## Vessel theory: RAOs and phases |

Vessel motions in waves can be defined by displacement RAOs (response amplitude operators) that are specified on the displacement RAOs page of the vessel type data form. Each displacement RAO consists of a pair of numbers that define the vessel response, for one particular degree of freedom, to one particular wave direction and period. The two numbers are an amplitude, which relates the amplitude of the vessel motion to the amplitude of the wave, and a phase, which defines the timing of the vessel motion relative to the wave.

Example: | A surge RAO of 0.5 in a wave of height 4m (and hence wave amplitude 2m) means that the vessel surges to and fro -1m to +1m from its static position; a pitch RAO of 0.5° per metre in the same wave means that the vessel pitches from -1° to + 1°. |

The vessel has 6 degrees of freedom: 3 translations (surge, sway, heave) and 3 rotations (roll, pitch, yaw), so the RAO data consist of 6 amplitude and phase pairs for each wave period and direction. The RAO amplitude and phase vary for different types of vessel, and for a given vessel type they vary with draught, wave direction, forward speed and wave period / frequency. It is important to obtain accurate values for the RAO amplitude and phase if the dynamics of the system are to be correctly modelled.

RAOs can be obtained either from model tests or from diffraction analysis software. The data may be presented in tabular or graphical form: tables of numbers are better for our purposes since they can be imported directly into OrcaFlex (see importing hydrodynamic data).

There are many different conventions for defining RAOs. There have been attempts at standardisation, but these have not been successful so there remain differences between the main computer programs and model basins: some establishments even use different conventions for reporting model and computed data. The only safe course is to obtain a complete description of the system used for the data in each case.

The Orcina convention is to use the amplitude of response (in length units for surge, sway, heave, in degrees for roll, pitch, yaw) per unit wave amplitude, and to use the phase **lag** from the time the wave **crest** passes the RAO origin until the maximum positive excursion is reached (in other words, the phase origin being at the RAO origin). Mathematically, this is given by
\begin{equation}
x = R\ a\ \cos(\omega t - \phi)
\end{equation}
where

$x$ is the vessel displacement (in length units for surge, sway, heave, in degrees for roll, pitch, yaw)

$a$, $\omega$ are wave amplitude (in length units) and frequency (in radians/second)

$t$ is time (in seconds)

$R$, $\phi$ are the RAO amplitude and phase.

However, OrcaFlex can accept RAO data using a wide range of different conventions, so you can input your RAO data in its original form and simply tell OrcaFlex what conventions apply to those data.

In addition to the actual RAO data you therefore also need to know:

- The coordinates, relative to the vessel origin, of the RAO origin and of the phase origin.
- The system used to define wave direction. In OrcaFlex 0° means waves approaching the vessel from astern and 90° means waves coming from the starboard side, but if a different convention applies to your data then you must allow for this when entering the data.
- The coordinate system used to define vessel motions and, in particular, which direction is positive. That is, whether surge is positive forward or aft, whether heave is positive up or down and whether pitch is positive bow up or bow down.
- Whether the rotational RAO data are in degrees (or radians) of rotation per metre (or foot) of wave amplitude, or in degrees (radians) per degree (radian) of wave slope or wave steepness.
- The reference time for phase angles, and the reporting convention used (e.g. whether phases are reported as lags or leads). Again, OrcaFlex allows a range of options.

Although OrcaFlex allows the RAO *input* data to use a wide range of systems, all OrcaFlex *results* use a right-handed system in which the positive movements are as follows:

Surge | positive forward |

Sway | positive to port |

Heave | positive up |

Roll | positive starboard down |

Pitch | positive bow down |

Yaw | positive bow to port |

As well as **displacement** RAOs, which directly specify the motion of a vessel in waves, OrcaFlex can apply **wave load** RAOs, which represent instead the load (force and moment) on a vessel due to waves. In this case, the amplitude represents the magnitude of the force (in the surge, sway or heave direction) or moment (in the roll, pitch or yaw direction); the meaning of the phase remains unchanged.

Example: | A surge force RAO of 300 kN/m in a wave of height 6m (and hence wave amplitude 3m) means that a vessel experiences a surge force varying harmonically between -900kN and +900kN over each wave cycle; a pitch moment RAO of 1e6 kN.m/m in the same wave means that the vessel experiences a moment about the y axis varying from -3e6 kN.m to +3e6 kN.m. |

Wave load RAOs do not completely define the vessel motion as do displacement RAOs: they merely define the force and moment which a wave exerts on the vessel. OrcaFlex uses these forces and moments, together with any other loads on the vessel and data on the vessel's mass and inertia, to determine the vessel motion from its equation of motion.

The description of RAO conventions above, for displacement RAOs, carries over to wave load RAOs with just one minor difference: rotational wave load RAOs must be expressed per unit of wave amplitude, and they will have dimensions of moment per unit length.