### Why time domain seakeeping analysis shows the signal in the noise (and where it can derail your work)

Imagine yourself busy cooking an evening meal for your family: after a flurry of work in the heat of the kitchen, you bring a big dish of food over to the dining room table for your family. Strangely, there are more people than usual, but you weren’t expecting visitors. Yet these new people look eerily like your own loved ones – the same hair colour, eye colour, height, weight – you can’t tell the difference! To add to the confusion, there’s a barrage of sound as everyone calls, shouts, and demands food for dinner. It’s overwhelming. You know there are strangers, but you can’t tell the difference. What do you do?

This isn’t some alien invasion scenario or a science fiction story; it’s the lived experience of many species of bird. This happens because some birds lay eggs in others’ nests, hoping to shift responsibility for feeding and raising their young on others. Even though they are entirely different species, there are genetic tricks that help the imposter chicks blend in with colouring, patterns, and voices that sound similar to the hosts. So, how can you tell the difference from your own offspring? The Fairy Wren has a trick that it uses. But it requires planning.

The Fairy Wren sings to its eggs. It teaches them a specific song, much like a family signature, that gets encoded in their memory even while they are still forming in the egg. The trick is teaching the subtlety of the song over time, so they have a chance to memorize it, even before they hatch.

So when faced with a series of begging chicks, the real offspring echo back the family call, and the Fairy Wren knows who gets the food and who gets left out. Through the din of screeching chicks, the family song means there’s a signal in the noise.

Looking for a signal in the noise is a common problem in the technical world. It comes up in ship motion analysis, too. You may be faced with a barrage of ship motion performance data in the design process. It can feel overwhelming, and you need to find a way to dig into the details. This helps you filter out what’s meaningful and real. One way to dig into the details is with time domain analysis. In this article, we’re going to cover what it means to do this and how you can explore a variety of subtleties in hydrodynamics.

### What is time domain ship motion analysis?

Generally, time domain analysis is a process in which all the forces on a structure are calculated at a specific point in time, and the resulting acceleration from these forces predicts the change in position and velocity with a tiny increment in time. The process is repeated to resolve how the forces and resulting motion of the structure evolve through time. For ship motion analysis, the structure is the entire ship, and all the forces include effects from wave excitation, radiation, buoyancy, gravity, viscous drag, propulsion, and so on.

### What’s so crucial about time domain analysis is that there are no assumptions about what will happen in time

In frequency domain analysis, all forces and resulting motions are assumed – and have to fit the form of – a sinusoidal function. There’s no such assumption or requirement in time domain analysis. This is so important because it makes it straightforward to incorporate a wide range of complexity in the physics model of the ship. For example, nonlinear forces like viscous drag are simple to include in resolving the forces on the hull: the viscous forces are a function of ship geometry and relative velocity, which you know at each instant in time, so you can gauge the effect on the acceleration at that moment. Another example is using the complete hull form to calculate the static and dynamic pressure on the hull, or in other words, the buoyancy and Froude-Krylov forces on the ship hull. When ship motions or wave conditions (or both!) start to get large, these forces may be highly nonlinear and not just sinusoidal. Other complexities that are easy to implement in time domain analysis are things like changing propulsion or rudder position over time, which is not possible to do any other way than in the time domain.

### The most common use for time domain ship motion analysis is for vessel maneuvering

It’s easy to schedule rudder position, or propeller RPM, at different points in time, allowing evaluation of typical ship maneuvers like turning circle or zig-zag tests. It’s impossible to model these effects in frequency domain analysis, which solves ship motion problems in steady-state and doesn’t allow for transient effects.

### How does it capture nuance in vessel motions?

But there are other reasons to use time domain analysis to uncover what happens to ship motion in moderate to extreme conditions. A central assumption in frequency domain analysis is that the wetted hull area doesn’t change much from the static displacement. But this may be off in moderate and extreme conditions, such as when either ship motion is significant in a resonance condition or a heavy sea state. The ability to capture the change in buoyancy and Froude-Krylov force based on the instantaneous wetted hull area in the time domain is a subtlety that becomes important in these conditions. This capability provides insight into the nuance of effective vessel speed in waves. Accounting for the changing static and dynamic pressure on the instantaneous wetted hull is necessary to gauge the effective forward speed of a vessel in a seaway. Yet another example is in parametric roll: this phenomenon is possible to see in ship motion analysis but only when both the forces from static and dynamic pressure is resolved on the actual wetted hull at each instant in the time domain analysis.

### Some of the nuances of time domain analysis can derail your work.

It can derail your work because the setup and post-processing can take up a lot of time. You may spend a lot of time comparing results and trying to understand what is happening when you get subtle differences in ship motion between similar scenarios. One example is in drifting effects of the floating system. In some maneuvers, this may not be an issue. But at lower speeds, and in certain degrees of freedom like yaw, the motion of the vessel can drift in time. This is a natural and expected process, especially when there aren’t any restoring forces from a mooring, thruster system, or autopilot and rudder control to keep things in position or on course. The result is that it can be frustrating if you want to characterize a relatively steady indication of ship motion with a specific heading over time. This leads to another challenge in working in the time domain – everything is transient.

### How do you know when you have convergence of motion or not?

The simulation tends to be always transient as there are no assumptions about steady state motion in time domain analysis. It usually means you need to run longer time domain simulations to get a reasonable characterization of the steady state motion of a system. This also leads to another challenge in determining what level of complexity to include in your analysis – in other words, how many and how much nonlinearity to include.

Some nonlinearities make sense to include all the time – such as viscous effects from appendages and hull resistance. But for other effects, it is not always clear. For example, it’s possible to calculate buoyancy and Froude-Krylov forces using a linear approach in the same fashion as in the Frequency Domain analysis: this means the wave excitation forces are sinusoidal and you are ignoring the effect of changing wetted hull area on these forces in the time domain. Computationally, resolving the time domain forces is much faster. However, determining when to make these changes in how you model the system is not always obvious.

### Why don’t we use time domain analysis for everything?

The computational cost is the primary reason it is only used for some things. Resolving forces and marching a simulation through time can be quick, but evaluating many different scenarios can add up to a lot of time. Typically, simulations also need to be resolved on a case-by-case basis, and it can be challenging to get results in parallel or as instantly as frequency domain analysis. Often, the specifics of the scenario require careful consideration to set up, too, such as the specifics for a particular maneuver or to post-process in understanding what is happening with the results. This leads to other challenges with working on time domain ship motion analysis.

### It’s example time

Letâ€™s look at an example to show how time domain seakeeping analysis can shed some light on ship motion in some severe circumstances. In this scenario, we used the ProteusDS ShipMo3D toolset to investigate the Generic Fishing Vessel in a 3m, 7sec sea state at a moderate forward speed. This is particularly challenging because of the steepness of the waves but also because the natural roll period of the Generic Fishing Vessel is 7 seconds. Generally, roll motion is an important area of focus in ship motion because of how much it dominates comfort and safety. In this scenario, with the sea state so close to the natural roll period, the roll motions are of even more significant interest.

We start off with a frequency domain analysis. With a calculation time of 1 second, we get a snapshot of the ship performance at all relative wave directions around the hull. It can feel like a bit of a barrage when looking at all this ship motion data, so how can we zero in on what might be a likely problematic scenario? The maximum roll is in a beam to stern quartering direction. While this might be the worst-case scenario, the reality is that in such harsh conditions, for safety, the ship may opt to run into forward into the waves, and so evaluating ship motion in head or just off head in a bow quartering condition are of more interest.

For a bow quartering condition with waves 30 degrees off head sea, the frequency domain ship motion results indicate a 4.9 degree roll RMS. A statistical analysis shows that with this RMS value, an extreme roll of 15 degrees may be possible in 1 hour of exposure.

However, because of the relatively small size of the vessel in an extreme condition near its roll natural period, there could be significant changes in wetted hull area. So a nonlinear time domain seakeeping analysis that helps account for large changes in wetted hull area may be worthwhile. In addition, a time domain analysis will show the full time series of ship motion, rather than relying on a statistical extrapolation from the RMS motion, which may help reduce uncertainty a bit further. We completed a nonlinear time domain seakeeping analysis. The calculation time for 1 hour of ship motion in this condition was 3 hours total. The resulting time domain seakeeping analysis revealed much more detailed ship motion behavior with a 6 degree roll RMS, and the maximum roll in 1 hour of simulation was 25 degrees.

In this example, the frequency domain ship motion calculations are still useful and give insight into the resulting motions, especially relative to other sea state conditions and directions. This produces a massive amount of information quickly. But a more detailed time domain analysis allowed for some additional subtle change in the hydrodynamics model by accounting for changes in wetted hull area. This added nuance revealed larger RMS and extreme ship motions in this specific scenario, showing it was well worth some additional investigation.

### It’s summary time!

Time domain analysis considers all forces acting on a structure and takes small incremental steps in time to resolve how the resulting motion evolves over time. Because there are no assumptions about the resulting ship motion form as there are in Frequency Domain analysis, it’s possible to incorporate a lot of complexity in operations or in forces acting on the vessel. In ship motion analysis, time domain analysis is exclusively useful for resolving vessel maneuvers. But it is also useful at exploring subtle but important hydrodynamic effects like forward speed in waves or parametric roll.

Much like the Fairy Wren looking for a particular song from their brood, you may need to use time domain analysis to tease out what is really going on with your ship at sea in special circumstances.

### Next step

Learn more about how the ProteusDS ShipMo3D toolset can help with seakeeping and maneuvering here.