Capt. Paul Butusina, Master of DSV Mamola Champion
1. Definition
Pivot Point (PP) is the point around a ship turns. It is of special interest for who is involved in handling of ships. As the vessel turns about the PP, it needs more space to turn if the PP is not located near midship.
For a ship of length L, if the PP is located at midship, the swept path required is about Pi(L²/4).
It takes four times as large swept path if the PP is located at the bow[4].
Few distinctions have to be done in terms which are using to operate with.
Obviously, there is a PP when a vessel is in a rotation movement only. Theoretically it is used, on a large scale, to talk about PP as a virtual point existing all the time inside of the shape of the vessel. The reality is not quite so.
We have from the beginning to make a distinction between PP and the Center of Turning circle.
PP is connected with ship’s rotation, it has nothing to do with the centers of circles or parts of circles, of the ship’s track.
To clarify few things regarding PP we have to introduce, between well known factors which react during ship handling, the hydrodynamic effects, pointing the Center of Water Resistance and Pressures around the vessel.
During straight fore ward movement, water-resistance force is applied right on the stem, somewhere at mid draft, depending of bow shape (classic or bulbous).In the same time it is recorded high pressure in front and around the bow.
Same judgment works for straight astern movement simplifying and do not considering influence of the propeller and rudder. Shape of underwater hull is very important of high pressure repartition.
As soon as, during movement ahead or astern, due to one of controlled or uncontrolled horizontal forces acting on the vessel, ship starts to turn and she will expose to the water flow another section, larger than going straight along fore-aft axis X-X’, the pick of water-resistance and pressure will shift from axe X-X’ to the geometrical center of underwater hull section area perpendicular on the new direction of the movement. The maximum of the ship’s section which can be exposed to the water flow is her longitudinal section for a lateral movement (drift) on transversal axis Y-Y’. In this case center of water-resistance is called Center of Lateral Resistance.
The direction of the water-resistance could be anywhere between axis X-X’ and Y-Y’.
Depending of the direction of the movement, the vessel’s speed, hull shape, trim and heel, the application point of the water-resistance force will be in different points along the vessel.
To analyze the influence of horizontal forces applied on the vessel (steering force, propeller force, lateral thrusters, tugs or pushers, wind and currents) and reducing phenomenon to classical mechanics, we have to report these forces to the water-resistance force or high pressure in the area where they act. The arm lever of these forces is distance between their supports and water-resistance force.
The action of a force or resultant of few forces acting on a vessel will generate two movements – a rotation and a sideway movement. A rotation movement has a center of rotation which it is used to call Pivot Point (PP).
The ship’s PP is the place from where fore and aft extremities of the vessel are turning with the same angular speed. This does not mean that PP is inside of ship’s shape in all situations.
Trying to be more accurate, PP is defined as a point Xp measured from the center of gravity of the ship that satisfies the relationship [4]:
ν + Xp • r = 0 (1)
Where: v - is sway speed at the center of gravity of the ship;
r - is the yaw rate.
PP is characterized by a zero drift angle. Fore ward of the PP, inflow appears to come from one side of the ship while aft of the PP, the inflow appears to come from the other side of the ship [8].
Specifically, due to combination of side motion and the rotation, causes the ship to appear to rotate about PP. An observer changing his position on a vessel in rotation, can find, by sight, a place from where the horizon seems to rotate with same speed – it is the PP of the ship at that moment.
It is possible to estimate the pivot point Xp from Eq.(1) if it can measure the yaw rate r and the sway speed v. It can be done either using a dual axes Doppler speed loch in conjunction with a gyro rate or a GPS receiver placed in Center of Gravity of the vessel and a compass (gyro/magnetic) either, independent of Center of Gravity, using two GPS receivers placed on ship’s extremities and measuring tangential speeds.
Keeping in mind that PP is determined by relation involving water-resistance respectively water-pressure, another method to find it is to measure the pressure around the vessel or a testing model and to find the picks (up and down) of pressures. This method still needs to be tested by experiments.
It follows, from Eq.(1) that:
Xp = -v/r (2)
Eq.(2) is ill defined when the yaw rate is zero, which corresponds to a straight line motion. When the vessel movies on a straight line ahead or astern or she is in a pure sway motion, it is reasonable to consider PP at infinity [4]. In other words saying, when the vessel moves along axe X-X’ or she drifts along axe Y-Y’, there
are not a PP and it is unfair to declare that PP is fore or aft function of direction of ship’s movement,
as it is used in present.
2. Water Resistance and Pivot Point of a vessel stopped
Considering a stopped ship, without movement through the water and rudder mid ship’s, we can find a point situated about at its mid length, from where if a tug will push with a force F (fig.1), fore and aft
extremities of the ship are moving with V1= V2. The force F is applied on the same support as water-resistance force WR. Its center of application is Center of Water Resistance, CWR. Arm lever of F and WR is
zero.
The ship will be translated from axis a-a’ to b-b’ (a-a’ II b-b’). In this case there is not rotation, nor PP, or it is situated at infinity.
If the force is applied closer to mid length but more to one side of the considered ship, let to say aft (fig.2), the arm lever of forces F and WR will be “d” and the ship will record a side movement and a rotation (V1>V2). The axis a-a’ will be intersected by b-b’ in PP. If the ship would be anchored, PP would be where the anchor was dropped. In this case PP is outside of ship’s shape.
If the tug will push with force F closer to aft (fig.3), the arm lever of F and WR , d1 > d (from fig.2) and rotation of the ship will be faster (V1>>V2) and PP will be located inside of ship’s shape close to application point of WR in ship fore part. During action of F, it appears a water flow (WF) around the bow which creates low pressure (LP) responsible for a slight fore ward movement of the vessel.
3. Water Resistance and Pivot Point of a moving vessel
If the ship will start to move ahead keeping her rudder mid ship’s (fig.4), due to drifting movement with speed D, vessel will move on the resultant of propulsion speed P and D, respectively with speed P1 on its direction. Water resistance force WR increases and it will shift fore ward. The result is a shift of PP fore ward in the direction of movement.
Arm lever of F and WR, d2 > d1 and in consequence V1 >> V2, it means the vessel will turn more quickly. Even with a kick ahead this increasing of rotation speed can be seen. This phenomenon is valid for astern movement if the tug acts fore ward. In such case, due to short distance between WF application point and PP, in practice, by ship handlers, it is used to consider PP as reference point for WR application point. In reality application point of WR depend of underwater shape of the vessel.
Fig.4.1a shows the measured sway speed v and 4.1b the yaw rate r for a Very Large Vessel for a 35 degrees turning maneuver[4]. PP computed with Eq.(2) is shown in fig. 4.1c. It is observed that during 35 degrees turning maneuver, the PP moves from midship to about 1/5 ship length aft bow.
Fig. 4.1
Once PP position is established during turning under rudder resistance force or engines forces, if another force is applied in PP, the effect of this latest force is only a sideways movement.
PP shifted fore ward, it results that most of the ship length will be situate aft of it and that part of the vessel will have a lateral movement. A large amount of water will be pushed on opposite side of force F,
pressure on that part will increase significantly and a water flow WF (fig.4a) is generated under ship’s keel and along the hull. On the side where is the tug, it appears a depression. These masses of water and their inertia can be responsible for an effect less understood and explained. This water flow will modifies also distribution of pressures along the ship’s hull affected by it.
To estimate effect of F it calculates its moment. Again, in practice ship handlers use to calculate this taking arm lever, d, between PP and position of the tug or thruster. Explanation is clear. PP can be appreciates by sight but WR application point can not.
Effects of changing the resistance through water and position of PP can be seen on a ship stopped with bow and stern thrusters engaged at equal power, thrusting in the same direction, let to say to starboard (fig.5). Rudder is set mid ship. Vessel is on even keel and no heel. Arm levers of thrusters are equals, d1 = d2 and consequently V1 = V2. Vessel will drift only.
As soon as the engine is engaged, even for a kick ahead (fig.6), the position of WR will shift in direction of movement and the ratio between arm levers of the thrusters will change, in our case d2 = 0 and d1 reaching its maximum length. V1 become bigger than V2 and vessel will turn to port, placing her on axis b-b’.
It is important not to overlook the fact that we are considering only the ship's headway or sternway through the water, not over the ground. If the vessel is stationary with respect to the shore, but is stemming in the current, the PP will be forward,since the vessel has headway with respect to the water. Likewise, if she is tied to the dock,with a current from astern, the pivot point will be aft the moment the lines are cast off.
Changing of arm lever of ship’s thrusters is very well known by ship handlers which use it to increase efficiency of these controllable forces. If bow thruster is weak it is enough a kick astern to increase its arm lever and consequently its efficiency (fig.6a).
Opposite effect can be obtained if it will try to turn the vessel using the thruster situated at same level as WR, in our case stern thruster (fig.6b). Its arm lever will be almost zero and the effect will be the drift of entire vessel only. In this case no PP.
4. Water Resistance ,Pivot Point and particular effects
4.1 Coming into wind
Another interesting effect against usual expectation of changing WR application point is coming into wind.
To be more evident, we will consider a vessel with accommodation aft and a right hand propeller, rudder mid ship’s, being in position along axis a-a’ and just start moving astern (fig.7). The wind blows from starboard quarter.
Expectation, due to propeller transversal thrust and wind is movement M1, a turning and eventually drifting to port. Due to shifting of WR application point aft, between wind force W and WR appear a couple which will bring ship’s stern in movement M2 on axis b-b’ in the wind, up to a certain angle when d will become zero.
4.2 “Donkey-like” effect
One of the most spectacular effect of application of an external force upon a vessel and getting an opposite result (donkey-like) is the movement of the vessel when a tug is acting on the support of water resistance force against it.
In fig.8 it was represented only the speed vectors of the vessel and the tug. If the tug will start to push in position M1 and vessel has an appreciable speed fore ward, it can not turn the vessel. She will keep initial direction as in position M2. As soon as the tug will stop pushing in position M3, the vessel will start to turn towards the tug. She will continue to turn in the same direction a certain time as is shown in position M4.
Due to reduced arm lever d of forces T and WR (fig.8a), tug and vessel will drift and no rotation will be recorded or even a slight rotation towards the tug against expected result – pos. M0. When the tug will stop and will leave the vessel in pos. M1, her center of gravity G (fig.8b) will continue by inertia its drifting with speed GI.Due to arm lever d between GI and WR, it appears a couple which will start to turn the vessel towards the tug. PP starts shifting from fore ward PP1to mid ship PP2 (fig.8a, pos.M2).
The masses of water moved during action of the tug have their own inertia and will continue to generate a water flow (WF) directed to port quarter. In the same time, fore of the vessel is passing relatively undisturbed water which will keep the bow, amplifying the rotation.
The effects of water flow and pressures around the vessel and how they influences ship’s manoeuvre is still a possible subject of research.
4.3 Sideways manoeuvre
An example of how ship handlers play with rotation and drift of the vessel is manoeuvre of sideways, in our example a berthing on port quarter at an oil rig (fig.9).
We consider a vessel with limited bow-thruster power. The thruster runs at constant speed. To move the vessel sideways to oil rig, rudders (Rd) will be set hard starboard, starboard engine astern (P1), port engine ahead (P2) and bow-thruster to port at maxim power (Tcst). Usually stern sideway speed is greater than bow sideways speed, M1>M2 (fig.9a) and the vessel will come with aft part closer to berthing place (fig.9b). In order to increase bow-thruster effect, it will increase P1 and it will stop P2 which it will reduce aft turning effect and it will give an astern movement for the vessel. Effect of astern speed will be a shift of center of
water resistance (CWR) and the pivot point in PP2. The arm lever of bow-thruster will increase, consequently its sideways effect and M2a become greater than M1a. The result will be astern movement for a while and a closer position of the bow to oil rig (fig.9c). In this moment, it will be reduced starboard engine power and port engine will be set ahead at a greater rate than starboard. Vessel will start, finally, to move fore ward and CWR will come again fore ward and the pivot point will take position PP3. Arm lever of bow-thruster will be reduced and M1b>M2b. Playing in above described manner, rudders can be kept in the same position.
For a stopped vessel, starting the engine ahead and setting rudder hard to one side, during turning, PP position will shift quickly, in evolution phase, from a location close to Center of Gravity and mid of the vessel to fore ward, keeping the position of 0.10-0.25 of length between perpendiculars (Lpp) from stem before stabilization state of turning. Time of PP setting is about half of turning stabilization time, respectively sideways speed and turning rate stabilization time. PP final position and time of stabilization depend of underwater shape. It is a particularity of each vessel or category of vessels.
Position of PP can be estimated by calculation even in design stage on basis of model tests but this is not the subject of this work.
Conclusions
Effect of controllable or uncontrollable forces upon a vessel is function of the arm lever between Water Resistance force and resultant of others forces.
If effect of the forces has a rotation motion, this rotation has a center, inside or outside of the ship’s shape, function of ratio between rotation and sideways motion, call Pivot Point.
Position of pivot point depends of position of Water Resistance application point which is function of direction of movement and ship’s shape.
There are particular positions reported to Water Resistance application point where if a force is applied the effect upon a vessel is special.
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