Page 269 - Introduction to Naval Architecture

254 MANOEUVRING

In these equations m is the mass of the ship, F and N are the lateral
force and yawing moment, (5 R is the rudder angle and subscripts denote
differentiation with respect to the quantity in the subscript. Other
terms have their usual meaning.
These equations look rather complicated but they are only equating
the rate of change of momentum to the applied force. The total force
and moment are then expressed as the sum of the components due to
each variable, that is the force due to lateral velocity is the product of
the velocity and the rate of change of force with velocity, and so on.
The equations can be made non-dimensional, the non-dimensional
terms being denoted by a prime, giving:

The coefficients Y w N v etc. are called the stability derivatives,
Since the directional stability of a ship relates to its motion with no
corrective action the equations defining it are as above with the rudder
terms removed. It can then be shown that the condition for positive
stability, or stability criterion, is:

This is the same as saying that the centre of pressure in pure yaw must
be ahead of that for pure sway. The centre of pressure for pure sway is
often called the neutral point. It is kL forward of the centre of gravity
where:

The value of k is typically \ so that the neutral point is about | of the
length aft of the bow. With a lateral force applied at the neutral point
the ship continues on its heading but with a steady sideways velocity.
That is it is moving at a small angle of attack such that the
hydrodynamic forces on the hull balance the applied moment and