ANGLE OF HEEL-TURNING

ANGLE OF HEEL WHEN TURNING Consider a ship turning turning to starboard . The

sequence of events is as follows:

1.

Rudder put over to starboard.

2.

The

athwartshi athwartships ps component component of thrust thrust (F) acts on the face of the rudder at P, P being the cen centr  tr e of pr essur e which coincides with the geometric centre of the rudder face.

FORWARD MOTION M OTION SHIP¶S FORWARD Thrust acting normal to rudder face at P P Athwartships component of thrust

F

Angle of heel when turning (MAR)

1

An equal and opposite reaction (F1) resists the athwartships motion at the cen centr  tr e of lat eral  r esistance sistance (CLR).

3.

(The CLR is at the centroid of the ship¶s longitudinal area below the the wat erli ne ne.)

OUTW OUTWARD

F

INWARD

CLR 

Q

1

P

4.

F

An inward heeling couple is set up for fo r which the heeling moment is: F v PQ F

Q

1

P

F

Angle of heel when turning (MAR)

2

5.

When

the ship achieves a st eady rat e of tur n the inward heel is overcome by the effect of  cen centrifugal  trifugal  force acting outwards throug through h the ship¶ ship¶ss centre centre of  gravity (G). Centrifugal Centrifugal Force Force = WV2 tonnes gR 

where:

W is

ship displacement; V is ship speed in metres per second; second ; g = 9.81 m/sec2; R is the radius of the turning circle.

The

centrifug centri fugal al for ce ce is opposed by an equal equal an and  opposit e centrip centripeetal for ce ce which acts through the CLR . The

CLR is assumed to be at the same height above the keel as the centre of buoyancy (B). Consider the diagram diagram.

Angle of heel when turning (MAR)

3

OUTW OUTWARD

INWARD Centrifugal force WV2 gR 

G Centripetal force WV2 gR 

CLR 

B

The

orig origin inal al inwa inward rd heeli heeling ng mome moment nt is over overcom comee by this this outward heeling couple which develops in the steady turn state.

In the turn the ship will settle at an angl e of st eady hee heel  l  when the outw outwar ard d heel heelin ing g mome moment nt balan balance cess the the norm normal al righ rightin ting g moment (RM = GZ v Displacement). At small angles of heel:

GZ = GM  v S i  ine   eU n

Angle of heel when turning (MAR)

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OUTW OUTWARD

INWARD M UHEEL

Z

WV2 gR 

G

d WV2 gR 

B B1

CLR 

B and B1 are assumed to be at the same depth. Z

G

UHEE L

Cos U = ADJ HYP

=

d BG

d Therefore:

d = BG CosU

(At small angles of heel) B1

B

Angle of heel when turning (MAR)

5

At the small an angl e of hee eel  l  shown: RIGHTING MOMENT = HEELING MOMENT W v GZ = WV2 v d gR 

Therefore:

If:

d = BG Cos U

then:

W v GM Sin U = WV2 v BG Cos U gR 

Transposing

and:

GZ = GM Sin U

gives:

gR v W v GM Sin U = WV2 v BG Cos U

Divide both sides by Cos U: gR v W v GM Sin U = WV2 v BG Cos U

Gives: gR v W v GM v Tan U = WV2 v BG Thus:

Finally:

Tan U =

WV2 v BG gR v W v GM

Tan U =

V2 v BG g v R v GM

Angle of heel when turning (MAR)

6

Note

In practice the outward angle of heel will be slightly l ess than that given by the formula because of the small inward heeling moment set by the athwartships component of thrust on the rudder .

Howeve However, r, if the the rudd er is r etur ned  ned  qui ck  ck ly l y to the the amidships position position, the the outwa outward rd angl e of hee eell due due to tur ni ng will  i nstan stantane taneously ously i ncr  ncr ease. ase. I f the the rudd er is sudd enly enly r evers everseed  i .e. .e. put hard-ahard-a-port port on a star starbo boar ard d tur ni ng  cir cl e, an ev even en mor e seriou riouss outw outwar ard d angl e of hee eell woul ould aris rise ( albe albeit  t emporarily) mporarily) whi ch could  cause ause excessi ve ve hee eeli  li ng and i n tr eme situation situations cargo shift or even or  even capsizi  ex tr  capsizi ng!

Angle of heel when turning (MAR)

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