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로드 중.... (전체 텍스트 보기)

전체 글

(1)

& Ocean Engineering

& Ocean Engineering

& Ocean Engineering

& Ocean Engineering

October, 2008 October, 2008 Prof.

Prof. Kyu Kyu- -Yeul Yeul Lee Lee

Department of Naval Architecture and Ocean Engineering Department of Naval Architecture and Ocean Engineering Department of Naval Architecture and Ocean Engineering, Department of Naval Architecture and Ocean Engineering,

Seoul National University of College of Engineering Seoul National University of College of Engineering

2008_Pressure Integration Technique (2) 2008_Pressure Integration Technique (2)

1

(2)

[06

[06- -2] Pressure integration technique (2) 2] Pressure integration technique (2)

2

(3)

유체 입자가 주위 유체 입자에 작용하는 정적인 압력

C P z

t + ∇ Φ + g + =

∂ Φ

∂ ρ

ρ

2

2 1

Bernoulli Eq.

P

Static

= − ρ gz + C

atm

Static

gz P

P = − ρ +

P

atm

P

atm

r

y z

O j

k

C z t g

P − ∇ Φ − +

∂ Φ

− ∂

= ρ

2

ρ

2 1

정수¨ 유체의 속도= 0

P

Static

atm

Fluid

z P

P ( ) + gz z

P

Fluid

( ) = − ρ S

B : 침수표면

유체 입자들이 유체 중에 잠긴 물체에 작용하는 정적인 힘과 모멘트

B

V

: 침수부피

유체 입자들이 유체 중에 잠긴 물체에 작용하는 정적인 힘과 모멘트

( )

∫∫

=

SB

d gz S

F ρ

=

∫∫ ( )

SB

dS z

g n

ρ

=

∫∫

SB

zdS

ρ g n = ∫∫∫

V

dV ρ g

k F

[ ]

∫∫∫

Divergence Theorem

( )

∫∫

×

=

SB

d gz S r

M ρ

=

∫∫

×

( )

SB

dS z

g r n

ρ

=

∫∫ (

×

)

SB

zdS g r n

ρ = ∫∫∫ [ ]

V

dV x y

g i j

M ρ

2008_Pressure Integration Technique (2) 2008_Pressure Integration Technique (2)

3

(4)

-

- Pressure Integration Technique Pressure Integration Technique essu e essu e eg a o eg a o ec ec que que

1) J.N.Newman, Marine Hydrodynamics, 1977, pp.290~295

선박이 유체 중에서 받는 정적인 힘과 모멘트

(Surge)

x

translation

z z′

1) J.N.Newman, Marine Hydrodynamics, 1977, pp.290 295

)

4

T ( z g

WP

k ρ ξ

F

∫∫∫

=

V

dV ρ g

k

F = ∫∫∫ [ ]

V

dV x y

g i j

M ρ

) , ,

(ξ3ξ4 ξ5 (ξ3,ξ4,ξ5)

(Roll)

x

rotation

x x′

y y′

ξ

1

z z′

P G

T

B

g I mg z g I

z V

g

4 0 0 4 4

)

4

( ρ ξ ρ ξ ξ j ρ ξ

i

M ≈ − ′ − + ′ +

) , ,

( ξ

3

ξ

4

ξ

5

F

F = M = M ( ξ

3

, ξ

4

, ξ

5

)

(Q) 침수부피(중심)가 변하는 자세는?

ξ

4

y′ y

x x′

(Q) ( )

Heel

대경사 경우 미소경사 경우

z z′

(Sway) (Heave)

z

y

translation

z

translation

z′

C

L

I i

y y′

x x′

ξ

2

x x′

y

y′ ξ

3

)

5

L

(

z g

WP

k ρ ξ F

)

(

5 0 0 5 5

5

ρ ξ ρ ξ ξ

ξ

ρ g I

P

+ − g V z

B

g I

L

+ mg z

G

i j

M

Immersion

CL

y

rotation

z

rotation (Yaw)

y′

y

ξ z z′

(Pitch)

) ( )

( L

T ′ ξ ′

ξ j

i M

)

3

A ( z g

WP

k ρ ξ F

z x′

z′

x ξ

6

x x′

y y′

ξ

5

M ≈ − i ρ g ξ

3

T

WP

( z ′ ) + j ρ g ξ

3

L

WP

( z ′ )

4

(5)

초기 자세( , , )에서

복원력( ), 횡 방향 복원 모멘트(3 ), 종 방향 복원 모멘트( )를 알고 있을 때,

ξ ξ

4

ξ

5

F M

T

M

L

복원력( ), 횡 방향 복원 멘 ( ), 종 방향 복원 멘 ( )를 알 있을 때, 미소 변화된 자세( , , )에서의

복원력( ), 횡 방향 복원 모멘트( ), 종 방향 복원 모멘트( )는?

3

3

ξ

ξ + Δ ξ

4

+ Δ ξ

4

ξ

5

+ Δ ξ

5

F

T L

F M

T

M

L

3 4 5

( , , ) F ξ ξ ξ

3 4 5

( , , ) M

T

ξ ξ ξ

( )

M ξ ξ ξ

3 3 4 4 5 5

( , , )

F ξ + Δ ξ ξ + Δ ξ ξ + Δ ξ

3 3 4 4 5 5

( , , )

M

T

ξ + Δ ξ ξ + Δ ξ ξ + Δ ξ

( )

M ξ + Δ ξ ξ + Δ ξ ξ + Δ ξ

?

3 4 5

( , , )

M

L

ξ ξ ξ M

L

( ξ

3

+ Δ ξ ξ

3

,

4

+ Δ ξ ξ

4

,

5

+ Δ ξ

5

)

Taylor series expansion

3 3 4 4 5 5 3 4 5 3 4 5

3 4 5

( , , ) ( , , ) F F F

F ξ ξ ξ ξ ξ ξ F ξ ξ ξ ξ ξ ξ

ξ ξ ξ

∂ ∂ ∂

+ Δ + Δ + Δ = + Δ + Δ + Δ +

∂ ∂ ∂ "

M M M

∂ ∂ ∂

선형화

선형화

3 3 4 4 5 5 3 4 5 3 4 5

3 4 5

( , , ) ( , , )

T T T

T T

M M M

M ξ ξ ξ ξ ξ ξ M ξ ξ ξ ξ ξ ξ

ξ ξ ξ

∂ ∂ ∂

+ Δ + Δ + Δ = + Δ + Δ + Δ +

∂ ∂ ∂ "

3 3 4 4 5 5 3 4 5 3 4 5

( , , ) ( , , )

L L L

L L

M M M

M ξ + Δ ξ ξ + Δ ξ ξ + Δ ξ = M ξ ξ ξ + Δ + ξ Δ + ξ Δ +" ξ

선형화

선형화

2008_Pressure Integration Technique (2) 2008_Pressure Integration Technique (2)

3 3 4 4 5 5 3 4 5 3 4 5

3 4 5

( , , ) ( , , )

L

ξ ξ ξ ξ ξ ξ

L

ξ ξ ξ ξ ξ ξ

ξ ξ ξ

∂ ∂ ∂

5

(6)

( ) ( ) F F F F ξ + Δ ξ ξ + Δ ξ ξ + Δ ξ = F ξ ξ ξ + Δ + ξ Δ + ξ Δ ξ

9 Taylor series 1차항까지 전개

3 3 4 4 5 5 3 4 5 3 4 5

3 4 5

( , , ) ( , , )

F ξ ξ ξ ξ ξ ξ F ξ ξ ξ ξ ξ ξ

ξ ξ ξ

+ Δ + Δ + Δ = + Δ + Δ + Δ

∂ ∂ ∂

3 3 4 4 5 5 3 4 5 3 4 5

3 4 5

( , , ) ( , , )

T T T

T T

M M M

M ξ ξ ξ ξ ξ ξ M ξ ξ ξ ξ ξ ξ

ξ ξ ξ

∂ ∂ ∂

+ Δ + Δ + Δ = + Δ + Δ + Δ

∂ ∂ ∂

3 3 4 4 5 5 3 4 5 3 4 5

3 4 5

( , , ) ( , , )

L L L

L L

M M M

M ξ ξ ξ ξ ξ ξ M ξ ξ ξ ξ ξ ξ

ξ ξ ξ

∂ ∂ ∂

+ Δ + Δ + Δ = + Δ + Δ + Δ

∂ ∂ ∂

( ) ( ) F F F

F ξ

3

+ Δ ξ ξ

3 4

+ Δ ξ ξ

4 5

+ Δ ξ

5

F ξ ξ ξ

3 4 5

= Δ + ξ

3

Δ + ξ

4

Δ ξ

5

3 4 5

( , , ) ( , , )

F ξ ξ ξ ξ ξ ξ F ξ ξ ξ ξ ξ ξ

ξ ξ ξ

+ Δ + Δ + Δ − = Δ + Δ + Δ

∂ ∂ ∂

3 3 4 4 5 5 3 4 5 3 4 5

3 4 5

( , , ) ( , , )

T T T

T T

M M M

M ξ ξ ξ ξ ξ ξ M ξ ξ ξ ξ ξ ξ

ξ ξ ξ

∂ ∂ ∂

+ Δ + Δ + Δ − = Δ + Δ + Δ

∂ ∂ ∂

3 3 4 4 5 5 3 4 5 3 4 5

3 4 5

( , , ) ( , , )

L L L

L L

M M M

M ξ ξ ξ ξ ξ ξ M ξ ξ ξ ξ ξ ξ

ξ ξ ξ

∂ ∂ ∂

+ Δ + Δ + Δ − = Δ + Δ + Δ

∂ ∂ ∂

F F F

F ξ ξ ξ

Δ Δ

3

Δ

4

Δ

5

3 4 5

F ξ ξ ξ

ξ ξ ξ

Δ = Δ + Δ + Δ

∂ ∂ ∂

3 4 5

T T T

T

M M M

M ξ ξ ξ

ξ ξ ξ

∂ ∂ ∂

Δ = Δ + Δ + Δ

∂ ∂ ∂

3 4 5

L L L

L

M M M

M ξ ξ ξ

ξ ξ ξ

∂ ∂ ∂

Δ = Δ + Δ + Δ

∂ ∂ ∂

3 4 5

ξ ξ ξ

∂ ∂ ∂

6

(7)

9 Taylor series 1차항까지 전개

F F F

∂ ∂ ∂ ⎛ ∂ FFF

⎜ ⎟

9 Matrix로 표현

b = Ax

3 4 5

3 4 5

F F F

F ξ ξ ξ

ξ ξ ξ

∂ ∂ ∂

Δ = Δ + Δ + Δ

∂ ∂ ∂

3 4 5

T T T

T

M M M

M ξ ξ ξ

ξ ξ ξ

∂ ∂ ∂

Δ = Δ + Δ + Δ

∂ ∂ ∂

3 4 5

3 4

T T T

T

F F F

F M M M

M

ξ ξ ξ ξ

ξ ξ ξ ξ

⎛ ∂ ∂ ∂ ⎞

⎜ ∂ ∂ ∂ ⎟

⎜ ⎟

Δ Δ

⎛ ⎞ ⎜ ∂ ∂ ∂ ⎟ ⎛ ⎞

⎜ Δ ⎟ = ⎜ ⎟ ⎜ Δ ⎟

⎜ ⎟ ⎜ ∂ ∂ ∂ ⎟ ⎜ ⎟

⎜ ⎟ ⎜ ⎟

변수3개, 식 3개

3 4 5

3 4 5

L L L

L

M M M

M ξ ξ ξ

ξ ξ ξ

∂ ∂ ∂

Δ = Δ + Δ + Δ

∂ ∂ ∂

3 4 5

3 4 5

T

∂ ξ ∂ ξ ∂ ξ

3 4 5

5

3 4 5

L

L L L

M M M M

ξ ξ ξ

ξ

ξ ξ ξ

⎜ ⎟ ⎜ ∂ ∂ ∂ ⎟ ⎜ ⎟

⎜ Δ ⎟ ⎜ Δ ⎟

⎝ ⎠ ⎜ ∂ ∂ ∂ ⎟ ⎝ ⎠

⎜ ⎟

⎜ ∂ ∂ ∂ ⎟

⎝ ⎠

F F F

⎛ ∂ ∂ ∂ ⎞

⎜ ⎟ ⎛ ∂ FFF

⎜ ⎟

1. 자세의 변화량이 주어져 있을 때, 힘(모멘트)의 변화량을 구하는 경우

2. 힘(모멘트)의 변화량이 주어져 있을 때, 자세의 변화량을 구하는 경우

3 4 5

3 4

3 4 5

T T T

T

F M M M

M M

ξ ξ ξ ξ

ξ ξ ξ ξ

ξ

⎜ ∂ ∂ ∂ ⎟

⎜ ⎟

Δ Δ

⎛ ⎞ ⎜ ∂ ∂ ∂ ⎟ ⎛ ⎞

⎜ Δ ⎟ = ⎜ ⎟ ⎜ Δ ⎟

⎜ ⎟ ⎜ ∂ ∂ ∂ ⎟ ⎜ ⎟

⎜ Δ ⎟ ⎜ Δ ⎟

⎝ ⎠ ⎝ ⎠

3 4 5

3 4

3 4 5

T T T

T

F M M M

M M

ξ ξ ξ ξ

ξ ξ ξ ξ

ξ

⎜ ∂ ∂ ∂ ⎟

⎜ ⎟

Δ Δ

⎛ ⎞ ⎜ ∂ ∂ ∂ ⎟ ⎛ ⎞

⎜ Δ ⎟ = ⎜ ⎟ ⎜ Δ ⎟

⎜ ⎟ ⎜ ∂ ∂ ∂ ⎟ ⎜ ⎟

⎜ Δ ⎟ ⎜ Δ ⎟

⎝ ⎠ ⎝ ⎠

Given Given

Find Given Given Find

3 4 5

5

3 4 5

L

L L L

M M M M

ξ ξ ξ

ξ

ξ ξ ξ

⎜ ⎟

⎜ Δ ⎟ ⎜ Δ ⎟

⎝ ⎠ ⎜ ∂ ∂ ∂ ⎟ ⎝ ⎠

⎜ ⎟

⎜ ∂ ∂ ∂ ⎟

⎝ ⎠

3 4 5

5

3 4 5

L

L L L

M M M M

ξ ξ ξ

ξ

ξ ξ ξ

⎜ ⎟

⎜ Δ ⎟ ⎜ Δ ⎟

⎝ ⎠ ⎜ ∂ ∂ ∂ ⎟ ⎝ ⎠

⎜ ⎟

⎜ ∂ ∂ ∂ ⎟

⎝ ⎠

Ax

b

를 풀면 됨

x A

1

x

를 풀면 됨

2008_Pressure Integration Technique (2) 2008_Pressure Integration Technique (2)

Ax

b =

를 풀면 됨

x = A x

를 풀면 됨

A

※ 가 선형화되어 있기 때문에 반복 계산(iteration)을 해야 함

7

(8)

( ) 3

( ) ( ) ( ) ( ) *

3 3 3 3

3

( ) ( )

k

k k k B k

B B B

F F F F

ξ

ξ ξ ξ ξ

ξ

+ Δ = + ∂ ⋅ Δ =

Taylor series 1차 항까지 전개

( )

3

F

B

ξ

3( )

3 k

F

B

ξ

ξ

ξ3

* ( ) ( )

3 3

(

k

)

B k

B B

F F ξ F ξ

ξ

− = ∂ ⋅ Δ

힘의 변화량과 자세의 변화량 관계 ξ3

( 1)

3 3k

F

B

ξ

ξ +

( )

(

3k

) F

B

ξ

( )

3 ξ3k

ξ

1

F

⎛ ∂ ⎞

자세의 변화량에 대해 표현

( 1)

(

3k

) F

B

ξ

+

( )

( ) 3

( ) * ( )

3 3

3

( )

k

k B k

B B

F F F

ξ

ξ ξ

ξ

⎛ ∂ ⎞

⎜ ⎟

Δ = −

⎜ ∂ ⎟

⎝ ⎠

자세 변화

*

F

B ( 2)

(

3k

)

F

B

ξ

+ 자세 변화

ξ

3(k+1)

= ξ

3( )k

+ Δ ξ

3( )k

No

ξ

3 ( )

3

ξ

k ( 1) 3

ξ

k+

( ) 3

ξ

k

Δ

( 1) 3

ξ

k+

Δ

*

ξ

3

( 1) ( )

3 3

k k

ξ

+

− ξ < ε k k = + 1

* (k 1)

ξ = ξ

+

종료

No

Yes

( 2) 3

ξ

k+ 종료

ξ

3

= ξ

3( )

8

(9)

* ( ) ( ) ( ) ( ) ( ) ( )

3 4 5 3 4 5

(

k

,

k

,

k

)

B k B k B k

B B

F F F

F F ξ ξ ξ ξ ξ ξ

ξ ξ ξ

∂ ∂ ∂

− = ⋅ Δ + ⋅ Δ + ⋅ Δ

∂ ∂ ∂

힘의 변화량과 자세의 변화량 관계

( ) ( ) ( )

4 5

3 ξ3k 4 ξ k 5 ξ k

ξ ξ ξ

∂ ∂ ∂

( )

( ) ( )

4 5

3

* ( ) ( ) ( ) ( ) ( ) ( )

3 4 5 3 4 5

3 4 5

( , , )

k

k k

k k k T k T k T k

T T

M M M

M M

ξ

ξ ξ

ξ ξ ξ ξ ξ ξ

ξ ξ ξ

∂ ∂ ∂

− = ⋅ Δ + ⋅ Δ + ⋅Δ

∂ ∂ ∂

( )

( ) ( )

4 5

3

* ( ) ( ) ( ) ( ) ( ) ( )

3 4 5 3 4 5

3 4 5

( , , )

k

k k

k k k L k L k L k

L L

M M M

M M

ξ

ξ ξ

ξ ξ ξ ξ ξ ξ

ξ ξ ξ

∂ ∂ ∂

− = ⋅Δ + ⋅ Δ + ⋅Δ

∂ ∂ ∂

( ) ( )

( )

4 4

3

1

3 4 4

( ) *

(

( ) ( ) ( )

)

k k

k

B B B

k k k k

F F F

F F

ξ ξ

ξ

ξ

ξ ξ

ξ ξ ξ ξ

⎡ ∂ ∂ ∂ ⎤

⎢ ⎥

∂ ∂ ∂

⎢ ⎥

⎢ ⎥

⎡ Δ ⎤ ⎡ ⎤

자세의 변화량에 대해 표현

자세 변화

(k 1) ( )k

3

( ) ( )

( )

4 4

3

( ) * ( ) ( ) ( )

3 3 4 5

( ) * ( ) ( ) (

4 3 4 5

3 4 4

( ) 5

( , , ) ( , ,

k k

k

k k k k

B B

k T T T k k k

T T

k

L L L

F F

M M M

M M

M M M

ξ ξ

ξ

ξ ξ ξ ξ

ξ ξ ξ ξ

ξ ξ ξ

ξ

⎢ ⎥

⎡ Δ ⎤ −

⎢ ∂ ∂ ∂ ⎥

⎢ Δ ⎥ = ⎢ ⎥ −

⎢ ⎥ ⎢ ∂ ∂ ∂ ⎥

⎢ Δ ⎥

⎣ ⎦ ⎢ ⎢ ∂ ∂ ∂ ⎥ ⎥

)

* ( ) ( ) ( )

3 4 5

) (

k

,

k

,

k

)

L L

M M ξ ξ ξ

⎡ ⎤

⎢ ⎥

⎢ ⎥

⎢ − ⎥

⎣ ⎦

( 1) ( ) ( )

3 3 3

k k k

ξ

+

= ξ + Δ ξ

( 1) ( ) No

3 3

k k

ξ

+

− ξ < ε k k = + 1

( ) ( ) ( )

3 3 3

3 k 3 k 3 k

L L L

M M M

ξ ξ ξ

ξ ξ ξ

∂ ∂ ∂

⎢ ⎥

⎢ ∂ ∂ ∂ ⎥

⎣ ⎦

* ( 1)

3 3

ξ = ξ

k+

종료

Yes

2008_Pressure Integration Technique (2) 2008_Pressure Integration Technique (2)

3 3

ξ ξ

9

(10)

미소

미소 자세와 자세와 미소 미소 힘 힘, , 모멘트와의 모멘트와의 관계식 관계식

F F F

⎛ ∂ ∂ ∂ ⎞

⎜ ⎟

( ) ( ) ( )

3 3 5 3

3 4

3

4

k k k

T T T

T

F F F

F M M M

M

ξ

ξ

ξ

ξ

ξ ξ

ξ

ξ ξ ξ ξ

∂ ∂ ∂

⎜ ⎟

∂ ∂

⎜ ⎟

⎜ ⎟

Δ Δ

⎛ ⎞ ⎛ ⎞

⎜ ∂ ∂ ∂ ⎟

⎜ Δ ⎟ = ⎜ ⎟ ⎜ Δ ⎟

⎜ ⎟ ∂

( )

( )

( )

⎜ ⎟

4

4 4

( )

( ) ( )

5

5 5

4

3 4 5

5

3 4 5

k

k k

k

k k

T

L

L L L

M

M M M

ξ ξ ξ

ξ

ξ ξ

ξ ξ ξ ξ

ξ

ξ ξ ξ

⎜ ⎟

⎜ ⎟ ∂ ∂ ∂ ⎜ ⎟

⎜ Δ ⎟ ⎜ ⎟ ⎜ Δ ⎟

⎝ ⎠ ⎜ ⎟ ⎝ ⎠

∂ ∂ ∂

⎜ ⎟

⎜ ∂ ∂ ∂ ⎟

⎝ ⎠

Jacobian matrix

(자세 변화량을 힘과 모멘트의 변화량으로 변환)

5

5 ξ 5

ξ ξ

⎝ ⎠

( ) ( ) ( )

3 3 3 3

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( ) ( )

k k k

WP WP WP

k k k k k k

gA gT gL

gT gI gV z mg z gI

ρ ξ ρ ξ ρ ξ ξ

ρ ξ ρ ξ ρ ξ ρ ξ ξ

⎛− − ⎞ ⎛Δ ⎞

⎜ ⎟ ⎜ ⎟

=⎜ + ⋅ ⎟ ⎜⋅ Δ ⎟

(자세 변화량을 힘과 모멘트의 변화량으로 변환)

4 4 4 4 4

( ) ( ) ( ) ( ) ( ) ( )

5 5 5 5 5

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

WP T B G P

k k k k k k

WP P L B G

gT gI gV z mg z gI

gL gI gI gV z mg z

ρ ξ ρ ξ ρ ξ ρ ξ ξ

ρ ξ ρ ξ ρ ξ ρ ξ ξ

= −⎜ − − + ⋅ ⎟ ⎜⋅ Δ ⎟

⎜ ⎟

⎜ − − + ⋅ ⎟ ⎝Δ ⎠

⎝ ⎠

10

(11)

을 이용한 이용한 -

- Pressure Integration Technique Pressure Integration Technique을 을 이용한 이용한 부력

부력, , 횡 횡 방향 방향 모멘트 모멘트, , 종 종 방향 방향 모멘트 모멘트 계산 계산

J.N.Newman, Marine Hydrodynamics, 1977,

J.N.Newman, Marine Hydrodynamics, 1977,

pp.290~295

(12)

z

x '

z z′ z

( )k

ξ Δ

( )

( 4k )

V Δξ

P

( )

(

4k

) V ξ

x y

'

y x

'

( ) 4

ξ

k

Δ

( ) 4

ξ

k ( )

(

4k

) V ξ

y

z x′

y′

z′

( )k ( )k

P P

x = x

CL

z x x

z

( 1) ( ) ( ) ( ) ( )

( 1) ( ) ( ) ( ) ( )

4 4

4 4

cos sin

cos sin

k k k k k

k k k k k

P P P

P P P

y y z

z z y

ξ ξ

ξ ξ

+

+

= Δ − Δ

= Δ + Δ

sol) 좌표의 회전 변환에 의해

ex) 배가 x축 중심으로 회전하고 있다. (k)단계에서 (k+1)단계로 넘어갈 때,

만큼 회전한다고 한다. (k)단계에서 (30,10)의 위치는 (k+1)단계에서 어떻게 보이는가?

( )

4k

/ 60( rad )

ξ π

Δ =

(회전변환)

sol) 좌표의 회전 변환에 의해

( 1) ( ) ( ) ( ) ( ) ( ) ( ) ( )

4 4 4 4

( 1) ( ) ( ) ( ) ( ) ( ) ( ) ( )

4 4 4 4

cos sin cos sin

sin cos sin cos

k k k k k k k k

P P P P

k k k k k k k k

P P P P

y y y z

z z y z

ξ ξ ξ ξ

ξ ξ ξ ξ

+ +

⎡ ⎤ ⎡ Δ − Δ ⎤ ⎡ ⎤ ⎡ Δ − Δ ⎤

= =

⎢ ⎥ ⎢ Δ Δ ⎥ ⎢ ⎥ ⎢ Δ + Δ ⎥

⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦

29.44

12

30 cos 3 10sin 3

° °

⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦

⎡ − ⎤ ⎡ ⎤

= ⎢ ⎥ ⎢ = ⎥

12

(13)

좌표 변환

( )k ( )k

P P

x

=

x

⎛ ⎞

⎜ ⎟

y y′

(

4

) V ξ

CL

ξ

4 ( )

(

4k

) V ξ

( 1) ( ) ( ) ( ) ( )

( 1) ( ) ( ) ( ) ( )

4 4

4 4

cos sin

cos sin

k k k k k

k k k k k

P P

P P P

P P P

y y z

z z y

ξ ξ

ξ ξ

+

+

⎛ ⎞

⎜ = Δ − Δ ⎟

⎜ ⎟

⎜ = Δ + Δ ⎟

⎝ ⎠

x

z x′

z′

( ) ( )

( ) ( )

4 4

( )

k k

k k

B

g dV

ξ ξ

ρ

+ Δ

= ∫∫∫

F k

( )k ( )k

g dV dV

ρ

= ⎨ + ⎬

⎪ ⎪

⎩ ∫∫∫ ∫∫∫ ⎭

k

(k)번째 상태의 부피

(k)번째 상태의 부피와 변화된 부피에 의한 힘으로 분리

( ) ( )

4 4

( k k )

V ξ +Δξ

( ( )k ) V

dV

∫∫∫

ξ

= V ( ξ

4( )k

)

( ( )k )

V

dxdydz

ξ

= ∫∫∫

( ) ( )

4 4

( k ) ( k )

V ξ V Δξ

⎪ ⎪

⎩ ⎭

( )

( 4 )

V ξ V4( )k )

2008_Pressure Integration Technique (2) 2008_Pressure Integration Technique (2)

13

(14)

힘 ((부력 부력)) (2) (2)

x

' x '

Δ ξ

4

y y′

( )

(

4k

)

V ξ y ' y x '

CL

( ) 4

ξ

k ( )

(

4k

) V ξ

( )k

y

좌표 변환

( )k ( )k

P P

x

=

x

⎛ ⎞

⎜ ⎟

z x x′

dx z′

y

P

( )k

y

S

( )k

x

A

x

S( )k

( 1) ( ) ( ) ( ) ( )

( 1) ( ) ( ) ( ) ( )

4 4

4 4

cos sin

cos sin

k k k k k

k k k k k

P P

P P P

P P P

y y z

z z y

ξ ξ

ξ ξ

+

+

⎛ ⎞

⎜ = Δ − Δ ⎟

⎜ ⎟

⎜ = Δ + Δ ⎟

⎝ ⎠

y′

z′ z

( ) ( )

( ) ( )

4 4

( )

k k

k k

B

g dV

ξ ξ

ρ

+ Δ

= ∫∫∫

F k

( )k ( )k

g dV dV

ρ

= ⎨ + ⎬

⎪ ⎪

⎩ ∫∫∫ ∫∫∫ ⎭

k

(k)번째 상태의 부피와 변화된 부피에 의한

힘으로 분리 변화된 부피

( )k

∫∫∫ dV = ∫∫∫ dxdydz = ∫∫∫ dzdydx

적분하기 편리하게

적분 순서를 변경 ∇

y

( )

y

4

ξ

k

Δ

( ) 4

ξ

k

( ) ( )

4 4

( k k )

V ξ +Δξ

⎪ ⎩

V4( )k ) V(Δξ4( )k )

⎪ ⎭

( )

( 4k )

V Δξ V(Δξ4( )k ) V(Δξ4( )k )

( )k 0 0

xF

d d d

∫ ∫ ∫

CL

우현만 보면,

( )k

cos

( )k

y Δ ξ

만약

Δ ξ

4( )k 가 작다면,

( ) ( ) ( ) ( )

4 4

cos tan

k k k k

A S

x y y

dzdydx

ξ ξ

Δ Δ

∫ ∫ ∫

tan

( )k

z = y Δ ξ

( )k

y

S

( ) ( )

cos

4

y

S

Δ ξ

y

( ) 4

ξ

k

좌현도 이와 같이, Δ

( )k ( )k cos ( )k tan ( )k

x y Δξ y Δξ

∫ ∫

( )( ) 0 ( )cos 4( )

0 tan 4( )

y

S

z = y tan Δ ξ

4( )

F P

k A

x y y

x Δξ Δξ

dzdydx

∫ ∫ ∫

14

(15)

y y′

(

4

) V ξ

CL

ξ

4 ( )

(

4k

) V ξ

( )k

y

좌표 변환

( )k ( )k

P P

x

=

x

⎛ ⎞

⎜ ⎟

z x x′

dx z′

y

P

( )k

y

S

( )k

x

A

x

S( )k

( 1) ( ) ( ) ( ) ( )

( 1) ( ) ( ) ( ) ( )

4 4

4 4

cos sin

cos sin

k k k k k

k k k k k

P P

P P P

P P P

y y z

z z y

ξ ξ

ξ ξ

+

+

⎛ ⎞

⎜ = Δ − Δ ⎟

⎜ ⎟

⎜ = Δ + Δ ⎟

⎝ ⎠

( ) ( )

( ) ( )

4 4

( )

k k

k k

B

g dV

ξ ξ

ρ

+ Δ

= ∫∫∫

F k

( )k ( )k

g dV dV

ρ

= ⎨ + ⎬

⎪ ⎪

⎩ ∫∫∫ ∫∫∫ ⎭

k

(k)번째 상태의 부피와 변화된 부피에 의한

힘으로 분리 변화된 부피

( )k

∫∫∫ dV

( )( ) ( ) ( ) ( ) ( ) 4( ) 4( )

4 4

0 0 cos tan

cos tan 0 0

k k k k

F P

k k k k

A S

x y y

x y y

dzdy

ξ ξ

dzdy dx

ξ ξ

Δ Δ

Δ Δ

⎛ ⎞

≈ ∫ ∫ ⎜ ⎝ ∫ − ∫ ∫ ⎟ ⎠

( ) ( )

4 4

( k k )

V ξ +Δξ

⎪ ⎩

V4( )k ) V(Δξ4( )k )

⎪ ⎭

만약

Δ ξ

4( )k 가 작다면,

( )

( 4k )

V Δξ

∫ ∫

xA

yS cosΔξ4

ytanΔξ4

0

0

( ) ( )

( ) ( ) ( )

4

( ) ( ) ( )

4

0 ( ) cos ( )

4 4

cos

tan

0

tan

k k k

F P

k k k

A S

x k y k

x y

y dy

ξ

y dy dx

ξ

ξ

Δ

ξ

Δ

⎛ ⎞

= ∫ ∫ ⎜ ⎝ − Δ − ∫ Δ ⎟ ⎠

( ) ( )

( ) 4

( )

( ) ( )

4

0 cos

2 ( ) 2 ( )

4 4

cos 0

tan tan

2 2

k k

k P F

k

A k k

S

k k y

x x

y

y y

dx

ξ

ξ

ξ ξ

Δ

Δ

⎛ − Δ Δ ⎞

⎜ ⎟

= −

⎜ ⎟

⎝ ⎠

2008_Pressure Integration Technique (2) 2008_Pressure Integration Technique (2)

⎝ ⎠

15

(16)

힘 ((부력 부력)) (4) (4)

x

' x '

Δ ξ

4

y y′

( )

(

4k

)

V ξ y ' y x '

CL

( ) 4

ξ

k ( )

(

4k

) V ξ

( )k

y

좌표 변환

( )k ( )k

P P

x

=

x

⎛ ⎞

⎜ ⎟

z x x′

dx z′

y

P

( )k

y

S

( )k

x

A

x

S( )k

( 1) ( ) ( ) ( ) ( )

( 1) ( ) ( ) ( ) ( )

4 4

4 4

cos sin

cos sin

k k k k k

k k k k k

P P

P P P

P P P

y y z

z z y

ξ ξ

ξ ξ

+

+

⎛ ⎞

⎜ = Δ − Δ ⎟

⎜ ⎟

⎜ = Δ + Δ ⎟

⎝ ⎠

( ) ( )

( ) ( )

4 4

( )

k k

k k

B

g dV

ξ ξ

ρ

+ Δ

= ∫∫∫

F k

( )k ( )k

g dV dV

ρ

= ⎨ + ⎬

⎪ ⎪

⎩ ∫∫∫ ∫∫∫ ⎭

k

(k)번째 상태의 부피와 변화된 부피에 의한

힘으로 분리 변화된 부피

( )k

∫∫∫ dV

( ) ( )

( ) 4

( )

0 cos

2 ( ) 2 ( )

4 4

tan tan

2 2

k k

k P F Ak

k k y

x x

y y

dx

ξ ξ

Δξ

⎛ − Δ Δ ⎞

⎜ ⎟

= −

⎜ ⎟

( ) ( )

4 4

( k k )

V ξ +Δξ

⎪ ⎩

V4( )k ) V(Δξ4( )k )

⎪ ⎭

( )

( 4k )

V Δξ A

2

( )k cos 4( )k

2

0

S

x

y Δξ

⎝ ⎠

( ) ( )

( )

( )

2 2

( ) ( )

2 ( ) ( )

4 4

cos tan

2 2

k F

k A

k k

x S P

k k

x

y y

ξ ξ dx

= Δ Δ −

⎜ ⎟

cos

2 4( )

tan

4( ) ( )A( )Fkk

1 2 ( ( ) ( )

( ) 2 ( ) 2

)

k k x k k

P S

x

y y dx

ξ ξ

= − Δ Δ ∫ −

2 2

A

⎜ ⎟

⎝ ⎠

( )

( ) ( )

( ) ( )

2

k k

k k

P S

P S

y y

y y dx

⎛ + ⎞ × −

⎜ ⎟

⎝ ⎠

횡 방향 중심 수선면적

( )

2 ( ) ( ) ( )

4 4 4

cos ξ

k

tan ξ

k

T

WP

ξ

k

= − Δ Δ

(k)번째 상태의

수선면 횡 뱡향1차 모멘트

( )

2

A

만약

Δ ξ

4( )k 가 작다면, 수선면 횡 뱡향 차 횡 방향 중심 수선면적

( )

( ) ( )

4 4

k k

T

WP

ξ ξ

≈ −Δ

ξ

4 가 작다 ,

16

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