Download - sk 1 knjiga
-
8/13/2019 sk 1 knjiga
1/88
( )
. . , ...
2011
-
8/13/2019 sk 1 knjiga
2/88
:. . , ..., ([email protected]) : , ..., ([email protected]).
:1. : ,2. : ,3. , : ,4. : ,5. : 1,6. : .
:1. : ,2. : .
....
. . . .
:o ( )o ( )
, , .
, , .
. .
3 . . , (Plondgar).
, .
15 .
, , .
. , , .
18 19 20 21 :
o o
o
o
o o
-
8/13/2019 sk 1 knjiga
3/88
1
1 2
: . . , ...
:
, (
) .
A (F) .
.
:o ( ),o ( ).
.
, .
j , .
-
8/13/2019 sk 1 knjiga
4/88
2
,
.
, , .
.
, .
:
(u, v, , ) ,
() .
-
8/13/2019 sk 1 knjiga
5/88
3
,
, .
-
8/13/2019 sk 1 knjiga
6/88
4
: C, C1, C, C1,
C1 C .
... ... ()
-
8/13/2019 sk 1 knjiga
7/88
5
[ (Bernoulli) ]
. , ,
.
, ! , !
, !
: ,
.
, ,
.
-
8/13/2019 sk 1 knjiga
8/88
6
(z),
.
ds(z), d
, ds=d, ds(z)=( z) d
.
-
8/13/2019 sk 1 knjiga
9/88
7
, , cosT=1.0
-
8/13/2019 sk 1 knjiga
10/88
8
-
8/13/2019 sk 1 knjiga
11/88
9
. ()
().
, .
:
, .
!
-
8/13/2019 sk 1 knjiga
12/88
10
:
!
!
! !!!!
-
8/13/2019 sk 1 knjiga
13/88
11
, .
, .
.
, !
-
8/13/2019 sk 1 knjiga
14/88
12
.
.
.
...
-
8/13/2019 sk 1 knjiga
15/88
13
,
.
;
:
-
8/13/2019 sk 1 knjiga
16/88
14
,
-
8/13/2019 sk 1 knjiga
17/88
15
-
8/13/2019 sk 1 knjiga
18/88
-
8/13/2019 sk 1 knjiga
19/88
17
( )
,
3
( )
1
2
3
( )
,
2
( )
1
2
3
-
8/13/2019 sk 1 knjiga
20/88
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
-
8/13/2019 sk 1 knjiga
21/88
1
3 4
: . . , ...
:
.
,
.
. ,
.
.
.
( ,...) . .
,
. , Zo, Cu. ,
(m-1)
-
8/13/2019 sk 1 knjiga
22/88
2
(Zo+ Cu).
.
:
( , ,
), :
.Co
.. Cu
m
m=3
Zo ,
Zu ,
Zs Sik,Zk+m Mik, Mki
.
-
8/13/2019 sk 1 knjiga
23/88
3
,
.
.
.
.
,
2
( ),
.
, :
1. , 2. .
Uj:..................... 2K Zs
ui, vi:.............. np=2K
:
-
8/13/2019 sk 1 knjiga
24/88
-
8/13/2019 sk 1 knjiga
25/88
5
-
8/13/2019 sk 1 knjiga
26/88
6
:
1.
.
2.
.
3. 4.
-
8/13/2019 sk 1 knjiga
27/88
7
, Zs K .
, : H=0, V=0, M=0
: H=0, V=0, M=0
:
-
8/13/2019 sk 1 knjiga
28/88
8
( ):
:
, :
. . ,
.
. ,
.
, .
.
-
8/13/2019 sk 1 knjiga
29/88
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
!
.
()
-
8/13/2019 sk 1 knjiga
30/88
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 ,
.
. .
!!!
-
8/13/2019 sk 1 knjiga
31/88
11
:
, .
.
: , ,
, . ,
, .
.
. .
.
. m m1
, Zg Zz,
:
3Zg3=2Zz Zg
, .
:
:
:
-
8/13/2019 sk 1 knjiga
32/88
12
.
,
() .
,
. .
. 3Zg3=2Zz,
Zs=2K3,
.
-
8/13/2019 sk 1 knjiga
33/88
13
.
Zk+Zs=2K3 3Zg3=2Zz, . , . .
.
.
,
,
. ,
() .
.
-
8/13/2019 sk 1 knjiga
34/88
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. .
.
-
8/13/2019 sk 1 knjiga
35/88
15
.
. .
.
. .
.
-
8/13/2019 sk 1 knjiga
36/88
16
. .
, 2 m
Sik Mik, Mki,
2K+m, 2+m
.
-
8/13/2019 sk 1 knjiga
37/88
17
Coi, Cui, Sik, Mik, Mki. . .
,
Zp . , .
3Zp Zo , Zu , 2Zz
. ,
.
.
. ,
.
. ,
. ,
.
,
.
. .
.
.
-
8/13/2019 sk 1 knjiga
38/88
-
8/13/2019 sk 1 knjiga
39/88
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,
.
-
8/13/2019 sk 1 knjiga
40/88
2
.
, , .
.
.
-
8/13/2019 sk 1 knjiga
41/88
3
1.
2.
-
8/13/2019 sk 1 knjiga
42/88
4
3. ,
. ..
. =
=
F .
F .
F F ,
FI F
I . F
I
FI - FF
I, .
-
8/13/2019 sk 1 knjiga
43/88
5
F F, FI F
I
.
-
8/13/2019 sk 1 knjiga
44/88
6
1.
-
8/13/2019 sk 1 knjiga
45/88
7
2.
3.
-
8/13/2019 sk 1 knjiga
46/88
-
8/13/2019 sk 1 knjiga
47/88
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,
, , , , , , .
-
8/13/2019 sk 1 knjiga
48/88
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
, . .
: ,
.
-
8/13/2019 sk 1 knjiga
49/88
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.
-
8/13/2019 sk 1 knjiga
50/88
4
.
, , .
.
Va', Vb'
, .
Va=Vb=0
Ha=Hb=H
Va=Vb=0Ha=Hb=H
-
8/13/2019 sk 1 knjiga
51/88
5
:
I I, I, (g)
bg.
:
, c.
c :
c ,
, , :
, :
cos + + = 0
, :
sin + = 0
, = , :
: =0:
-
8/13/2019 sk 1 knjiga
52/88
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 , :
-
8/13/2019 sk 1 knjiga
53/88
7
.
Zz=4, .
n n+1 n+1. ,
3(n+1) .
n+3 , 2n . 3(n+1)=n+3+2n, .
:
-
8/13/2019 sk 1 knjiga
54/88
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)
. .
.
.
-
8/13/2019 sk 1 knjiga
55/88
9
Zo+Zu=Zp+2, :
1. , , 2. , .
,
. . . .
, .
,
. , .
V'm+1 V(m+1)
(m+1) m+1:
-
8/13/2019 sk 1 knjiga
56/88
10
m A, B,
, :
V'm+1 V(m+1) Hm+1 H(m+1) Vm+1 V(m+1) :
) () , )
) )
-
8/13/2019 sk 1 knjiga
57/88
1
o e 8
: . . , ...
:
, , , , ,
, ,
. ,
. :
. .
. . .
, ,
, .
.
, .
. . , :
, , , .
: , , , .
:
, .
-
8/13/2019 sk 1 knjiga
58/88
2
.
," " .
( )
. =0 =0. . :
....Zo Zs
Zs+Zo=2K, ,
. .
(
-
8/13/2019 sk 1 knjiga
59/88
3
). .
.
:
1.
"0",2. ,
,
3. , , "0",
4. ,
. .
.
5.
4
, , ,
6. 4 ,
.
:
.
1. ,2.
3. .
-
8/13/2019 sk 1 knjiga
60/88
4
4.
5. ,
6. . 7. .
, :
:
1. .
2. .3. ,
.
4. , 1, 2, 3 .5. (. ) =0
"0".
6. =0 H=0 V=0.
-
8/13/2019 sk 1 knjiga
61/88
5
. "", "U".
"D", "V".
"" , "" , "" . .
m+1=0, :
=1
1
1
m=0, :
+1 =
1
+1
=0, :
-
8/13/2019 sk 1 knjiga
62/88
6
=0, :
m=0, :
m-1=0, :
=0, :
-
8/13/2019 sk 1 knjiga
63/88
7
m-1=0, : +
+ 1 +1 +1 1
= 0
Vm,
(m), m+1=0.
,
. ,
.
Vm=0, :
-
8/13/2019 sk 1 knjiga
64/88
8
m-1=0, :
m=0, :
tm t .
, :
-
8/13/2019 sk 1 knjiga
65/88
1
9 10 11
: . . , ...
: ( )
:
,
-
8/13/2019 sk 1 knjiga
66/88
2
,
:
, , , , , , , . , , , ,
,, . ,
+ = 0 ,
+
+ = 0 ,
= 0 ,
, 1 , 2 , 3a ( ) , , i
k :
= ( ) +
+ ( )
=
+
pn, pt Pi Mi Coi , Cui N, T, M
() () .
, ,T, coi cui () lik
ik
, ().
-
8/13/2019 sk 1 knjiga
67/88
3
, .
:,
,
.
:
.
:
( )
, .
-
8/13/2019 sk 1 knjiga
68/88
4
1. , (Betti ) , :
:
:
:
:
, , , :
, Pm Ci
Pn Ci .
-
8/13/2019 sk 1 knjiga
69/88
5
2. , (Maxwell )
P1 P2,
P1 P2 P2 P2 P1.
3. , ( Rayleigh )
1 2 2
1.
-
8/13/2019 sk 1 knjiga
70/88
6
4. , ( Rayleigh )
1
1.
-
8/13/2019 sk 1 knjiga
71/88
-
8/13/2019 sk 1 knjiga
72/88
-
8/13/2019 sk 1 knjiga
73/88
-
8/13/2019 sk 1 knjiga
74/88
10
= 0, = 0 =
1 = 0 + + = 0 = +
,
=
( ) =
=
= + =
+ =
= = +
1
= + =
+ +
v "" , ( ) .
. . .
:
= 1
= 1
-
8/13/2019 sk 1 knjiga
75/88
11
.
:
, , ,
.
:
= = =
:
.
() ,
.
. ,
.
.
, .
-
8/13/2019 sk 1 knjiga
76/88
12
.
-
8/13/2019 sk 1 knjiga
77/88
-
8/13/2019 sk 1 knjiga
78/88
14
-
8/13/2019 sk 1 knjiga
79/88
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)
-
8/13/2019 sk 1 knjiga
80/88
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.
:
,
,
.
:
:
:
-
8/13/2019 sk 1 knjiga
81/88
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 , ,
-
8/13/2019 sk 1 knjiga
82/88
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
-
8/13/2019 sk 1 knjiga
83/88
5
Xn. , Cj
, Cj0 X1 Xn.
1.
, .
, 1.
, .
, 1b.
, .
, 1c.
, .
, , 1d. = 0
, , = /2
, .
.
. , 2 2b
.
X1 X2.
, . ,
. .
, . , 2c.
, ,
, 2d.
-
8/13/2019 sk 1 knjiga
84/88
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)
-
8/13/2019 sk 1 knjiga
85/88
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)
-
8/13/2019 sk 1 knjiga
86/88
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
-
8/13/2019 sk 1 knjiga
87/88
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.
-
8/13/2019 sk 1 knjiga
88/88