, , . , (. 134), .. . . : , 3 4 , - .
( tj) , tjo, τj . , , (), . ( ) .
, . , .. tj, . , α s , , , ..
, .
:
(14.105)
. (14.106)
, . (14.106) x y:
. (14.107)
lki
, (14.108)
i k, ; αkio αki' . ( (), ).
:
. (14.109)
ξ η. , s - . ,
|
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, (14.110)
, (14.111)
(14.112)
.
, vki αki, Δ αki - αkio.
:
, (14.113)
ki ; zk () k.
αki (14.105) (14.113)
(14.114)
. (14.115)
zko δzk, δzk.
Mki :
, (14.116)
(14.110). () (.. ), (14.116) δzk.
,
(14.117)
αki - , ( ); Mki' - ; zkio - k; zko - () :
. (14.118)
(14.118) n k.
k:
1. .
2. - :
. (14.119)
3. .
4. δz .
5. ) i , k , ) i , k , ) , :
)
) (14.120)
)
6. ( ):
- ;
- , ; , ;
- ;
- ;
- ;
- ( ).
, :
(14.121)
. (14.122)
(14.116) ,
|
|
(14.123)
. (14.124)
lijk W :
lijk + lkji + likj = - W, (14.125)
lijk = (αkjo αkio) βijk .. (14.126)
β.
(14.127)
. (14.128)
(14.128) ,
. (14.129)
:
, (14.130)
, (14.131)
. (14.132)
, , ski - .
(14.130) .
, . , , n k . r , .. . , = . :
= 180 ( + );
(14.133)
(r) , b , .
j. : = 1, = 2, = 3. (14.133) :
= 180 (1 + 2)
(14.134)
, , 1, 2, , n ( ) , 1, 2, , n, k , r =(n k) . xi pi.
j (j = 1, 2, , k), xi (i = 1, 2,, n) , k .
(14.135)
(14.135) .
j , , .
Tj tj,
x1' = x1 + ν1 = f1 (t1, t2, , tj,, tk)
x2' = x2 + ν2 = f2 (t1, t2, , tj,, tk)
xi' = xi + νi = fi (t1, t2, , tj,, tk) (14.136)
xn' = xn + νn = fn (t1, t2, , tj,, tk)
(14.136) ,
νi = fi (t1, t2, , tj,, tk) - xi. (14.137)
(14.136) , .
(14.136) tj tj ( ), (14.136) .
tj ,
tj = tj + τj, (14.138)
τj tj.
νi = fi (t1+τ1, t2+τ2, , tj+τj,, tk+τk) - xi . (14.139)
(14.139) , :
. (14.140)
:
(14.141)
( ), t ( τ.
. (14.142)
xio tio xi. li :
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|
li = xio xi = fi (t1o, t2o, , tjo, , tko) - xi. (14.143)
C (14.143) (14.139) :
(14.144)
..
..
n (14.144) (n+k) , . , , , .. [ pv2 ] = min.
( ), , k k τj pi li :
[ 11 ] τ1 + [ 12 ] τ2 + [ 13 ] τ3 + + [ 1k ] τk + [ pa1l ] = 0
[ 21 ] τ1 + [ 22 ] τ2 + [ 23 ] τ3 + + [ 2k ] τk + [ pa2l ] = 0
(14.145)
[ i1 ] τ1 + [ i2 ] τ2 + [ i3 ] τ3 + + [ ik ] τk + [ pail ] = 0
[ n1 ] τ1 + [ n2 ] τ2 + [ n3 ] τ3 + + [ nk ] τk + [ panl ] = 0
(14.145) aij (14.144).
(14.145) aij pi li (. 14.15).
C . 14.15 (14.145) .
14.15
,
j i | j | k | li | pi | |||||
a11 | a12 | a13 | a1j | a1k | l1 | p1 | |||
a21 | a22 | a23 | a2j | a2k | l2 | p2 | |||
a31 | a32 | a33 | a3j | a3k | l3 | p3 | |||
i | ai1 | ai2 | ai3 | aij | aik | li | pi | ||
n | an1 | an2 | an3 | anj | ank | ln | pn |
1.
τ1 ( ) 1- ( ), ..
[ pa1a1 ] = c11 = p1a112 + p2a212 + p3a312 +...+ pnan12.
τ2 1- 2- , ..
[ pa1a2 ] = c12 = p1a11a12 + p2a21a22 + p3a31a32 + + pnan1an2.
τ 1- τ.
1 i, li a , ..
[ pa1l ] = d1 = p1a11l1 + p2a21l2 + p3a31l3+ + pnan1ln.
2.
τ1 2- 1- , ..
[ pa2a1 ] = c21 = p1a12a11 + p2a22a21 + p3a32a31 + + pnan2an1.
τ2 2- ( ), ..
[ pa2a2 ] = c22 = p1a122 + p2a222 + p3a322 +...+ pnan22.
τ3 2- 3- , ..
[ pa2a3 ] = c23 = p1a12a13 + p2a22a23 + p3a32a33 ++ pnan2an3.
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2- .
2 i, li a , ..
[ pa2l ] = d2 = p1a12l1 + p2a22l2 + p3a32l3++ pnan2ln.
. k .
, , . , , , , .
, τj:
(14.146)
(14.146) τj tj0, νi (14.144) Tj (tj = tj0 + τj).
.
1. (k), (r) n xi, pi.
2. tj , , .. . k . tj.
3. xi tj, .. (14.135).
4. tj0 tj. () . , .
5. (14.141), (14.142) (14.144), aij li vi (14.143), .. (14.135) .
6. aij, li pi (. 14.15) (14.145), τj tjo.
7. vi xi τj (14.143) .
8. B xi' =(xi + vi) tj =(tjo + τj) (14.135).
- tj tjo. - , . - . (, : .. - , ; , , ).
, , (14.135), , .. tjo, .
, , . , , . ..