Parametric Rolling

In certain conditions a ship will roll heavily, if mariners are not aware of the phenomena of parametric rolling it will seem inexplicable. For example the seas can be from dead ahead and the ship will take quick, heavy rolls.

Parametric rolling is an unstable phenomenon, which can quickly generate large roll angles that are coupled with significant pitch motions. The rolling occurs in phase with pitch

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Parametric rolling from a container ship’s POV.

Well timed, Captain. I’m sitting my stability exam tomorrow. Its not intuitive, is it?

That model test video looks familiar…

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It wasn’t to me, it was a complete surprise.

I was hove to into a steep head sea and starting rolling so heavy the main engine tripped out on low lube oil pressure. It wasn’t till I got home and researched it did figure out what happened.

When the ship is supported only at the ends there’s not much buoyancy to hold it upright and over it goes. Then when it’s supported only at the mid-section it snaps upright. If that wave length is synced up it’ll roll like you’ve never seen, and you won’t know why.

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This is really interesting stuff. But I think the decreased stability actually occurs when the wave crest is amidships. When the forward and after sections of the ship are in successive wave crests with a trough amidships, the waterplane is, on average, wider than it is when in still water as a result of the flared section shape. This results in increased metacentric height and heeled righting moments compared to still water. When the crest is amidships the mean waterline width, therefore the metacentric height and righting moments, are generally less because of the narrowing waterline at the ends and the stability is diminished compared to its value in still water. Here is a link to a great journal article.


Recommended reading material on the subject of parametric rolling can be found here:

Parametric rolling is mentioned only since the fifties. More or less the same happened with the dangers of quartering seas. It is remarkable that it took such a long time before these important and sometimes crucial phenomena were taught in nautical schools

Hoppe Marine sells her MOCON, monitoring and control system, based on the U-Tank Anti-rolling system which makes use of the ship’s double bottom and was developed by Frahm already in 1910.

Hoppe stabilization system.

Furthermore there is the Arrow software tool for avoiding synchronous or parametric resonance.

The word resonance is the right expression to describe parametric rolling because that is what it is, just simple physics. In physics, resonance is a phenomenon in which a vibrating system or external force - the waves - drives another system - the ship - to oscillate with greater amplitude at specific frequencies. The waves are so to speak in total tune with the ship.

The most prominent feature of the frequency response of a resonant circuit is a sharp resonant peak in its amplitude characteristics which can, under certain circumstances, can take on catastrophic proportions as can be witnessed in the video below.

As was proven already long ago resonance can even bring a bridge down.

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That seems to be a description of synchronous rolling. Parametric rolling is different

This is from the Mutual Steamship paper.

The term describes a state of motion that results not from direct excitation by a time-varying external force or moment but from the periodic variation of certain parameters of the oscillating system.

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The extra element in the case of parametric resonance, as compared to synchronous rolling, is the pitch of the vessel. The pitch in the case of synchronous rolling is not absent but very small. Therefore parametric rolling will take place with head on, or near head on, seas which means operating in longitudinal waves.

It is explained rather well in the paper here under. I refer especially to figures 2.2 and 2.3.

In the example of the pendulum parametric resonance occurs when instead of actuating the pendulum by a moment around turning point A, now point A is periodically translated in vertical direction see figure 2.2.

The paper also mentions that decreasing the ship’s speed always helps to counteract the parametric rolling, but increasing it only sometimes, Well, I suppose that the natural instinct in such cases will be to decrease the speed, not increase it.

The extra element I think is the change in righting arm.

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Yes, you’re right.

On the mid-body less change in water plane area with draft compared to the ends.

The video shows by means of a simple pendulum action, which is excited in the vertical plane with a force that has twice the frequency, the effect of parametric resonance. A second example is added.

The pendulum stands for the rolling motion of the ship. The vertical movement is the pitch and heave motion of the ship.


So parametric rolling is, more or less, a simple transfer of energy from vertical to horizontal? The change in waterplane and righting arm are not involved?

Feeling my way here – appears to me that the change in center of buoyancy relative to center of mass is what’s pumping the system.

Yes, the moment created by the two forces, center of buoyancy up and center of mass down is called the “righting arm”, aka GZ.fig3-8

Waterplane and the GM are all part of the ship’s susceptibility to parametric rolling.

Longitudinal waves (head and following) cause the most change in stability and, therefore, create maximum parametric excitation. Parametric roll resonance develops when the frequency of stability change is nearly twice that of natural roll frequency or when the frequency of encounter is nearly twice that of natural roll frequency. The value of natural roll frequency mostly depends on GM value (transversal distribution of weight also may have an influence). Therefore, whether parametric roll resonance may occur in following or head seas depends mostly on current GM value. Wave length also has an influence because it is related to the wave frequency on which the frequency of encounter is dependent. While the physical basis of parametric roll resonance in following and head seas is essentially identical, head seas parametric roll is more likely coupled with, or at least influenced by, the heave and pitch motion of the ship, as these motions are typically more pronounced in head seas.

A ship that is sailing in longitudinal seas experiences a time-varying transverse stability characterized by increased stability when the ship has a wave trough amidships and decreased stability in the wave crest. If such a ship experiences a small arbitrary roll disturbance, then that disturbance can grow provided that the stability fluctuations occur at a frequency approximately twice the natural frequency of roll and provided that the roll damping is less than some threshold value. This rolling motion, under certain circumstances, can grow to quite large amplitudes and is referred to as “autoparametrically excited roll” or simply “parametric roll”.

The change in waterplane and righting arm is not part of the ships susceptibility, it’s the reason for the rolling, not simply a transfer of energy from heave to roll.

Otherwise why is the hull shape so critical? Containerships with fine lines on the ends will roll when a tanker will not. If it was driven by a transfer of energy tankers and containerships alike would roll.

First half of this goes through the math for a pendulum model; the second half is graphical.


I see that now, it’s like a child on a swing “pumping” which is (?) similar to the changing GZ. Pushing a child on a swing is like synchronous rolling.

What I understand is that hull forms with a pronounced bow flare, flat transom stern and non vertical ship sides near the waterline are most vulnerable to parametric roll. GM-variation of a ship in waves seems to be an important factor.

This link gives some more information on the parametric criteria for large container vessels.