Planning Procedure of Naval Architecture and Ocean Engineering

Ship Stability

September 2013

Myung-Il Roh

Department of Naval Architecture and Ocean Engineering Seoul National University

1 Planning Procedure of Naval Architecture and Ocean Engineering, Fall 2013, Myung-Il Roh Ship Stability

þ Ch. 1 Introduction to Ship Stability þ Ch. 2 Review of Fluid Mechanics þ Ch. 3 Transverse Stability þ Ch. 4 Initial Transverse Stability þ Ch. 5 Free Surface Effect þ Ch. 6 Inclining Test þ Ch. 7 Longitudinal Stability þ Ch. 8 Curves of Stability and Stability Criteria þ Ch. 9 Numerical Integration Method in Naval Architecture þ Ch. 10 Hydrostatic Values þ Ch. 11 Introduction to Damage Stability þ Ch. 12 Deterministic Damage Stability þ Ch. 13 Probabilistic Damage Stability (Subdivision and Damage Stability, SDS)

2 Planning Procedure of Naval Architecture and Ocean Engineering, Fall 2013, Myung-Il Roh Ch. 6 Inclining Test

3 Planning Procedure of Naval Architecture and Ocean Engineering, Fall 2013, Myung-Il Roh The Problem of Finding an Accurate Vertical Center of Mass(KG)

The problem of finding an accurate KG for a ship is a serious one for the ship’s designer.

F G How can you get the value of the KG?

G ü Any difference in the weight of structural parts, equipment, or welds in different ship will produce a different KG.

K

K: Keel G: Center of mass

There is an accurate method of finding KG for any particular ship and that is the inclining test.

4 Planning Procedure of Naval Architecture and Ocean Engineering, Fall 2013, Myung-Il Roh t r=F B × GZ Required Values to Find the KG (1/3) GZ» GM ×sinf (at small angle f ) GM= KB +BM - KGKG The purpose of the inclining test is to determine the position of the center of mass of the ship in an accurately known condition. z z¢ t e

Required values to find the KG M f d • Draft FG

Z • Total weight(FG) G 1 y • Hydrostatic values(KB, BM) • Weight(w) y¢ • Distance(d) B 1 • Angle of inclination(f )*

FB

G: Center of total mass B: Center of buoyancy at initial position * Angle of inclination should be within the range of inclinations t r G1: New position of center of total mass after the weight has been moved where the metacenter(M) is not changed(about 7°~10°). B1: New position of center of buoyancy after the ship has been inclined Z: The intersection of vertical line through the center of buoyancy with the transversally parallel line to a waterline through center of total mass M: The intersection of a vertical line through the center of buoyancy at previous position(B) with a vertical line through the present position of the center of buoyancy(B ) after the τ : Heeling moment 1 e 5 Planningship Procedurehas been ofinclined Naval Architecturetransversally and through Ocean a Engineering, small angle Fall 2013, Myung-Il Roh τr : Righting moment t r=F B × GZ Required Values to Find the KG (2/3) GZ» GM ×sinf (at small angle f ) GM= KB +BM - KGKG

z Shift of center of total mass ¢ z w× d t e GG1 = FG M Heeling moment produced by total weight f d th=F G × GG1 cos f FG Z Righting moment produced by buoyant force G 1 y tr=F B × GZ » F B × GM sin f Static equilibrium of moment y¢ FGB× GG1 cosf = F × GM sin f B1 (QStatic equilibrium: FGB= F )

FB GG w× d \GM =1 = tanfFG × tan f

t r Inclining test formula

6 Planning Procedure of Naval Architecture and Ocean Engineering, Fall 2013, Myung-Il Roh t r=F B × GZ Required Values to Find the KG (3/3) GZ» GM ×sinf (at small angle f ) GM= KB +BM - KGKG Inclining experiment formula z w× d z¢ GM = t e FG × tanf KG M f d KG= KB + BM - GM FG Z w× d G 1 y =KB + BM - Known Known FG × tanf

Known y¢

B1 The angle of inclination can be measured when we perform the inclining test. FB

t r

7 Planning Procedure of Naval Architecture and Ocean Engineering, Fall 2013, Myung-Il Roh t r=F B × GZ Derivation of Inclining Test Formula (1/5) GZ» GM ×sinf (at small angle f ) GM= KB +BM - KGKG Suppose that the center of B d yB d zB mass is located at zKn . Then the zn B z¢ KN represents the righting arm. 1 M f dy cos f dz sin f KBsinf B B f KG, N NZ, w 0

y y

d y B B B y¢ B1 FG F d zB G F B FB FB KG, N NZ, KG, 0

GZ = KN

=KNNN0 + 0

= KB ×sinf +d yBB ×cosf + d z × sin f 8 Planning Procedure of Naval Architecture and Ocean Engineering, Fall 2013, Myung-Il Roh These terms have positive effect to the restoring moment arm. t r=F B × GZ Derivation of Inclining Test Formula (2/5) GZ» GM ×sinf (at small angle f ) GM= KB +BM - KGKG

KG= d zG z z z¢ z¢ M M

f f

FG

G Z y y

d zG d y f d y B B B B y¢ y¢ F B1 B1 G F d zB G d zB FB FB FGB ' FB NZ, 현재 이 이미지를 표시할 수 없습니다. KG, N0 K N N0 GZ = KN GZ =KNKG - ' =KNNN + 0 0 = KB ×sinf +d yBB ×cosf + d z × sin f = KB ×sinf +d y ×cosf + d z × sin f BB -dzG ×sin f

This term has negative effect to the restoring moment arm. 9 Planning Procedure of Naval Architecture and Ocean Engineering, Fall 2013, Myung-Il Roh t r=F B × GZ Derivation of Inclining Test Formula (3/5) GZ» GM ×sinf (at small angle f ) GM= KB +BM - KGKG KG= d z G d y G z G z GG= d y z¢ Z n z¢ 1 G M M f G1 Z1 dyG cos f f f d FG G ' G1 ' N FG FG G Z G Z d y G Z1 y G 1 y

d z G f d y d y B B B B y¢ y¢ B1 B1 FG d zB d zB dyG cos f FGB ' FB FGB ' G1 ' FB K dz sin f N K dz sin f N NG0 NG 0

GZ =KNKG - ' GZ1 1 =KNKGGG -''' - 1

= KB ×sinf +d yBB ×cosf + d z × sin f = KB ×sinf +d yBB ×cosf + d z × sin f

-dzG ×sin f -dzG ×sin f -dyG ×cos f

This term has negative effect to the restoring moment arm. 10 Planning Procedure of Naval Architecture and Ocean Engineering, Fall 2013, Myung-Il Roh t r=F B × GZ Derivation of Inclining Test Formula (4/5) GZ» GM ×sinf (at small angle f ) GM= KB +BM - KGKG

ü Rotational transformation matrix (b to n frame) KG= d zG z zn n n b GG1 = d yG zb zn ¢ rPz=b R b r P z M M écosf- sin f ù n ê ú = Rb ()f ësinf cos f û yb f f f yn d FG FG ü Rotational transformation matrix (b to n frame) FG z n z b G Z G Z Z1 cos(---f ) sin( f ) d y G1 é ù n G y ê ú =Rb () -f yn ësin(--f ) cos( f ) û d z y G f -f n d y B B y y¢ b yb B ü In n-frame, because the forces are acting on 1 d zB vertical direction, moment arm is yn-component. dyG cos f G ' FGB ' G1 ' FB n b 현재 이 이미지를 표시할 수 없습니다. édyBB ùKécosf sinN f ù é d y ù K N N0 ên ú= ê ú ê b ú ëdzBB ûë-sinf cosf û ë d z û GZ1 1 =KNKGGG -''' - 1 n b b dyBBB= d ycos f + d z sin f = KB ×sinf +d yBB ×cosf + d z × sin f n b b -dz ×sin f -dy ×cos f nd y dyGGG= d ycos f + d z sin f G G B n 11 Planning Procedure of Naval Architecture and Ocean Engineering, Fall 2013, Myung-Il Roh - d yG w× d t r=F B × GZ Derivation of GG1 = FG GZ» GM ×sinf (at small angle f ) KG= d z Inclining Test Formula (5/5) G GM= KB +BM - KGKG GG1 = d yG z Calculation of moment using “GZ” z¢ GZ =KNKGGG -''' - t e 1 1 1

=KB ×sinf + d yBB × cos f + d z × sin f M -dz ×sin f -dy ×cos f f d G G = KN -KG sinf -GG cosf F 1 G = GZ Z G 1 y = GM sinf : Assume that f <<1

=GMsinf - GG1 cos f Moment produced by total weight & buoyant force y¢ d MGZ= D × B1 1 1

= D ×(GM sinf - GG1 cos f ) FB Static equilibrium of moment

GMsinf= GG1 cos f

t r GG w× d \GM =1 = : Inclining test formula tanfFG × tan f 12 Planning Procedure of Naval Architecture and Ocean Engineering, Fall 2013, Myung-Il Roh Precautions to be Taken During the Inclining Test

Certain precautions must be taken to ensure that accuracy is obtained. (1) The ship must be floating upright and freely without restraint from ropes. (2) There should be no wind on the beam. (3) All loose weights should be fixed. (4) All cross-connections between tanks should be closed. (5) Tanks should be empty or pressed full. If neither of these conditions is possible, the level of liquid in the tank should be such that the free surface effect is readily calculable and will remain sensibly constant through the experiment.

(6) The number of men on board should be kept to a minimum and they should be on the center line. (7) Any mobile equipment used to move the weights across the deck must return to a known positions for each set of readings.

13 Planning Procedure of Naval Architecture and Ocean Engineering, Fall 2013, Myung-Il Roh w× d Inclining Method of Measuring GM = test F × tanf the Angle of Inclination (1/2) G formula

How can you measure the angle of inclination when you perform the inclining test? z z¢

f y Deflection L

y¢ Mark ! Mark ! Deflection

Deflection L: Length of plumb line \tanf = Plumb: The line which is exactly vertical or perpendicular to a level horizontal line L Batten: Long and thin strip of wood

14 Planning Procedure of Naval Architecture and Ocean Engineering, Fall 2013, Myung-Il Roh Method of Measuring the Angle of Inclination (2/2)

* Reference: http://www.mchl.fr 15 Planning Procedure of Naval Architecture and Ocean Engineering, Fall 2013, Myung-Il Roh w× d Inclining GM = test F × tanf Example G formula KG= KB + BM - GM A ship is inclined by moving a weight of 40 tons a distance 8 m from the center line. A 12 m pendulum shows a deflection of 0.3 m. Displacement of the ship is 3,700 tons. If the KB is 5 m and BM is 14 m, what is the KG ? z z¢ Solution) 0.3 tanf = = 0.025 M d= 8 m 12 w× d 40 × 8 GM = = = 3.46 Z FG ×tanf 3700 × 0.025 G f 1 y 12m KG= KB + BM - GM =5 + 14 - 3.46 y¢ =14.54

B1 0.3m \KG =14.54[ m ]

16 Planning Procedure of Naval Architecture and Ocean Engineering, Fall 2013, Myung-Il Roh w× d Inclining GM = test F × tanf List Problems G formula

The inclining test formula can be used in list problem as follows:

(1) To find the angle of heel f, a ship will take by moving a weight a transverse distance d. w× d tanf = GM× FG 2) To find the weight w necessary to remove or produce a heel by moving it a transverse distance d. GM×tanf × F w = G d 3) To find the distance d necessary to move a weight in order to remove or produce a heel. GM×tanf × F d = G w

17 Planning Procedure of Naval Architecture and Ocean Engineering, Fall 2013, Myung-Il Roh