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.
Editor's note:
Opinion pieces reflect the personal opinion of individual authors. They do not allow any conclusions to be drawn about a prevailing opinion in the respective editorial department. Opinion pieces might be deliberately formulated in a pronounced or even explicit tone and may contain biased arguments. They might be intended to polarise and stimulate discussion. In this, they deliberately differ from the factual articles you typically find on this platform, written to present facts and opinions in as balanced a manner as possible.

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Louis Vest Houston Pilots, USA
on 30 November 2020, 19:33 UTC

Sean - My experience has been that the pivot point changes rapidly as the ship changes from headway to sternway or vice versa and the position is right away near the bow or stern. You can see this when docking starboard side to with a right hand prop and no tug/thruster or when turning a ship with a bow tug/thruster and the ship stops and begins coming astern during the turn. If you take your attention away from the bow as you get sternway you can be surprised by how much difference that big lever arm makes. Used to scare the poop out of myself until I learned to anticipate that.

Sean O’Toole United Kingdom
on 30 November 2020, 17:46 UTC

The photograph of the Panamax ships are actually manned models from Solent University Ship Handling Centre https://www.solent.ac.uk/facilities/ship-handling-centre .
Undoubtedly it can be seen the PP is forward when initially coming astern when the rotation is caused by transverse thrust. However the PP will start moving astern when the vessel starts moving astern, how far is the question. Should the only force being transverse thrust I would say the PP when moving astern is possibly just aft of amidships. Add a bow thrust or fwd tug to increase the rotation, the PP will move further astern, use an aft tug then it will push the PP more forward. The PP is very dynamic and always gradually changing and as practical ship handlers we are in reality balancing all forces and driving the resultant to have a good outcome.

Jim Wright Southwest Alaska Pilots Association, USA
on 21 November 2020, 18:17 UTC

Before tugs were available, Alaska pilots used this principle to undock vessels up to 100K DWT from berths where the mooring dolphins were inside the face of the berth. Often with little or no opposing current.
When all lines were reported in we would use an astern bell to get sternway on the vessel. As the midpoint of the vessel passed the knuckle on the berth, an ahead bell together with hard over rudder to begin rotating the stern toward the catwalk would be ordered. This order would also begin rotating the bow off the fenders while maintaining a foot or so clearance between the shell plating and the knuckle. From this point, judicious use of the engine and rudder kept the vessel rotating off and clear of the knuckle as forward speed was increased. This maneuver was effective with up to 30 knots of wind on the berth.
In later years it occurred to me that aft movement of the pivot point with sternway made this maneuver possible without laying back on the knuckle. Absence of stress was not guaranteed.

Wade Armstrong Hawaii Pilots' Association, USA
on 19 November 2020, 03:26 UTC

Concur completely with Capt Lou Vest. I have also attended manned model schools at Grenoble and two in the US, and experienced the same conclusive demonstrations. My 30+ years of handling vessels of all sizes supported the traditional view of how the pivot point will act.

I fully agree with one statement towards the end that "A bigger block coefficient will cause the pivot point to be closer to the bow."

Louis Vest Houston Pilots, USA
on 19 November 2020, 02:02 UTC

I attended the manned model ship handling schools in Grenoble twice, Poland once and Australia once. They all teach and demonstrate conclusively that when stopped the pivot point is at the center of lateral resistance. When the vessel begins to move forward the pivot point rapidly moves forward to a point about 1/5 to 1/4 from the bow. When coming astern the pivot point moves back to a point about the same distance from the stern. I spent 30 years docking ships of all sizes and used that information daily. After docking maneuvers I always graded myself by looking to see how the spirograph produced by my pilot software conformed to the docking plan I had mentally rehearsed before the maneuver. Those graphics confirmed over and over again that the pivot point acted as I had been taught.

Your article did not convince me that the pivot point acted any differently from that.


Opinion Scientific Fact: The ‘traditional’ understanding of the ship’s pivot point is wrong!

by Tim Cummins, Harbour Pilot, Portsmouth International Port - published

In fact, the pivot point that we “see” is a trick of the eye, it looks like the ship is rotating about this point but in fact it is elsewhere, a point that you cannot see.


Article A review of the ship’s pivot point: Science, Maths and Observation’ Where is the centre of a ship’s rotation?

by Tim Cummins, Harbour Pilot, Portsmouth International Port - published

This my attempt to do just that. This is a summary of all scientific papers and articles that I could find about the ship’s pivot point. I have added links to any videos associated with the publications and have also included any useful diagrams or equations that the authors published to help explain their research.


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