Resonance Avoidance in Heave Compensation

Q: How much can resonance amplify the motion?

At resonance the load moves √(1 + (2ζ)²)/(2ζ) times the crane-tip motion, close to 1/(2ζ) when damping is light, where ζ is the damping as a fraction of critical: about 5 times at 10 % damping and 2.2 times at 25 %.

What resonance is, why a passive heave compensator isolates the load only from waves well shorter than its natural period, what sets that period, and how a lift is kept clear of it from the deck to the seabed.

By Tord Martinsen, CEO · March 2026 · Updated September 2026 · 12 min read

Blender illustration of a construction vessel lifting a subsea module over the side in a long swell, seen from above.

In brief

Resonance is the large response of a system driven near its own natural period. The motion builds up far beyond the input, limited only by damping: with 10 % damping, a load on a spring moves about five times as much as the point it hangs from.
A passive compensator isolates the load from crane-tip motion only for waves shorter than its natural period divided by √2. Near its natural period it amplifies the motion instead.
√2 is the break-even, not a target. A sea spreads its energy past its peak period, so isolating it well takes a natural period of about twice the peak period.
With the gas charged to carry the load, the natural period in air is set by the gas volume per unit of piston area, and it shortens as the charge pressure rises, because nitrogen stiffens. In water, added mass lengthens it once the charge matches the load’s weight in water.
In deep water the wire adds a stretch mode whose period grows as wire is paid out, so the lift model checks resonance along the whole descent.

A passive heave compensator is a gas spring, with a damper, between the crane and the load. Like any spring and mass, the lift then has a natural period, and waves near that period make the load move more, not less. Keeping the lift away from it is the first rule of sizing a passive compensator.

What resonance is

A mass on a spring has a natural frequency, and a natural period:

\omega_n = \sqrt{\frac{k}{m}}, \qquad T_n = 2\pi\sqrt{\frac{m}{k}}

Drive it at that frequency and each cycle adds energy in step with the motion, so the response builds up until damping takes out as much energy per cycle as the driving puts in. For a load hanging from a moving crane tip through a spring and damper, the load’s motion at resonance is

\frac{\text{load motion}}{\text{crane-tip motion}} = \frac{\sqrt{1 + (2\zeta)^2}}{2\zeta} \approx \frac{1}{2\zeta}

where $\zeta$ is the damping as a fraction of critical: about 5 at 10 % damping and 2.2 at 25 %. A lift has several such modes, from the compensator’s gas spring, the wire’s stretch and the crane’s own flexibility, and each is checked.

Isolation, and the √2 rule

How much of the crane-tip motion reaches the load depends on the ratio of the natural period to the wave period, $r = T_n / T$ :

\frac{\text{load motion}}{\text{crane-tip motion}} = \sqrt{\frac{1 + (2\zeta r)^2}{(1 - r^2)^2 + (2\zeta r)^2}}

Whatever the damping in the unit, the ratio falls below 1 only when $r > \sqrt{2}$ (Figure 1). A passive compensator therefore isolates the load only from waves shorter than its natural period divided by √2. The period that counts is the lift’s: the gas spring, the wire and the rigging in series.

Figure 1. How much of the crane-tip motion reaches the load, for a spring and damper between them, at three damping levels, for a crane tip moving at a single period (logarithmic scale). Where the natural period is close to the wave period the load moves more than the crane tip, about five times at 10 % damping. Every curve crosses 1 at √2: only waves shorter than the natural period divided by √2 are isolated, and there less damping isolates better.

Damping trades one region against the other. More damping lowers the resonant peak, but in the isolation region it lets more motion through: where the natural period is twice the wave period, 10 % damping passes 36 % of the crane-tip motion and 50 % damping passes 62 %. A unit is tuned so that the operation’s wave periods sit well inside the isolation region, with enough damping to pass through resonance safely when it must.

In a real sea: twice the peak period, not √2

Figure 1 is for a crane tip moving at one period. A sea spreads its energy over a band of periods around its peak period $T_p$ , and close to a third of it lies at periods longer than $T_p$ , nearer the natural period. √2 is only the break-even point, so a unit set to √2 times $T_p$ removes almost nothing from a real sea: the load’s significant motion comes out at 0.9 to 1.3 times the crane tip’s, and more than the crane tip’s at light damping.

At twice $T_p$ , the load keeps about a third to a half of the crane-tip motion, depending on the damping. For a sea with an 8 s peak period, that means a natural period of about 16 s, as in the worked example on our passive heave compensation page. A vessel’s own response moves the crane-tip motion to longer periods than the waves, which makes isolation harder still, so the lift model works from the crane-tip motion given by the vessel’s RAOs.

What sets a passive compensator’s natural period

The gas spring’s stiffness is $k = \gamma\, p A^2 / V$ , with $p$ the gas pressure, $A$ the piston area, $V$ the gas volume and $\gamma$ the gas’s stiffness exponent. At wave periods the gas has no time to exchange heat, and at a compensator’s pressures nitrogen is a real gas: $\gamma$ is 1.4 at atmospheric pressure but about 1.6 at 10 MPa, 2.0 at 20 MPa and 2.6 at 35 MPa. When the gas carries the load, $pA = mg$ , and the mass cancels:

T_n = 2\pi \sqrt{\frac{V}{\gamma\, g\, A}}

So in air the natural period is set by the gas length $V/A$ at the working point, not by the load’s mass. A longer period needs more gas, and a higher charge pressure needs more still:

Gas length	Natural period in air, charged to 10 MPa	Charged to 30 MPa
$V/A$ = 20 m	7.0 s	5.8 s
$V/A$ = 50 m	11.1 s	9.2 s
$V/A$ = 100 m	15.8 s	13.0 s
$V/A$ = 150 m	19.3 s	15.9 s

The load still matters in two ways. With a fixed charge, a heavier load pulls the rod out and compresses the gas, which shortens the period; a lighter one lengthens it until the rod reaches its stop. And the load sets the charge pressure: for the same gas length, a unit charged to 30 MPa has a natural period about 17 % shorter than one charged to 10 MPa.

In water the gas carries the load’s weight in water, but the load moves with its full mass plus the added mass of the water around it. Once the charge is set for the weight in water, the period grows by $\sqrt{(m + A_{33}) / m_{\text{wet}}}$ , with $A_{33}$ the added mass and $m_{\text{wet}}$ the weight in water divided by $g$ : for a structure that traps a lot of water, often by a factor of two or more. A unit still charged for the weight in air sees a lighter load, and its rod moves toward the retract stop; at depth, the sea pressure on the rod pushes the same way. CYGNUS and ANTARES re-centre by releasing gas, and ANTARES can also add it. RIGEL’s charge is set on deck for the lift. The wire and rigging in series with the gas spring soften the system further.

When the risk rises

Too little gas. A small gas volume, or a high charge pressure, makes the spring stiff and the natural period short, into the wave band.
A load well below the charge. If the gas force exceeds the load’s weight, the rod sits on its stop, the unit goes rigid, and the wire and rigging alone set the natural period.
Long-period swell. Swell pushes the wave period up toward the natural period, and it can arrive with a wind sea as a second peak in the wave spectrum.
Deep water. As wire is paid out, its stiffness falls and the natural period of the lowering system grows. DNV-RP-N103 gives it as $T_0 = 2\pi\sqrt{(M + A_{33} + \theta\, m_{\text{w}} L)/K}$ , with $K$ the stiffness of the whole line, compensator included. A line with no working compensator, or with the rod on its stop, starts below the wave band and can grow into it at some depth. A soft unit keeps the lift’s period above the band and carries the load past that depth, as in the film below. What then climbs toward the band is the wire’s own, shorter stretch mode.

Designing around resonance

Size the gas volume and stroke. Set the natural period well above the operation’s wave periods, about twice the longest peak period, for the loads the unit will carry, with the stroke to take the crane-tip motion. At √2 times the peak period a unit isolates almost nothing.
Set the damping. Enough to cap the response if the lift has to pass through resonance, no more than the isolation region can afford.
Check the whole range. Every load, every depth and every gas state along the operation, from lift-off through the splash zone to the seabed.
Change the setting where one is not enough. ANTARES can change its setting during the lift: stiffer and better damped through the splash zone, softer below it, and a landing mode for a soft set-down, with automatic damping control throughout. It also releases and adds gas to stay centred as the load’s weight changes, which keeps the rod off its stops.

A time-domain model of the vessel, crane, wire, compensator and payload confirms it. CONSTELLATION runs the lift at each depth and sea state and reports the motion, the line tension and the stroke used.

Film: a 120 t open-top suction pile, with about 430 t of water moving with it, lowered through the depth near 1,300 m where a rigid line would resonate, at Hs 2.9 m and Tp 10 s. On ANTARES the DAF stays at 1.02, against 1.94 on the rigid line. Simulated in CONSTELLATION.

Figure 2. The film’s run in numbers, from CONSTELLATION: wire tension while the flooded pile is lowered from 1,250 to 1,350 m at 0.16 m/s (H_s 2.9 m, T_p 10 s). Each band spans the lowest to the highest tension in one 10 s wave cycle. Around 1,300 m the rigid line’s natural period meets the waves, and its tension swings between 36 and 234 t with every wave group; through ANTARES it stays between 138 and 144 t.

Sources and assumptions

The resonance and isolation relations are those of a single spring-mass-damper excited at its support, with the damping in the unit (standard vibration theory). Drag on a submerged load damps it against the water instead, which helps isolation. A lift has several coupled modes; the lift model checks each.
The sea-state ratios integrate the same relation over Pierson-Moskowitz and JONSWAP spectra (peak factor 3.3), with the crane tip moving like the waves, at 10, 25 and 50 % damping: 0.87 to 1.26 at √2 times $T_p$ , and 0.35 to 0.57 at twice $T_p$ . Each ratio is the load’s significant motion over the crane tip’s, each four times its root-mean-square motion, as $H_s$ is for the waves.
The gas-spring period assumes one gas volume carrying the load through one piston area, small motions, and nitrogen’s real stiffness exponent at 15 °C from the Span-Wagner reference equation: 1.62 at 10 MPa and 2.38 at 30 MPa. Where a unit connects more gas, as ANTARES does in its modes, $V$ is the volume connected. Large strokes stiffen the spring by a few per cent. Valve damping grows with the square of the flow, and too much of it locks the unit and hands the motion to the wire. Seal friction, a fraction of a per cent of the load, matters only for very small motions. The lift model uses the actual unit and a real-gas model of the nitrogen.
The lowering system’s natural period is from DNV-RP-N103, formerly DNV-RP-H103 (April 2010, 4.3.3.3 and 5.3.5). In $T_0$ , $M$ is the load’s mass in air, $A_{33}$ its added mass, $m_{\text{w}}$ the wire’s mass per metre and $L$ its length, $\theta$ the share of the wire’s mass that moves with the load (about 1/3 when the wire is the softest part, near 0 when a soft unit hangs just above the load), and $K$ the stiffness of the whole line, compensator included. The RP lets wave-induced resonance be disregarded when the peak period is well above $T_0$ , typically $T_p > 1.6\,T_0$ for an object in the wave zone.
The film’s case and its numbers are from our ANTARES page.

Frequently asked questions

What is resonance in a heave compensation system?

The large motion of a spring and a mass driven near their own natural period. The load on the compensator’s gas spring, on the wire and on the crane each has a natural period, and a wave period close to one of them makes the load move more than the crane tip instead of less.

How much can resonance amplify the motion?

At resonance the load moves $\sqrt{1 + (2\zeta)^2}/(2\zeta)$ times the crane-tip motion, close to $1/(2\zeta)$ when damping is light, where $\zeta$ is the damping as a fraction of critical: about 5 times at 10 % damping and 2.2 times at 25 %.

What natural period does a passive compensator need?

More than √2 times the wave period to isolate at all, because at √2 the load moves exactly as much as the crane tip. A real sea’s energy reaches past its peak period, so isolating it well takes about twice the peak period: for an 8 s peak period, about 16 s.

What gas exponent should be used for the nitrogen?

At wave periods the gas is compressed adiabatically, and at a compensator’s charge pressures nitrogen is a real gas: its stiffness exponent is about 1.6 at 10 MPa, 2.0 at 20 MPa and 2.4 at 30 MPa, not the ideal-gas 1.4. A lower value makes the natural period look longer than it is.

Why can a lighter load raise the risk?

Not because it is lighter as such: with the gas charged to carry the load, the natural period in air is set by the gas volume, not the load’s mass. But if the gas force exceeds the load’s weight, the rod sits on its stop, the unit goes rigid, and the wire and rigging alone set a much shorter natural period, which in deep water can fall in the wave band.

Does resonance only concern the compensator?

No. The wire has a stretch mode, and in deep water its period grows as wire is paid out; the crane and rigging add their own. The lift model checks every mode at every depth and sea state of the operation.

Worried about resonance in your lift?

Send us the load range, the depths, the wire and rigging, and the site’s sea states. We’ll model the lift in CONSTELLATION, find its natural periods at each stage, and size the compensator to keep clear of them.

Send your lift case →