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__Introduction__

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As it is known to us that a surface ship has 6 degrees of freedom, viz. surge, sway, heave, roll, pitch and yaw. Surge, sway and heave represent the translational motion of a ship along the x-axis, y-axis and z-axis respectively. Roll, pitch and yaw are associated with the rotational motion of a ship about the x-axis, y-axis and z-axis respectively. The picture shown below depicts the 6 degrees of freedom of a surface ship.

__6 degrees of freedom of a ship (Courtesy- Google images)__

Ship rotational motions just seem normal. Almost each and every vessel experiences motion in the seaway and the magnitude of the motion depends on the efficiency of the designer behind it.

Just have a look at this picture.

__Image courtesy-__

__https://www.ilwu19.com/photos/china/apl28.jpg__

Guess what would have caused such a macabre to the containers?

Well, obviously, one would say it's due to extreme rolling motion the ship has encountered, probably due to the resonance of the ship’s natural frequency with the encountered wave frequency in the seaway.

Generally, beam waves (waves perpendicular to the ship’s centerline axis or the x-axis) are the reason behind roll motion of the ship.

But what if this is rolling is not caused due to beam waves? It seems illogical, but here is what we will be discussing in this article, rolling due to head waves (along the ship’s centerline axis or the x-axis). This phenomenon is also called as Parametric Rolling resulting due to resonance and certain special processes as discussed below.

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__Parametric Rolling – The phenomenon__

Now, assume a ship is moving in a straight line with zero drift angle and a wave is encountered opposite to the direction of its motion. Now if the wavelength of the waves is same as the length of the ship then major transient changes will occur in the water-plane of the ship, which would consequently change the transverse stability of the ship. Generally for a typical ship design, looking at the plan view, the midship is made fuller and the aft and forward ends are fine-form to balance with the cargo carrying capacity (or space requirement) and streamlined shape of the vessel. Also, the local breadth of the ship at any station decreases as one goes down the waterline to the keel.

When the wave crest is at amidship, due to the shape of the ship, the aft and forward ends of the ship contribute less to waterplane area. So stability decreases as we know the transverse stability of a ship is directly proportional to the waterplane area. And when the trough is amidship, stability increases as crests prevail at the forward and aft end of the ship. This transition in transverse stability takes place with a certain frequency which gives rise to parametric rolling. The picture below illustrates the change in waterplane area with respect to wave position.

__Image courtesy – Team LSD__

For a particular location and loading condition, natural roll period of the ship is a function of waterplane area, mass and added moment of inertia and hence remains constant. Assuming the frequency of the head wave remains constant; hence we know that there are particular bands of frequencies which cause parametric rolling. But the fact that we have ignored here is that there is a relative motion between ship and waves, which consequently changes the frequency with which a wave hits the ship. The actual frequency with which the waves hit the ship in a seaway is known as Encounter frequency.

ω

_{e}=ω- (ω

^{2}*U*cos μ)/g

ω

_{e}is Encounter frequency
μ is wave heading angle of the ship

U is velocity of ship

ω is angular frequency of the wave

ω is angular frequency of the wave

One of the fundamental conditions for the parametric roll to set up is that the encounter frequency of waves should be twice that of natural roll frequency of the ship. So, there will be a certain set of ship speeds, heading angle and wave frequencies which will cause parametric rolling. Under this frequency condition, when the crest is amidships, the stability decreases, so the ship would roll, but after half the natural frequency of roll of the ship, stability increases abruptly due to increase in waterplane area, so the restoring force contributes to further rolling. This small roll angle slowly turns to a large parametric roll.

The image below shows parametric rolling of a ship (damping neglected). The changing GM values pertain to changing waterplane which we have already discussed. More water plane area means more GM. As the transverse stability and natural roll come near in phase, the roll angle increases. If the two are in-phase then resonance occurs. The image depicts the above-mentioned concept.

__Image courtesy- of Thesis paper; Parametric roll instability of ships by Irfan Ahmad Sheikh, University of Oslo__

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__Hull Form Impact on Parametric Roll__

Also, there are certain types of hull forms, which are susceptible to parametric rolling and same can be inferred by analyzing previous voyage data. Due to improvement in hull designs for better cargo capacity and flow efficiency, the bow flare, stern overhangs and fuller amidship are introduced in ship design. Due to this when waves travel along the length of the ship, gradients in waterplane area are very significant. This results in parametric rolling.

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__Parametric Roll- Havoc and Prevention__

The picture of the container ship we saw at the beginning is of container vessel APL China. Due to parametric roll, 60% of her cargo was lost to waters.

To tackle this serious issue, most of the ships are now being engaged with sensor systems which alarm the crew about the possibility of parametric roll and paves a way to take immediate actions. Although, this phenomenon can never be eliminated, so other measures are adopted to reduce rolling amplitude of the ship such as bilge keel, stabilizing fins, anti-roll systems, etc. Since the phenomenon depends upon the encounter frequency, a ship can simply change its speed or heading angle to counter parametric roll especially to avoid resonance.

Although this phenomenon takes place very rarely, it’s consequences force the naval architects to incorporate preventive measures. With the help of simulation software, the response characteristics of the ship in different wave conditions can be easily predetermined, which helps in techno-economical motion based designing of ships.

The following video shows an experimental demonstration of parametric roll for a ship model in a wave flume. Hope it helps you to visualize the havoc it creates in large vessels.

(Video courtesy- YouTube channel tupsumato)

__Article by:__**Kartik Garg**
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