A Corrected Version on Positioning of Pivot Point

by Capt. Santosha K Naya - published -

A Corrected Version on Positioning of Pivot Point

Text and pictures by Capt. Santosha K. Nayak, Marine Pilot, Krishnapatnam Port  from his book "Theory and Practices of Marine Pilotage"

Understanding the fundamentals of the pivot point is highly required for understanding the alteration of the courses. Pivot point is an imaginary point on the vessel which turns on a circular path on the perimeter of vessel’s turning circle when the vessel makes a turn. The knowledge about the position of the pivot point in a manoeuvring situation provides the ship handler with the information on the geometry of motion of the ship. When sway and yaw occur simultaneously, a ship handler can only perceive the combined effect of drift and turn, which gives him a false impression that only a rotational motion happened about a certain point on the ship’s centreline. This apparent centre is called the Pivot Point of the ship. This is a simplified perception of two motions down to one motion.

It is at the same point as the longitudinal centre of gravity of the vessel when vessel is stopped and making no movement. It starts moving towards the bow when the vessel increases her speed. The distance of the PP from the longitudinal COG varies with the speed of the vessel.

We can understand the existence of pivot point mathematically as an imaginary point. Among all the points in the ship in planar motion, there is only one point on the centreline at which the sway and yaw completely cancel each other, thus making this point seem to be stationary. All other points appear to be turning about this point. This point is the Pivot Point.

Sway means the linear transverse (port to starboard) motion. This motion is generated directly either by the water and wind or currents exerting forces against the hull or by the ship’s own propulsion or indirectly by the inertia of the ship while turning. Yaw is the rotational motion of the vessel about the vertical axis. If the sway speed and yaw speed are known, the position of the pivot point can be obtained as the distance from the centre of mass (GP) using equation:

V + (GP x ROT) = 0

Where,   V(m/s) = Sway Speed;

                       G = Centre of Gravity;

                        P = Pivot Point;

               GP(m) = distance between P and G;

       ROT (rad/s) = Yaw Speed.

There are some traditional views held by ship handlers of the Pivot Point and also mostly found in the literature of books on ship manoeuvring. These views about pivot point are:

  • It moves towards the bow or stern depending on the direction of the longitudinal motion of the vessel
  • When making sternway, the pivot point moves aft and establishes itself approximately 1/4L from the stern
  • It is the centre of rotation of the vessel
  • It has instantaneous movement from the COG to its position

There have been many experiments carried out to understand the existence of Pivot Point and how it moves with the motion of the vessel. There are some new findings related to Pivot Point and the some of the traditionally held views about PP are incorrect. All the above mentioned views are incorrect. The corrected facts about PP are:

  • It is independent of direction of motion,
  • It is only an imaginary point
  • It moves gradually towards or away from the COG depending on the application of forces on the vessel.

However, ship handling professionals, particularly the seasoned practitioners, find it very difficult to accept these findings.


Verification Experiment to establish the location of Pivot Point

A verification experiment was done in for a panamax vessel. The ship’s turning force was provided by setting the engine half astern. The propeller is right handed with fixed pitch. For the purpose of analysis, the whole experiment was divided into 8 time intervals. In each interval, the result was analysed calculating the position of the pivot point as the average in the interval. The positions are given as percentage lengths between the bow and the pivot point, to the length of the ship. The experiment shows plainly that the pivot point was at around 17% of the ship length from the bow. Near the end of the experiment, it is obvious that the pier is interfering with the water flow being created by the propeller.

This experiment conclusively proves that the traditional teachings and leanings about the pivot point for centuries are incorrect

Position of Pivot Point as deducted from the experiment                           

The exact position of the pivot point may be deducted from the following formulae. GP is the distance of the Pivot Point (P) from the longitudinal Centre of Gravity (G).

GP = - (L2 +B2)/ 12GFr, is a simple equation for a box shaped vessel

Where, GP – Distance between the GOG and PP,

              Fr – Position of Resultant force on the vessel,

               L – Length and B- Breadth of vessel

The interpretations of above findings are essential knowledge for the ship handlers. Above equation correlates some practical points which can be used by ship handlers during manoeuvring of ships are as follows:

  • The minus (‐) sign indicates that the pivot point appears on the other side of G from Fr.
  • A bigger GFr yields a smaller GP, which means that an external force farther away from G causes the pivot point to be closer to G.
  • A bigger block coefficient will cause the pivot point to be closer to the bow.
  • The direction of the longitudinal motion is irrelevant with the pivot point location.
  • If the propeller and rudder combination at the stern is used as the only propulsion system, the pivot point will always appear near the bow


A-Vessel moving ahead,
B-Vessel moving astern, 
C-Vessel experiencing external force 
D- Vessel being pushed by one tug (In case 2 tugs, the tug applying more force decides)

A-Vessel moving ahead,
B-Vessel moving astern, 
C-Vessel experiencing external force 
D- Vessel being pushed by one tug (In case 2 tugs, the tug applying more force decides)

A-Vessel moving ahead,
B-Vessel moving astern, 
C-Vessel experiencing external force 
D- Vessel being pushed by one tug (In case 2 tugs, the tug applying more force decides)

A-Vessel moving ahead,
B-Vessel moving astern, 
C-Vessel experiencing external force 
D- Vessel being pushed by one tug (In case 2 tugs, the tug applying more force decides)

The pivot point settles down at one point when the motion of a vessel becomes steady. All above observations are confirmed by pilots carrying out manoeuvring regularly on day to day basis. The understanding of above facts can be understood very well when vessel is making a astern movement at a certain speed aided by the tugs while keeping in a restricted and narrow channel.
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