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Inclining experiments and lightship: from measured GM to reliable centre of gravity

A worked guide to inclining moments, free-surface correction, lightship adjustment and measurement uncertainty.

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Ship stability calculations depend on reliable displacement and centre-of-gravity data. An inclining experiment is an important way of establishing these data, but its result is more than a pendulum reading. The mass, tank condition, missing or surplus equipment and external influences present during the experiment must be resolved before a measured GM can become a lightship centre of gravity.

A lightship survey and an inclining experiment differ

Draughts and water density, combined with hydrostatic data, establish the vessel's displacement during the experiment. Longitudinal equilibrium and suitable hydrostatics help establish longitudinal centre of gravity. Vertical centre of gravity cannot be obtained simply from how deeply the ship floats. A small heel in response to a known transverse weight shift provides additional measurement information.

MCA's stability-survey explanation for fishing vessels makes this distinction: the lightship survey establishes mass and longitudinal centre, while the added inclining operation establishes vertical centre. The fishing-vessel thresholds and repeat-test conditions in that source are not universal requirements for every vessel type.

The meaning of small-angle equilibrium

K is the chosen keel or baseline datum, G the centre of gravity and M the transverse metacentre. KG and KM are the vertical heights of G and M above that datum; GM is the metacentric height between G and M. A mass w moved transversely through distance d creates a heeling moment w g d. At small angles the equilibrium can be written w d = Δ GM effective tan θ, where Δ is the displacement mass during the experiment, expressed in the same unit as w. The factor g cancels. Using tonnes for both masses gives GM in metres; tonnes and kilonewtons must not be mixed indiscriminately.

The relation assumes free flotation and conditions suitable for the small-angle approximation. Contact with a quay, a taut line, bottom contact or changing wind can introduce another moment. Taking more readings does not necessarily remove those effects. MCA's large-commercial-yacht guidance illustrates the importance of free heeling, calm conditions and the experiment inventory. These methodological principles should be distinguished from yacht-specific numerical acceptance conditions.

An example connecting GM, free surface and KG

Assume an illustrative experiment with Δ = 5,000 t, shifted mass w = 20 t and transverse movement d = 8.0 m. An ideal pendulum with a vertical suspension-to-reading-plane distance of 5.0 m and a horizontal displacement of 0.100 m gives tan θ = 0.100 / 5.0 = 0.020, corresponding to approximately 1.15°. The effective GM is 20 × 8 / (5,000 × 0.020) = 1.60 m.

Free liquid in a slack tank shifts toward the low side as the ship heels, adding a heeling effect. For this mechanism, described in MCA MSIS43, May 2023, the small-angle model gives free-surface correction FSC = ΣFSM / Δ. FSM is the free-surface moment: its total in t·m divided by displacement in t gives metres. Actual tank geometry, filling and liquid density determine FSM. Now assume hydrostatic KM = 6.40 m for the experiment condition and a total free-surface correction of 0.10 m. The measured GM including free-surface effects is smaller than the GM of the solid-weight system. Thus GM solid = 1.60 + 0.10 = 1.70 m, and KG = 6.40 − 1.70 = 4.70 m. Reversing the correction sign would incorrectly raise KG. The example does not calculate tank free-surface moments; 0.10 m is an explicit assumption.

The experiment condition is not lightship

Temporary equipment, test weights and personnel present during the experiment require mass and position accounting. Missing permanent items are added, temporary items removed and items due for relocation receive moment corrections. A consistent sign convention should be maintained in the inventory. Correcting total tonnage without correcting vertical moment leaves the centre-of-gravity calculation incomplete.

In the example, assume the test condition includes 40 t of temporary items at KG = 8.0 m and that no other adjustments are needed. Lightship mass is 4,960 t and vertical moment is 5,000 × 4.70 − 40 × 8.0 = 23,180 t·m. Corrected KG is approximately 4.673 m. The 40 t is the entire temporary inventory, including the 20 t shifted test mass; that test mass must not be subtracted again.

Why one attractive reading is insufficient

The relation between heeling moment and tan θ can be examined over several shifts. Returns to the starting condition reveal zero drift and differences resembling hysteresis. Agreement between two measurement systems is useful, but systems affected by the same external moment can agree while both are biased. A straight-looking plot also cannot repair an incorrect displacement input.

A 1 mm reading error in the example's 100 mm pendulum displacement produces approximately a 1-percent relative error in the angle ratio. With other inputs fixed, the effect on GM is approximately 0.016 m. This is a sensitivity calculation, not a total uncertainty or acceptance tolerance. Weight measurement, shift distance, water density, the hydrostatic model and free-surface correction also enter the uncertainty budget.

Transferring the result to stability information

The lightship data established by the experiment underpin loading conditions; they do not by themselves demonstrate compliance with every seagoing stability requirement. The GZ curve, downflooding angles, loading restrictions and applicable criteria need separate evaluation. Positive GM does not establish adequate righting ability over the entire heel range.

IMO's explanation of the 2008 IS Code distinguishes mandatory and recommendatory provisions within its framework; amendments making the Code mandatory entered into force on 1 July 2010. Application to a particular vessel depends on applicable amendments and Administration requirements. A crane, battery installation or added superstructure can change actual lightship mass and centre even though the original test certificate remains unchanged.

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