Глава 13. Оптимизация конструкций 1

13.
13.1.
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1
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u1
θ
u2
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- . 13. 1. &
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b1 , b2 , b3 (
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13.
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F ( x ) = aρ( 2b1 + b2 + 2b3 ) ,
ρ –
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(13.1)
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( b1 , b2 , b3 ),
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u1
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{R} .
u2
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[ K ]{u} − {R} = 0 .
(13.2)
0
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σ1 =
E (u1 + u2 )
Eu2
E (u2 − u1 )
, σ2 =
, σ3 =
.
2a
a
2a
σia ,
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0
E (u1 + u2 )
− σ1a ≤ 0 ,
2a
(13.3)
g2 ≡
Eu2
− σa2 ≤ 0 ,
a
(13.4)
g3 ≡
E (u2 − u1 )
− σ3a ≤ 0 .
2a
(13.5)
:
−b1 ≤ 0 , −b2 ≤ 0 , −b3 ≤ 0 .
(13.6)
,
b1 , b2 ,
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(13.1)
u1
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R = 20000
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θ 2 = 3π / 4 . 2
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2
(13.2), (13.3)–(13.6). 1
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g1 ≡
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b1 = b3 . &
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u1 =
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1
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a
σ = 20000 . '
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(13.2)
σ
P , a, b
5.1,
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a
g1 ≡
P
b1 + 2b2
,
,
2
,
(13.7)
(13.4):
P
P
+
− σa ≤ 0 ,
2b1 2(b1 + 2b2 )
g2 ≡
-
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aP
aP
, u2 =
.
b1E
(b1 + 2b2 ) E
(13.3)
2.
:
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θ1 = π / 4
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(13.8)
− σa ≤ 0 .
(13.9)
:
−b1 ≤ 0 , −b2 ≤ 0 .
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aρ
(13.1),
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F ( x ) = 2 2b1 + b2 .
(9.10)
13.
-
g1 = 0
Excel. /
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g2 = 0
b2 = f1 (b1 )
b2 = f 2 (b1 ) ,
2b1 (1 − b1 )
,
1 − 2b1
b2 =
2 (1 − b1 )
.
2
b2 =
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g1 g 2 ( . 13.2). %
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Fk ,
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b2
0.9000
0.8000
0.7000
0.6000
0.5000
0.4000
F(x)=2.8
0.3000 F(x)=2.4
F(x)=2.63
0.2000
g1=0
g2=0
0.1000
0.0000
0.700
0.750
0.800
0.850
F (x )
- . 13. 2. 2
b1
0.950
0.900
g1
!
g2
b2
b2 = Fk − 2 2b1 .
Fk
$
b1
{b1 > 0, b2 > 0} !
b2 = Fk − 2 2b1
b2 = f1 (b1 ) . $
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{ b1 = 0.79 , b2 = 0.4045 },
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g1 = 0 ,
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F ( x ) ≡ 2 2b1 + b2 = 2.639
. 13.2.
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g1 –
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g2 –
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13.
F ( x + ∆x1 ) − F ( x )
∆x1
∂F
∂x1
∂F
∂xi
∇F ({x}) =
=
∂F
∂xn
∆xi
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n-
F ( x + ∆xi ) − F ( x )
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∆xi
F ( x + ∆x n ) − F ( x )
∆x n
xi ,
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.. 13.3 . 0
x2
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x2
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g1 ( x )
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g 2(x)
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. 13.3
F (x ) .
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g 2 ( x)
g2(x)
∇F(x*)
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g1(x)
g1(x)
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∇g 2(x*)
xO
x1
∇g1(x*)
∇g2(x*)
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λ2 > 0 . &
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13.
∇F(x*)
λ1∇g1(x*)
∇g2(x*)
λ2∇g2(x*)
∇g1(x*)
−∇F(x*)
- . 13. 4.
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