Directional Stability And Control
It was seen in an earlier chapter that when a ship at rest in still water is disturbed in the horizontal plane there are no hydrostatic forces to return it to its original position or to increase the movement. The ship is in neutral equilibrium. When a moving ship is disturbed in yaw it is acted upon by hydrodynamic forces which may be stabilizing or destabilizing. If stabilizing, the ship will take up a new steady line of advance but unless some corrective action is applied, by using the rudder for example, this will not be the original line of advance. The vessel is said to be directionally stable in these conditions but clearly this stability differs from that discussed in considering inclinations from the vertical. A ship is said to be directionally stable if, after being disturbed in yaw, it takes up a new straight line path.
An arrow is an example of a directionally very stable body. If gravity is ignored the flight of an arrow is a straight line. If it is disturbed, say by a gust of wind, causing it to take up an angle of attack relative to its line of motion, the aerodynamic forces on its tail feathers will be much greater than those on the shank. The disturbing force will push the arrow sideways and the moment from the force on the tail will reduce the angle of attack. The arrow will oscillate a litde and then settie on a new straight line path. The arrow, like a weathercock, has a high degree of directional stability.
For a ship form it is not clear from the lines whether it will be stable or not. By analogy with the arrow, good stability requires that the resultant hydrodynamic moment following a disturbance should tend to reduce yaw. The disturbing force is said to act at the hull's centre of lateral resistance. For stability this must be aft of the centre of gravity and it is to be expected that a cut away bow, a large skeg aft and trim by the stern would all tend to improve stability. That is about as much as one can deduce from the general shape at this stage. A degree of directional stability is desirable otherwise excessive rudder movements will be needed to maintain a straight course. Too much stability makes a ship difficult to turn.
Ignoring any longitudinal components, a disturbing force on a ship will lead to a small sideways velocity, v, an angular velocity in yaw, r, and linear and angular accelerations. In addition, in the general case, there will be forces and moments due to the use of the rudder. For small deviations second order terms in the equations of motion can be ignored and the equations become:
(/ Nr)r = Nvv+Nrr+NdRdft in these equations m is the mass of the ship, Y and N are the lateral force and yawing moment, <5R 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 nondimensional, the nondimensional terms being denoted by a prime, giving:
Cm'  rjv' = ry + (rr  mv + *V'r (/A/V)/ = A/V + NV + Ar6vd'R As an example:
1 BY
w \pV2L2 BV
The coefficients Y v, Nv 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:
must be greater than
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 I 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 force. If the applied force is F then the resulting sideways velocity is v = F/Yy. There will be a short period of imbalance before the ship settles down to its new steady state.
If the sideways force is applied aft of the neutral point and to starboard the ship will turn to port. If it is applied forward of the neutral point the ship turns in the direction of the force. The greater the distance the force is from the neutral point the greater the turning moment on the ship. Thus rudders placed aft are more effective than rudders at the bow by a factor of about five for typical hull forms. Aft they can benefit from the propeller race aft as well and are less vulnerable in a collision.
TURNING A SHIP
From simple mechanics it will be appreciated that to cause a ship to move in a circle requires a force to act on it, directed towards the centre of the circle. That force is not provided by the rudder. The rudder exerts a moment on the ship which produces an angle of attack between the ship's heading and its direction of advance. This angle of attack causes relatively large forces to act on the hull and it is the component of these directed towards the centre of the circle that turns the ship. The fore and aft components will slow the ship down which is a noticeable feature of a ship's behaviour in turning.
MEASURES OF MANOEUVRABILITY
These are not easily quantified although there has been much discussion on the matter. Large ocean going ships spend most of their transit time in the open seas, steering a steady course. They can use tugs to assist with manoeuvring in confined waters so the emphasis will probably be on good directional stability. Poor inherent directional stability can be compensated for by fitting an auto pilot but the rudder movements would be excessive and the steering gear would need more maintenance. For ships such as short haul ferries the designer would aim for good rudder response to help the ships avoid collision and to assist berthing and unberthing.
If possible the parameters used to define manoeuvrability should be direcdy related to the performance the master desires. This is not easy and use is made of a number of standard manoeuvres which can be carried out full scale and during model experiments. Other movements can be created in a model for measuring the stability derivatives, that cannot be direcdy simulated at full scale. The measures commonly studied are now described.
The turning circle
The motion of a ship turning in a circle is shown in Figure 10.1.
As the rudder is put over there is a force which pushes the ship sideways in the opposite direction to which it wishes to turn. As the hydrodynamic forces build up on the hull the ship slows down and
starts to turn in a steadily tightening circle until a steady state speed and radius of turn is reached. A number of parameters are used to define the turning performance. They are:
(1) the drift angle, which at any point is the angle between the ship's head and its direction of motion. This varies along the length, increasing the further aft it is measured. Unless otherwise specified the drift angle at the ship's centre of gravity is to be understood.
(2) the advance, which is the distance travelled by the ship's centre of gravity, in the original direction of motion, from the instant the rudder is put over. Usually the advance quoted is that for a 90° change of heading although this is not the maximum value.
(3) the transfer which is the lateral displacement of the ship's centre of gravity from the original path. Usually transfer is quoted for 90° change of heading.
(4) the tactical diameter which is the value of the transfer for 180° change of heading although this is not the maximum transfer. It is usual to quote a tactical diameter to length ratio, TD/L. Modern frigates at high speed and full rudder turn with a TD/L of about 3. For smaller turning circles such as may be required of a mine countermeasures vessel lateral thrust units or azimuthing propellers would be used. A value of 4.5 would be regarded as good for most merchant ships but a value greater than 7 as very poor.
(5) the diameter of the steady turning circle. The steady state is typically reached at some point between 90° and 180° change of heading.
(6) the steady speed on turn. Due to the fore and aft component of the hydrodynamic forces the ship slows down during the turn. Unless engine power is increased it may be only 60 per cent of die approach speed. The steady speed is reached as the diameter steadies. If a ship does need to reverse direction, as might be the case of a frigate hunting a submarine, the time to turn through 180° is likely to be more important than a really small diameter of turn. Because of the loss of speed on turn such ships would choose a lesser rudder angle to get round quickly and to avoid the need to accelerate so much after the turn.
(7) the turning rate. The quickest turn might not be the tightest. A frigate would turn at about 3° per second. Half this rate would be good for merchant ships and values of 0.51 would be more typical.
(8) the pivoting point. This is the foot of the perpendicular from the centre of the turning circle to the middle line of the ship, extended if necessary. This is the point at which the drift angle will be zero and it is typically about  of the length from the bow.
(9) the angle of heel during the turn. A ship typically heels in to the turn as the rudder is initially applied. On the steady turn it heels outwards, the heeling moment being due to the couple produced by the athwartships components of the net rudder and hull hydrodynamic forces and the acceleration force acting at the centre of gravity which is caused by the turning of the ship. It is countered by the ship's stability righting moment.
If the steady radius of turn is R, Figure 10.2, and the steady heel is <p and the transverse components of the forces on the hull
Figure 10.2 Ship heeling in turn and rudder are Fh and Fn acting at KH and KR above the keel then:
and the heeling moment is: AV2
For most ships (KH  KR) will be small and the heeling moment becomes (Fh  Fr) GH. This leads to an angle of heel such that:
AV2 GH V2
AGM sin <p = (Fh  Fr)GH = GH, giving sin <p =X —
Rg GM Rg
This is only an approximation to the angle as it is difficult to estimate the centre of lateral resistance for a heeled hull. In some high speed turns the heel can be quite pronounced. It is important in passenger carrying ships and may influence the choice of metacentric height.
The zigzag manoeuvre
A ship does not often turn through large angles and seldom through even a half circle. Thus the turning circle is not realistic in terms of movements of a ship in service. It is also difficult to measure the initial reaction to the rudder accurately in this manoeuvre. On the other hand a ship does often need to turn through angles of 10° to 30". It is
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