Dynamical stability (MAR) 1

DYNAMICAL STABILITY

The ‘dynamical stability’ of a ship at any particular angle of heel is defined as the work done by wind/waves or any other external forces to heel the ship to that angle.

G

B

B B1

G Z

bb1

A ship is initially upright with Bf and Wf acting through B and G respectively.

BG = vertical separation of points of application of Bf and Wf.

When the ship is heeled to some angle , B moves to B1

parallel to bb1.

BG < B1Z

i.e. the vertical separation between B and G has now increased. Work has been done by the heeling force to drive these points of application of the forces of Bf and Wf apart.

DYNAMICAL STABILITY = WORK DONE, and;WORK DONE (by wind/waves etc.) = W (B1Z - BG) t-m

Bf

Wf

Dynamical stability (MAR) 2

b

hh1

b1

G Z

B B1

P R

Considering the formula:WORK DONE (by wind/waves etc.) = W (B1Z - BG) t-m

(B1Z - BG) represents the increase in vertical separation of B and G as a result of the ship being heeled. Moseley’s formula considers the transferred wedge of buoyancy and dynamical stability could be calculated thus:

Dynamical Stability = W [v(bh + b1h1)] + BG Cos - BG V

Dynamical stability (MAR) 3

i.e. Dynamical Stability = Sum of all righting moments 0 to

or: Dynamical Stability = W (Area 0 to )

It should be noted that the M.S. (Load Line) Regulations 1998 specify minimum areas under the curve, up to specified angles of heel, to ensure that minimum dynamical stability requirements are met; for these purposes area is expressed in metre-radians. Hence, dynamical stability is expressed in:

tonnes-metre-radians (since radians are simply a ratio of to 57.3, the radians are often ignored, so dynamical stability units may be expressed as tonnes-metres.)

Since a curve of statical stability (GZ curve) should be available, it is more practical for dynamical stability to be calculated by consideration of the area under the curve up to the angle of heel concerned.

GZ

(m)

Heel ()

0

Dynamical stability (MAR) 4

Important points to note1. Transverse statical stability is given by:

Righting moment (t-m) = W GZwhere GZ is a measure of how far G and B are horizontally separated.

It is a measure of the ‘work available’ to right the ship at a particular angle of heel (assuming ‘still water’ conditions - no external forces.)

2. Dynamical stability is given by:Dynamical stability (t-m) = W (B1Z - BG)

where (B1Z - BG) is a measure of the increase in vertical separation of B and G at the angle of heel concerned.

3. Consider the curve of statical stability shown.

At both angles of heel 1 and 2 statical stability is the same.But, dynamical stability at 2 is greater than at 1 (consider the areas under the curve up to the angles of heel concerned).

1 2

GZ

Dynamical stability (MAR) 5

4. Compare the curves of statical and dynamical stability.

GZ(m)

DS(t-m)

MAXIMUM GZ

Angle of vanishing stability

Inflexion of DS curve

The curve of dynamical stability has a point of inflexion at the angle of heel when GZ is maximum.

Since dynamical stability is a measure of the work to be done by external forces to heel the ship over to a particular angle of heel, dynamical stability continues to increase with heel as long as the ship continues to resist i.e. so long as the ship has positive righting levers.