Пвлов_PROCHNOST_2_FULL+PROTECTION
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ξT , $ ($ )
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2. 2.1.1
* (2.1.1)
. . ξ ζ
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Mη , |
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Qξ ξ ζ1 , . . ..-. 1, : |
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η |
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æ |
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(2.1.2) |
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η |
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η |
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S − , $ ;
τ− $ , Qξ ;
G − .
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1 |
(2.1.1) |
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..-. 1, |
!.!. !, 4.4. 6 ., |
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GJζ |
[5]. |
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*
$ ,
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ξ |
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Mη |
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æ = |
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; æ |
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(2.1.3) |
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ξ |
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η |
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Mζ = GJζ τζ
1 , 8,
, . -
$ " "
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[5].
*, , $
( ) ξηζ 1 ξη1ζ ,
. * xyz
$
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% ! & ' ( "
2. 2.1.2
2 , . 2.1.2,a
− $ $ i , j,k,
i j $ ,
$ , k
$ . 1 $
i , j,k .
! $
S. 1 * S. 1
$ , , ξη1 ,
i *, |
j *, k k*. |
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* . |
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i jk xyz i * j *k * |
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- α (0 ≤ α ≤ 2π) i. '
i1 |
j1k1 , , - |
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( . 2.1.2 ): |
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i |
1 = [1, 0, 0] ; |
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j1 = |
[0, c s α, sin α] ; |
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k1 = [0, −sin α, c s α] . |
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1 , $ ijk |
i1 j1k1 $ : |
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+(α ) = |
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cosα |
sinα |
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− sinα |
cosα |
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[ x, y, z ] = +(α) [ , , z] . |
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- j1 β (0 ≤ β ≤ 2π) $ : |
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cosβ |
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- sinβ |
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+(β ) = |
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sinβ |
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i2 j2k2 :
[ x2, y2, z2 ] = + (β) + (α) [ , , z] .
1 ϕ ( 0 ≤ ϕ ≤ 2π) k2
i jk i * j*k * , , xyz
ξηζ .
"$ : |
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cosϕ |
sinϕ |
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+(ϕ ) = - sinϕ |
cosϕ |
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1 |
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% α, β ϕ , $
+ = +(ϕ) +(β) +(α) ,
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sinα sin β cosϕ + |
sinα sinϕ − |
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cos β cosϕ |
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+ cosα sinϕ |
− cosα sin β cosϕ |
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cosα cosϕ − |
cosα sin β sinϕ + |
(2.1.4) |
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= - cos β sinϕ |
− sinα sin β sinϕ |
+ sinα cosϕ |
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sin β |
− sinα cosβ |
cosα cos β |
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- , i * j*k *
S ( . 2.1.2,a). %
j *, , ξ ξ ,
i * η − η i * j * ζ −
τζ . ,
( , $ $ ).
! α, β ϕ. 2
. - α, β ϕ, α
+ δ α, β + δ β, ϕ + δ ϕ. "
, :
θ1 = i δα , θ2 = j1 δβ , θ3 = k2 δϕ.
6,
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θ = i δα + j1 δβ + k2 δϕ
:
ω = dθ = α ′i + β ′j1 + ϕ′k2. ds
!
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7 S, . .
dt = ds, .
- $ ω , $ ( æ9, æ:, ;<), :
æ9 = ωi *, |
æ: = ω j , |
τς = ωk |
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æ9 = α' cos β cos ϕ + β' sin ϕ; |
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æ: = − α' cos β sin ϕ + β' cos ϕ; |
(2.1.5) |
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τζ |
= α' sin β − ϕ'. |
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- - |
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R= Xi + Yj |
+ Zk . |
(2.1.6) |
6 , Y , Z , , ds
k*:
dX = sinβ , |
dY |
= −sinα cosβ , dZ = cosα cosβ . |
(2.1.7) |
ds |
ds |
ds |
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' α, β ϕ, (2.1.7),
(2.1.3), (2.1.5)
Mξ |
T |
MX T |
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Mη |
= MY |
(2.1.8) |
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Mζ |
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MZ |
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" x , y , z ,Y,Z .
1 ,
. " $ , , ( ,
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sin α = α, sin β = β, sin ϕ = ϕ ,
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cos α = cos β = cos ϕ = 1.
! $ (2.1.4) : |
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1 |
ϕ |
− β |
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= − ϕ 1 |
α . |
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+ |
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β |
− α |
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(2.1.7) |
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α = − dY ,β = dX ,ds = dz , ds ds
1 |
ϕ |
− X ′ |
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− Y ′ |
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+ = − ϕ |
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X ′ |
Y ′ |
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2 |
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α ′ = − |
d Y |
,β ′ = |
d X |
, |
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dz |
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dz |
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æξ = − Y" + X "ϕ ;
æη = Y"ϕ + X ";
τ ζ = − Y" X ′ + ϕ ′ .
, ,
, .
" ,
- . 1
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ϕ |
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+ = − ϕ |
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Y′ |
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% &, ,,
- . , 2.1.4 2.1.5
α, β ϕ, ,
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θ = i α + j β + k ϕ .
1
æ ξ = − Y" + X "ϕ − ϕ 'X '; æ η = Y"ϕ + X " − ϕ 'Y '; τ ζ = − Y" X ' + X "Y ' + ϕ '.
6 - !.-. !
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..-. 1 "
" $ . -
. 2
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% M x, M y M z − z t x
t y, , xyz,
. * . 2.1.3
$ . - $. .
, t x t y Y ,
c − X z Yz, −
ϕ ϕ z.
!
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2. 2.1.3
x y ,
, . .
l |
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Mx = − ty dzdz; |
My = tx dzdz. |
(2.1.9) |
z |
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" z , z
z l :
l |
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M z = [− t x (Y − Yz )+ t y ( X − X z )+ mz ]dz. |
(2.1.10) |
z |
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mz - , .
- X Y (2.1.7)
- Qξ Qη $ ,
$ +
Qξ |
Q |
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x |
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Qη |
= Qy . |
(2.1.11) |
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0 |
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Qζ Qz.
!
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! (2.1.3)-(2.1.10)
. 2 0,
, $[5].
-
F(X, P) = 0 |
(2.1.12) |
X = (α,β,ϕ,c)
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