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https://er.chdtu.edu.ua/handle/ChSTU/8369| Title: | Автоматизований вимірювальний стенд контролю космічних апаратів |
| Authors: | Трембовецька, Руслана Володимирівна Ротаєнко, Віталій Сергійович |
| Keywords: | автоматизований вимірювально-обчислювальний комплекс;метрологічне забезпечення;параметри геометрії мас космічних апаратів;центр мас;тензор інерції;точність вимірювань |
| Issue Date: | 15-Dec-2025 |
| Abstract: | У роботі розглянуто підвищення рівня метрологічного забезпечення автоматизованих вимірювально-обчислювальних комплексів для контролю параметрів геометрії мас космічних апаратів шляхом поєднання вимірювань маси, положення центру мас і компонентів тензора інерції на єдиному обладнанні. The work addresses improving the metrological support of automated measurement and computing systems for monitoring spacecraft mass geometry parameters by combining measurements of mass, center-of-mass position, and inertia tensor components on a single piece of equipment. |
| URI: | https://er.chdtu.edu.ua/handle/ChSTU/8369 |
| Appears in Collections: | 174 Автоматизація, комп'ютерно-інтегровані технології та робототехніка (Робототехнічні системи та автоматизація) |
Files in This Item:
| File | Description | Size | Format | |
|---|---|---|---|---|
| Диплом-магистр_Ротаєнко В.pdf Restricted Access | КРМ Ротаєнко В. | 8.01 MB | Adobe PDF | View/Open Request a copy |
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vi - :
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:
[aa]xΣ + [ab] yΣ + [ac]zΣ = [al],
[ab]xΣ + [bb] yΣ + [bc]zΣ = [bl], (2.6)
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.
,
:
x = Dx , y = Dy
Σ D Σ D , z = Dz
Σ D (2.7)
D - (2.6), Dx, Dy, Dz -
(2.6)
.
i ,
.
.
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2=90 , 3=180 , 4=270
37
D, Dx, Dy, Dz
:
:
D = 2((a 2 2
1 + a3) + (a2 + a4) );
Dx = 2((a1 + a3)(l1 + l3) + (a2 + a4 )(l2 + l4 ));
Dy = (a2 + a4 )((a2 + a4 )(l1 + l3) + (l2 + l4 )(a3 − a1)) + 2(a3l1 + a2l3)(a1 + a3); (2.8)
Dz = (a1 + a3)((a1 + a3)(l2 + l4 ) + (l1 + l3)(a4 − a2 )) + 2(a4l2 + a2l4 )(a2 + a4 );
:
li = −Hai − S
(2.8) ,
:
38
D = 2((a1 + a 2
3) + (a2 + a4 )2 );
Dx = −4S2(a1 + a2 + a3 + a4 ) − 2H ((a1 + a3)2 + (a + a )2
2 4 );
(2.9)
Dy = 2S(a1 + a3)(a1 + a2 + a3 + a4 );
Dz = 2S(a2 + a4 )(a1 + a2 + a3 + a4 );
(2.9) (2.7)
:
x = −2S (a1 + a2 + a3 + a4 )
Σ (a + a )2 − H ;
1 3 + (a 2
2 + a4 )
y = S (a1 + a3)(a1 + a2 + a3 + a4 )
Σ ;
(a1 + a 2
3) + (a2 + a )2 (2.10)
4
z (a2 + a4 )(a1 + a2 + a3 + a4 )
Σ = S ;
(a1 + a3)2 + (a 2
2 + a4 )
i- : H, S, pi.
H S (Hc
Sc), , (Hd
Sd), ,
.
H -
( OX). S -
OY. (2.11)
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S S S (2.11)
= c + d
1.
. i-
pi :
pui
.
ci.
39
,
, - .
d,
.
.
:
α pi = α pui + αci + αd (2.12)
1.
2.5
pui.
Oc
( - ). ij
( - ).
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( - ),
. (P j)
. ( F)
2.5.
'
i- j-
:
M Σ ((RΣi + S )cosαij − (xΣ + H )sinα ij ) − Pαij (Lcosα ij − Qsinα ij ) = 0 (2.14)
ij - , P ij - , M -
, L, Q-
.
40
2.5 -
(2.1) (2.14)
:
P
tgα − R L − tgα αijQ
pui αij = tgα
(xΣ + H )M ij
Σ (xΣ + H )M ij
Σ (2.15)
Q
, (2.15)
:
Q = (xΣ min + H )M Σ min / Pα max
P max , M min,
x min - ,
H - :
,
tg ij,
P ij tg pui.
41
.
(2.15) vj -
.
ti − Pα ijki + v j = tgα ij , j = 1,n, (2.16)
ti ki (2.17):
ti = tgα pui , (2.17)
ki = L / (xΣij + Hi ) / MΣ
n
.
:
[tk ]t i + [kk ]ki = [kw];
(2.18)
[tt]t i + [tk ]ki = [tw],
:
[tt] = n,
n
[tk ] = − Pαij
j=1
n
[kk ] = P 2
αij ,
j=1 (2.19)
n
[tw] = tgαij ,
j=1
n
[kw] = − Pαijtgαij
j=1
:
D′ = [tt][kk ] − [tk ]2 ,
Dt′ = [tw][kk ] − [tk ][kw], (2.20)
Dk′ = [tt][kw] − [tk ][tw]
:
t = Dt′ ,k = Dk′
i D′ i D′, (2.21)
:
42
n n n n
( tgαij )( P 2
αij ) − ( Pαij )( Pαijtgαij )
α pui = arctg j=1 j=1 j=1 j=1
n n (2.22)
n( P 2 ) − ( P )2
αij αij
j=1 j=1
( u, yu, zu) :
xΣ (ma + mu ) = (xu + ha )mu + xama ,
yΣ (ma + mu ) = yumu + yama , (2.23)
zΣ (ma + mu ) = zumu + zama ,
m - , ha - , xa, ya, za, ma -
.
(2.23) ( u, yu, zu)
:
xu (xΣ (ma + mu ) − xama ) / mu − ha ,
yu (yΣ (ma + mu ) − yama / mu , (2.24)
zu (zΣ (ma + mu ) = zama / mu
.
- , (m )
(O X Y ) ,
.
-
, .
1
.
2 - .
2 .
, .
43
.
2.6 -
1 - , 2 -
2.7. 1 - 2.
, .
2.7 - .
1 - , 2 -
44
2.8
i- .
( Cm) pmi
OcCm OcXc.
2.8 -
:
MΣ (S + RΣi ) − y m M
= Σ (H + xΣ ) − x m
tgα pmi M + m M (2.25)
Σ Σ + m
:
y + x tgα
MΣ = pmi m
(S + RΣi ) − (H + x )tgα (2.26)
Σ pmi
, S, H, R , x , pmi
, .
M ma
45
mu = M Σ − ma (2.27)
.
[24] [46]
,
[78].
.
S (x ):
4
S (x ∂xΣ 2 2
Σ ) = (( ) S (α pi ))
∂α (2.28)
i=1 pi
∂xΣ , ∂ Σ , ∂z
Σ
∂α ∂α ∂α
pi pi pi
; S( pi)–
.
S(y ), S(z ) .
(x ), (y ), (z ):
ε (xΣ ) = txS(xΣ );ε ( yΣ ) = tyS (yΣ );ε (zΣ ) = tzS(zΣ ); (2.29)
tx, ty, tz – p=0, 5
f :
4
(( ∂xΣ )2 S 2 (α pi ))
i=1 ∂α
f (x ) = (n +1) pi
Σ 4 − 2
∂x (2.30)
(( Σ )2 S 2 (α ))
i=1 ∂α pi
pi
f b dbpyfxf.nmcz s .
' S(xu), S(yu), S(zu):
S(xu ) = (ma + mu )S(xΣ) / mu ;
S(yu ) = (ma + mu )S(yΣ ) / mu ; (2.31)
S(zu ) = (ma + mu )S(zΣ ) / mu
(xu), (yu), (zu):
46
ε (xu ) = (ma + mu )ε (xΣ ) / mu ;
ε (yu ) = (ma + mu )ε (yΣ ) / mu ; (2.32)
ε (zu ) = (ma + mu )ε (zΣ ) / mu
:
∂α i −1 / cos2 α pi , j = i
=
∂α pj 0, j ≠ i
x
:
∂x 2 2
Σ = 2S (a1 + a3 − a2 − a4 ) − 2(a1 + a3) ;
∂α p1 cos2 α p1 ((a + a 2 2 2
1 3) + (a2 − a4 ) )
∂xΣ = 2S (a1 + a3 − a2 − a4 )2 − 2(a1 + a 2
3) ;
∂α 2
p2 cos α p2 ((a 2 2 2
1 + a3) + (a2 − a4 ) )
(2.33)
∂x 2S (a + a − a − a )2 − 2(a + a )2
Σ = 1 3 2 4 1 3
2 2 2 2 ;
∂α p3 cos α p3 ((a1 + a3) + (a2 − a4 ) )
∂xΣ = 2S (a1 + a3 − a − a 2 2
2 4 ) − 2(a1 + a3) ;
∂α p4 cos2 α p4 ((a + a )2 + (a − a )2 2
1 3 2 4 )
y
:
∂y 2S (a − a )((a + a − a 2 2
Σ = 1 3 1 3 2 − a4 ) − 2(a1 + a3) ) + (a1 + a3 − a2 − a4 ) ;
∂α p1 cos2 α p1 ((a1 + a3)2 + (a2 − a4 )2 )2 (a1 + a 2
3) + (a 2
2 − a4 )
∂yΣ = −S(a1 + a3) (a1 + a3 − a2 − a )2
4 − 2(a2 + a 2
4 ) ;
∂α p2 cos2 α p2 ((a1 + a )2 + (a − a )2 2
3 2 4 )
(2.34)
∂y −S (a 2 2
Σ = 1 − a3)((a1 + a3 − a2 − a4 ) − 2(a1 + a3) ) + (a1 + a3 − a2 − a4 ) ;
∂α p3 cos2 α ((a 2 2 2
p3 1 + a3) + (a2 − a4 ) ) (a1 + a 2 2
3) + (a2 − a4 )
∂yΣ = −S(a1 + a3) (a + a 2 2
1 3 − a2 − a4 ) − 2(a1 + a3) ;
∂α 2 2 2 2
p4 cos α p4 ((a1 + a3) + (a2 − a4 ) )
z
47
∂zΣ = −S(a2 − a4 ) (a + a − a 2
1 3 2 − a4 ) − 2(a1 + a3)2
∂α 2 2 2 2 ;
p1 cos α p1 ((a1 + a3) + (a2 + a4 ) )
∂z −S (a 2 2
Σ = 2 − a4 )((a1 + a3 − a2 − a4 ) − 2(a1 + a3) ) + (a1 + a3 − a2 − a4 ) ;
∂α p3 cos2 α p2 ((a1 + a3)2 + (a 2 2
2 − a4 ) ) (a1 + a )2
3 + (a2 − a )2
4
(2.35)
∂z −S(a + a ) (a + a − a − a )2 2
Σ = 2 4 1 3 2 4 − 2(a1 + a3) ;
∂α cos2
p3 α p3 ((a1 + a 2 2 2
3) + (a2 − a4 ) )
∂zΣ = −S (a2 − a4 )((a1 + a 2 2
3 − a2 − a4 ) − 2(a1 + a3) ) + (a1 + a3 − a2 − a4 )
∂α cos2 α ((a + a )2 2 2 2 2 ;
p4 p4 1 3 + (a2 − a4 ) ) (a1 + a3) + (a2 − a4 )
S(α 2
pi ) = S (α pui ) + S 2(αkj ) (2.36)
S ( pui) -
, S ( ki) -
.
,
. (2.18)
,
t i , k i (2.17) ' j
v j = tgα ij − ti + Pα ijki , j = 1,n, (2.37)
n
:
n
u(v j ) = 1 v2
j (2.38)
n − 2 j=1
uA(ti)
:
uA(ti ) = u(v j ) [kk ] / D′ (2.39)
(2.36)
pui
' ti:
n
v2
j
2 (2.40)
S(α pui ) = cos α pi [kk ] / D′ j=1
n − 2
48
, (2.40):
S(αki ) = S(α pui ) (2.41)
(xu), (yu), (zu) [46]
. ,
,
, ,
.
ci - c,
ij - ;
Pij- p;
Sc, Hc - Sc, Hc;
ha- ha;
ma- ma;
xa, ya, za - xa, ya , za;
x , y , z - x , y
, z ;
dYA, dXA, dXB, dXC -
d;
i - .
,
, , ,
,
, ,
,
49
.
xu, yu, zu
(2.4):
θ (xu ) =1.1 b2 2
xxΣθxΣ + b2 θ 2 2 2 2 2 2
xxa xa + bxmaθma + bxmuθmu +θha ;
θ (y ) =1.1 b2 2 2 2 2 2 2 2
u yyΣθ yΣ + byyaθ ya + bymaθma + bymuθmu ; (2.42)
θ (z ) =1.1 b2 θ 2 2 2
u zzΣ zΣ + bzzaθza + b2 θ 2 + b2 2
zma ma zmuθmu ;
bxxΣ ,bxxa ,bxma ,bxmu , ,byyΣ ,byya ,byma ,bymu ,bzzΣ ,bzza ,bzma ,bzmu -
x , xa, ma, mu, ha –
.
b = ma
xxΣ +1;b = ma ;b xΣ − xa xa − xΣ
m xxa m xma = ;bxmu = ma 2 ;
u u mu mu
b = ma
yyΣ +1;b = ma ;b = yΣ − ya
yya yma ;bymu = m ya − yΣ ;
mu mu m a m2 (2.43)
u u
b = ma +1;b = ma ;b = zΣ − za ;b = m za − zΣ
zzΣ m zza m zma zmu a 2 ;
u u mu mu
,
( xa, ya, za, mu, ma, ha)
.
:
4
θ ∂xΣ 2 2
xΣ = (( ) θα pi + (∂xΣ )2 (θ 2
Sc +θ 2 ) + (∂xΣ
Sd )2 (θ 2 +θ 2 );
i=1 ∂α pi ∂S ∂H Hc Hd
4
θ ∂yΣ 2 2 ∂yΣ 2 2 2
yΣ = (( ) θα pi + ( ) (θHc +θ
∂α ∂H Hd ); (2.44)
i=1 pi
4
θ zΣ = (( ∂zΣ )2θ 2
α pi + (∂zΣ )2 (θ 2 2
Hc +θHd );
i=1 ∂α pi ∂H
∂xΣ , ∂yΣ , ∂zΣ -
∂α pi ∂α pi ∂α pi
(2.10)
∂xΣ , ∂yΣ , ∂zΣ , ∂xΣ -
∂S ∂S ∂S ∂H
.
50
∂xΣ = xΣ + H ; ∂yΣ = yΣ ; ∂zΣ = zΣ ; ∂xΣ = −1 (2.45)
∂S S ∂S S ∂S S ∂H
Sc, Hc -
,
.
Sd, Hd -
,
:
θSd = θHd = θd (2.46)
pi -
(2.12):
θ = θ 2
α pi α pui +θ 2
αc +θ 2
αd +θ 2
αki (2.47)
pui
p.
:
∂D′
cαij = = 0
∂αij
n n
P2 −P P
c = ∂D′ αij αij αij
t j=1 j=1
tαij =
∂αij cos2 αij (2.48)
′ n
cPij = ∂D = 2nPαij − 2 P
∂α αij
ij j=1
c = ∂D n n n
t′
tPij = 2 ⋅ P
∂P αij tgαij − Pαijtgαij − tgαij Pαij
αij j=1 j=1 j=1
:
n 2 n 2
θ = cos2 α θ 2 ctαij 2 ctPijD′ − cPijDt′
α pui pi α ( ) +θP ( 2 ) (2.49)
j=1 D′ j=1 D′
c
.
51
d [46]
(dXB dXC dXA),
:
3 / 2θ 2
θαd = d (2.50)
Rd cos(ϕd )
ki
[46]
θ 2 = θ 2
αki α +θ 2 +θ 2 +θ 2
α u αc αd (2.51)
pi -
:
θ = b2 θ 2 + b2 2
α αSi s αHiθH + b2 θ 2 + b2 2 2 2
αxΣi x Σ α yΣiθy Σ + bα zΣiθz Σ (2.52)
baaSi, baHi, bax , bay , baz :
∂θ
b α 1 1
αSi = = ;
∂S 1+ tg 2ã + Σ
∂θ 2
b α tg ã 1
α i = = ;
∂ 1+ tg 2ã + Σ
∂θ 2
b = α tg ã
= 1
α i 2 ; (2.53)
∂ Σ 1+ tg ã + Σ
∂θ
b α
α i = = 1 osϕi
2 ;
∂ Σ 1+ tg ã + Σ
∂θ
b = α = 1 sinϕi
α z i ;
∂z Σ 1+ tg 2ã + Σ
S, H, x , y , z -
:
x -
:
θ = b2 θ 2 + b2 θ 2
x Σ xx x xm m + b2 θ 2 2 2 2 2
xxa xa + bxmaθma + bxhaθha (2.54)
bxx , bxm , bxxa, bxma, bxha - x , m , xa,
ma, ha - .
52
bxx = m / (m + m );
bxx = m / (m + m );
b 2
xha = m / (m + m ) ;
(2.55)
bxm = (x + ha − x 2
a )ma / (m + m ) ;
bxma = −(x + ha − xa )ma / (m + m )2;
y -
:
θ = b2 θ 2 2 2
z Σ yyΣ yΣ + byyaθ ya + b2 2
ymaθma + b2 2
ym θm (2.56)
byy , bym , byya, byma - y , ya -
.
byy = m / (m + m );
byy = m / (m + m );
bym = y m / (m + m )2; (2.57)
byma = y m / (m + m )2;
z -
:
θ = b2 2 2 2 2 2
z Σ zzΣθ zΣ + bzzaθ za + bzmaθma + b2
zm θ 2
m (2.58)
bzz , bzm , bzza, bzma - z , za -
.
bzz = m / (m + m );
bzz = m / (m + m );
(2.59)
bzm = z m / (m + m )2;
bzma = z m / (m + m )2;
xu, yu, zu
(xu), (yu), (zu)
(xu), (yu), (zu):
Δ xu = Kx (ε (xu ) +θ (xu ));
Δ yu = K y (ε (yu ) +θ (yu )); (2.60)
Δ zu = Kz (ε (zu ) +θ (zu ))
53
Kx, Ky, Kz, 2
(xu) / S (xu), (yu) / S (yu), (zu) / S (zu)
P = 0,95 [46].
2.1
/S 00,5 00,75 11,0 22,0 33,0 44,0 55,0 66,0 77,0 88,0
00,81 00,77 00,74 00,71 00,73 00,76 00,78 00,79 00,80 00,81
, -
.
e
ax, - ay,
yc, mu.
0,5ax. ax = 1,0 ... 2,0
0,1 ; ay = 0,2 ... 1,0 , 0, 2 yc = 0 ... 0,1 0,05 ; mu = 50 ...
1000 50
mu 200 , 100 mu> 200 .
, mu,
xu yu
P p,
H, S, c, d.
e : P 0,1 ; p
30, H 0,1 ; S 0,05 ; c 10; d 0,01 .
2.2
2.2.1
,
54
.
.
.
,
.
-
2.9.
2.9 -
1 - , 2- , 3 - ,
4 -
4 -
, , -
3. ,
.
,
.
55
:
JΣ ⋅ ä + (C ⋅ a − MΣgrc ⋅sin a) = MΣmp (2.61)
: J – ; -
;å -
; C - ; mu - ; rc-
; g - M m -
.
'
,
- T
J :
JΣ = C T 2 (2.62)
T C ;
C = (C − mu grc ) / 4π 2 (2.63)
' ,
.
.
,
C,
. ,
C . ,
- .
- ' ,
, .
1,
.
56
. [13]
.
:
- 6;
– ;
,
.
(1.5). (
sin cos s ) :
J P c2ϑ + J P 2 2 P 2 2 P 2 P P 2
xx i yys ϑic ϕi + Jzzs ϑis ϕi − Jxyc (2ϑi )cϕi − Jxzs(2ϑi )sϕi − J yzs ϑis(2ϑi ) = Ji (2.64)
i - i- X,
i - OYZ OY
( 2.10).
J0x 0 0
[J0 ] = 0 J0 y 0 (2.65)
0 0 J0 z
2.10 -
57
.
OcXcYc.
OcXcZc ,
[41, 17].
2.11.
2.11 -
2.2.2
24].
i- Ji
:
Ji = (C − mu grc cosϑ )T 2 / 4π 2 − J 2 2
ai + M Σrc cos ϑ − J0 (2.66)
Jai - i-
, Jo - , i -
i- OXZ.
J
(J) (J):
Δ J = K (ε (Ji ) +θ (Ji )) (2.67)
[46]:
58
ε (J ) = (C − mu grc cosϑ )
i 2 (2.68)
2π
( )- .
:
θ (J ) =1,1 b2 θ 2 + b2 2 2 2 2 2 2 2 2 2 2 2
i mu mu maθma + bcθc + bgθg + brθr + bϑθϑ + bJaθJo (2.69)
bmu, bm , bc, bT, bg, br, b -
; mu, m , c, T, g, r, , Ja, Jo -
.
b = −gr cosϑ T 2 / 4π 2 2
mu c − rc cos2 ϑ ;
bma = r2
c cos2 ϑ ;
bc = T 2 / 4π 2;
b 2 2
T = (C − mu grc cosϑ )T / 2π ; (2.70)
b = −m 2 2 2 2
r u g cosϑ T / 4π − 2MΣrc cos ϑ ;
bg = −murc cosϑ T 2 / 4π 2;
b = m gr cosϑ T 2 / 4π 2 + M r2 sin2
θ u c Σ c 2ϑ ;
,
( mu, m , c, T, g, r, , Ja , Jo),
(mu, ma, r ),
(T), ( ), ( ),
(g) (J)
.
(2.64)
:
JP
j = det(Fj ) / det(F) (2.71)
JP
j - (j = 1 Jxx, j = 2 Jyy . .), F -
(2.64), Fj, -
(2.64), j-
' .
JP
j [46]
2
ε (J P
j ) = (Gi
j / det(F ) ⋅ε (Ji )) (2.72)
59
Gi
j - detFj Ji (
fij Fj), s (Ji) -
Ji.
, ,
:
6
θ (J P ) = 1.1
j ((H i det(F ) + Ei det(F ))2 ⋅θ 2 + ( ((H i det(F ) + Ei det(F )) ⋅θ 2 (2.73)
det(F ) ϕ j ϕ j ϕ ϑ j ϑ j ϑ
i=1
Hi
j Ei
j - detFj detF i i. j, j -
.
Jj
s (JP
j) (JP
j)
Δ = K (ε (JP
Jj j ) +θ (JP
j )) (2.74)
λ3 − I ⋅ λ 2 + L ⋅ λ − D (2.75)
I, L, D –
:
I = Jxx + J yy + J zz
L = J xx ⋅ J yy + J yy ⋅ J zz + Jxx ⋅ Jzz − (J2
xy + J2 2
xz + J yz ) (2.76)
D = det[J ]
k ,
(Jxx, Jyy,
Jzz), m . m - ,
1,5.
,
,
. 0,1%
0,01 · 2.
J0k (k = 1 J0x, k = 2 J0y,
k = 3 J0z)
60
JPj:
6
θ (J0k ) = c2
λ jΔ
2
Jj (2.77)
j=1
j - k j- ,
, k, .
λ 2
k − (J yy + Jzz ) ⋅ λk + J yy ⋅ Jzz − J 2
c yz
λxx =
3 ⋅ λ 2 (2.78)
k − 2 ⋅ I ⋅ λk + L
c = λ 2
k − (J 2
xx + J zz ) ⋅ λk + J xx ⋅ Jzz − J xz
λ yy λ 2
λ (2.79)
3 ⋅ k − 2 ⋅ I ⋅ k + L
λ 2
k − (J yy + Jxx ) ⋅ λ + J 2
c = k yy ⋅ Jxx − Jxy
λzz (2.80)
3 ⋅ λ 2
k − 2 ⋅ I ⋅ λk + L
λ 2
k − 2 ⋅ Jxy ⋅ λ
c = k + 2 ⋅ J xy ⋅ Jzz
λxy (2.81)
3 ⋅ λ 2
k − 2 ⋅ I ⋅ λk + L
λ 2
k − 2 ⋅ J xz ⋅ λk + 2 ⋅ J
c = xz ⋅ J yy
λxz (2.82)
3 ⋅ λ 2
k − 2 ⋅ I ⋅ λk + L
λ 2
k − 2 ⋅ J yz ⋅ λk + 2 ⋅ λ 2
c k
λ yz = (2.83)
3 ⋅ λ 2
k − 2 ⋅ I ⋅ λk + L
,
. [J] [J0]
- , - , - .
[M] :
[J ] = [M ]T ×[J0 ]×[M ] (2.84)
,
k.
nk.
:
([J ] − λk [E])nk = 0 (2.85)
xn, yn, zn
' [N],
[J]:
61
[N ] = adj([J ] − λk [E]) (2.86)
- nk ,
λk = J0k:
xn = (J yy − J0k ) ⋅ (J zz − J0k )
yn = J yz ⋅ Jxz + Jxy ⋅ (J zz − J0k ) (2.87)
zn = Jxy ⋅ J yz + J xz ⋅ (J yy − J0k )
k,
rn - xn (2.88)
nk αk = ar cos( )
rn
r = x2 2 2 (2.89)
n n + yn + zn
k
JPj
J0k :
6 (2.90)
θ (α ) = d 2 θ 2 (J ) + d 2 Δ2
x α 0 0k α j Jj
j=1
d j - k j- j = 1 ... 6
J0k j = 0.
xnj ⋅ rn − xn ⋅ rnj (2.91)
dα j = − 2
rn ⋅sinαk
xnj, rnj - xn rn j-
j = 1 ... 6 j = 0:
xn0 = 2J0k − J yy − J zz , xn2 = J zz − J0k , xn3 = J yy − J0k , xn1 = xn4 = xn5 = xn6 = 0 (2.92)
yn0 = −J xy , yn3 = J xy , yn4 = J zz − J0k , yn5 = J xz , yn6 = J yz , yn1 = yn2 = 0 (2.93)
zn0 = −J xz , zn2 = J xz , zn4 = J yz , zn5 = J xy , zn6 = J yy − J0k , zn1 = zn3 = 0 (2.94)
xnj ⋅ xn + y
r = nj ⋅ yn + znj ⋅ zn (2.95)
nj rn
62
, ,
J0k.
2.2.3
1,
.
, ,
.
[80].
(« »), , , ( « »)
.
Ji,
, , .
,
, ,
'
( ).
Ox
Δα (ϕi ,θi ) → min (2.96)
a, b, c,
(Oxyz) - ( 2.12).
O
. (Oxyz)
(Ox'y'z ') O.
, , .
63
i - i-
Ox, i - i- Oyz Oy.
2.12 -
,
,
.
O.
, .
,
( 2.2).
( 1) ,
.
64
2.2
1 2 3 4 5
1 45 0 90 90 0 0 90 0 0 0
2 45 90 51 0 63 0 90 120 90 180
3 45 180 51 72 63 72 90 240 90 270
4 45 270 51 144 63 144 35 60 45 0
5 90 45 51 216 63 216 35 180 90 45
6 90 135 51 288 63 288 35 300 45 90
, ’ 3,49 3,58 4,21 3,70 5,75
oJx0, % 1,10 1,20 0,89 0,93 0,92
, 2.13. ,
, , - , - .
.
( i, i) ,
.
.
j
i , i j.
- , Ox,
O 140 ( i = 20 ); - (Oyz).
65
) )
) )
2.13 - 1 ( ), ( ), 4 ( ), 5 ( )
70 90
. ,
,
.
– ' ( 2.14):
1: 1=90 ; 0 1<360 ;
2: 2=80 ; 0 2< 3;
3-5: i=80 ; i-1< i< i+1;
6: 6=80 ; 5 < 6<360 ;
66
2.14 –
- ( 2.15):
1: 1=90 ; 0 1< 2;
2: 2=90 ; 1< 2<360 ;
3: 3=80 ; 0 3< 4;
4-5: i=80 ; i-1< i< i+1;
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