Month: June 2021

Why an RAO is the dynamic fingerprint of a hull

The Great Blue Hole in Belize looks mysterious from above. It stands out as a perfect dark blue circle – almost black – amid a shallow water atoll. It’s such a dark blue because it is a marine sinkhole. In other words, it’s a large cavern that expands over 100m below the surface. Yet, though it looks mysterious, there’s a lot we know about how it formed.

Thousands of years ago, during an ice age and shallow sea levels, a cavern formed. As it aged, this cavern grew in size and formed many giant stalagmites and stalactites. But eventually, the ice age waned, and sea levels rose. Finally, the weight of seawater collapsed the cavern roof and submerged it. How can we tell all this happened?

The key is the stalagmites and stalactites: they can not form underwater. They tell us the historical characteristics of the cave that became a marine sinkhole and act as a historical fingerprint.

Fingerprints reveal a lot of information. They can even capture a unique identity. When it comes to floating systems, a Response Amplitude Operator (RAO) acts just like a fingerprint. But in this case, the fingerprint gives a hint about how these floating systems respond to ocean waves. All the details of a hull go into making this unique dynamic fingerprint. In this article, we’re going to talk about an RAO and what it tells us about ships and floating structures.

The Great Blue Hole of the coast of Belize looks mysterious, but we know a lot about its history from its geological fingerprints.

What is an RAO?

The RAO shows how much a floating hull responds to ocean waves of different periods. A motion RAO shows how much the hull moves in each degree of freedom. For example, a heave motion RAO will show how much a hull will move up and down across a wide range of ocean wave periods. There will be a different RAO curve for each degree of freedom of the hull – the linear motions surge, sway, heave, and the orientation motions roll, pitch, and yaw.

An RAO gives a hint about motions at sea

The RAO can show you in a single view just how sensitive the hull will be to different ocean wave periods. Depending on the hull type, a few degrees of freedom are often susceptible to a dangerous resonance condition. A resonance condition is when the hull motions or accelerations may get very large. Either of those conditions can lead to injury, damaged equipment, or damage to the vessel itself.

An RAO allows comparison between two different vessels. RAOs can be helpful to understand what might be a more suitable ship for a particular operation. But it can also be beneficial feedback in the design process, where a designer can see how subtle changes to the hull form or load out will affect the overall ship response.

How is an RAO calculated?

A common way to produce RAOs is with a seakeeping analysis based on potential flow theory, such as ShipMo3D. These tools take a wide range of input on the hull shape, hull appendages, system mass and inertia, and then calculate the RAO.

Seakeeping software typically assembles the RAO from calculated steady-state sinusoidal ship movement in sinusoidal ocean waves. The magnitude of the RAO is then the steady-state ship motion amplitude in individual sinusoidal ocean waves. Ultimately, the RAO illustrates the variation in ship motion amplitude across a range of ocean periods.

Since the goal is to show the variation independent of wave height, the RAO may be nondimensionalized by individual sinusoidal wave heights for linear motions and wave slope for rotational movements.

While you get a lot of information from an RAO, it isn’t necessarily representative of motion in an actual sea state.

Ship motion in a sea requires more than the RAO

An actual wave state in the ocean is rarely just a single sinusoidal wave. Typically, there is a combination of many different ocean waves, each with a slightly different direction and period. One way to compute the expected ship motion response in a realistic sea state is with the RAO. But to do this, you need the RAO in combination with the ocean wave state spectrum.

The mathematical combination of the sea state and RAO produces the ship response spectrum in that sea. The ship motion spectrum then tells you precisely the characteristics of the ship’s motion in that sea state.

So the RAO is not in itself the absolute ship motion in a specific sea state. But it is very much a unique dynamic fingerprint. So this fingerprint is then what you can use to determine how the ship will move in a wide range of ocean wave states.

US Navy Ship in a heavy sea state. An RAO tells us a lot about the characteristics of a ship hull. But you need both the RAO and the sea state spectrum to predict ship motion at sea.

Are there other ways to compute ship motion response in the ocean?

Absolutely: physical scale model tests predict ship motion response. Commercial Computational Fluid Dynamics software tools resolve fluid physics much more accurately than potential flow methods. But there are advantages and disadvantages to each approach. Calculating an RAO and ship motion response with a potential flow tool like ShipMo3D is typically very fast – on the order of minutes – and generally cost-effective.

On the other hand, physical tank tests may reach tens or even hundreds of thousands of dollars depending on the testing required. Commercial Computational Fluid Dynamics software tools are sophisticated and powerful but incur high computational costs – and that means more time to compute ship motions or higher prices in using a more powerful computational facility to get answers faster.

Example time

In a previous article on seakeeping, we used a Generic Frigate to showcase ship motions in a particular sea state. Part of this process includes calculating the Generic Frigate motion RAO. The RAO for this Generic Frigate configuration at 10kts forward speed in a beam condition is below. The roll motion RAO, the middle plot on the right side, shows a peak around 0.6 rad/s or 10 seconds. The roll RAO peak reaches almost 3 here, which hints that the ship is pretty sensitive to waves around a period of 10 seconds in a beam loading condition.

Generic Frigate motion RAO summary at 10kts in beam condition. Note the roll motion RAO shows a peak around 0.6 rad/s or 10 seconds. This shows the ship is fairly sensitive to roll from ocean wave periods around 10 seconds.

Remember, these values indicate the ship hull response to a single sinusoidal wave. A ship motion time-series requires a combination of the RAO and a specific sea state spectrum. With a particular sea state spectrum, you can compute a ship motion spectrum and time series to predict specific motions and accelerations. A sample time series created from these RAOs in an irregular sea state is below.

A typical sample time series of the Generic Frigate motion in a beam sea condition with short crested, irregular seas.


The RAO shows how a particular hull will respond to a wide range of ocean wave periods. It’s a helpful calculation that helps with a comparison of different ships or how design or configuration changes affect a ship’s response. Most often, it’s a standard computation from commercial seakeeping analysis tools, like ShipMo3D. Calculating the ship motion response in a specific sea state needs the RAO combined with the sea state spectrum, producing the overall ship response spectrum. In this way, the RAO represents a dynamic fingerprint of a specific ship.

A marine sinkhole may look mysterious, but we have many clues about how they formed through their detailed rock formations. Similarly, RAOs provide valuable clues about predicting ship motion in any sea state – so you don’t leave safety at sea as a mystery.

Next step

ShipMo3D is an example of a seakeeping tool that you can use to calculate motion RAOs and better understand all ship motions in various sea conditions. Read more and apply for a free demo of ShipMo3D here.


Read more about the Great Blue Hole near Belize here.

How seakeeping analysis reveals surprising differences about ship seaworthiness

Grocery stores are places of scientific discovery. It’s true: biologists wanted to know just what was in a package of store-bought dried mushrooms. Taking an extremely close look, they sequenced the DNA of each mushroom and were stunned at what they found. They were mushrooms all right – but what was surprising was that they found three entirely new species of Porcini mushroom. A mushroom might look like any other, but a closer look reveals surprising differences.

The closer you look, the more often you find these surprising differences. In ship design, one hull of a specific ship size and class may look like many others. But only a few invisible changes can make all the difference in how they respond in the ocean.

At first glance, you might expect ships of a similar class to have similar motions at sea – but a closer look will show how different they can be. Like discovering new mushroom species in the grocery store!

Seakeeping analysis is a prediction of how a ship will move in a particular sea state

Any floating vessel moves in linear – heave, surge, sway – and rotational – roll, pitch, and yaw – fashion in different amounts based on many factors. Chiefly, the amount of movement depends on the ship’s forward speed and orientation to the prevailing sea, the size of the waves, the range of wave period.

Ship motion is a crucial factor of seaworthiness

Seaworthiness is how safe and easy it is to live and work on a ship in different sea conditions. In the worst-case extreme seas, you might think the hull flipping over or capsizing is the biggest concern. It certainly is, but many more things can go wrong before that extreme condition.

Seakeeping analysis reveals deck accelerations and motion at any point on the ship in a particular sea state. These accelerations and how severe they are affects comfort and safety. The larger the accelerations are, the more difficult it is to work and operate the ship properly. Ship motions also drive sea sickness for people on board.

A US Navy ship in a heavy sea state. The motion of the ship affects both the health of the ship and of the crew.

Health and safety are not the only factors

There is also the possibility of damage to the ship. Large accelerations may damage specific equipment. It’s also possible to damage or lose cargo at sea.

In large pitch motions, massive slamming forces are likely from the impact of the hull on the water. Large pitch motions may also mean the bow is partially submerged, and water can wash over the deck. The weight and pressure of water on the deck from these kinds of events are considerable. These are much more extreme scenarios, but they can create tremendous forces and dangerous stresses on the hull structure.

Seakeeping analysis is possible through a few methods

Physical tests involve a scale ship model in a wave basin. But test tank facilities are expensive and physical models take a lot of time and cost to create. Numerical tests are possible through a variety of commercial software tools.

One of the most common approaches is a numerical approach based on potential flow theory. These tools calculate how a specific hull form interacts with an ocean wave field. There are a few limitations to what is possible with this approach. But it is a robust technique that has been in use for decades. The software program ShipMo3D is an example of a software program that is purpose-built for seakeeping analysis.

Most often, seakeeping analysis is part of new ship design

The shape of the hull and the configuration of equipment and cargo are all factors that play into how the ship will move in certain sea conditions.

Once the vessel is constructed, there are also many reasons to check the ship’s motions. Depending on the vessel configuration and possible sea state intensity in an upcoming voyage, a specific seakeeping analysis may be necessary to evaluate risks.

Even though two ship hulls may look similar, there are many factors in addition to the hull shape that goes into a ship motion analysis. The load out of the hull can affect the inertia and shift the center of mass. The inertia and location of the center of mass alone have a significant influence on the ship’s motions. All these details need to be carefully considered for an accurate assessment.

Do we always have to complete a seakeeping analysis?

Seakeeping analysis is one of many tools that fit into the design process. It depends on the risk involved and how much concern there is for ship performance. Naval architects, engineers, and ship designers build up a tremendous amount of experience working with certain styles of ships. They can get a feel for how similar vessels will respond.

In this way, seakeeping analysis is a complementary tool to experience. But at the same time, many details affect a ship’s seakeeping ability. Changing only one of many parameters on a vessel can make a surprising change in the motion response. Tools like ShipMo3D can quantify those differences. But whether a seakeeping analysis is necessary or not depends on the risks involved and consequences of damage. There may also be a regulatory requirement for specific analysis depending on a particular jurisdiction.

It’s time for an example

A ShipMo3D model of a Generic Frigate is shown in the picture below. The wet and dry hull mesh is shown in yellow and green, respectively, and highlights the draft for the specified vessel weight. Hull appendages like bilge keels, skegs, and rudders appear in red that round out the details necessary for the ship motion evaluation.

A ShipMo3D model of a Generic Frigate. This shows the numerical mesh of the wet (yellow) and dry (green) portion of the hull. Appendages like skegs and bilge keels are shown in red. All the hull geometry, appendages, and mass properties of the ship factor into the resulting ship motion calculations in a seakeeping analysis.

A software tool like ShipMo3D calculates the hydrodynamics and then resulting motions of a vessel like this in a variety of sea conditions. So how much can these parameters change? To illustrate these changes, we increased the roll inertia of the Generic Frigate by 15% and recalculated the performance metrics in ShipMo3D.

In a pure roll-resonance condition, the Generic Frigate with larger roll inertia has a 20% larger amplitude. But roll resonance from a pure sinusoidal ocean wave is not always the most common sea state. What are the differences in motion in a more realistic sea?

A more realistic sea might look like a short-crested sea state with a spectrum of different wave frequencies. We picked a short-crested sea state 5 condition with a spectrum peak period at 10 seconds to investigate. The two Generic Frigate models were set to a 10-knot forward speed in beam sea condition. This time, the Generic Frigate with larger roll inertia reduced the peak roll accelerations by 10%. In this specific case, the added inertia moved the natural roll period farther away from the spectrum peak period, helping reduce roll activity.

A short time history of roll motion in a short-crested sea state in the figure below gives an idea of the output of a seakeeping program. There are more detailed outputs that show statistics like maximum motions, accelerations and probabilities for motion sickness and interrupted work – all important factors to consider for safety at sea. This is only one specific sea state and ship condition that illustrates how one key parameter can make a significant difference – in spite of all the physical similarities of hull shape and appendages.

A typical sample time series of the Generic Frigate motion in a beam sea condition with short crested, irregular seas.

It’s summary time

Seakeeping analysis is about predicting how a particular ship will move in a specific sea state. The ship hull form and specific sea conditions all factor into this analysis. A host of information on the ship’s motions feeds into indicators like interruptions of work to seasickness. There are also many indicators for large loads, the potential for damage to the hull, or cargo loss.

Ships of a particular type may look one and the same, like mushrooms at the grocery store. Seakeeping analysis will help you take a deeper dive and reveal how each one differs from the other.

Next step

We talked a lot about what seakeeping is and how it helps better understand ships. A ship motion Response Amplitude Operator (RAO) is a fundamental parameter in seakeeping analysis. But what is an RAO and how do they work? Read more about RAOs here.


Read more on how Mycologists discovered new species of mushroom at a grocery store here.