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( )

. . , ...

2011

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:. . , ..., (milicic@gf.uns.ac.rs) : , ..., (aborkovic@agfbl.org).

:1. : ,2. : ,3. , : ,4. : ,5. : 1,6. : .

:1. : ,2. : .

....

. . . .

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, , .

. .

3 . . , (Plondgar).

, .

15 .

, , .

. , , .

18 19 20 21 :

o o

o

o

o o

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1 2

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, (

) .

A (F) .

.

:o ( ),o ( ).

.

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j , .

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,

.

, , .

.

, .

:

(u, v, , ) ,

() .

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,

, .

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: C, C1, C, C1,

C1 C .

... ... ()

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[ (Bernoulli) ]

. , ,

.

, ! , !

, !

: ,

.

, ,

.

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(z),

.

ds(z), d

, ds=d, ds(z)=( z) d

.

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, , cosT=1.0

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8

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. ()

().

, .

:

, .

!

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:

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, .

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, !

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.

.

.

...

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,

.

;

:

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,

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17

( )

,

3

( )

1

2

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,

2

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1

2

3

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18

,

, .:

1. ,2. ,3. .

3, 1,

6 6 (, , M, , , ).

(, , ) .

6

1, 2, 3 .

: [ x(s), y(s), (s), (s) ], [ h(s), F(s), I(s), k(s) ], [ E, G, t],

[ p, t, t ], 6 .

3 , 3 .

3 .

( )

,

1

( )

1

2

3

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3 4

: . . , ...

:

.

,

.

. ,

.

.

.

( ,...) . .

,

. , Zo, Cu. ,

(m-1)

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2

(Zo+ Cu).

.

:

( , ,

), :

.Co

.. Cu

m

m=3

Zo ,

Zu ,

Zs Sik,Zk+m Mik, Mki

.

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3

,

.

.

.

.

,

2

( ),

.

, :

1. , 2. .

Uj:..................... 2K Zs

ui, vi:.............. np=2K

:

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5

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:

1.

.

2.

.

3. 4.

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, Zs K .

, : H=0, V=0, M=0

: H=0, V=0, M=0

:

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( ):

:

, :

. . ,

.

. ,

.

, .

.

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9

.

=2, , . det D, (u, v),

(lik, ikki)

.

, , l ik=0,

ik=ki=0, Coi=Cou=0, (u,v). , ,

.

Zo+Zu+Zs+Zk>2K,

2K det D .

1

=14, Zo=3, Zu=0, Zs=25, Zk=0, Zo+Zu+Zs+Zk=2K, det D=0

!

.

()

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10

2

1, 2.!.

1, ,

, 2, ( ) .

( lik) (v).

,

.

, :

Zs+Zk2K 3 .

: Coi (Zo), Cui (Zu), Sik (Zk), Mik, Mki (Zk+m), 2K+m

. Zo+Zs+Zu+Zk+m=2K+m, Zo+Zs+Zu+Zk=2K ,

.

. .

!!!

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11

:

, .

.

: , ,

, . ,

, .

.

. .

.

. m m1

, Zg Zz,

:

3Zg3=2Zz Zg

, .

:

:

:

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12

.

,

() .

,

. .

. 3Zg3=2Zz,

Zs=2K3,

.

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.

Zk+Zs=2K3 3Zg3=2Zz, . , . .

.

.

,

,

. ,

() .

.

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14

. ,

.

. .

. 3Zp3=2Zz.

.

.

3 4 1 2 1 2 3 4. 3 4 ( ) 1 3 . 1 2

, 1 2.

1 2, 3 4, 1 2, 3 4 1 2

3 4. .

.

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15

.

. .

.

. .

.

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16

. .

, 2 m

Sik Mik, Mki,

2K+m, 2+m

.

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17

Coi, Cui, Sik, Mik, Mki. . .

,

Zp . , .

3Zp Zo , Zu , 2Zz

. ,

.

.

. ,

.

. ,

. ,

.

,

.

. .

.

.

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1

5

: . . , ...

: ,

, ,

, . Zp=1, Zz=0, Zo+Zu=3. , : Zo=3

a Zu=0, Zo=2a Zu=1. .

.

,, .

1, 2, 3 .

.

,, .

4, 1, 1 4,

.

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2

.

, , .

.

.

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3

1.

2.

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4

3. ,

. ..

. =

=

F .

F .

F F ,

FI F

I . F

I

FI - FF

I, .

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F F, FI F

I

.

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1.

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2.

3.

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1

6 7

: . . , ...

: ,

, , ,

., Zp=2, Zz=1,

3Zp=2Zz+Zu+Zo, Zo+Zu=4. :

1. , , 2. .

:

Zo=4 Zu=0,

Zo=3 Zu=1,

Zo=2 Zu=2,

, , , , , , .

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2

, (g)

. 3, 1,

(, , , ) 2 : D, (g) Vg Hg. 1,

(g) 3 A, B, C.

1, D, (g), 2, G. .

(, , )

1, 2, .

1 2, .

,

. () A, B, C, D

, . .

: ,

.

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3

, :

,

( )

.

, , (g) (f). yc,

.

, :

1+2 = 0, 1+2 = 0,

=

=

Ha, Ha, Hb, Hb : 1 = 0, 2 = 0

= 0 = 0 I I I Hg, Vg.

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4

.

, , .

.

Va', Vb'

, .

Va=Vb=0

Ha=Hb=H

Va=Vb=0Ha=Hb=H

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5

:

I I, I, (g)

bg.

:

, c.

c :

c ,

, , :

, :

cos + + = 0

, :

sin + = 0

, = , :

: =0:

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6

. I, I I, , I I I, I V

g. 3Zp=2Zz+Zo+Zu

. 43=12 , 4 24=8 .

I I I, :

I V, :

I, H=0, V=0, :

I I I, I V , :

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7

.

Zz=4, .

n n+1 n+1. ,

3(n+1) .

n+3 , 2n . 3(n+1)=n+3+2n, .

:

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8

, , , ., .

Zp, Zz=Zp1.

, 2Zz+Zo+Zu=3Zp,, Zo+Zu=3Zp2Zz= 3Zp2(Zp1)= 3Zp2Zp +2 = Zp+2. , Coi, Cui,

Zp1,

. 3 4,(. ), .

(I+II , V).

.

1, 2, 3, 4 .

( I I I, . )

( I V) (I+ I I). , (I I I I V)

. .

.

.

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9

Zo+Zu=Zp+2, :

1. , , 2. , .

,

. . . .

, .

,

. , .

V'm+1 V(m+1)

(m+1) m+1:

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10

m A, B,

, :

V'm+1 V(m+1) Hm+1 H(m+1) Vm+1 V(m+1) :

) () , )

) )

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1

o e 8

: . . , ...

:

, , , , ,

, ,

. ,

. :

. .

. . .

, ,

, .

.

, .

. . , :

, , , .

: , , , .

:

, .

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2

.

," " .

( )

. =0 =0. . :

....Zo Zs

Zs+Zo=2K, ,

. .

(

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3

). .

.

:

1.

"0",2. ,

,

3. , , "0",

4. ,

. .

.

5.

4

, , ,

6. 4 ,

.

:

.

1. ,2.

3. .

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4

4.

5. ,

6. . 7. .

, :

:

1. .

2. .3. ,

.

4. , 1, 2, 3 .5. (. ) =0

"0".

6. =0 H=0 V=0.

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5

. "", "U".

"D", "V".

"" , "" , "" . .

m+1=0, :

=1

1

1

m=0, :

+1 =

1

+1

=0, :

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6

=0, :

m=0, :

m-1=0, :

=0, :

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7

m-1=0, : +

+ 1 +1 +1 1

= 0

Vm,

(m), m+1=0.

,

. ,

.

Vm=0, :

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8

m-1=0, :

m=0, :

tm t .

, :

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1

9 10 11

: . . , ...

: ( )

:

,

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2

,

:

, , , , , , , . , , , ,

,, . ,

+ = 0 ,

+

+ = 0 ,

= 0 ,

, 1 , 2 , 3a ( ) , , i

k :

= ( ) +

+ ( )

=

+

pn, pt Pi Mi Coi , Cui N, T, M

() () .

, ,T, coi cui () lik

ik

, ().

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3

, .

:,

,

.

:

.

:

( )

, .

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4

1. , (Betti ) , :

:

:

:

:

, , , :

, Pm Ci

Pn Ci .

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2. , (Maxwell )

P1 P2,

P1 P2 P2 P2 P1.

3. , ( Rayleigh )

1 2 2

1.

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6

4. , ( Rayleigh )

1

1.

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= 0, = 0 =

1 = 0 + + = 0 = +

,

=

( ) =

=

= + =

+ =

= = +

1

= + =

+ +

v "" , ( ) .

. . .

:

= 1

= 1

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11

.

:

, , ,

.

:

= = =

:

.

() ,

.

. ,

.

.

, .

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12

.

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14

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1

12 13 14

: . . , ...

: , ,

Zs

. ,

. .

, .

X1X2,X3.

Nik=Sik+Rx/2 (1)Nki=Ski+Rx/2 (2)

Tik=(MikMki)/lik+RyR (3)

Tki=(MkiMik)/likRyR (4)

(1) (4) Mi k,Mki Si k.

:

+ + = 0 + + + = 0

+ + = 0(5)

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2

(5) Nik (1), Tik

(3), :

+ + = 0

+ + + = 0 + + = 0

(6)

:

= + 12

= + 12 +

(7)

(6) .

2K + m. :

Zo Coi,Zu Cui,

Zs Si k,

Zk+ m Mi k Mki.

:

,

,

.

:

:

:

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3

:

. X1,X2,. . . Xn.

Cj{CojCuj),Sik,Mik :

X , F

(6). n (8) 2K + m (6) zo + zu + zs+ zk +m

zo+ zu+ zs+ zk+m Cj,Sik, Mik.

(6) Hi, Vi ,Mi, p, (8) X1, X2, . . . Xn, (6) (8)

p X1, X2, . . . Xn.

:

,

(6) (8) :

:

Cj0, Sik,0, Mik,0 , ,

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4

p, .

Xi= 0. Cj,m, Mik,m, Sik,m, ( m

= 1, 2, . . . n) , n , . n

.

. X1= 1, X2= 1.... Xn= 1.

:

0 1 1 2 2

0 1 1 2 2

0 1 1 2 2

...... ,

...... ,

...... ,

n n

n n

n n

N N X N X N X N

T T X T X T X T

M M X M X M X M

(10)

:

0 0 0

1 1 1

2 2 2

, , ;

, , ;

, , ;

.................

, , ,n n n

N T M

N T M

N T M

N T M

(11)

Xi = 0,

X1 = 1, X2= 1 Xn= 1. Cj ,Si k, Mi k,

,

, (9) (10)

, .

Xi= 0, Cj ,0, Sik,0, Mik,0, N0, T0, M0, p

,

a Xm= 1, Cj,m, Sik,m,Mik,m Nm,Tm,Mm,

, , Xm

= 1.

. Xi= 0

p,

Xm= 1 Xm= 1.

X1 Xn.

(9) (10) , . (9) (10)

,

, p X1

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5

Xn. , Cj

, Cj0 X1 Xn.

1.

, .

, 1.

, .

, 1b.

, .

, 1c.

, .

, , 1d. = 0

, , = /2

, .

.

. , 2 2b

.

X1 X2.

, . ,

. .

, . , 2c.

, ,

, 2d.

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6

,

.

(9). .

:

X1, X2. . . Xn

, 1X1+ + 2X2+ . . . nXn

. , . .

X1 Xn .

.

, . , ,

. . Xi = 1

, :

ji j i i i T s

C c M N T ds (1)

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7

, i T ( )

:

, ,

o

o

t t T

M t N Tt k

EI h EF GF (2)

N, TiM, :

, ,

n n n

0 k k 0 k k 0 k k

k 1 k 1 k 1

N X N T T X T M M X M (3)

:

,

on

ji j i 0 k k t

k 1s

n no

i 0 k k t i 0 k k

k 1 k 1

1 tC c M M X M

EI h

1 kN N X N t T T X T ds

EF GF

(4)

, :

i= 1, 2, .. . n n nX1

Xn. :

,

= 0 + + (7)

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8

:

=1 + = 0, = 1,2, , (8)

:

1 11 +2 12 ++ 1 + 1 = 01 21 +2 22 ++ 2 + 2 = 0. .. .. .. .. .. .. .. .. .

1 1 +2 2 ++ + = 0(9)

,

Xi= 1 :

= = ,

=

(17)

Si Xi=1,

l : = + (18)

S

, (4), :

= +

=1 (19)

(17), (18), (19) :

= 0 + =1 + :

+ 0 + (20)

i= 1, 2, , n

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9

:

=

0 =

0

= = (21)

(7), (20) (8)

.

(9)

, (3), .

, , .

,

.

, ,

. a a

, ac

Z :a0 p,

at t t,

ac c

ak Xk= 1, k = 1,2, . . ., n, : = 0 + + +1 1 +2 2 ++ (28):

= +

=1 (29)

:

= 0 + + (22),

(23)

i(i =1, 2, . . ., n) Xi(i = 1, 2,. . . n) .

1.

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