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[2009] [09] [10]

Innovative ship design

- Ship Motion & Wave Load -

April, 2009

Prof. Kyu-Yeul Lee

Department of Naval Architecture and Ocean Engineering, Seoul National University of College of Engineering

서울대학교 조선해양공학과 학부4학년 “창의적 선박설계” 강의 교재

Hydrodynamics

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Chap.1 Loads acting on a ship

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Ship Structural Design

 Ship Structural Design

what is designer’s major interest?

 Safety :

Won’t ‘IT’ fail under the load?

L

x

y ^{f x} ( )

( ) f x

x

y

react

F

( ) V x

( ) M x

( ) y x

Differential equations of the defection curve 4

4

( ) ( ) d y x

EI f x

dx = − what is our interest?

: ( ) : ( ) : ( ) Shear Force V x Bending Moment M x Deflection y x

: ( ) Load f x

cause

( ) ( ) dV x f x

dx = − ( )

, dM x ( ) dx = V x

2

2

, d y x ( ) ( )

EI M x

dx =

‘relations’ of load, S.F., B.M., and deflection

 Safety :

Won’t it fail under the load?

 Geometry :

How much it would be bent under the load?

,

_act

y i

M M

where

I y Z

σ = =

act l

σ

≤

σ

Stress should meet :

a ship a stiffener a plate

global local

σ

act

: Actual Stress

σ

t

: Allowable Stress

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Ship Structural Design

 Ship Structural Design

what is designer’s major interest?

 Safety :

Won’t ‘IT’ fail under the load?

a ship a stiffener a plate

L

x

y ^{f x} ( )

global local

a ship

Actual stress on midship section should be less than allowable stress

l

act σ

σ _. ≤

Allowable stress by Rule (for example):

2

, σ

_l

= 175 f

1

[ N mm / ]

, _act . ^{S W} mid

M M

σ Z

= + Hydrostatics

L

x

y ^{f x} ( )

( ) f x

x

y

react

F

( ) V x

( ) M x

( ) y x

Differential equations of the defection curve 4

4

( ) ( ) d y x

EI f x

dx = − what is our interest?

: ( ) : ( ) : ( ) Shear Force V x Bending Moment M x Deflection y x

: ( ) Load f x

cause

( ) ( ) dV x f x

dx = − ( )

, dM x ( ) dx = V x

2

2

, d y x ( ) ( )

EI M x

dx =

‘relations’ of load, S.F., B.M., and deflection

 Safety :

Won’t it fail under the load?

 Geometry :

How much it would be bent under the load?

,

_act

y i

M M

where

I y Z

σ = =

Stress should meet :

σ

act

: Actual Stress σ

t

: Allowable Stress

act l

σ

≤

σ

M

_S

= Still water bending moment

M

_W

= Vertical wave bending moment Hydrodynamics

z

x

what kinds of load

cause hull girder moment?

f

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Ship Structural Design

 Ship Structural Design

what is designer’s major interest?

 Safety :

Won’t ‘IT’ fail under the load?

a ship a stiffener a plate

global local

a ship

L

x

y ^{f x} ( )

( ) f x

x

y

react

F

( ) V x

( ) M x

( ) y x

Differential equations of the defection curve 4

4

( ) ( ) d y x

EI f x

dx = − what is our interest?

: ( ) : ( ) : ( ) Shear Force V x Bending Moment M x Deflection y x

: ( ) Load f x

cause

( ) ( ) dV x f x

dx = − ( )

, dM x ( ) dx = V x

2

2

, d y x ( ) ( )

EI M x

dx =

‘relations’ of load, S.F., B.M., and deflection

 Safety :

Won’t it fail under the load?

 Geometry :

How much it would be bent under the load?

,

_act

y i

M M

where

I y Z

σ = =

Stress should meet :

σ

act

: Actual Stress σ

t

: Allowable Stress

act l

σ

≤

σ

S ( ) f x

,

_act . ^{S W} mid

M M

σ Z

= +

l

act σ

σ _. ≤

: load in still water

( )

0^x

( )

S S

V x = ∫ f x dx

S ( ) V x

S ( ) M x

( )

0^x

( )

S S

M x = ∫ V x dx

Hydrostatics Hydrodynamics

. F K diffraction added mass

mass inertia

damping

: still water shear force

: still water bending moment

, M

_S

= Still water bending moment M

_W

= Vertical wave bending moment what kinds of load f cause hull girder moment?

weight

buoyancy

f _S (x) : load in still water

= weight + buoyancy

W ( )

f x : load in wave

( )

0^x

( )

W W

V x = ∫ f x dx

W ( ) V x

W ( )

M x

( )

0^x

( )

W W

M x = ∫ V x dx

: wave shear force

: vertical wave bending moment

f

_W

(x) : load in wave

= added mass + diffraction

+ damping + Froude-Krylov + mass inertia ( )

_S

( )

_W

( ) f x = f x + f x

( )

_S

( )

_W

( ) V x = V x + V x

( )

_S

( )

_W

( )

M x = M x + M x

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1. Loads acting on a ship (1) - Loads in still water

z

x

x ( ) z

w x w(x):weight

x ( ) z

b x

+

b(x):bouyancy

In still water

( ) ( ) ( ) f S x = b x − w x

what kinds of load f _S cause ? M _s

( ) _S ( ) _W ( ) f x = f x + f x

x

( ) z f S x

f

_S

(x)= b(x) – w(x) : Load

=

b(x) : buoyancy distribution in longitudinal direction w(x) = LWT(x) + DWT(x)

- w(x) : weight distribution in longitudinal direction - LWT(x) : lightweight distribution

- DWT(x) : deadweight distribution

(M

_S:

Still water bending Moment in midship)

f(x) : distribution load in longitudinal direction

f _S (x) : distribution load in longitudinal direction in still water

f _W (x) : distribution load in longitudinal direction in wave

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1. Loads acting on a ship (2)

- Example of 3,700TEU Container carrier

Load Curve, f

_S

(x) Still water

Shear Force, V

_S

(x)

Still water

Bending Moment, M

_S

(x) Weight, w(x)

Buoyancy, b(x)

- Frame space : 800mm

 Example of 3,700 TEU Container Ship in Homogeneous 10t Scantling Condition - Principal dimensions & drawings

- Loading Condition (Sailing state) in homogeneous 10t scantling condition

( )

0^x

( )

S S

V x = ∫ f x dx M

_S

( ) x = ∫

0^x

V x dx

_S

( )

- principal dimension - profile & plan drawing - midship section

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LIGHTWEIGHT DISTRIBUTION DIAGRAM

FR. NO 0 25 50 74 99 125 150 175 200 226 251 276 301 326 0.0

20.0 40.0 60.0 80.0 100.0 120.0 140.0 160.0 180.0 200.0 220.0 240.0 TONNES

Crane

Bow Thruster Emergency

Pump Engine

(Example of 3,700TEU Container carrier)

1. Loads acting on a ship (2) - Lightweight

AP FP

E/R

A.P F.P

Load Curve, f

_S

(x) Still water

Shear Force, V

_S

(x)

Still water

Bending Moment, M

_S

(x) Weight, w(x)

Buoyancy, b(x)

( )

0^x

( )

S S

V x = ∫ f x dx M

_S

( ) x = ∫

0^x

V x dx

_S

( )

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- Loading Plan in homogenous 10t scantling condition

1. Loads acting on a ship (3) - Deadweight

(Example of 3,700TEU Container carrier)

Deadweight distribution in longitudinal direction in homogenous 10t scantling condition

- Deadweight distribution curve in homogenous 10t scantling condition

A.P FR.Space : 800 mm F.P

Load Curve, f

_S

(x) Still water

Shear Force, V

_S

(x)

Still water

Bending Moment, M

_S

(x) Weight, w(x)

Buoyancy, b(x)

( )

0^x

( )

S S

V x = ∫ f x dx M

_S

( ) x = ∫

0^x

V x dx

_S

( )

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(2) 면적 길이 방향으로 적분하여 부피 계산

(1) 수선면 아래의 선박 단면적계산

x

A

' y

' z

 Buoyancy Curve in Homogeneous 10ton Scantling Condition

1. Loads acting on a ship (4) - Buoyancy curve

(Example of 3,700TEU Container carrier)

100 ton

FR.No

A.P F.P

FR.Space : 800 mm

 Buoyancy 계산방법

Load Curve, f

_S

(x) Still water

Shear Force, V

_S

(x)

Still water

Bending Moment, M

_S

(x) Weight, w(x)

Buoyancy, b(x)

( )

0^x

( )

S S

V x = ∫ f x dx M

_S

( ) x = ∫

0^x

V x dx

_S

( )

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11 /203 LIGHTWEIGHT DISTRIBUTION DIAGRAM

FR. NO 0 25 50 74 99 125150175 200226 251276301 326 0.0

20.0 40.0 60.0 80.0 100.0 120.0 140.0 160.0 180.0 200.0 220.0 240.0 TONNES

=Lightweight + Deadweight

Load Curve

Still Water shear Force

Load Curve

=Weight+ Buoyancy

+

+

=

Bouyancy Curve Weight curve

S ( ) f x

1. Loads acting on a ship (5) - Load/Shear/Moment curve

(Example of 3,700TEU Container carrier)

Lightweight Distribution Curve Deadweight Distribution Curve

FR.No

A.P F.P A.P F.P

A.P FR.Space : 800 mm F.P

100 ton

FR.No

A.P F.P

FR.Space : 800 mm A.P F.P

= Weight w(x) + Buoyancy b(x)

Load Curve, f

_S

(x) Still Water Shear Force, V

_S

(x)

Still Water

Bending Moment, M

_S

(x) Weight, w(x)

Buoyancy, b(x)

( )

0^x

( )

S S

V x = ∫ f x dx M

_S

( ) x = ∫

0^x

V x dx

_S

( )

in homogenous 10t scantling condition

in homogenous 10t scantling conditi on in homogenous 10t scantling condition

in homogenous 10t scantling condition

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Shear force

Permissible shear force

Still Water Shear Curve

Bending Moment Permissible Bending Moment

Still Water Bending

Moment Curve Load Curve

Actual still water shear force is lower than permissible Shear.

→ O.K

Actual still water bending moment is lower than permissible bending.

→ O.K

( ) ( ) ( ) f S x = b x − w x

( ) 0 ^x ( )

S S

V x = ∫ f x dx

1. Loads acting on a ship (6) - Load/Shear/Moment curve

(Example of 3,700TEU Container carrier)

( )

0 ^x

( )

S S

M x

=

∫ V x dx

Load Curve, f

_S

(x) Still water

Shear Force, V

_S

(x)

Still water

Bending Moment, M

_S

(x) Weight, w(x)

Buoyancy, b(x)

( )

0^x

( )

S S

V x = ∫ f x dx M

_S

( ) x = ∫

0^x

V x dx

_S

( )

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33 3 33 3

( ) ( ) ( )

f R x = − a x ξ  − b x ξ 

How to know ?

ξ ξ ^{ } ₃ , ₃ z

x

z

x

In still water

In wave

( ) ( ) ( ) f S x = b x − w x

( ) ^{( )}

( ) _{S W}

f x = f x + f x

3 .

( ) ( ) ( ) (

( ) ( ) m x f _D x f _{F K} x f _R x )

b x w x − ξ + + +

= − 

x

( ) z

f

W

x

?

2. Loads in wave

• for example, consider heave motion

• from 6DOF motion of ship

[ ξ ₁ ^, ξ ₂ ^, ξ ₃ ^, ξ ₄ ^, ξ ₅ ^, ξ ₆ ] ^T

= x

ξ 3

x

( ) z f

S

x

f

_S

(x)= b(x) – w(x) : Load

+

f

_D

(x) : Diffraction force in a unit length f

_R

(x) : Radiation force in a unit length f

_F.K

(x) : Froude-Krylov force in a unit length

Where,

Roll ξ

4

Pitch ξ

5

Hea ve ξ

3

Yaw ξ

6

O

x y

z

 ref.> 6 DOF motion of ship

additional loads in wave

 Loads in wave

In order to know loads in wave, we have to know ξ ξ ^{ }

3

,

3

f(x) : Distribution load in longitudinal direction

f

_S

(x) : Distribution load in longitudinal direction in still water

f

_W

(x) : Distribution load in longitudinal direction in wave

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3. 6DOF Equation of motion of ship

R D I

Φ Φ

Φ : Incident wave velocity potential : Diffraction wave velocity potential : Radiation wave velocity potential F

_F.K

: Froude-krylov force

F

_D

: Diffraction force F

_R

: Radiation force

matrix coeff.

restoring 6

6 :

matrix coeff.

damping 6

6 :

matrix mass

added 6 6 :

×

×

×

C B M

_A

 선박에 작용 하는 유체력

.

Static + F K + D + R

F F F F

Gravity +

= F x M  

Restoring

F F Wave exciting F _R = − Ax Bx  − 

, ,

( _{Gravity Static} ) Wave exciting External dynamic External static

= + + − − + +

Mx  F F F Ax Bx F   F

( M + A x Bx Cx )  +  + = F Wave exciting + F External dynamic , + F External static ,

Linearization

, (

F _Restoring

= (

F _Gravity

+

F _Static

)≈ −

Cx

)

added

mass Damping Coefficient Surface force

Body force

, ,

External dynamic External static

+ F + F

Wave exciting force를 제외한 외력(ex. 제어력 등)

( )

B B B

I D R

Fluid

S P dS ρgz dS S S dS

t t t

ρ

^{∂Φ ∂Φ ∂Φ}

= = − − + +

∂ ∂ ∂

∫∫ ∫∫ ∫∫

F n n

 선박의 6자유도 운동방정식

Newton’s 2

^nd

Law

=

= ∑ ^F

x M  

Gravity Fluid

= F + F

F _Body

+ F

_Surface

External

+F

선박의 Surface force로 작용

How to know ?

3 , 3

ξ ξ ^{ }

Fluid

Pρgz = − − ρ ^∂Φ t

∂ 



 





∂ Φ + ∂

∂ Φ + ∂

∂ Φ

− ∂

−

= ρgz ρ t

^I

t

^D

t

^R

By solving equation of motion, we could know the velocities, accelerations!!

 선박에 작용하는 압력

Static

=

F

+

F _{F K .}

+

F _D

+

F _R

Linearized Bernoulli Eq.

By solving equation of motion,

we could know the velocities,

accelerations.

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Chap 2.

6 DOF Equations of Ship Motion

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Coordinate system ^{[ ]}

T 6 1 , , ξ ξ 

=

(변위 : ) x

O-xyz : Global coordinate system

① right-handed coordinate system (O-x,y,z)

origin in the plane of the undisturbed free-surface

② if body moves with a mean forward speed, coordinate moves with the same speed

③ body have the x-z plane as a plane of symmetry

O’-x’ y’ z’ : Body-fixed coordinate system

 G : Position of center of gravity y

z

G O

y′ c

z′ c

x

_c

,y

_c,

z

_c

: distance from O’-x’y’z’ to center of gravity

U ^{Roll ξ}

⁴

Pitch ξ

5

Hea ve ξ

3

Yaw ξ

₆

O G

x y

z z′

2

y′

ξ

ξ

3

ξ

4

O′

x z

O

x′

z′

O′ G

x′ c

z′ c

ξ 1

ξ 3

ξ

5

1) Newman, J.N. , Marine Hydrodynamics, The MIT Press, Cambridge, 1977, pp 285~290

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Uncoupled Heave motion equation

 Heave 운동 방정식 유도

B W

restore damping F

F

z

Z

X

평형상태에서

더 들어간 부피

= F ^gravity + F ^static + F ^F _. ^K + F ^D + F ^R

Fluid gravity

Surface Body

F F

F F

F M

+

=

+

=

= ∑

ξ ^{ } 3

exciting,3

F

3 33 3

33 ξ ^{ } B ξ ^

A −

3 −

0 ρ ξ

ρ gV − gA _wp ⋅

− Mg

3 33 3

33 3

, 3

0 )

( ρ gV ρ gA ξ ^ F A ξ ^{ } B ξ ^

Mg + − _wp + _exciting − −

−

=

3 , 3

3 33 3

33 )

( M + A + B + gA _wp = F _exciting

∴ ξ ^{ } ξ ^ ρ ξ

 Surge 운동 방정식 유도

(Heave에서 복원력 성분만 제외)

1 , 1

11 1

11 )

( M + A + B = F _exciting

∴ ξ ^{ } ξ ^

 Sway 운동 방정식 유도

(Heave에서 복원력 성분만 제외)

2 , 2

22 2

22 )

( M + A + B = F _exciting

∴ ξ ^{ } ξ ^

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Uncoupled Heave motion equation 풀이

ex) For Heave motion

( ^{M + A} ₃₃ ) ^{ξ  } ₃ ^{+ B} ₃₃ ^{ξ } ₃ ^{+ C} ₃₃ ^ξ ₃ ^{= F} _{exciting , 3}

t i A t

i A t

i A t

i

A e B i e C e f e

A

M ₃₃ )( ω ² ξ ₃ ^ω ) ₃₃ ( ω ξ ₃ ^ω ) ₃₃ ( ξ ₃ ^ω ) η _{0 3} ^ω

( + − + + =

t i A e t ξ ^ω ξ ₃ ( ) = ₃

t i A e i

t ω ξ ^ω ξ  ₃ ( ) = ₃

t i A e t ω ξ ^ω ξ   ₃ ( ) = − ² ₃

t i A t

i A

exciting F e f e

F _{, 3} = ₃ ^ω = η _{0 3} ^ω

( : Wave Amplitude, Real), (ξ η _{0 3} ^A : Amplitude of heave motion, Complex) ( : 1m 파고에 대한 Wave exciting force Amplitude, Complex) f ₃ ^A

Assumption :

^{시간이 충분히 흘러}

steady 상태에서, 선박이 외력의 주파수

ω 와 같은 운동을 함

(Harmonic Motion) -> 초기 Transient Motion은 고려하지 않음.

{ ^{ω M A} 33 ^{i ω B} 33 ^C 33 } ^ξ 3 ^{A e i ω t η} 0 ^f 3 ^{A e i ω t}

2 ( + ) + + =

−

= D

{ ^{M A} 33 ^{i B} 33 ^C 33 } 3 ^A 0 ^f 3 ^A

2 ( ) ω ξ η

ω + + + =

− ¹

3 0 3

= f ^A D −

A η

ξ ₃ ¹

0

3 = ^A D − A

η f ξ

 RAO (Response Amplitude Operator)

: 1m wave Amplitude 를 가지는 주파수 ω인 wave에 대한

heave운동 변위의 진폭

( Complex)

1) Newman, J.N. , Marine Hydrodynamics, The MIT Press, Cambridge, 1977, pp 307~310

동일한 항으로 정리가능

( : Wave exciting frequency) ω

,( A

₃₃

: heave motion에 의한 heave 방향 added mass)

,( B

₃₃

: heave motion에 의한 heave 방향 damping coefficient)

,( C

₃₃

: heave motion에 의한 heave 방향 복원력 coefficient)

,( M :Mass of ship)

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Uncoupled roll motion equation

 Roll 운동 방정식 유도

Fluid gravity

Surface Body

xx

M M

M M

M I

+

=

+

=

= ∑

ξ ^{ } 4

R D

K F static

gravity M M M M

M + + + +

= _.

exciting,4

M

4 44 4

44 ξ ^{ } B ξ ^

A −

− k

r × ∆

B

1

k r _G × W

4 44 4

44 4

, A ξ ^{ } B ξ ^ M

GZ + _exciting − −

⋅

∆

=

,

GZ

= −

y _G

+

y _B

1

O

y z

τ

(+ )

B G

z

′

y

′

K

C L

y z

F _∆

B ₁

B

1

r

g ₁ g

W

M _T

τ

e

r G

Z

W r

G

γ

G

O

restoring

τ

ξ 4

ξ 4

F

_∆

B1

r

B1

γ

γ π − O

γ γ π ) sin sin( − =

+ π

γ 2 π

γ − π − x sin

x

M

_T

: B Global

₁

을 지나는 부력 작용선과 선체 중심선과의 교점 coordinate oyz

coordinate fixed

Body z

oy :

:

'

'

Hydrodynamics

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Uncoupled Roll motion equation

 Roll 운동 방정식 유도

Fluid gravity

Surface Body

xx

M M

M M

M I

+

=

+

=

= ∑

ξ ^{ } 4

R D

K F static

gravity M M M M

M + + + +

= _.

exciting,4

M

4 44 4

44 ξ ^{ } B ξ ^

A −

− k

r × ∆

B

1

k r _G × W

4 44 4

44 4

, A ξ ^{ } B ξ ^ M

GZ + _exciting − −

⋅

∆

=

4 44 4

44 4

,

sin ξ 4 M A ξ ^{ } B ξ ^

GM + _exciting − −

⋅

∆

−

=

4 44 4

44 4

,

4 ξ ξ

ξ M A ^{ } B ^

GM + _exciting − −

⋅

∆

−

≈

44 44 4 44 4 4 ,4

( I A ) ξ B ξ GM T ξ M exciting

∴ +  +  + ∆ ⋅ =

 Pitch 운동 방정식 유도 (Roll 운동 방정식과 동일)

 Yaw 운동 방정식 유도

(RollHeave에서 복원력 성분만 제외)

sin 4

GZ

GM ξ

= −

4

sin ξ 4 ≈ ξ

55 55 5 55 5 5 ,5

( I A ) ξ B ξ GM L ξ M exciting

∴ +  +  + ∆ ⋅ =

66 66 6 66 6 ,6

( I A ) ξ B ξ M _exciting

∴ +  +  =

O

B G

z ′

y ′

K

C L

y z

F _∆

B ₁

g ₁ g

W M _T

Z

W

F

_∆

B1

r

B1

γ

γ O π −

γ γ π

) sin sin( − =

+ π

γ 2π

γ − π− x sin

x

M

_T

: B Global

₁

을 지나는 부력 작용선과 선체 중심선과의 교점 coordinate oyz

coordinate fixed

Body z oy

:

:

'

'

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Uncoupled Pitch motion equation

For Pitch motion

( A 55 + I 55 ) ξ  5 + B 55 5 ξ  + C 55 5 ξ = F exciting ,5

2

55 55 5 55 5 55 5 0 5

( A + I )( − ω ξ ^{A i t} e ^ω ) + B ⋅ ( i ωξ ^{A i t} e ^ω ) + C ⋅ ( ξ ^A e ^{i t ω} ) = η f e ^{A i t ω}

5 ( ) t 5 ^{A i t} e ^ω ξ = ξ

5 ( ) t i 5 ^{A i t} e ^ω ξ  = ωξ

2

5 ( ) t 5 ^{A i t} e ^ω ξ  = − ω ξ

,5 5 0 5

A i t A i t

exciting

F = F e ^ω = η f e ^ω

( : Wave Amplitude, Real), (ξ η 0 ₅ ^A : Amplitude of pitch motion, Complex) ( : 1m 파고에 대한 Wave exciting force Amplitude, Complex) f ₅ ^A

Assumption :

^{시간이 충분히 흘러}

steady 상태에서, 선박이 외력의 주파수

ω 와 같은 운동을 함

(Harmonic Motion) -> 초기 Transient Motion은 고려하지 않음.

{ ^{− ω 2 ( A 55 + I 55 ) + i B ω 55 + C 55} } ^{ξ 5 A i t e ω = η 0 f e 5 A i t ω}

= D

{ ^{− ω 2 ( A 55 + I 55 ) + i B ω 55 + C 55} } ^{ξ 5 A = η 0 f 5 A} ξ ₅ ^A = η ₀ f D ₅ ^{A − 1 5} ₅ ¹

0 A

f A

ξ η

= D −

 RAO (Response Amplitude Operator)

: 1m wave Amplitude 를 가지는 주파수 ω인 wave에 대한

pitch운동 변위의 진폭

( Complex)

1) Newman, J.N. , Marine Hydrodynamics, The MIT Press, Cambridge, 1977, pp 307~310

동일한 항으로 정리가능

( : Wave exciting frequency) ω

,( A

55

: pitch motion에 의한 pitch 방향 added mass)

,( B

₅₅

: pitch motion에 의한 pitch 방향 damping coefficient)

,( C

₅₅

: pitch motion에 의한 pitch 방향 복원력 coefficient)

,( I

55

: Mass moment of inertia of ship with respect to y axis)

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Motion of any point on the body - Acceleration of any point

 Position vector of any point about the Origin(O)

T

′

R

′

= + + ×

x ξxξx

 Translatory displacements of body fixed coordinate in the x-,y-, and z-directions with respect to the origin(O)

 Angular displacements of rotational motion in the x-,y-, z- axis with respect to the origin(O)

4 5 6

R = ξ + ξ + ξ

ξi j k

,(Surge, Sway, Heave)

,(Roll, Pitch, Yaw)

4 5 6

,

_R

x y z

ξ ξ ξ

 

 × = ′ 

 

 ′ ′ ′ 

 

i j k

 ξx   

x-,y-,z- acceleration are coupled with other motions

G′

G

ξ

5 x′

z′

' O O

z

x′

Heave motion caused by pitch motion Similarly, roll motion cause heave motion

( ^ξ

¹

^{z ′ ξ}

⁵

^{y ′ ξ}

⁶

) ( ^ξ

²

^{z ′ ξ}

⁴

^{x ′ ξ}

⁶

) ( ^ξ

³

^{y ′ ξ}

⁴

^{x ′ ξ}

⁵

)

= + − + − + + + −

x    i    j    k



 ex.) Acceleration of heave motion :

x′ ^G

G′

ξ

5

O

5 1

x

c

ξ ≈ k z

1

s z

If ξ

5

is small,

s ≈ k z

1

k i

( ^{ξ }

³

^{+ y ′ ξ }

⁴

^{− x ′ ξ }

⁵

) ^k ( ^{ξ }

³

^{+ y ′ ξ }

⁴

^{− x ′ ξ }

⁵

) ^k

① : heave acceleration ② heave acceleration by roll

③ : heave acceleration by pitch

① + ② + ③

Find : Acceleration of any point  x

 Motion of equation :

1 2 3

T = ξ + ξ + ξ

ξi j k

( : position vector with respect to O’-x’y’z’ coordinate) ( : position vector with respect to O-xyz coordinate)

x y z

′ = ′ + ′ + ′

x i j k

x y z

= + +

x i j k

restoring exciting radiation

M  x = F + F + F

T

′

R

′

R

′

= + + × + ×

x



ξxξxξx

   

 Velocity vector

 Acceleration vector

( )

2

T R R R R

T R R R

′ ′ ′ ′ ′

= + + × + × + × + ×

′ ′ ′ ′

= + + × + × + ×

x ξxξxξxξxξx ξxξxξxξx

   

    

     

 선박을 강체로 가정하면, 선박 위의 한 점은 시간에 따라 변하지 않는다

T R

′

= + ×

x ξξx 



( ^ξ

¹

^{z ′ ξ}

⁵

^{y ′ ξ}

⁶

) ( ^ξ

²

^{z ′ ξ}

⁴

^{x ′ ξ}

⁶

) ( ^ξ

³

^{y ′ ξ}

⁴

^{x ′ ξ}

⁵

)

=  +  −  i +  −  +  j +  +  −  k

1) Newman, J.N. , Marine Hydrodynamics, The MIT Press, Cambridge, 1977, pp 307~310

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Motion of any point on the body

- (참조) Acceleration of any point Find : Acceleration of any point  x

 Motion of equation :

restoring exciting radiation

M  x = F + F + F

 Position vector of any point about the Origin(O)

T

′

R

′

= + + ×

xξxξx

( Translatory displacements : ξ

T

= ξ

1

i+ ξ

2

j+ ξ

3

k , (Surge, Sway, Heave) ) ( Angular displacements : ξ

R

=ξ

4

i+ ξ

5

j+ ξ

6

k , (Roll, Pitch, Yaw) )

4 5 6

,

_R

x y z

ξ ξ ξ

 

 × = ′ 

 

 ′ ′ ′ 

 

i j k

 ξx   

O

T O′

′

= +

xξR x

 Position vector of any point about the Origin(O)

6 6

1

6 6

2

cos sin sin cos

x x

y y

ξ ξ

ξ

ξ ξ

ξ

  ′

 − 

 

  =   +    

  ′

         

6 6

1

6 6

2

cos sin

sin cos

x y

x

x y

y

ξ ξ

ξ

ξ ξ

ξ

′ −

 

 

  =   +  

  ′ + ′

     

 If we consider 2-dimensional motion

 If we consider 2-dimensional motion

1

6 2

0 0 0

x x

y y

x y

ξ ξ

ξ

  ′

 

  =   +   +

  ′

        ′ ′

i j k

6 1

6 2

y

x x

x

y y

ξ ξ

ξ ξ

  ′  − ′ 

   

=   +    + 

  ′ + ′

         

6 1

6 2

x y x

y x y

ξ ξ

ξ ξ

′ − ′

 

   

=   +  

  ′ + ′

     

Linearize (cosθ→1, sinθ→θ)

x′

O′

y′

P′

Inertial frame (O-frame)

y

O x

[ ^{x y ,} ]

^T

′ = ′ ′ x

i O

j O

k O

i O′

j O′

k O′

ξ

6

[ ]

^{x y ,}

^T

= x

[ 1 , 2 ] ^T

T = ξ ξ

ξ

(병진운동)

(회전운동)

x′

O′

y′

P′

Inertial frame (O-frame)

y

O x

[ ^{x y ,} ]

^T

′ = ′ ′ x

i O

j O

k O

i O′

j O′

k O′

ξ

6

[ ]

^{x y ,}

^T

= x

[ 1 , 2 ] ^T

T = ξ ξ

ξ

(병진운동)

(회전운동)

× ′ ξx

P

If summing the body motions ξ j are small and neglecting,

Linearize

ξ

R

x ′

R

× ′

ξx

Hydrodynamics

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Motion of any point on the body

- (참조) Acceleration of any point Find : Acceleration of any point  x

 Motion of equation :

restoring exciting radiation

M  x = F + F + F

 Position vector of any point about the Origin(O)

T

′

R

′

= + + ×

xξxξx

T

′

R

′

R

′

= + + × + ×

xξxξxξx     

 Velocity vector

 Acceleration vector

T

′

R

′

R

′

R

′

R

′

= + + × + × + × + ×

x ξxξxξxξxξx    

    

O

T O′

′

= +

x ξR x

O O

T O O

O O

T O O

d dt

ω

′ ′

′ ′

′ ′

= + +

′ ′

= + × +

x ξR x R x ξR x R x

  

 

( ) ( )

2

2

O O O O O

T O O O O O

d

dt x = + ω ×

_′

′ + × ω

_′

′ + × ω

_′

′ +

_′

′ +

_′

′ ξR x   R x  R x R x  R x   

 Position vector of any point about the Origin(O)

 Velocity vector

 Acceleration vector

①E-frame 에 대한

점 P의 가속도

( Translatory displacements : ξ

T

= ξ

1

i+ ξ

2

j+ ξ

3

k , (Surge, Sway, Heave) ) ( Angular displacements : ξ

R

=ξ

4

i+ ξ

5

j+ ξ

6

k , (Roll, Pitch, Yaw) )

2

2 _T ^O _O ^O _O

2

^O _O ^O _O

d

dt x = + × ω

_′

′ + × × ω ω

_′

′ + ω ×

_′

′ +

_′

′

ξR x   R x R x R x   ^{ x ξxξxξxξx =  T + +  ′  R × + ′ 2} ( ^{ R ×  ′} ) ^{+ R ×  ′}

x′

O′

y′

P′

Inertial frame (O-frame)

y

O x

[ x y , ]

^T

′ = ′ ′ x

i

O

j

O

k

O

i

O′

j

O′

k

O′

ξ

6

[ ]

^{x y, T}

x=

[

1

,

2

]

^T

T

= ξ ξ

ξ

(병진운동)

(회전운동)

x′

O′

y′

P′

Inertial frame (O-frame)

y

O x

[ ^{x y ,} ]

^T

′ = ′ ′ x

i

O

j

O

k

O

i

O′

j

O′

k

O′

ξ

6

[ ]

^{x y, T}

= x

[

1

,

2

]

^T

T

= ξ ξ

ξ

(병진운동)

(회전운동)

× ′ ξx

P

Linearize

② E-frame 에 대한

A-frame의 원점

A의 가속도 ⑥ A-frame 를 고정시켜 놓았을 때,

A-frame 에 대한

점 P의 가속도

③A-frame이 각 가속도를 가지고 회전하고 있을 때, 점 P의 접선 방향의 가속도

④A-frame이 회전하고 있 을 때, 점 P의 회전 중심 방향의 가속도 (구심력)

⑤ Coriolis Acceleration (Coriolis Effect) 회전 좌표계에서 기술된움직이는

점을 고정 좌표계에서 바라봤을 때,발생하는 효과

O O O O O

T

ω

O_′

′ ω ω

O_′

′ ω

O_′

′ ω

O_′

′

O_′

′

= ξR x  + ×  + × × R x R x + ×  R x + × R x  +  ⑥ A-frame 를 고정시켜 놓았을 때,

A-frame 에 대한

점 P의 가속도

⑤ Coriolis Acceleration (Coriolis Effect) 회전 좌표계에서 기술된움직이는

점을 고정 좌표계에서 바라봤을 때,발생하는 효과

③A-frame이 각 가속도를 가지고 회전하고 있을 때, 점 P의 접선 방향의 가속도

② E-frame 에 대한

A-frame의 원점 A의 가속도

①E-frame에 대한

점 P의 가속도

(선형화한 가정으로 인해 구심가속도 성분이 나타나지 않음)

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( ^{2 64 4 66 6} )

yaw = M x c I I

τ ′ + ξ  ξ  + ξ  k

( ^{1 1 2 3 55 5} )

= +

=

pitch pitch pitch

c c

M z M x I

τ τ τ

ξ ξ ξ

′  − ′  +  j

( ^{2 44 4 46 6} )

roll = M z c I I

τ − ^′ ξ  + ξ  + ξ  i

2 = 1 3

pitch z c M x c M

τ k × ξ  i + i × ξ  k

Moment in pitch motion

1) Journee, J.M.J. , Adegeest, L.J.M. ,Theoretical Manual of Strip Theory program“ Seaway for Windows”, Delft University of Technology, 2003, pp38~42 2) 구종도 역, 선체와 해양구조물의 운동학, 연경문화사, pp68~71

 Moment in pitch motion

(with respect to y-axis)

Find : Moment

restoring exciting radiation

I

ω

 = M + M + M I

ω

^

 Motion of equation for roll, pitch, yaw

x′

z′

O′

G

x′ c

z′ c

M

ξ

^ 3

M

ξ

^ 1

5

I yy

ξ

^

1

55 5 pitch = I

I τ

ξ

= ω

j





 Moment by mass moment of inertia

( ^{1 3} )

= M z ^{moment arm} _c ξ  − M x _c ξ  j

force

 Moment by inertia force

, ( I : mass moment of inertia ) , ( : Angular acceleration ) ω

 Moment in pitch motion

 Moment in roll motion

(with respect to x-axis, y

_c

=0 according to Lateral Symmetric)

 Moment in yaw motion

(with respect to z-axis , y

_c

=0 according to Lateral Symmetric)

[ ξ ₁ , ξ ₂ , ξ ₃ , ξ ₄ , ξ ₅ , ξ ₆ ] ^T

= x

heave sway surge

: : :

3 2 1

ξ ξ ξ

yaw pitch roll

: : :

5 4 3

ξ ξ ξ

x'

_c

,y’

_c,

z’

_c

: distance from O’-x’y’z’ to center of gravity

O

z

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Mass moment of inertia

1) Journee, J.M.J. , Adegeest, L.J.M. ,Theoretical Manual of Strip Theory program“ Seaway for Windows”, Delft University of Technology, 2003, pp38~42 2) 구종도 역, 선체와 해양구조물의 운동학, 연경문화사, pp68~71

 mass moment of Inertia

2 2

44 ( )

V

I = ∫∫∫

ρ

y + z dV ^{, 55 ( 2 2 )}

V

I = ∫∫∫

ρ

x + z dV ^{, 55 ( 2 2 )}

V

I = ∫∫∫

ρ

x + y dV

46 V

I = ∫∫∫

ρ

xzdV ^{, 45 ...}

V

I = ∫∫∫

ρ

xydV

 관성 모멘트는 계산하기 위해 많은 정보가 필요하여 , 계산이 복잡하다.

따라서 자료가 주어지거나 추정식을 사용하여 구한다 . (m=ρ▽)

2 2 2

44 44 , 55 55 , 66 66 ,

I = k

ρ

∇ I = k

ρ

∇ I = k

ρ

∇

- 추정식① ¹⁾

44 55 66

0.30 to 0.40 0.22 to 0.28 0.22 to 0.28

k B B

k L L

k L L

 ≈

 ≈

  ≈



- 추정식② 1) (Proposal of Bureau Veritas)

2

44

0.289 1.0 2 KG

k B

B

   

 

≈  +   

 

 

 

- 추정식③ ²⁾ K ₄₄

- 객선 : 0.38 ~ 0.43 - 화물선 : 0.32 ~ 0.35(만재)

0.375 ~ 0.4(Ballast) - 석탄운반선 : 0.31~0.33(만재)

0.35~0.39(Ballast) - 전함 : 0.34 ~ 0.38

- 순양함 : 0.39 ~ 0.42 - 어선 : 0.38 ~ 0.44

Find :Mass moment of inertia I

 관성모멘트를 Radius of gyration ( 관동반경 k ₄₄ , k ₅₅ , k ₆₆ )을 이용하여 표현하면 다음과 같다.

 관성모멘트 추정식

restoring exciting radiation

Mx  = F + F + F

 Equation of motion :

restoring exciting radiation

I

ω

 = M + M + M