Vessel Motion RAOs: From Wave Spectrum to Crane-Tip Motion

By Tord Martinsen, CEO · Method and sources checked August 2026 · 14 min read

For a vessel-based lift, wave height is only the environmental input. The useful engineering quantity is the motion of the actual lifting point — and that requires the vessel response, its phase, the crane geometry and the sea spectrum to be treated together.

A tall sea can produce modest crane-tip motion when most of its energy sits outside the vessel’s responsive periods. A lower sea can be worse when its energy overlaps a heave, pitch or roll peak. Loading condition, heading and crane position can change the answer again.

The link between the sea and the vessel is the response amplitude operator, or RAO. The link between the vessel and the lift is the crane-tip RAO. That distinction matters: the compensator does not see wave height directly. Its primary kinematic excitation is motion at the crane tip.

What a vessel-motion RAO actually contains

An RAO describes a first-order linear response to a unit regular wave, as a function of wave period or frequency and wave heading. A heave RAO of 0.80 m/m at T = 10 s means that a regular wave with 1 m single amplitude produces 0.80 m of heave single amplitude at that frequency, for the stated heading and vessel condition.

A complete response carries both magnitude, how large the response is, and phase — when it occurs relative to the wave. The file normally contains both for all six rigid-body motions: surge, sway, heave, roll, pitch and yaw. It must also identify the RAO origin, axes, positive rotations, loading condition, heading convention, vessel speed and frequency convention.

RAOs may come from a radiation–diffraction analysis combined with the vessel mass, inertia, restoring and damping model; from model testing; or from an approved vessel motion manual. They are specific to the modelled condition. A transit-draft set does not automatically describe the same vessel at working draft with different ballast, trim, deck cargo or crane configuration.

Read the overlap, not the tallest curve

Three regions commonly appear in a heave response. At very long wavelengths relative to the vessel, the zero-speed heave RAO in m/m generally approaches its quasi-static limit near 1. At very short periods, rigid-body displacement response generally tends towards zero. Between them, inertia, hydrostatic restoring and damping produce peaks or shoulders in one or more motions.

Roll and pitch curves require more care because their usual units are angle per metre of wave. A rotational RAO of 1 deg/m does not mean the vessel moves “more than the wave”. The useful question is what that angle does at the lifting point.

These are tendencies, not operating rules. Heading can be an effective operability lever, but it is constrained by stationkeeping, wind and current, thruster limits, crane offlead and sidelead, lift path, nearby assets and the approved procedure.

From vessel RAOs to crane-tip motion

The hydrodynamic reference point is rarely the lifting point. A crane mounted forward, aft or off the centreline turns angular motion into vertical travel.

For small rigid-body motions, the complex vertical crane-tip RAO is:

Here H_3, H_4 and H_5 are the heave, roll and pitch RAOs; x_P and y_P are signed offsets from the RAO origin; omega is frequency and beta is heading. Rotational RAOs must be in radians per metre before multiplying by a lever arm.

Every term is complex. Magnitude and phase are combined first; only then is the final magnitude taken. Adding the three plotted magnitudes throws away the timing.

From a wave spectrum to motion statistics

A real sea contains many frequency components. For a long-crested sea at one heading, the first-order crane-tip response spectrum is:

For a directional sea, the response is integrated over direction:

This matters when wind sea and swell arrive from different directions, or when spreading reaches both pitch-sensitive and roll-sensitive headings. The wave spectra, H_s and T_p article covers the environmental side of the calculation.

The response statistics then come from spectral moments:

^* Narrow-band Gaussian approximation. A design maximum also needs a stated exposure duration, response bandwidth, statistical model and probability criterion.

If the vessel has meaningful forward speed, the analysis must distinguish wave frequency from encounter frequency. Station-kept construction vessels are often assessed near zero speed; transit, towing and some installation cases are not.

Where RAOs stop and the lift model starts

A crane-tip motion spectrum is not yet a hook load, compensator stroke, DAF or allowable sea state. It becomes the excitation for the crane, winch, wire, rigging, compensator and payload.

Where waves also act directly on the payload, generate wave elevation, particle kinematics and vessel response from the same directional components, or the same time-domain realisation, so their relative phase is preserved.

This is the boundary between vessel response and the dynamic amplification factor (DAF): RAOs describe imposed motion; DAF reports a named peak load response relative to a stated reference after the load path has responded. For marine-lift hydrodynamics and dynamic analysis, see the DNV-RP-N103 marine-operations guide.

The input pack a vessel-motion screening needs

A usable vessel-motion package contains more than a PDF plot. Before calculating, check that the data answer each of the questions below.

An offshore lift operability screening follows this chain across the H_s–T_p plane. It reports the workable region by heading and identifies the gate that closes it, rather than returning one unqualified motion number.

Generic or class-prescribed motions may support concept screening only where the governing method permits and its validity range is satisfied. They are not inherently conservative across period, heading, loading condition or crane location. Final weather limits and equipment sizing should use the project-accepted vessel basis, or document why a substitute is adequate.