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.