2x

    2m 1m

    2q 1q

    1.5:

    = jig 2

    nm1=n(nriqnr iq

    )= ixnr

    i =6 n 0i = n 1= ]g[

    [0 1m

    2m 0

    ].

    q 1x = 1q1x 2x = 2

    = T1

    21m(1q2)

    +1

    22m(2q + 1q

    2).

    nriq

    1r1q

    1 =1r2q

    0 =2r1q

    1 =2r2q

    1 =

    = 11g2

    1=n

    nm

    (nr1qnr 1q

    )2m+ 1m =

    = 21g2

    1=n

    nm

    (nr1qnr 2q

    )2m =

    = 12g2

    1=n

    nm

    (nr2qnr 1q

    )2m =

    = 22g2

    1=n

    nm

    (nr2qnr 2q

    )2m =

    = T1

    2

    [2q 1q

    2m 2m+ 1m [ ]2m 2m

    []1q

    2q

    ]

    22hcalaF roiL dna vegeS nevueRc

  • snoitrabiV fo yroehT

    2.5 : )y ,x(f = z

    3.1 y = 2q ,x = 1q

    k)2q ,1q(f + j2q +i1q = r

    r

    1q+i =

    f

    1q,k

    r

    2q+ j =

    f

    2q.k

    m = 11gr

    1qr 1q

    m =

    [+ 1

    (f

    1q

    ]2)

    m = 21gr

    1qr 2q

    m =f

    1qf

    2q

    m = 22gr

    2qr 2q

    m =

    [+ 1

    (f

    2q

    ]2)

    6

    : vm = p i

    ixm = ivm = ip

    = ipT

    ix=

    ix

    j

    m

    2j2x

    .ixm = , nP

    32 hcalaF roiL dna vegeS nevueRc

  • Theory of Vibartions

    Pn =T

    qn=

    1

    2

    Nk,m=1

    gkmqk

    qnqm +

    Nk,m=1

    gkmqk q

    m

    qn

    .=

    1

    2

    [Nm=1

    gnmqm +

    Nk=1

    gknqk

    ]

    =Nm=1

    gnmqm

    Pn =T

    qn=

    Nm=1

    gnmqm.

    Lagrange 7

    Lagrange

    d

    dt

    (T

    qn

    ) (T )

    qn=

    d

    dt

    (Nm=1

    gnmqm

    ) 1

    2

    Nm=1

    Nk=1

    gkmqn

    qmqk

    =Nm=1

    d (gnm)

    dtqm +

    Nm=1

    gnmqm 1

    2

    Nm=1

    Nk=1

    gkmqn

    qmqk

    =Nm=1

    Nk=1

    gnmqk

    qkqm +Nm=1

    gnmqm 1

    2

    Nm=1

    Nk=1

    gkmqn

    qmqk.

    Lagrange

    d

    dt

    (T

    qn

    ) (T )

    qn=

    Nm=1

    Nk=1

    gnmqk

    qkqm +Nm=1

    gnmqm 1

    2

    Nm=1

    Nk=1

    gkmqn

    qmqk

    =Nm=1

    Nk=1

    [gnmqk

    12

    gkmqn

    ]qkqm +

    Nm=1

    gnmqm = Qn (7.1)

    N

    nkm(q1, . . . , qN ) =

    gnmqk

    12

    gkmqn

    cReuven Segev and Lior Falach 24

  • snoitrabiV fo yroehT

    N1=m

    N1=k

    qmkn+ mqk

    N1=m

    qmngnQ = m

    , 0q 0 = )0q(nQ.

    N1=m

    N 1=k

    nqmk

    mqk )0q(mng )q(mng N1=m

    q)0q(mng.nQ = m

    1.7

    . 0q = q

    .N , . . . ,1 = n ,0 = nq ,0 = nq

    , " . ". N , . . . ,1 = n ,0 = nq ,0 = nq

    0 = )0q(nQ ".

    N1=m

    N1=k

    qmkn+ mqk

    N1=m

    qmng m

    [N1=m

    N1=k

    qmkn+ mqk

    N1=m

    qmngm

    ]0=q=q,0q

    +N1=l

    [

    lq

    (N1=m

    N1=k

    qmkn+ mqk

    N1=m

    qmngm

    ])0=q=q,0q

    lq

    +N1=l

    [

    lq

    (N1=m

    N1=k

    qmkn+ mqk

    N1=m

    qmngm

    ])0=q=q,0q

    lq

    +N1=l

    [

    lq

    (N1=m

    N1=k

    qmkn+ mqk

    N1=m

    qmngm

    ])0=q=q,0q

    lq

    [ N1=m

    N1=k

    qmkn+ mqk

    N1=m

    qmngm

    ]0=q=q,0q

    0 =

    52 hcalaF roiL dna vegeS nevueRc

  • snoitrabiV fo yroehT

    N1=l

    [

    lq

    (N1=m

    N1=k

    qmkn+ mqk

    N1=m

    qmngm

    ])0=q=q,0q

    = lq

    N1=l

    ([N1=m

    N1=k

    mknlq

    + mqkqN1=m

    mnglq

    mq

    ])0=q=q,0q

    .0 = lq

    [

    lq

    (N1=m

    N1=k

    qmkn+ mqk

    N1=m

    qmngm

    ])0=q=q,0q

    =

    ([N1=m

    N1=l

    qmln2m

    ])0=q=q,0q

    .0 =

    N1=l

    [

    lq

    (N1=m

    N1=k

    qmkn+ mqk

    N1=m

    qmngm

    ])0=q=q,0q

    = lqN1=l

    ,lq 0q| lng

    = nQN1=m

    N1=k

    qmkn+ mqk

    N1=m

    qmng m

    N1=m

    )2.7( .mq 0q| mng

    0q|g = M. " 0q 0q + = q 0 = ",

    0 = q "

    = nmMP1=i

    im

    (irnqir mq

    )0=q

    .

    egnargaL

    d

    td

    (T

    nq

    )) T(

    nq=

    N1=m

    qmnM.nQ = m

    = T1

    2

    N1=m,n

    1 = nqnq 0=q| mng2

    N1=m,n

    qmnMnqn

    = mnMT2

    mqnq. 0=q|

    0 > T] M[ = ] M[.

    62hcalaF roiL dna vegeS nevueRc

  • snoitrabiV fo yroehT

    2.7

    , , , ,

    .

    "

    1.5

    = T1

    2

    N1=m,k

    qmkgmqk

    + )0 = q( T = Tm1=l

    T

    lqT.O.H+ lq 0q|

    1 = )0 = q( T T2

    N1=n

    N1=m

    P1=i

    im

    (irnqir mq

    )mqnq 0q|

    = mnMP1=i

    im

    (irnqir mq

    )0=q|

    1 T2

    N1=m,n

    qmnM)3.7( .mqn

    "

    . "0 = 0q 0 =0| nqU = nQ 0 = )0q(U )

    C = )0q(U C U = U U = U(. 0q U

    ,0 = )0q(UU

    nq0 = nQ =0q|

    72hcalaF roiL dna vegeS nevueRc

  • snoitrabiV fo yroehT

    . 0q

    . .

    : .

    : 0U ,0T

    U + T = 0U + 0T

    )0U U( = 0T T 0U < U 0 > 0T T . 0 < 0T T 0 T 0 ) T (.

    . 1.7 ".

    A

    B

    C

    1.7: A " ,B " , C "

    + )0 = q(U = ) Nq , . . . ,1q(UN1=l

    U

    lq1 + lq 0=q|

    2

    N1=k,l

    U2

    kqlqT.O.H+ kqlq 0=q|

    82 hcalaF roiL dna vegeS nevueRc

  • snoitrabiV fo yroehT

    0 = )0 = q(U 0 = nQ = nqU

    = ) Nq , . . . ,1q(U1

    2

    N1=k,l

    U2

    kqlq)4.7( .T.O.H+ kqlq 0=q|

    1 ) Nq , . . . ,1q(U2

    N1=k,l

    U2

    kqlq.kqlq 0=q|

    ]K[

    = mnKU2

    mqnq,0=q|

    1 U2

    N1=m,n

    qmnK)5.7( .mqn

    = noc,nQN1=m

    qmnK.m

    2.7 N1=l

    qlnM+ l

    N1=l

    qlnK.nocnon,nQ = l

    5.7 3.7

    1 = U T = L2

    N1=l,k

    qlkM1 lqk

    2

    N1=l,k

    qlkK.lqk

    L

    nq=

    T

    nq=

    nq

    12

    N1=l,k

    qlkMlqk

    =

    1

    2

    N 1=l,k

    lkMkq

    nq+ lq

    N1=l,k

    qlkMq k

    l

    nq

    =

    1

    2

    [N1=l

    qlnM+ l

    N1=k

    qnkMk

    ]=

    N1=l

    qlnMl

    92 hcalaF roiL dna vegeS nevueRc

  • snoitrabiV fo yroehT

    d

    td

    (L

    nq

    )=

    d

    td

    (N1=l

    qlnMl

    )=

    N1=l

    qlnM.l

    L nq

    =U

    nq=

    nq

    12

    N1=l,k

    qlkKlqk

    =

    1

    2

    N 1=l,k

    lkKkq

    nq+ lq

    N1=l,k

    qlkKq k

    l

    nq

    =

    1

    2

    [N1=l

    qlnK+ l

    N1=k

    qnkKk

    ]=

    N1=l

    qlnKl

    d

    td

    (L

    nq

    )L nq

    =N1=l

    qlnM+ l

    N1=l

    qlnK.nocnon,nQ = l

    , . M K .

    .

    3.7

    2m 1m2k 1k

    2x 1x

    )t(y

    )t(y.

    = T1

    2x1m

    2+ 1

    1

    2x2m

    22

    = U1

    21 + 2)y 1x(1k

    22)1x 2x(2k

    03hcalaF roiL dna vegeS nevueRc

  • snoitrabiV fo yroehT

    )y ,2x ,1x(U = U y

    3 .

    = M

    T21x

    2T2

    2x1xT2y1x

    T22x

    2T2y2x

    mysT22y

    = 0 2m0 0 1m

    0 mys

    ,

    = K

    U212x

    U22x1x

    U2y1x

    U222x

    U2y2x

    mysU22y

    = 0 2k1k 2k 2k + 1k

    1k mys

    , 0 2m0 0 1m

    0 mys

    2x1x

    y

    +0 2k1k 2k 2k + 1k

    1k mys

    2x1x

    y

    = 00

    0

    . 2 2

    [, 0 1m

    2m

    []1x

    2x

    ]+

    [2k 2k + 1k

    2k

    []1x

    2x

    ]=

    [y1k

    0

    ].

    4.7

    2m,1m l 3m . 3m )t( F. 54 = .

    .

    13hcalaF roiL dna vegeS nevueRc

  • Theory of Vibartions

    m1 m2

    m3

    m3

    k1 k2k3

    F (t)

    l l

    l l

    m3 m1 ( )m1 q1 = x () m3 .( )

    r3 = xi+ k (l cos()i+ l sin()j)

    m2

    r2 = r3 + 1k (l cos()i l sin()j)= xi+ k (l cos()i+ l sin()j) + 1k (l cos()i l sin()j)= i

    (x l sin() + l sin()1

    )+ j

    (l cos() + l cos()1

    ) = 1 i m2

    r2 = i(x 2l sin()

    ),

    r3 =(x l sin()

    )i+ l cos()j.

    m3

    r3d =(x l sin()

    )i l cos()j.

    T =m1r1 r1

    2+m2r2 r2

    2+ 2

    m3r3 r32

    =m1 (x)

    2

    2+m2(x 2l sin()

    )22

    +m3

    [(x l sin()

    )2+(l cos()

    )2] [

    m1 +m2 + 2m3 2l sin() (m2 +m3)sym 4l2 sin2()m2 + 2m2l

    2

    ].

    cReuven Segev and Lior Falach 32

  • snoitrabiV fo yroehT

    = Ux1k

    21

    2+x2k

    22

    2+)3y2( 3k

    2

    2

    .)(soc l = 3y ,)(nis l2 x = 2x ,x = 1x

    = Ux1k

    2

    2+2))(nis l2 x( 2k

    2+))(soc l2( 3k

    2

    2

    [ )(nis 2kl2 2k + 1k2l4 mys

    (nis 2k

    soc 3k + )(2)(2

    . ] )

    3r F = xQx

    j)t( F =(i)

    0 =

    3r F = Q

    j)t( F =(j)(soc l +i)(nis l

    ).)(soc l)t( F =

    8

    . 6

    = T1

    2rT+ rM

    1

    2.cI T

    cI .

    ji f j i iJ , . . . ,1 = j.

    ir i.

    jir j i.

    i M J , . . . ,1 = j.

    33 hcalaF roiL dna vegeS nevueRc

  • snoitrabiV fo yroehT

    iA i.

    i i, ) R = )R( TAA R(.

    N P .

    ,) Nq , . . . ,1q(ir = ir

    .) Nq , . . . ,1q(iA = iA

    iJ1=j

    )1.8( P , . . . ,1 = i ,irim = ji f

    iJ1=j

    )2.8( .P , . . . ,1 = i ,iH = i M+ ji f jir

    i

    = iuN1=n

    irnq

    )3.8( ,nS

    i iv

    = ivN1=n

    inq

    )4.8( .nS

    1.8 3.8 2.8 4.8 i.

    P1=i

    iJ 1=j

    ji f jir

    N 1=n

    inq

    i M+ nSN1=n

    inq

    + nS

    iJ1=j

    ji f

    N1=n

    irnq

    nS

    = N1=n

    P1=i

    iJ 1=j

    ji f jir

    i nq

    +

    iJ1=j

    ji firnq

    i i M+nq

    = nSN1=n

    P1=i

    iJ 1=j

    ji f(inqjir

    )+

    iJ1=j

    ji firnq

    i i M+nq

    = nSN1=n

    P1=i

    iJ 1=j

    ji f(irnq

    +inqjir

    )i i M+

    nq

    .nS

    43 hcalaF roiL dna vegeS nevueRc

  • Theory of Vibartions

    f ij riqn +

    iqn rij

    .qn i iqn qn

    Pi=1

    [miri

    Nn=1

    riqn

    Sn + Hi Nn=1

    iqn

    Sn

    ]=

    Nn=1

    {Pi=1

    [miri ri

    qnSn + Hi i

    qn

    ]}Sn.

    Nn=1

    Pi=1

    Jij=1

    f ij (riqn

    +iqn rij

    )+M i i

    qn

    Sn =Nn=1

    {Pi=1

    [miri ri

    qnSn + Hi i

    qn

    ]}Sn,

    Sn

    Pi=1

    Jij=1

    f ij (riqn

    +iqn rij

    )+M i i

    qn

    = Pi=1

    [miri ri

    qn+ Hi i

    qn

    ].

    T =Pi=1

    [miri ri

    2+Ti Ii i

    2

    ] 3 . i Ii

    d

    dt

    (

    qn

    (Pi=1

    miri ri2

    )) qn

    (Pi=1

    miri ri2

    )=

    Pi=1

    miri riqn

    d

    dt

    (

    qn

    (Pi=1

    Ti Ii i2

    )) qn

    (Pi=1

    Ti Ii i2

    )=

    Pi=1

    Hi iqn

    cReuven Segev and Lior Falach 35

  • snoitrabiV fo yroehT

    H, ,I , I

    d

    td

    (

    nq

    (P1=i

    i iI iT2

    ))=

    P1=i

    1

    2

    d

    td

    (inq

    T

    + iiITiI i

    inq

    ),

    =P1=i

    d

    td

    (inq

    T

    iiI

    ),

    =P1=i

    d

    td

    (inq

    T

    iH

    )=

    P1=i

    (d

    td

    (inq

    T)i + iH

    nq

    T

    iH ),

    =P1=i

    d

    td

    (inq

    T

    iH

    )=

    P1=i

    (inq

    T

    i + iHnq

    T

    iH ),

    .

    nq

    (P1=i

    i iI iT2

    )=

    P1=i

    1

    2

    [inq

    T

    + iiITiI i

    inq

    ]

    =P1=i

    [inq

    T

    iiI

    ]

    =P1=i

    [inq

    T

    iH

    ]

    d

    td

    (T

    nq

    )T nq

    .N , . . . ,1 = n nQ =

    = nQI1=i

    iJ 1=j

    ji fjirnq

    +J1=j

    i i Mnq

    nq.

    4.

    1.8

    M L 2k ,1k )t( F )t( T 1.8.

    .

    63 hcalaF roiL dna vegeS nevueRc

  • Theory of Vibartions

    k1 k2

    L

    a

    F (t)

    T (t)

    M

    :8.1

    k1 k2

    L

    a

    F (t)

    T (t)

    M

    x1 x2

    k1 k2

    L

    a

    F (t)

    T (t)

    M

    y

    I II

    8.1 :8.2

    cReuven Segev and Lior Falach 37

  • snoitrabiV fo yroehT

    I ,

    = Tcv cvM

    2+cI

    2

    2.

    j1x = 1r j2x = 2r

    iL k + j1x = 21r + 1r = 2r

    = 1x 2xL

    2x + 1x = c1r + 1r = cv2

    .j

    = TM

    2

    (2x + 1x

    2

    2)+cI2

    (1x 2xL

    2),

    = U1k2

    )1x(+ 2

    2k2

    )2x(. 2

    = a1r + 1r = ar[+ 1x

    (1x 2xL

    )a

    ]j

    = 1x 2xL

    k

    ar F = 1Q1x

    T +1x

    j)t( F =(

    a 1L

    ) k)t( T + j

    (1 Lk

    )=)a L()t( F

    L)t( T

    L

    ar F = 2Q2x

    T +2x

    j)t( F =a (L

    ) k)t( T + j

    (1

    Lk

    )=)t( Fa

    L+)t( T

    L

    II

    = T2)y( M

    2+cI(2)

    2

    = U1k2

    (L y

    2

    2)+2k2

    ( + y

    L

    2

    2)

    83 hcalaF roiL dna vegeS nevueRc

  • snoitrabiV fo yroehT

    ar F = yQy

    T +y

    k)t( T + j j)t( F =(

    k0)

    )t( F =

    ar F = Q

    T +

    j)t( F =((L

    2a ))

    k)t( T + j(k)

    )t( F =

    (L a

    2

    ))t( T +

    9

    M 0ml ,ml ,mk , m, . m 2)0ml ml( 2mk = mU

    = UM1=m

    = mUM1=m

    mk2

    . 2)0ml ml(

    = jiKU2

    jqiq=0=q|

    M1=m

    mU2jqiq

    =0=q|M1=m

    , jimK

    = jimKmU2jqiq

    =0=q|2

    jqiq

    (mk2

    2)0ml ml()0=q|

    =

    iq

    (ml )0ml ml( mk

    jq

    )0=q|

    mk =

    ( )0ml ml(

    ml2iqjq

    +mljq

    mliq

    )0=q|

    mk =mljq

    mliq. 0=q|

    1.9

    93hcalaF roiL dna vegeS nevueRc

  • Theory of Vibartions

    rk

    rs

    um

    kmrs

    rk

    :9.1

    lm = (rs rk) um, ,

    um

    um =rs rkrs rk

    um um = 0 um

    lm =(rs rk

    ) um + (rs rk) um = (rs rk) um. lm

    qi= lm

    qi rs

    qi= rs

    qi

    lmqi|q=0= lm

    qi|q=0=

    ((rs rk

    ) um)qi

    |q=0=(rsqi rkqi

    ) um |q=0 .

    Kij =Mm=1

    kmlmqj

    lmqi|q=0, lm

    qi=

    (rsqi rkqi

    ) um

    cReuven Segev and Lior Falach40

  • Theory of Vibartions

    9.1

    k1

    k2

    a,m2

    m1

    x

    u

    r2

    r1

    :9.2

    2 9.2

    q1 = x, q2 =

    l1 = l10 + q1 l1 l1q1

    = 1,l1q2

    = 0

    K1 =

    [k1 0

    0 0

    ]

    l2 = u2 (r2 r1

    ) l2

    q1, l2q2

    k2

    u2 |q=0= cos i+ sin jr1 = q

    1i

    r2 = aq2[ sin ( + q2) i+ cos ( + q2) j]

    r1q1|q=0= i , r1

    q2|q=0= 0,

    r2q1|q=0= 0, r2

    q2|q=0= a

    [ sini+ cosj

    ].

    l2q1|q=0 = u2

    (r2q1 r1q1

    )q=0

    = cos ,

    l2q2|q=0 = u2

    (r2q2 r1q2

    )q=0

    = a (cos sin sin cos ) = a sin( ).

    cReuven Segev and Lior Falach 41

  • snoitrabiV fo yroehT

    2k = 2K

    [ soc ) (nis a 2soc

    ) (2nis 2a soc ) (nis a

    ]

    = T1m2

    (1q2)

    +2I2

    (2q2)

    = M

    [0 1m

    2I 0

    ].

    01

    , nq . .

    = TM1=n

    1

    2q( nm

    + 2)nN

    1+M=n

    1

    2q( nI

    2)n

    , . \ \ .

    )(

    U = nFnq

    =N1=m

    qmnKm

    = mq lnK = nF. l =6 m 0l = m 1

    lnK , , nq lq.

    :

    24hcalaF roiL dna vegeS nevueRc

  • snoitrabiV fo yroehT

    m . .

    .

    n m mnK.

    1.01 1

    m

    1k

    2k

    3k

    x

    m

    1f

    2f

    3f

    1 = 3 = 2 = 1

    = K31=i

    3k + 2k + 1k = )3k 2k 1k( = if

    2.01 2

    a

    b

    ak

    bk

    34 hcalaF roiL dna vegeS nevueRc

  • snoitrabiV fo yroehT

    .1 b = 2 ,1 a = 1

    .2bkb = 2M ,1aka = 1M

    = K21=i

    aak = iMbbk + 2

    .2

    3.01 3

    aRbR

    cR

    aL ,aJ ,aGcL ,cJ ,cG

    bL ,bJ ,bG

    bm amcm

    cR ,bR ,aR . iL ,iJ ,iG c ,b ,a = i. a .

    44hcalaF roiL dna vegeS nevueRc

  • snoitrabiV fo yroehT

    bRaR

    cR

    1 = b= c

    bRcR= b

    bRcR

    = abRaR= b

    bRaR

    bT

    cT

    cF

    cFaT

    aF

    aF

    a

    = K

    )bRaF bRcF bT( = M

    = bT, bJbGbL

    = bbJbGbL

    = aTaJaGaL

    = aaJaGaL

    bRaR

    = baJaGaL

    bRaR

    = cTcJcGcL

    = ccJcGcL

    bRcR= b

    cJcGcL

    bRcR

    . aF ,cF c ,a

    = cFcTcR

    =cJcGcL

    bRc2R

    = aFaTaR

    =aJaGaL

    bRa2R

    )bRaF bRcF aT( = K=

    bJbGbL

    +cJcGcL

    b2Rc2R

    +aJaGaL

    b2Ra2R

    .

    . ): (.

    4.01 4

    54 hcalaF roiL dna vegeS nevueRc

  • Theory of Vibartions

    R

    k1k2 k3

    m1m2

    x

    a

    = 1, x = 0 .q2 = x q1 = , 2

    1 = (a+R), 2 = R, 3 = 0

    K11 =

    M = ((R + a)k11 +Rk22) = k1(R + a)2 + k2R2K21 = f = (k22) = k2R

    = 0, x = 1

    1 = 0, 2 = 1, 3 = 1

    K12 =

    M = (Rk22) = k2RK22 =

    f = (k22 k33) = k3 + k2,

    K =

    [k1(R + a)

    2 + k2R2 k2R

    k2R k3 + k2

    ].

    5 10.5

    m

    k1

    k2 k1

    k2

    x

    y

    cReuven Segev and Lior Falach 46

  • Theory of Vibartions

    2

    q1 = x, q2 = y.

    M =

    [m 0

    0 m

    ].

    x

    k1 1

    1

    k2

    1

    2

    k22

    k22

    k11

    k11

    .1 = cos,2 = cos

    K11 =

    Fx = [2k11 cos 2k22 cos ] = 2k1 cos2 + 2k2 cos2 K21 =

    Fy = [2k11 sin + 2k22 sin ] = k1 sin(2) k2 sin(2)

    y

    k1

    1

    1

    k2

    1

    2

    k22

    k22

    k11

    k11

    1 = sin,2 = sin

    K12 =

    Fx = [2k11 cos + 2k22 cos ] = k1 sin(2) k2 sin(2)K22 =

    Fy = [2k11 sin 2k22 sin ] = 2k1 sin2 + 2k2 sin2 ,

    K =

    [2k1 cos

    2 + 2k2 cos2 k1 sin(2) k2 sin(2)

    k1 sin(2) k2 sin(2) 2k1 sin2 + 2k2 sin2

    ]

    cReuven Segev and Lior Falach47

  • snoitrabiV fo yroehT

    11

    . , .

    . nF nq

    = nG, N

    qmnK1=m nG nF = nG, m

    = nFN1=m

    qnmKm

    )q( K = F

    1K = ) F( = q

    = nqN1=m

    mFmn

    mn nq mq mF.

    Q = qK+ qM

    M(q)

    Q = q +

    M(q)

    0 = q +

    84hcalaF roiL dna vegeS nevueRc

  • snoitrabiV fo yroehT

    1.11 1

    m

    2k 1k

    x

    f x f = 2, 1k1 = 1 x

    12k

    = 2 + 1 = 1

    1k+

    1

    2k=2k + 1k2k1k

    ,

    = K1

    =

    2k1k2k + 1k

    .

    2.11 2

    a b

    m

    1k 2k

    x

    f x .

    94hcalaF roiL dna vegeS nevueRc

  • snoitrabiV fo yroehT

    a b

    2T1T f

    21

    ,1 = f = 2T + 1T

    .0 = b2T a1T

    = 1Tb

    b +a= 2T ,

    a

    b +a,

    = 11T1k

    =b

    )b +a( 1k= 2 ,

    2T2k

    =a

    )b +a( 2k.

    + 1 = a)1 2(

    b +a=

    b

    )b +a( 1k+

    a1k b2k)b +a( 1k2k

    a2

    =b2k

    2a1k ba2k1k2 + 22)b +a(2k1k

    ,

    = K1

    =

    )b +a(2k1k2

    . 2a1k ba2k1k2 + 2b2k

    3.11 3

    .

    05hcalaF roiL dna vegeS nevueRc

  • snoitrabiV fo yroehT

    m

    1k

    2k

    x

    f x f2 = 2f = 1f

    = 11f1k

    =2

    1k= 2 ,

    2f2k

    =2

    2k.

    = 22 + 12 = )2k + 1k(4

    2k1k,

    = K1

    =

    2k1k)2k + 1k(4

    .

    4.11 4

    L3

    L6

    L2

    2m 1m

    :

    15hcalaF roiL dna vegeS nevueRc

  • Theory of Vibartions

    a bf

    L

    y(x) =

    fbx

    6EIL

    [L2 b2 x2] x a

    fb6EIL

    [Lb (x a)3 +

    (L2 b2)x x3] x a

    a = L3 , b =2L3 m1 f = 1

    11 = y

    (x =

    L

    3

    )=

    4L3

    243EI,

    21 = y

    (x =

    L

    2

    )=

    23

    1296

    L3

    EI.

    a = L2 , b =L2 m2 f = 1

    12 = y

    (x =

    L

    3

    )=

    23

    1296

    L3

    EI,

    22 = y

    (x =

    L

    2

    )=

    1

    48

    L3

    EI.

    cReuven Segev and Lior Falach52

  • snoitrabiV fo yroehT

    II

    .

    1

    .)t(f = xk +xc +xm

    , 0 = )t(f . 0 = c

    .

    ,0 = xk +xm

    ,)t(nis B+ )t(soc A = )t(x

    )t(soc B + )t(nis A = )t(x)t(nis B2 )t(soc A2 = )t(x

    0 = ))t(nis B+ )t(soc A()2m k(

    = 0 = m2 kk

    m.

    cesdar = ][ .

    = f

    pi2

    = ] f[ f1 = T.1

    zH = ces

    35 hcalaF roiL dna vegeS nevueRc

  • snoitrabiV fo yroehT

    0x

    0v

    D

    1.1:

    ,0v = )0(x ,0x = )0(x

    + )t(soc 0x = )t(x0v

    .)t(nis

    1nat = (0x/0v

    )= D ,

    + 02x

    0v(

    2).

    = )(nis0xD= )(soc ,

    /0v

    D,

    ])t(nis )(soc + )t(soc )(nis[ D = )t(x

    . ) +t( nis D =

    D .

    2

    .

    ,)t(nis 0F = xk +xm

    45 hcalaF roiL dna vegeS nevueRc

  • snoitrabiV fo yroehT

    mk

    )t(nis 0F = )t( F

    1.2:

    ,)t(px + )t(hx = )t(x

    )t(hx )t(px .

    )t(nis D+ )t(soc C = )t(px

    ( .)t(nis 0F = )t( nis D)m2 k( + )t(soc C)m2 k

    = D0F

    ,0 = C ,m2 k

    = )t(x0F

    )t(nis B+ )t(soc A + )t(nis m2 k

    B,A . ) (

    ,0x = )0(hx + )0(px = )0(x

    .0v = )0(hx + )0(px = )0(x

    55 hcalaF roiL dna vegeS nevueRc

  • snoitrabiV fo yroehT

    1.2 ecnanoseR

    0 = 0x0 = 0v

    = B ,0 = A

    0F= m2 k

    0F

    . )2 2( m

    )t(x

    = )t(x0F

    )t(nis )2 2( m0F

    = )t(nis )2 2( m0Fm

    ()t(nis )t(nis /

    2 2)

    )t(x mil latipoH'l

    mil

    mil = )t(x

    0Fm

    ()t(soc t )t(nis /1

    2

    )=

    0F2m2

    ))t(soc t )t(nis(

    001 08 06 04 02 0001

    08

    06

    04

    02

    0

    02

    04

    06

    08

    001

    ]ces[t

    1= rof ecnanoseR

    2.2:

    ) (

    .

    3

    65 hcalaF roiL dna vegeS nevueRc

  • snoitrabiV fo yroehT

    1m3k 2k 1k

    2m

    2x 1x

    1.3: 2

    2x2k + 1x)2k + 1k( = )2x 1x(2k 1x1k = 1x1m2x)3k + 2k( 1x2k = 2x3k )1x 2x(2k = 2x2m

    [ 0 1m

    2m 0

    []1x

    2x

    ]+

    [2k 2k + 1k3k + 2k 2k

    []1x

    2x

    ]=

    [0

    0

    ])1.3( .

    .) +t(nis 2A = 2x ,) +t(nis 1A = )t(1x

    ) +t(nis 2A2 = 2x ) +t(nis 1A2 = )t(1x[ 1.3

    0 1m

    2m 0

    []) +t(nis 1A2) +t(nis 2A2

    ]+

    [2k 2k + 1k3k + 2k 2k

    []) +t(nis 1A

    ) +t(nis 2A

    ]=

    [0

    0

    ][ t

    2k 21m 2k + 1k22m 3k + 2k 2k

    []1A

    2A

    ]=

    [0

    0

    ].

    . ) (.

    ted

    [2k 21m 2k + 1k

    22m 3k + 2k 2k]

    =(21m 2k + 1k

    ( )22m 3k + 2k

    0 = 22k )75hcalaF roiL dna vegeS nevueRc

  • snoitrabiV fo yroehT

    , .

    k = 3k = 2k = 1k m = 2m = 1m

    0 = ]k + 2m k2[ ]k 2m k2[ = 2k 2)2m k2(

    = 12k

    m= 22 ,

    k3

    m.

    ]2A ,1A[ = A 2 ,1 A . T

    [ k 2m k22m k2 k

    []1A

    2A

    ]=

    [0

    0

    ].

    = 12 2A = 1A

    k 2A ,1A m

    mk3 = 22 2A = 1A. , A R A .= 1 2x ,1x

    k m

    = 2 2x ,1x .

    k3 m

    1.3

    jiA i j . 1+n < n.

    = 1

    k

    m= 2 ,

    k3

    m.

    )2 +t2(nis 21A + )1 +t1(nis 11A = )t(1x

    )2 +t2(nis 22A + )1 +t1(nis 12A = )t(2x

    12A = 11A 22A = 21A B = 12A = 11A A = 22A = 21A

    )2 +t2(nis A + )1 +t1(nis B = )t(1x

    )2 +t2(nis A )1 +t1(nis B = )t(2x

    85 hcalaF roiL dna vegeS nevueRc

  • snoitrabiV fo yroehT

    0 = )0(2x = )0(1x 1 = )0(2x = )0(1x

    )2(nis A + )1(nis B = 1

    )2(nis A )1(nis B = 1)2(soc A2 + )1(soc B1 = 0

    )2(soc A2 )1(soc B1 = 0

    )1(soc B = 0 )1(nis B = 1

    2pi = 1 1 = B.

    )2(soc A = 0 )1(nis A = 0

    0 = A 2.

    nis = )t(2x = )t(1x(+t1

    pi

    2

    ).)t1(soc =

    .

    1 = )0(2x = )0(1x 0 = )0(2x = )0(1x

    )2(nis A + )1(nis B = 0

    )2(nis A )1(nis B = 0)2(soc A2 + )1(soc B1 = 1

    )2(soc A2 )1(soc B1 = 1

    0 = B 0 = 2 21 = A,

    = )t(1x1

    21 = )t(2x )t2(nis

    2.)t2(nis

    .

    95 hcalaF roiL dna vegeS nevueRc

  • snoitrabiV fo yroehT

    0 = )0(2x = )0(1x 0 = )0(2x 1 = )0(1x

    )2(nis A + )1(nis B = 1

    )2(nis A )1(nis B = 0)2(soc A2 + )1(soc B1 = 0

    )2(soc A2 )1(soc B1 = 0

    0 = )1(soc B )1(nis B2 = 1

    = 1 21 = B. pi 2

    0 = )2(soc A )2(nis A2 = 1

    = 2 21 = A pi 2

    = )t(1x1

    2nis(+t1

    pi

    2

    )+

    1

    2nis(+t2

    pi

    2

    )1 =

    2])t2(soc + )t1(soc[

    = )t(2x1

    2nis(+t1

    pi

    2

    )1

    2nis(+t2

    pi

    2

    )1 =

    2. ])t2(soc )t1(soc[

    4

    0 = x]K[ +x ] M[

    )+t(nis A = )t(x )+t(nis iA = )t(ix . )+t(nis A2 = x

    0 = A]K[ ) +t(nis +A] M[ ) +t(nis 2

    t

    [)1.4( 0 = A]] M[ 2 ]K[06hcalaF roiL dna vegeS nevueRc

  • snoitrabiV fo yroehT

    1] M[ 2 ]K[ 1] M[.

    1.4

    1. n n nA , . . . ,1A.

    2. .

    3. .

    ted n 2 2 n [ 0 = ]] M[ 2 ]K[

    ) ]K[ 1] M[ etined evitisoP ( ,

    ,n2 < < 12

    j2 1 n n jiA i j. 0 =6 j1A juj1A = jA ju

    (Aj)

    1A j1A =

    j

    = ju

    1

    j1A/j2A.

    .

    .

    j1A/j,1nAj1A/jnA

    = ] U[[nu . . . 1u

    ]=

    1 . . . 1 1

    n1A/n2A 21A/22A 11A/12A.

    .

    .

    .

    .

    .

    .

    .

    .

    .

    .

    .

    n1A/nnA 21A/2nA 11A/1nA

    .

    ju j. ] U[ .

    = )t(ixn1=j

    = )j +tj(nis jiAn1=j

    ,)j +tj(nis jiUj1A

    jiU n2 j1A j, n2 .

    16 hcalaF roiL dna vegeS nevueRc

  • snoitrabiV fo yroehT

    1.4

    n

    A]K[j

    A] M[ j2 =j

    juj1A = jA

    u] M[ j2 = ju]K[j

    Tiu

    uj2 = ju]K[ Tiu)2.4( ju] M[ Ti

    ju ,iu

    u i2 = iu]K[ Tju)3.4( iu] M[ Tj

    : D,C D C , D C

    . TC TD = T]D C[

    [ iu]K[ Tju

    T][ju]K[ Tiu = juT]K[ Tiu =

    iu] M[ TjuT]

    ju] M[ Tiu = juT] M[ Tiu =

    ] M[ ,]K[ . 2.4 3.4

    .iu] M[ Tju i2 ju] M[ Tiuj2 = iu]K[ Tju ju]K[ Tiu

    [ R iu] M[ Tju ,iu]K[ Tju iu]K[ Tju

    T][,iu]K[ Tju =

    iu] M[ TjuT]

    .iu] M[ Tju =

    26 hcalaF roiL dna vegeS nevueRc

  • snoitrabiV fo yroehT

    ju]K[ Tiu = iu]K[ Tju ju]K[ Tiu[iu]K[ Tju

    T]0 = ju]K[ Tiu ju]K[ Tiu =

    = 0(i2 j2

    ), ju] M[ Tiu

    j =6 i 0 =6 i2 j2

    .j =6 i 0 = ju] M[ Tiu

    " ". ]I[ ] M[ n .

    u] M[ j2 = ju]K[j

    Tju

    uj2 = ju]K[ Tju.0 = ju] M[ Tj

    ju] M[ Tiu ) ]I[ = ] M[ (.

    u 1 = 1ju j

    ju] M[ Tiu

    .1 = iu] M[ Tiu

    5

    , N , , n lx ,mx N , . . . ,1 = l ,m.

    .: N , . . . ,1 = j ju .

    36 hcalaF roiL dna vegeS nevueRc

  • snoitrabiV fo yroehT

    : , Na , . . . ,1a

    N1=j

    uja,0 = j

    [iuT] M

    [iuT]M

    N 1=j

    ujaj

    N = 1=j

    ja[iuT]ia = juM

    [iuT]0 = iuM

    0 = ia N , . . . ,1 = i

    N1=j

    ujaN , . . . ,1 = i 0 = ia 0 = j

    nR nR y

    = yN1=i

    uin.n] U[ = i

    in y .

    in , ] M[ Tju

    = y] M[ TjuN1=i

    uiniu] M[ Tj

    0 = iu] M[ Tju j =6 i

    = iny] M[ Tiu

    iu] M[ Tiu.

    1.5

    )t(f

    .)t(f = x]K[ +x] M[

    46 hcalaF roiL dna vegeS nevueRc

  • snoitrabiV fo yroehT

    = )t(xN1=i

    u)t(in,i

    = )t(x n

    u)t(in 1=i N , . . . 1 = i ,iu . i

    ] M[N1=i

    u)t(in]K[ + i

    N1=i

    u)t(in)t(f = i

    N1=i

    u] M[)t(in+ i

    N1=i

    u]K[)t(in.)t(f = i

    Tju

    N1=i

    u)t(in+ iu] M[ Tj

    N1=i

    u)t(in)t(f Tju = iu]K[ Tj

    i =6 j 0 = iu] M[ Tju = iu]K[ Tju

    u)t(jnu)t(jn + ju] M[ Tj

    )t(f Tju = ju]K[ Tj

    n , . . . ,1 = j

    )t(jn + )t(jnju]K[ Tju

    ju] M[ Tju=

    )t(f Tju

    ju] M[ Tju

    ju 0 = ju] M[ j2 ju]K[

    ju]K[ Tju

    ju] M[ Tju=u] M[ j2Tju

    j

    ju] M[ Tju, j2 =

    + )t(jn2= )t(jnj

    )t(f Tju

    ju] M[ Tju,

    n + )tj(soc jD+ )tj(nis jC = )t(jnp,)t(j

    )t(jpn .

    ,0v = )0(x ,0x = )0(x

    56 hcalaF roiL dna vegeS nevueRc

  • snoitrabiV fo yroehT

    = 0xN1=i

    u)0(ini

    ] M[ Tju

    u = 0x] M[ Tju] M[ Tj

    u)0(in

    = iN1=i

    u)0(inu)0(jn = iu] M[ Tj

    ju] M[ Tj

    = )0(jn0x] M[ Tju

    ju] M[ Tju.

    = )0(jn0v] M[ Tju

    ju] M[ Tju.

    2.5

    mmk k k

    2x 1x

    )t(2F )t(1F

    1.5: 2 .

    [ 0 m

    m 0

    []1x

    2x

    ]+

    [k k2k2 k

    []1x

    2x

    ]=

    [)t(1F

    )t(2F

    ].

    66 hcalaF roiL dna vegeS nevueRc

  • Theory of Vibartions

    [m 0

    0 m

    ][x1

    x2

    ]+

    [2k kk 2k

    ][x1

    x2

    ]=

    [0

    0

    ]

    2 =

    3km 1 =

    km

    u1 =

    [1

    1

    ], u2 =

    [1

    1

    ].

    u1T [M ]u1 = 2m u2T [M ]u2 = 2m.

    u1T f(t) = F1(t) + F2(t),

    u2T f(t) = F1(t) F2(t).

    n1 + 21n1 =

    F1(t) + F2(t)

    2m,

    n2 + 22n2 =

    F1(t) F2(t)2m

    .

    x0 =

    [x1

    x2

    ], v0 =

    [v1

    v2

    ],

    n1(0) =u1T [M ]x0

    u1T [M ]u1=x1 + x2

    2,

    n1(0) =u1T [M ]v0

    u1T [M ]u1=v1 + v2

    2,

    n2(0) =u2T [M ]x0

    u2T [M ]u2=x1 x2

    2,

    n2(0) =u2T [M ]v0

    u2T [M ]u2=v1 v2

    2.

    F2(t) = D2 sin(2t) F1(t) = D1 sin(1t)

    nP1 (t) =D1

    2m(21 21)sin(1t) +

    D22m(21 22)

    sin(2t)

    nP2 (t) =D1

    2m(22 21)sin(1t) D2

    2m(22 22)sin(2t)

    cReuven Segev and Lior Falach 67

  • Theory of Vibartions

    nH1 (t) = A1 sin(1t) +B1 cos(1t),

    nH2 (t) = A2 sin(2t) +B2 cos(2t).

    n1(0) = nP1 (0) + n

    H1 (0) = B1 =

    x1 + x22

    n2(0) = nP2 (0) + n

    H2 (0) = B2 =

    x1 x22

    n1(0) = nP1 (0) + n

    H1 (0) = 1A1 +

    1D12m(21 21)

    +2D2

    2m(21 22)=v1 + v2

    2

    n2(0) = nP2 (0) + n

    H2 (0) = 2A2 +

    1D12m(22 21)

    2D22m(22 22)

    =v1 v2

    2

    A1 =v1 + v2

    21 1D1

    2m(21 21)1 2D2

    2m(21 22)1,

    A2 =v1 v2

    22 1D1

    2m(22 21)2 2D2

    2m(22 22)2.

    5.3

    L2

    L2

    L2

    L2

    k 2k 3k

    x1 x2

    F0 sin(t)

    m 2m

    =1

    k

    [3/8 1/8

    1/8 5/24

    ], M = m

    [1 0

    0 2

    ], f(t) =

    [0

    F0 sin(t)

    ]

    [] [M ] x+ x = [] f(t)

    cReuven Segev and Lior Falach 68

  • snoitrabiV fo yroehT

    m

    k

    [8/2 8/3

    42/01 8/1

    []1x

    2x

    ]+

    [1x

    2x

    ]=

    [8/1

    42/5

    ]0Fk

    .)t(nis

    ] M[ ][ .

    ted[0 = ]] M[ 2 I

    km2 =

    1

    891 2

    42.0 = 1 +

    47.1 = 12k

    m95.4 = 22 ,

    k

    m.

    = 1u

    [1

    8.0

    ]= 2u ,

    [1

    826.0

    ].

    m87.1 = 2u] M[ T2u m82.2 = 1u] M[ T1u

    ,)t(nis 0F8.0 = )t(f T1u

    .)t(nis 0F826.0 = )t(f T2u

    47.1 + 1nk

    m= 1n

    0F8.0m82.2

    ,)t(nis

    95.4 + 2nk

    m= 2n

    0F826.0m87.1

    .)t(nis

    96hcalaF roiL dna vegeS nevueRc

  • snoitrabiV fo yroehT

    4.5

    .

    54

    k2

    k4k6

    )t(nis 0F = )t( F

    x

    y

    m

    .y = 2q ,x = 1q

    .

    0 = y ,1 = x 1 = y ,0 = x

    k2

    )54(soc k3 )54(soc k3

    k4

    07 hcalaF roiL dna vegeS nevueRc

  • snoitrabiV fo yroehT

    0 = y ,1 = x

    = 11K

    = xFk7 = ))54(2soc k6 k4(

    = 12K

    = yFk3 = ))54(2soc k6(

    1 = y ,0 = x

    = 22K

    = yF.k5 = ))54(2soc k6 k2(

    = M

    [0 m

    m 0

    ]= F ,

    [)(soc )t( F

    )(nis )t( F

    ].

    ted[k3 m2 k7 [ted = ]M2 K

    m2 k5 k3

    ]0 = 2k62 + km221 4 =

    38.2 = 12k

    m61.9 = 22 ,

    k

    m.

    [ Mj2 K

    ]0 = ju

    = 1u

    [1

    27.0

    ]= 2u ,

    [1

    83.1

    ]

    17 hcalaF roiL dna vegeS nevueRc

  • snoitrabiV fo yroehT

    m29.2 = 2uMT2u ,m25.1 = 1u] M[ T1u

    FT1u

    1u] M[ T1u])( nis 27.0 )(soc[ )t( F66.0 =

    FT2u

    2uMT2u])( nis 83.1 + )(soc[ )t( F43.0 =

    + 1n2)t(1F = )t(nis 0F])( nis 27.0 )(soc[ 66.0 = 1n1

    + 2n2)t(2F = )t(nis 0F])( nis 83.1 + )(soc[ 43.0 = 2n2

    2.45 = 0 = )t(1F )t(2F , 2u 7.53 =

    1u 0 = )t(2F )t(1F .

    27hcalaF roiL dna vegeS nevueRc

  • snoitrabiV fo yroehT

    III

    .

    1

    )t ,x(v

    L

    x

    1.1:

    T , T ) (. )t ,x(v x t.

    x . .

    x

    x + x x

    )x()x + x(

    T

    T

    2.1: "

    x = ))x((nis T ))x +x((nis Tv2

    2t,

    37 hcalaF roiL dna vegeS nevueRc

  • snoitrabiV fo yroehT

    v

    x,)(nis ' )(nat =

    ( Tv

    xv )x +x(

    xx = )

    v2

    2t

    x 0 x

    mil0x

    (Tv)t ,x(xv )t ,x +x( x

    x

    )T =

    v2

    2x =

    v2

    2t

    T = 2c

    v2

    2t2c =

    v2

    2x,

    . A = A EA = T= AEA = 2c

    E E

    c .

    2

    L)t ,x + x(N

    E,A

    x

    )t ,x(N

    x + x x

    )t ,x(f

    )t ,x(f

    1.2:

    47 hcalaF roiL dna vegeS nevueRc

  • snoitrabiV fo yroehT

    x

    x = )t ,x(N)t ,x +x(N+x)t ,x(fu2

    2t

    . x 0 x

    + )t ,x(fN

    x =

    u2

    2t

    )t ,x(EA = )t ,x(N xu =

    u2

    2t=AE

    u2

    2x+)t ,x(f

    = AE = 2c E A =

    u2

    2t2c =

    u2

    2x+)t ,x(f

    .

    3

    L

    x + x x

    )x( T)x + x( T

    R

    )t ,x( J ,G

    )t ,x(

    1.3:

    57hcalaF roiL dna vegeS nevueRc

  • snoitrabiV fo yroehT

    )t ,x( x t

    R

    L

    G

    R2pi = JJ 4

    = x .T= JG

    LT JG

    x

    I = x)t ,x( + )x( T )x +x( T2

    2tJx =

    2

    2t

    x 0 x

    mil0x

    [)x( T )x +x( T

    x)t ,x( +

    ]=T

    x= )t ,x( +

    (x JG

    )x

    J = )t ,x( +2

    2t

    G = 2c

    2

    2t2c =

    2

    2x+)t ,x(

    J.

    4

    2

    2t2c =

    2

    2x)1.4(

    . ,

    )t( T)x(X = )t ,x(

    )x( X)t( T2c = )x(X)t( T

    67 hcalaF roiL dna vegeS nevueRc

  • snoitrabiV fo yroehT

    )t( T)x(X

    )t( T)t( T

    2c =)x( X)x(X

    .

    , t x

    )t( T)t( T

    2c =)x( X)x(X

    . =

    . )t( T ,)x(X )0 ,,+( . 0 < ,

    T.)t( = )t( T

    0 > 2 = teB + teA = )t( T )t( T tmil.

    0 = B+tA = )t( T .

    0 < 0 < 2 = )t(soc D+ )t(nis C = )t( T.

    )t( T)t( T

    2c =)x( X)x(X

    .2 =

    , t

    0 = )t( T2 + )t( T

    x

    + )x( X(c

    2).0 = )x(X

    nis A = )x(X(cx)

    soc B+(cx))2.4( .

    .

    77 hcalaF roiL dna vegeS nevueRc

  • snoitrabiV fo yroehT

    1.4

    0 = )t ,L( = )t ,0(

    )t( T)x(X = )t ,x(

    ,0 = )t( T)0(X = )t ,0(

    .0 = )t( T)L(X = )t ,L(

    0 = )L(X = )0(X 2.4

    0 = B = )0(X

    nis A = )L(X(cL)

    soc B+(cL)

    0 =

    0 = B

    c. . . ,2 ,1 = n ,npi = L

    n = ncpi

    L,

    nis nA = )x(nXn(cx)

    nis nA =npi(Lx),

    )x(nX )( n.

    = )t ,x(1=n

    )t(nT)x(nX

    =1=n

    nisn(cx)

    . ])tn(soc nD+ )tn(nis nC[

    87hcalaF roiL dna vegeS nevueRc

  • snoitrabiV fo yroehT

    1 9.0 8.0 7.0 6.0 5.0 4.0 3.0 2.0 1.0 0

    1

    8.0

    6.0

    4.0

    2.0

    0

    2.0

    4.0

    6.0

    8.0

    1

    L/x

    n X)x(

    1=n2=n3=n4=n

    1.4:

    2.4

    L

    x

    L

    x

    nn

    )t ,L = x( T

    )t ,L = x(N

    2.4:

    L = x 0 = )t ,L = x(N 0 = )t ,L = x( T,

    JG

    x 0 =L=x|

    x,0 =L=x|

    AEu

    xu 0 =L=x|

    x.0 =L=x| 0 = x

    .0 = )t ,0 = x(u ,0 = )t ,0 = x(

    97 hcalaF roiL dna vegeS nevueRc

  • snoitrabiV fo yroehT

    = )t ,0(

    x.0 = )t ,L(

    0 = B = )0(X

    = )L( X

    csoc A

    (cL)

    +

    cnis B

    (cL)

    0 =

    soc (Lc)

    0 =

    cpi = L

    2. . . ,2 ,1 = n npi +

    c = n)1 n2( pi

    L2. . . ,2 ,1 = n

    nis A = )x(nXn(cx).

    = )t ,x(1=n

    )t(nT)x(nX

    =1=n

    nisn(cx)

    . ])tn(soc nD+ )tn(nis nC[

    1 9.0 8.0 7.0 6.0 5.0 4.0 3.0 2.0 1.0 0

    1

    8.0

    6.0

    4.0

    2.0

    0

    2.0

    4.0

    6.0

    8.0

    1

    L/x

    n X)x(

    1=n2=n3=n4=n

    3.4:

    08hcalaF roiL dna vegeS nevueRc

  • snoitrabiV fo yroehT

    3.4 x

    Tk

    k

    x

    L = x .

    )t ,L = x(uk = sF = )t ,L = x(N

    xuAE = )t ,x(N

    AEu

    x)t ,L = x(uk = )t ,L = x(

    AEu

    x.0 = )t ,L(uk + )t ,L(

    )t ,L = x( Tk = sM = )L = x( T

    x JG = )t ,x( T

    JG

    x)t ,L = x( Tk = )t ,L = x(

    JG

    x.0 = )t ,L( Tk + )t ,L(

    ,

    0 = )0(X 0 = )t( T)0(X = )t ,0(JG

    xXJG = )t ,L( Tk + )t ,L(

    0 = )L(XTk + )L( XJG 0 = )t( T)L(XTk + )t( T)L(

    18 hcalaF roiL dna vegeS nevueRc

  • snoitrabiV fo yroehT

    5.4 4 5.3 3 5.2 2 5.1 1 5.0 05

    4

    3

    2

    1

    0

    1

    2

    3

    4

    5

    c/

    4.4: 3.4

    0 = B = )0(X

    JG = )L(XTk + )L( XJG

    csoc A

    (cL)

    nis ATk +(cL)

    0 =

    nat c(Lc)

    JG =

    Tk)3.4( .

    5

    1.4

    2n

    2cX = )x(nX

    )x(n

    L

    )x(mX xd 0

    2n

    2c

    L 0= xd)x(mX)x(nX

    L 0xd)x(mX)x(n X

    2n

    2c

    L 0= xd)x(mX)x(nX

    [X)x(mX

    )x(n

    L]0L

    0X)x(n X

    .xd)x(m

    28 hcalaF roiL dna vegeS nevueRc

  • snoitrabiV fo yroehT

    n,m

    2m

    2c

    L 0= xd)x(mX)x(nX

    [X)x(nX

    )x(m

    L]0L

    0X)x(n X

    .xd)x(m

    n2 m22c

    L 0= xd)x(mX)x(nX

    [X)x(mX

    )x(n

    L]0)1.5( 0L])x(m X)x(nX[

    , .

    1.5

    [ 0 = )L(mX ,0 = )0(nX N m,n X)x(mX

    )x(n

    L]0

    ,0 =

    [ 0 = )L(m X ,0 = )0(nX N m,n X)x(mX

    )x(n

    L]0

    .0 =

    ( 1.5 n2 m2

    2c

    L )0.0 = xd)x(mX)x(nX

    m =6 n m =6 n L

    0, 0 = xd)x(mX)x(nX

    n = m L

    00 =6 xd)x(nX)x(nX

    0 )x(nX)x(nX.

    2.5

    , L = x

    0 = )L(XTk + )L( XJG

    38 hcalaF roiL dna vegeS nevueRc

  • snoitrabiV fo yroehT

    )L(XJGTk = )L( X

    = )L(n XTkJG

    .)L(nX

    [ 0 = x 0 = )0(mX X)x(mX

    )x(n

    L]0)L(nXJGTk [ )L(mX = 0L])x(m X)x(nX[

    ][Tk JG

    )L(mX

    ],0 = )L(nX

    L 0.0 = xd)x(mX)x(nX

    : , .

    0b + )t ,0(0a)t ,0(

    x,0 =

    Lb + )t ,L(La)t ,L(

    x.0 =

    L

    0.m =6 n ,0 = xd)x(mX)x(nX

    3.5 x

    x

    m0I

    , "

    48hcalaF roiL dna vegeS nevueRc

  • snoitrabiV fo yroehT

    x

    x

    m

    )t ,L(N)t ,L( T0I

    m = )t ,L(Nu2

    2t)t ,L(

    xuAE = )t ,x(N

    uAEx

    m = )t ,L(u2

    2t)t ,L(

    u22t

    2c =u22x

    uAEx

    2cm = )t ,L(u2

    2x)t ,L(

    2cmu2

    2xEA + )t ,L(

    u

    x.0 = )t ,L(

    JG = )t ,L( Tx

    0I = )t ,L(2

    2tc0I = )t ,L(

    22

    2x)t ,L(

    c0I 2

    2

    2xGJ + )t ,L(

    x.0 = )t ,L(

    .

    0 = )0(X 0 = )t( T)0(X = )t ,0(c0I

    22

    2xGJ + )t ,L(

    xc0I = )t ,L(

    0 = )L( XGJ + )L( X2c0I 0 = )t( T)L( XGJ + )t( T)L( X2

    58 hcalaF roiL dna vegeS nevueRc

  • Theory of Vibartions

    X = 2c2X .X(0) = B = 0

    I0c2X (L) + JGX (L) = I02X(L) + JGX (L)

    = I02A sin(cL)

    +

    cJGA cos

    (cL)

    = 0

    tan(cL) JGI0c

    = 0.

    X(0) = 0(

    2m 2nc2

    ) L1Xn(x)Xm(x)dx = Xm(L)X

    n(L)X m(L)Xn(L).

    I02X(L) + JGX (L) = 0

    X n(L) =I0

    2n

    JGX(L)

    (2m 2n

    c2

    ) L0Xn(x)Xm(x)dx =

    (2n 2m

    ) I0JG Xm(L)Xn(L)

    (2n 2m

    ) [ 1c2

    L0Xn(x)Xm(x)dx+

    I0JG

    Xm(L)Xn(L)

    ]= 0,

    n 6= m [1

    c2

    L0Xn(x)Xm(x)dx+

    I0JG

    Xm(L)Xn(L)

    ]= 0

    2n

    c2

    L0Xn(x)Xm(x)dx =

    L0X n(x)Xm(x)dx

    =[Xm(x)X

    n(x)

    ]L0 L

    0X n(x)X

    m(x)dx

    = Xm(L)Xn(L)

    L0X n(x)X

    m(x)dx

    =I0

    2n

    JGXn(L)Xm(L)

    L0X n(x)X

    m(x)dx

    cReuven Segev and Lior Falach 86

  • snoitrabiV fo yroehT

    n2

    [1

    2c

    L 0+xd)x(mX)x(nX

    0IGJ

    )L(mX)L(nX

    ]=

    L 0X)x(n X

    xd)x(m

    L 0X)x(n X

    .0 = xd)x(m

    6

    N , ) (.

    = )t ,x(n

    )t(nT)x(nX

    )t(nT )t(nX .

    nX , , )x(X

    = )x(Xn

    ,)x(nXng

    ng . ellivuoiL-mrutS.

    1.6 .

    2

    2t2c =

    2

    2x)t ,x(g +

    = )t ,x(

    )t ,x(g . )t(nT)x(nX1=n

    n

    T)x(nXc = )t( n

    2n

    )t ,x(g + )t(nT)x(n X

    78 hcalaF roiL dna vegeS nevueRc

  • snoitrabiV fo yroehT

    = n X 2n2c nX

    n

    )x(nX[ + )t( n T

    2)t(nTn

    ].)t ,x(g =

    L

    )x(mX xd 0

    L 0

    [n

    )x(nX[ + )t( n T

    2)t(nTn

    ]]= xd)x(mX

    L 0xd)x(mX)t ,x(g

    + )t( n T2= )t(nTn

    L L xd)x(nX)t ,x(g 0

    X 02xd)x(n

    T + )tn(soc nD+ )tn(nis nC = )t(nTP.)t( n

    nD ,nC

    )x(h = )0 ,x(

    t)x(v = )0 ,x(

    = )0 ,x(n

    )x(h = )0(nT)x(nX

    L

    )x(mX xd 0

    = )0(nT

    L L xd)x(nX)x(h 0

    X 02xd)x(n

    T + nD =P)0( n

    L

    L xd)x(nX)x(h 0X 0

    2xd)x(n

    T + nD =P)0( n

    = )0(nT

    L L xd)x(nX)x(v 0

    X 02xd)x(n

    + nCn =.)0( nPT

    88hcalaF roiL dna vegeS nevueRc

  • snoitrabiV fo yroehT

    2.6

    L

    ,E ,0A

    )t(nis 0Y = )t(y

    0A L, E. )t(nis 0Y = )t(y.

    .

    u2

    2t=

    0A0AE

    u2

    2x2c =

    u2

    2x,

    E = 2c,

    ,)t(y = )t ,0(u

    u

    x,0 = )t ,L(

    ,0 = )0 ,x(uu

    t.0 = )0 ,x(

    ,

    )t(y + )t ,x(v = )t ,x(u

    u2

    2t=

    v2

    2t+)t(y2d

    2td=v2

    2t)t(nis 0Y2

    u2

    2x=

    v2

    2x

    v2

    2t2c =

    v2

    2x)t(nis 0Y2 +

    .

    0 = )t ,0(v )t(y = )t(y + )t ,0(v = )t ,0(uu

    x= )t ,L(

    v

    x0 = )t ,L(

    98 hcalaF roiL dna vegeS nevueRc

  • snoitrabiV fo yroehT

    .

    0 = )0 ,x(v 0 = )0(nis 0Y + )0 ,x(v = )0 ,x(uu

    t= )0 ,x(

    v

    tv 0 = )0(soc 0Y + )0 ,x(

    t.0Y = )0 ,x(

    v2

    2t2c =

    v2

    2x)t(nis 0Y2 +

    0 = )t ,0(vv

    x0 = )t ,L(

    0 = )0 ,x(vv

    t0Y = )0 ,x(

    nis = )x(nXn(cx)= n ,

    cpi

    L2. . . ,2 ,1 = n )1 n2(

    1=n

    )t(nT)x(nX

    1=n

    )x(nX[ + )t( n T

    2)t(nTn

    ])t(nis 0Y2 =

    L )t(mX xd 0

    + )t( n T2 = )t(nTn

    )t(nis 0Y2

    L L xd)x(nX 0X 0

    2xd)x(n

    L

    0= xd)x(nX

    L 0

    nisn(cx)= xd

    [c n

    socn(cxL])

    0

    =c

    n,

    L 0= xd)x(n2X

    L 0

    2nisn(cx)= xd

    L 0

    [)x cn( soc 1

    2

    ]= xd

    L

    2.

    + )t( n T2 = )t(nTn

    )t(nis 0Y2c2

    Ln

    09 hcalaF roiL dna vegeS nevueRc

  • Theory of Vibartions

    TPn (t) = Gn sin(t) +Hn cos(t)

    Gn =2c2Y0

    nL (2n 2)

    Tn(t) = Gn sin(t) + Cn sin(nt) +Dn cos(nt).

    Tn(0) =

    L0 v(x, 0)Xn(x)dx L

    0 X2n(x)dx

    = 0 = Dn

    Tn(0) =

    L0

    vt (x, 0)Xn(x)dx L

    0 X2n(x)dx

    = 2cY0Ln

    = Gn + nCn

    Cn = n

    [2cY0

    Ln+Gn

    ].

    v(x, t) =n=1

    [Gn sin(t) + Cn sin(nt)] sin(ncx).

    u(x, t) = v(x, t) + Y0 sin(t).

    cReuven Segev and Lior Falach91

  • snoitrabiV fo yroehT

    7

    x

    1.7:

    . . 2.7.

    x

    )t ,x(v

    x x

    y

    )x( V z

    x x

    )x(M

    2.7:

    )t ,x(v )( x t.

    )t ,x( V x t.

    )t ,x(M x t.

    29 hcalaF roiL dna vegeS nevueRc

  • snoitrabiV fo yroehT

    v2

    2x=M

    IE)1.7(

    I E .

    M

    x)2.7( V =

    3.7 x.

    x + x x

    xx

    )t ,x( V)t ,x + x( V

    )t ,x(M)t ,x + x(M

    )t ,x(q

    3.7:

    x

    xA = x)t ,x(q + )t ,x( V )t ,x +x( V = F

    v2

    2t,

    x 0 x mil

    mil0x

    [)t ,x( V )t ,x +x( V

    x

    ]xA = )t ,x(q +

    v2

    2t,

    V

    xA = )t ,x(q +

    v2

    2t,

    39 hcalaF roiL dna vegeS nevueRc

  • snoitrabiV fo yroehT

    1.7 2.7

    V

    x =

    M2

    2x =

    2

    2x

    (IE

    M2

    2x

    ) IE =

    v4

    4x

    IEv4

    4xA = )t ,x(q +

    v2

    2t

    Av2

    2tIE +

    v4

    4x)t ,x(q =

    1.7

    Av2

    2tIE +

    v4

    4x0 =

    AIE = 2c

    v2

    2t 2c =

    v4

    4x

    )t( T)x(X = )t ,x(v

    T2d

    2tdd T2c = X

    X4

    4xd

    )t( T)x(X

    T2d

    2td1

    T1 2c =

    X

    X4d

    4xd

    ,

    T2d

    2td1

    T1 2c =

    X

    X4d

    4xd =

    4 2 = . T2d

    2td0 = T2 +

    X4d

    4xd

    2

    2c0 = X

    49 hcalaF roiL dna vegeS nevueRc

  • snoitrabiV fo yroehT

    ( xeB = )x(X 4

    (c

    )2)0 = teB

    c = 0 = 2) c( 4 i = 4,3 , = 2,1

    e1B = )x(Xe2B+ x

    xie1B+ xie3B+ x

    .)x(soc 4D+ )x(nis 3D+ )x(hsoc 2D+ )x(hnis 1D = )x(X

    )t ,x(v x t.

    )t ,x(xv = )t ,x( )( x t.

    IE = )t ,x(M x t.v22x

    )t ,x(

    v3 IE = )t ,x( V x t.3x

    x

    4.7:

    59 hcalaF roiL dna vegeS nevueRc

  • Theory of Vibartions

    7.4

    v(0, t) = 0, v(L, t) = 0, M(0, t) = 0, M(L, t) = 0.

    v(x, t) = X(x)T (t)

    v(0, t) = 0 X(0) = 0 D2 +D4 = 0M(0, t) = 0 d

    2X

    dx2(0) = 0 2D2 2D4 = 0

    v(L, t) = 0 X(L) = 0 D1 sinh(L) +D2 cosh(L) +D3 sin(L) +D4 cos(L) = 0M(L, t) = 0 d

    2X

    dx2(L) = 0 2D1 sinh(L) + 2D2 cosh(L) 2D3 sin(L) 2D4 cos(L) = 0

    ,D2 = D4 = 0

    D1 sinh(L) +D3 sin(L) = 0

    2D1 sinh(L) 2D3 sin(L) = 0

    D3 6= 0 D3 sin(L) = 0 D1 = 0 D1 sinh(L) = 0

    L = npi n = 1, 2, . . .

    n = c(npiL

    )2.

    Xn(x) = sin(npixL

    ).

    x

    :7.5

    cReuven Segev and Lior Falach 96

  • Theory of Vibartions

    7.5

    v(0, t) = 0, (0, t) = 0, M(L, t) = 0, V (L, t) = 0

    v(x, t) = X(x)T (t)

    X(x) = D1 sinh(x) +D2 cosh(x) +D3 sin(x) +D4 cos(x).

    v(0, t) = 0 X(0) = 0 D2 +D4 = 0(0, t) = 0 dX

    dx(0) = 0 D1 + D3 = 0

    V (L, t) = 0 d3X

    dx3(L) = 0 3D1 cosh(L) + 3D2 sinh(L) 3D3 cos(L) + 3D4 sin(L) = 0

    M(L, t) = 0 d2X

    dx2(L) = 0 2D1 sinh(L) + 2D2 cosh(L) 2D3 sin(L) 2D4 cos(L) = 0

    D1 = D3D2 = D4 D1 [cosh(L) + cos(L)] +D2 [sinh(L) sin(L)] = 0D1 [sinh(L) + sin(L)] +D2 [cosh(L) + cos(L)] = 0.

    [cosh(L) + cos(L) sinh(L) sin(L)sinh(L) + sin(L) cosh(L) + cos(L)

    ]{D1

    D2

    }= 0

    det

    [cosh(L) + cos(L) sinh(L) sin(L)sinh(L) + sin(L) cosh(L) + cos(L)

    ]=

    (cosh(L) + cos(L))2 (sinh(L) sin(L)) (sinh(L) + sin(L)) =cosh2(L) + 2 cosh(L) cos(L) + cos2(L) sinh2(L) + sin2(L) =

    2 cosh(L) cos(L) + 2 = 0

    cosh(L) cos(L) = 1.

    1L = 1.875, 2L = 4.694, 3L = 7.855, 4L = 10.99, 5L = 14.137

    Xn(x) = sinh(nx) sin(nx) cosh(nL) + cos(nL)sinh(nL) sin(nL) [cosh(nx) cos(nx)] .

    cReuven Segev and Lior Falach 97

  • Theory of Vibartions

    7.2

    (nc

    )2Xn(x) =

    d4Xndx4

    L

    0 dx Xm(x) (nc

    )2 L0Xm(x)Xn(x)dx =

    L0

    d4Xndx4

    Xmdx

    =

    L0Xmd

    (d3Xndx3

    )=

    [Xm

    d3Xndx3

    ]L0

    L

    0

    d3Xndx3

    d (Xm)

    =

    [Xm

    d3Xndx3

    ]L0

    L

    0

    d3Xndx3

    dXmdx

    dx

    =

    [Xm

    d3Xndx3

    ]L0

    L

    0

    dXmdx

    d

    (d2Xndx2

    )=

    [Xm

    d3Xndx3

    ]L0

    [dXmdx

    d2Xndx2

    ]L0

    +

    L0

    d2Xndx2

    d2Xmdx2

    dx

    (nc

    )2 L0Xm(x)Xn(x)dx =

    [Xm

    d3Xndx3

    dXmdx

    d2Xndx2

    ]L0

    +

    L0

    d2Xndx2

    d2Xmdx2

    dx

    (mc

    )2 L0Xm(x)Xn(x)dx =

    [Xn

    d3Xmdx3

    dXndx

    d2Xmdx2

    ]L0

    +

    L0

    d2Xndx2

    d2Xmdx2

    dx

    2n 2mc2

    L0Xm(x)Xn(x)dx =

    [Xm

    d3Xndx3

    Xnd3Xmdx3

    (dXmdx

    d2Xndx2

    dXndx

    d2Xmdx2

    )]L0

    n 6= m L

    0Xm(x)Xn(x)dx.

    cReuven Segev and Lior Falach 98

  • snoitrabiV fo yroehT

    3.7

    Av2

    2tIE +

    v4

    4x)t ,x(q =

    = )t ,x(v1=n

    )t(nT)x(nX

    )x(nX

    A1=n

    [)t(nT2d

    2td)x(nX

    ]IE +

    1=n

    [)t(nT

    )x(nX4d

    4xd

    ])t ,x(q =

    (nc

    2)= )x(nX

    nX4d4xd

    1=n

    [A

    )t(nT2d

    2td+ )x(nX

    n(c

    2))t(nTIE

    ])t ,x(q = )x(nX

    )t(nT2d

    2td= )t(nTn2 +

    L L xd)x(nX)t ,x(q 0

    X 02xd)x(n

    .

    4.7

    .

    xL

    a)t(nis 0F

    ,A,E

    99 hcalaF roiL dna vegeS nevueRc

  • snoitrabiV fo yroehT

    )x(a)t(nis 0F = )t ,x(q

    c = npin(L

    2)nis = )x(nX ,

    xpin(L

    ).

    L

    0nis = )a(nX = xd)x(a)x(nX

    apin(L

    )L

    0= xd)x(n2X

    L

    2

    )t(nT2d

    2td= )t(nTn2 +

    L L xd)x(nX)t ,x(q 0

    X 02xd)x(n

    =nis 0F2

    (apinL

    )L

    )t(nis nG = )t(nis

    = nG ) Lapin(nis 0F2

    L

    + )tn(soc nD+ )tn(nis nB = )t(nTnG

    2 n2)t(nis

    ,0 = )0 ,x(vv

    t.0 = )0 ,x(

    0 = )0(nT ,0 = )0(nT

    = nB ,0 = nDn

    nG2 n2

    .

    5.7

    6.7.

    001hcalaF roiL dna vegeS nevueRc

  • snoitrabiV fo yroehT

    )x( V)x + x( VP

    P)x(xv = )x()x + x(xv = )x + x(

    7.7: x

    P P

    6.7:

    x 7.7. y

    v P )x( V )x +x( Vx

    P + )x +x(v

    xAx = )x(

    v2

    2t

    P )x( V )x +x( V[v

    xv )x +x(

    x)x(

    ]Ax =

    v2

    2t

    = xV M22x

    x 0 x

    M2

    2x P

    v2

    2xA =

    v2

    2t

    v22x

    IEM =

    IEv4

    4x P

    v2

    2xA =

    v2

    2t

    v2

    2t=IE

    A

    v4

    4x+

    P

    A

    v2

    2x

    101 hcalaF roiL dna vegeS nevueRc

  • Theory of Vibartions

    v(x, t) = X(x)T (t)

    d2T

    dt2X =

    EI

    A

    d4X

    dx4T +

    P

    A

    d2X

    dx2T

    XT

    d2T

    dt21

    T=EI

    A

    d4X

    dx41

    X+

    P

    A

    d2X

    dx21

    X= 2

    T .

    T = A cos(t) +B sin(t).

    X

    EId4X

    dx4+ P

    d2X

    dx2 2AX = 0

    X(x) = Cex

    EI4 + P2 2AX = 0

    2 =P

    P 2 + 4EI2A

    2EI

    21 =P +

    P 2 + 4EI2A

    2EI= 2 > 0

    22 =P

    P 2 + 4EI2A

    2EI= 2 < 0

    = {,, i,i}

    X(x) = C1ex + C2e

    x + C3eix + C4eix

    X(x) = D1 cosh(x) +D2 sinh(x) +D3 cos(x) +D4 sin(x)

    cReuven Segev and Lior Falach 102

  • snoitrabiV fo yroehT

    0 = )t ,L(M ,0 = )t ,L(v ,0 = )t ,0(M ,0 = )t ,0(v

    0 = 3D = 1D 0 = 2D

    pin = L

    + PA2IE4 + 2 P

    IE2=2pi2n

    2L

    = n2AIE4

    (IE2pi2n2

    2LP

    2)2I2E4 = 2 P +

    pin(L

    4)4

    pin(L

    2)PIE

    = n2IE

    A

    pin(L

    4)P A

    pin(L

    2) 0 < P

    .

    301hcalaF roiL dna vegeS nevueRc

  • snoitrabiV fo yroehT

    VI

    .

    1

    : .

    c

    x

    m

    1.1:

    1.1

    .c = cM xc = cF

    2.1

    cm

    x

    k

    )t(f

    2.1:

    401 hcalaF roiL dna vegeS nevueRc

  • snoitrabiV fo yroehT

    ,xm = xk xc )t(f = xF

    +xc

    m+x

    k

    m0 = x

    n mn2c = 2= n

    k m

    c = . 2km

    +xn2 +x2,0 = xn

    , teA = )t(x

    +n2 + 220 = n

    = 2,1 n2

    n24 n2242

    n n =n = 1 2

    (

    1 2

    ).

    1.1 1 >

    pxe A = )t(x(n(+

    1 2

    )t)

    pxe B+(n(

    1 2

    )t)

    )tn(pxe =[pxe A

    (nt1 2

    )pxe B+

    (n

    t1 2

    ])

    = )(hnise e

    2= )(hsoc ,

    e + e

    2,

    )tn(pxe = )t(x[hsoc C

    (nt1 2

    )hnis D+

    (n

    t1 2

    ])

    0v = )0(x 0x = )0(x

    )tn(pxe = )t(x[hsoc 0x

    (nt1 2

    )+0xn + 0v

    nhnis 1 2

    (n

    t1 2

    ]).

    .

    501 hcalaF roiL dna vegeS nevueRc

  • snoitrabiV fo yroehT

    01 9 8 7 6 5 4 3 2 1 00

    2.0

    4.0

    6.0

    8.0

    1

    t

    (X)t

    0=0V,1=0X1=0V ,0=0X

    3.1:

    2.1 1 <

    ni n = 2,1

    2 1

    )tn(pxe = )t(x[pxe A

    (ni

    t2 1)

    pxe B+(ni

    t2 1

    ]),

    C B,A R )t(x

    )(nis i + )(soc = )i( pxe

    )tn(pxe = )t(x[soc )B+A(

    (n

    t2 1)

    nis )BA(i +(n

    t2 1]),

    )BA(i = D ,B+A = C

    R D,C B A.

    n = d

    ,2 1

    601 hcalaF roiL dna vegeS nevueRc

  • snoitrabiV fo yroehT

    01 9 8 7 6 5 4 3 2 1 01

    8.0

    6.0

    4.0

    2.0

    0

    2.0

    4.0

    6.0

    8.0

    1

    t

    4.1:

    . ])td( nis D+ )td( soc C[ )tn(pxe = )t(x

    0v = )0(x ,0x = )0(x

    )tn(pxe = )t(x[+ )td( soc 0x

    0xn + 0vd

    )td( nis

    ].

    = hX

    + 02x

    (0xn + 0v

    d

    2)= )(nat ,

    0x0xn+0v

    d

    . ) +td( nis )tn(pxe hX = )t(x

    tnemerced cimhtiragoL

    , , 2.0 = . 2.0 =

    = d

    ,n89.0 = n22.0 1

    701 hcalaF roiL dna vegeS nevueRc

  • Theory of Vibartions

    .d n x(t) .

    x(t) = Xh [n exp(nt) sin (dt+ ) + d exp(nt) cos (dt+ )] = 0

    tan (dt+ ) =dn

    tk = t +

    kpi

    d, k = 1, 2, . . .

    x(tk)

    x(t) = Xh {n exp(nt) [n sin (dt+ ) + d cos (dt+ )]d exp(nt) [n cos (dt+ ) d sin (dt+ )]}

    x(tk) = ndXh exp(ntk) cos (dtk + ) .

    x(tk) = 0, x(tk) < 0

    d =2pi

    d=

    2pi

    n

    1 2 .

    i Ai

    Ai = X0 exp(nti)

    Ai+kAi

    =X0 exp(ntj)X0 exp(nti)

    = exp (n (ti+k ti))

    = exp (n (kd)) = exp(nk 2pi

    n

    1 2

    )= exp

    (k 2pi

    1 2

    )

    cReuven Segev and Lior Falach108

  • Theory of Vibartions

    ln

    ln

    (Ai+kAi

    )= 2pik

    1 2 2pik

    12pik

    ln

    (AiAi+k

    )

    0 1 2 3 4 5 6 7 8 9 101

    0.8

    0.6

    0.4

    0.2

    0

    0.2

    0.4

    0.6

    0.8

    1

    t

    x(t)

    Ai

    Ai+k

    kd

    :1.5

    = 1 1.3

    d 0 1

    limd0

    x(t) = limd0

    exp(nt)[x0 cos (dt) +

    v0 + nx0d

    sin (dt)

    ]= exp(nt) lim

    d0

    [x0 cos (dt) +

    v0 + nx0d

    sin (dt)

    ]

    limd0

    [x0 cos (dt)] = x0

    limd0

    [v0 + nx0

    dsin (dt)

    ]= (v0 + nx0) lim

    d0

    [sin (dt)

    d

    ]= (v0 + nx0) lim

    d0

    [t cos (dt)

    1

    ]= t (v0 + nx0)

    cReuven Segev and Lior Falach 109

  • Theory of Vibartions

    x(t) = exp(nt) [x0 + (v0 + nx0) t] .

    . < 1

    2

    mx+ cx+ kx = F0 sin(t)

    x+ 2nx+ 2nx =

    F0m

    sin(t) = G0 sin(t)

    xP (t) = A sin(t) +B cos(t)

    xp(t) = A cos(t) B sin(t)xp(t) = 2A sin(t) 2B cos(t),

    [2A 2nB + 2nA] sin(t) + [2B + 2nA+ 2nB] cos(t) = G0 sin(t)

    2A 2nB + 2nA = G02B + 2nA+ 2nB = 0

    A =G0(2n 2

    )(2n 2)2 + (2n)2

    =

    F0m

    (km 2

    )(km 2

    )2+(cm)2 = F0

    (k m2)

    (k m2)2 + (c)2,

    B = G02n(2n 2)2 + (2n)2

    = F0m

    cm(

    km 2

    )2+(cm)2 = F0c(k m2)2 + (c)2 .

    cReuven Segev and Lior Falach 110

  • snoitrabiV fo yroehT

    = pX= 2B+ 2A

    0F2)c( + 2)2m k(

    c = )(nat2m k

    .) +t(nis pX = )t(px

    , ) +td( nis )tn(pxe hX+ ) +t(nis pX = )t(x

    ,hX . )tn(pxe )t(px = )t(ssx

    = pmA0F

    2)c( + 2)2m k(=

    (k/0F2)k/m( 1

    2)+(ck

    2)

    = n2k

    m,

    c

    mn2 =

    c

    k=m

    k

    c

    m=

    1

    n2= n2

    2

    n

    = pmA(k/0F

    2)n/( 12)

    24 +(n

    . 2)

    1.2

    . )(

    = )(S(2 n2

    2))n2( +

    2

    Sd

    d,0 = n228 +)2 n2( 4 =111 hcalaF roiL dna vegeS nevueRc

  • snoitrabiV fo yroehT

    2 8.1 6.1 4.1 2.1 1 8.0 6.0 4.0 2.0 00

    1

    2

    3

    4

    5

    6

    /n

    mAp

    1.0=2.0=3.0=4.0=5.0=6.0=7.0=8.0=9.0=1=

    1.2:

    ,)22 1(n2 = 2

    n = ,

    22 1

    n = r

    22 1

    . r

    = pXk/0F

    )22 1( 24 + 2))22 1( 1(=

    k/0F

    2

    = 2 10Fcd

    2

    0F pXk

    1

    2.

    1 > .2

    3

    )t(f T ) T + t(f = )t(f t, T , T1 = f fpi2 = .

    k2 cd0F = pX.

    n2mc k2 = 2 1

    cn = 2 1

    n2mc = cd = 2 12

    211 hcalaF roiL dna vegeS nevueRc

  • Theory of Vibartions

    3.1

    sin(nt), cos(mt) = 2piT

    T0

    sin(nt) sin(mt)dt =

    0 n 6= mT/2 n = m T

    0cos(nt) cos(mt)dt =

    0 n 6= mT/2 n = m T

    0cos(nt) sin(mt)dt = 0.

    f(t)

    f(t) = a0 +n=1

    [an sin(nt) + bn cos(nt)] ,

    a0 =1

    T

    T0f(t)dt, an =

    2

    T

    T0f(t) sin(nt)dt, bn =

    2

    T

    T0f(t) cos(nt)dt.

    mx+ cx+ kx = f(t) = a0 +n=1

    [an sin(nt) + bn cos(nt)]

    : 3.2

    3.1

    F0

    t

    f (t)

    :3.1

    cReuven Segev and Lior Falach 113

  • Theory of Vibartions

    f(t) =F0t 0 t

    , = 2pi

    a0 =1

    0f(t)dt =

    F02

    0tdt =

    F02

    an =2

    0f(t) sin (nt) dt =

    2F02

    0t sin (nt) dt = 2F0

    21

    n

    0td cos (nt)

    = 2F02

    1

    n

    {[t cos (nt)]0

    0

    cos (nt) dt

    }= 2F0

    21

    n( cos (n))

    = 2F0

    1

    2pin( cos (2pin)) = F0

    pin

    bn =2

    0f(t) cos (nt) dt =

    2F02

    0t cos (nt) dt =

    2F02

    1

    n

    0td sin (nt)

    =2F02

    1

    n

    {[t sin (nt)]0

    0

    sin (nt) dt

    }=

    2F02

    1

    n( sin (n))

    =2F0

    1

    2pin( sin (2pin)) = 0

    0 0.5 1 1.5 2 2.5 3 3.5 40.5

    0

    0.5

    1

    1.5

    2

    2.5

    3

    3.5

    t[sec]

    f(t)

    f(t)n=5n=10n=15n=20

    :3.2

    cReuven Segev and Lior Falach 114

  • Theory of Vibartions

    3.3

    :

    mx+ cx+ kx = F0 sin(t)

    xp(t) =F0 sin(t+ )

    (k m2)2 + c22tan() = c

    k m2 .

    xp(t) =a0k

    +n=1

    an sin(nt+ n) + bn cos(nt+ n)(k m (n)2

    )2+ n2c22

    tan(n) = cnk mn22 .

    xp(t) =a0k

    +n=1

    a2n + b2n(k m (n)2

    )2+ n2c22

    sin(nt+ n)

    n = m + tan1( bnan

    ).

    4

    (t) =

    0 t <

    1 < t < +

    0 t > +

    f(t) . (t) (t) = lim0 (t)

    0f(t) (t)dt = lim

    0

    [ 0f(t)0dt+

    +

    f(t)

    dt+

    +

    f(t)0dt

    ]= lim

    0

    [1

    +

    f(t)dt

    ]= lim

    0

    [1

    f(c)( + )

    ]for some c[ ,+].

    = lim0 [f(c)] for some c[ ,+]

    = f()

    cReuven Segev and Lior Falach115

  • snoitrabiV fo yroehT

    )t ,x( f

    )t( Fa

    1.4:

    .

    . 1.4

    v2

    2t2c =

    v2

    2x+)t ,x(f

    )t ,x(f , a = x )x(a)t(0F = )t ,x(f

    = )t(nTn + )t(nT1

    L L xd)x(nX)t ,x(f 0

    X 02xd)x(n

    =1

    L L xd)x(nX)x(a)t(0F 0

    X 02xd)x(n

    =)a(nX)t(0F

    L

    X 02xd)x(n

    1.4 esnopser eslupmI

    0 = t,

    )t(00F = xk +xc +xm

    611 hcalaF roiL dna vegeS nevueRc

  • snoitrabiV fo yroehT

    = x0Fmc )t(0

    mk x

    m,x

    + )0(x = )t(x

    t 0.td)t(x

    0 = )0(x ,0 = )0(x, )(x xmil

    mil0mil = )(x x

    [+ )0(x

    0td)t(x

    ]mil =

    0

    [0

    (0Fmt(0

    c )m+x

    k

    mx

    )td]

    =0Fm

    0 = xk +xc +xm

    = )0(x ,0 = )0(x0Fm,

    tne = )t(x0Fdm

    . )td( nis

    )t( 0F = )t(f < t0 = )t(x = t

    = )(x ,0 = )(x0Fm,

    t

    )t(ne = )t(x0Fdm

    . )) t( d( nis

    2.4

    ) ( , .

    711 hcalaF roiL dna vegeS nevueRc

  • snoitrabiV fo yroehT

    )1.4( )t(2f + )t(1f = xk +xc +xm

    0 = )0(x ,0 = )0(x )t(1x

    )2.4( )t(1f = xk +xc +xm

    )t(2x

    )3.4( )t(2f = xk +xc +xm

    )t(2x + )t(1x 1.4 . )t(2x ,)t(1x . . )t(1x )t(2x 2.4 3.4 0x = )0(x 0v = )0(x )t(2x + )t(1x

    1.4 0x2 = )0(2x + )0(1x 0v2 = )0(2x + )0(1x. 0v = )0(x ,0x = )0(x )t(0x )t(2x ,)t(1x )t(2x + )t(1x + )t(0x 1.4 . )t(0x

    )tn( pxe = )t(0x[+ )td( soc 0x

    0xn + 0vd

    )td( nis

    ].

    2.4. = t d)(f

    0F

    = )t(x

    . > t )) t( d( nis dm/0F )) t( n( pxe < t 0 = t d)(f

    = )t( x

    d)(f < t 0dm

    . > t )) t( d( nis )) t( n( pxe= )t(1x

    )t( x t

  • Theory of Vibartions

    f (t)

    t

    d

    f ( )d

    :4.2

    x0(t) = exp (nt)[x0 cos (dt) +

    v0 + nx0d

    sin (dt)

    ]

    x(t) =

    t=0

    f()

    mdexp (n (t )) sin (d (t )) d Force responce

    + exp (nt)[x0 cos (dt) +

    v0 + nx0d

    sin (dt)

    ] Initial condition

    4.3

    f(t) =

    F0 sin(t), 0 t pi0, t > pi t < pi

    x(t) =

    t=0

    f()

    mdexp (n (t )) sin (d (t )) d

    =F0md

    t=0

    sin() exp (n (t )) sin (d (t )) d

    cReuven Segev and Lior Falach 119

  • Theory of Vibartions

    sin() sin() =1

    2[cos( ) cos( + )]

    sin() sin (d (t )) = 12

    [cos (( + d) dt) cos (( d) + dt)]

    x(t) = x1(t) x2(t)

    x1(t) =F0

    2md

    t=0

    exp (n (t )) cos (( + d) dt) d,

    x2(t) =F0

    2md

    t=0

    exp (n (t )) cos (( d) + dt) d.

    t0

    exp (a + b) cos(h + g) =a2

    a2 + h2

    [exp (a + b)

    (cos (h + g)

    a+h sin(h + g)

    a2

    )]=t=0

    .

    x1

    a = n, b = nt, h = + d, g = dt

    x1(t) =F0

    2md

    (n)2

    (n)2 + ( + d)

    2[exp (n (t ))

    (cos (( + d) dt)

    n+

    ( + d) sin (( + d) dt)(n)

    2

    )]=t=0

    =F0

    2md

    (n)2

    (n)2 + ( + d)

    2[(cos (t)

    n+

    ( + d) sin (t)

    (n)2

    ) exp (nt)

    (cos (dt)

    n ( + d) sin (dt)

    (n)2

    )].

    x2

    a = n, b = nt, h = d, g = dt

    cReuven Segev and Lior Falach 120

  • Theory of Vibartions

    x2(t) =F0

    2md

    (n)2

    (n)2 + ( d)2[

    exp (n (t ))(

    cos (( d) + dt)n

    +( d) sin (( d) + dt)

    (n)2

    )]=t=0

    =F0

    2md

    (n)2

    (n)2 + ( d)2[(

    cos (t)

    n+

    ( d) sin (t)(n)

    2

    ) exp (nt)

    (cos (dt)

    n+

    ( d) sin (dt)(n)

    2

    )].

    t > pi

    x(t) =

    pi

    =0

    f()

    mdexp (n (t )) sin (d (t )) d +

    t=pi

    f()

    mdexp (n (t )) sin (d (t )) d

    =

    pi

    =0

    f()

    mdexp (n (t )) sin (d (t )) d

    x1(t) =F0

    2md

    (n)2

    (n)2 + ( + d)

    2[exp (n (t ))

    (cos (( + d) dt)

    n+

    ( + d) sin (( + d) dt)(n)

    2

    )]=pi=0

    =F0

    2md

    (n)2

    (n)2 + ( + d)

    2[exp

    (nt+ pin

    )(cos(pi d

    (t pi

    ))n

    +( + d) sin

    (pi d

    (t pi

    ))(n)

    2

    )

    exp (nt)(

    cos (dt)

    n ( + d) sin (dt)

    (n)2

    )]

    .x2(t)

    cReuven Segev and Lior Falach121

  • snoitrabiV fo yroehT

    V

    .

    1 .

    a2

    x

    1.1:

    W .

    221hcalaF roiL dna vegeS nevueRc

  • snoitrabiV fo yroehT

    a2

    2N 1Nx

    2N 1NW

    2.1:

    0 = W 2N+ 1N = yFx +a1N = 2N 0 = )x +a(1N)x a(2N = cM

    x a

    1N

    (+ 1

    x +a

    x a)

    x a W = 1N W =a2

    W = 2N,x +a

    a2.

    x

    xW = )2N 1N(a

    xm =

    =

    W

    ma=

    g

    a.

    .

    321hcalaF roiL dna vegeS nevueRc

  • snoitrabiV fo yroehT

    2 .

    M

    )t(nis 0F m

    K

    k

    M

    m

    K

    k

    )t(nis 0Y

    X

    x

    1.2:

    [ 0 M

    m 0

    {]X

    x

    }+

    [k k + Kk k

    {]X

    x

    }=

    {)t(nis 0F

    0

    }

    )t(nis 2a = )t(x ,)t(nis 1a = )t(X

    0F = )k(2a + )k + K+ 2M(1a0 = )k + 2m(2a + )k(1a

    = sX0FK= 2 ,

    K

    M= 2 ,

    k

    m= ,

    m

    M

    K k

    1a

    (+ 1

    k

    K

    2

    2

    )k 2a

    KsX =

    2a + 1a(

    12

    2

    )0 =

    421 hcalaF roiL dna vegeS nevueRc

  • snoitrabiV fo yroehT

    = 1a

    (2 1

    2

    )(SX

    2 12

    () Kk + 1

    2

    2

    )Kk

    = 2a(SX

    2 12

    () Kk + 1

    2

    2

    )Kk

    =

    0 = 1a

    KSX = 2ak

    0F =k.

    = Kk

    m M

    k= m

    K M

    = 1a

    (2 1

    2

    )(SX

    2 12

    ()2 + 1

    2

    )

    = 2a(SX

    2 12

    ()2 + 1

    2

    )

    ( 1

    2

    2

    () + 1

    2

    2

    )=

    (

    4)(

    2)0 = 1 + ) + 2(

    (

    2)=(

    + 1

    2

    )+

    2

    4.

    2.0 = 52.1 ,8.0 = .

    521hcalaF roiL dna vegeS nevueRc