Learning objectives
By the end of this chapter you will be able to:
- explain why the light ship KG must be measured, and when the measurement is required;
- apply GM = (w × s × length) ÷ (∆ × deflection) to inclining data;
- state the conditions under which a valid experiment is conducted, and why each matters;
- average a set of port and starboard readings and reject a spoiled one;
- correct the measured GM for free surface and convert it, through KM, to the solid KG;
- strip the surplus weights from the inclining condition to reach the light ship displacement and KG;
- describe the contents of the ship's approved stability information, and when it must be renewed.
Nine chapters of this book have leaned on one number: the light ship KG of 8.86 m printed on the first page of MV Ninja's stability data booklet. Every moments table started from it; every GM, every list, every curve stood on it. Yet no formula in this volume can produce it, because no drawing knows exactly where the builders welded every tonne of steel. The number is measured, once, with the ship newly complete and floating quietly at a fitting out berth, by the oldest trick in this book run backwards. This chapter is that measurement, and the approved book of stability information that is built upon it.
10.1 Why the light ship must be weighed
Displacement can be found any morning from the draughts and the hydrostatic tables, as Chapter 2 showed. But KG cannot be read from a draught mark: two ships of identical displacement can carry their weight high or low, and float identically. For everything loaded aboard, the moments table handles the bookkeeping; the one item no table can supply is the starting entry, the empty ship herself. So the Administration requires the completed ship to be inclined: heeled deliberately, minutely, by known weights, and her resistance measured. The experiment is repeated only after a significant conversion, or when a periodic lightweight check finds the ship has drifted from her recorded figures.
10.2 The principle: the list formula run backwards
Chapter 7 predicted the list from a known GM: tan(List) = GGH ÷ GM, with GGH = (w × s) ÷ ∆. The inclining experiment simply swaps the known for the unknown. Shift a known weight w through a known distance s, and GGH is known exactly. Measure the tiny list that results, and the only unknown left standing is GM itself. The list is far too small for a clinometer, so it is taken from a long plumb line: a pendulum of measured length hanging in a hatchway, whose bob moves a measured deflection along a batten, giving tan(List) = deflection ÷ length. The MCA sheet folds the whole experiment into one line:
A small coaster of 8000 t displacement is inclined by shifting a 20 t weight 14.0 m across the deck. A plumb line 6.00 m long deflects 84 mm. Find her metacentric height at the time of the experiment.
By the sheet formula, keeping the plumb line in millimetres top and bottom:
GM = (w × s × length) ÷ (∆ × deflection) = (20 × 14.0 × 6000) ÷ (8000 × 84) = 2.50 m
Or in two familiar steps: GGH = (20 × 14.0) ÷ 8000 = 0.035 m; tan(List) = 84 ÷ 6000 = 0.014; GM = 0.035 ÷ 0.014 = 2.50 m. Same triangle, same answer.
10.3 The conduct of the experiment
The formula is trivial; the discipline is everything. The angles being measured are fractions of a degree, so anything else capable of heeling or restraining the ship, wind, wash, taut mooring lines, a gangway ashore, people wandering about, liquid slopping in a tank, contaminates the answer. The classic conditions are these:
The weights, typically four, sit in pairs on deck, port and starboard. Each in turn is shifted across and back, giving eight independent heels; two or three plumb lines, damped in troughs of oil so they settle rather than swing, are read for each. Averaging the port and starboard shifts cancels any steady bias, wind on one bow, a nipped line, and scatters random reading error.
MV Ninja is inclined at a displacement of 5954 t. Each shift moves an 11 t weight 16.0 m across the deck, and the deflections read on an 8.00 m plumb line are: 17.5, 17.3, 17.6, 17.2, 17.4, 17.5, 17.3 and 17.4 mm. Find the fluid metacentric height at the inclining condition.
mean deflection = (17.5 + 17.3 + 17.6 + 17.2 + 17.4 + 17.5 + 17.3 + 17.4) ÷ 8 = 139.2 ÷ 8 = 17.4 mm
GM = (w × s × length) ÷ (∆ × deflection) = (11 × 16.0 × 8000) ÷ (5954 × 17.4) = 1408000 ÷ 103599.6 = 13.59 m
A very large GM, and rightly so: the ship is nearly empty, floating high, and at so shallow a draught the metacentre stands enormously high (the booklet's hydrostatic table, which begins at 2.60 m draught, already shows KM above 19 m there, and the inclining draught is shallower still). Light ships are stiff ships. Note also that this measured value is the fluid GM of the ship that morning: one fresh water tank was unavoidably slack, and the next worked example pays for it.
10.4 From the pendulum to the booklet
The measured GM belongs to the ship as she floated that morning: inclining weights on deck, a shore gang's gear aboard, ballast pressed into the double bottoms to keep the propeller wet, a little fresh water, and one slack tank. Four steps carry it to the number the booklet prints.
At the inclining condition of Worked example 10.2 (∆ 5954 t, measured fluid GM 13.59 m), the Fresh Water (P) tank was slack; the tank table gives i = 145 m4, so with fresh water FSM = i × RD = 145 × 1.000 = 145 t m. The yard's inclining hydrostatics give KM = 21.484 m at the floating draught. Find the solid KG of the inclining condition, by two routes.
FSC = FSM ÷ ∆ = 145 ÷ 5954 = 0.024 m
Route A, through the fluid KG: fluid KG = KM − fluid GM = 21.484 − 13.59 = 7.894 m; solid KG = 7.894 − 0.024 = 7.870 m
Route B, through the solid GM: solid GM = 13.59 + 0.024 = 13.614 m; solid KG = 21.484 − 13.614 = 7.870 m
Two routes, one answer, and note the direction of the correction: the free surface made the ship seem more tender than her solid geometry, so removing it moves G back down. The inclining draught lies below the booklet table's first row, which is why the yard supplies the hydrostatics for this one calculation.
On board at the inclining, besides the ship herself, were: the inclining weights, 45 t at Kg 12.80 m; shore gang gear, 8 t at Kg 10.00 m; the No.5 D.B. ballast pair pressed full with salt water, 2 × 449.2 × 1.025 = 921 t at Kg 2.16 m; and 30 t of fresh water at Kg 11.86 m, the tabulated centre of the Fresh Water Tank (P). Strip the condition (∆ 5954 t, solid KG 7.870 m) back to the light ship.
| Item | w (t) | Kg (m) | moment (t m) |
|---|---|---|---|
| Inclining condition | 5954 | 7.870 | 46858 |
| inclining weights, landed | −45 | 12.80 | −576 |
| shore gang gear, landed | −8 | 10.00 | −80 |
| No.5 D.B. pair, pumped out | −921 | 2.16 | −1989 |
| fresh water, to empty | −30 | 11.86 | −356 |
| Light ship | 4950 | — | 43857 |
light KG = 43857 ÷ 4950 = 8.86 m, at a light displacement of 4950 t (check: 4950 × 8.86 = 43857 t m)
These are, to the digit, the figures printed on page one of MV Ninja's stability data booklet and used since Chapter 2. Notice that landing the low ballast raised the KG from 7.870 to 8.86: the Chapter 6 sign rules, working in reverse, one last time.
Suppose the slack fresh water tank of Worked example 10.3 had been overlooked, and no free surface correction applied. What light ship KG would have been recorded, and what would it have cost the ship?
The uncorrected calculation would take KG at the inclining condition as 7.894 m. Carried through the same stripping table: moments = 5954 × 7.894 = 47001; less the same 3001 t m of removals = 44000; light KG = 44000 ÷ 4950 = 8.89 m.
An error of +0.03 m, written into the booklet, inherited by every moments table, every GM and every list calculation for the rest of the ship's life. Small slack tank, permanent consequence: this is why the conditions of Section 10.3 are enforced so jealously, and why the inclining report records every tank's state.
10.5 The ship's stability information
The experiment's product is not one number but a book. The Administration requires the master to hold approved stability information sufficient to assess the ship's stability in any intended condition, and every part of it has already passed through your hands in this volume:
Years later, a lightweight check finds MV Ninja's light displacement to be 5010 t against the recorded 4950 t. Comment.
The drift is 60 t, which is 60 ÷ 4950 = 1.2 percent of the recorded light displacement.
The customary tolerances require a new inclining experiment when the light displacement is found to have drifted by more than 2 percent, or the longitudinal centre of gravity by more than 1 percent of the ship's length. At 1.2 percent this ship keeps her booklet, but the accumulation, paint, stores absorbed into the inventory, small additions never struck off, is recorded and watched: ships gain weight quietly with age, and usually high up.
Interactive: conduct the experiment yourself
This ship has a hidden GM. Shift the weights, watch her settle, read the pendulum, and average your readings, then compute GM with the sheet formula and check your answer. The drawn heel is exaggerated six times so you can see it; the deflection readout is true.
Interactive: the inclining pro forma
The whole of Section 10.4 as a live calculator, preloaded with the MV Ninja experiment. Change any figure and watch the light ship follow.
To strip off (weight, Kg):
Chapter summary
- The light ship KG cannot be calculated or read from draughts: it is measured, once, by the inclining experiment.
- GM = (w × s × length) ÷ (∆ × deflection): a known moment, a plumb line's tangent, GM the only unknown.
- The conditions, calm water, slack moorings, tanks pressed or empty, staff only aboard, guard angles of fractions of a degree.
- Eight port and starboard shifts are averaged; the measured GM is the fluid GM, corrected by the free surface moments (i × RD from the tank table) of any slack tank divided by ∆.
- KM from the inclining hydrostatics converts GM to KG; the moments table, run in reverse, strips the condition to the light ship.
- The result becomes the approved stability information: hydrostatics, tank tables and FSMs, maximum KG, cross curves, standard conditions and the inclining report, renewed after significant conversion or excessive drift at a lightweight check.
And with that, the foundation of Volume One is complete under its own weight: the booklet whose tables this book has drawn on since Chapter 1 has now been built, page by page, from a plumb line, a bag of weights, and the formulas of the chapters behind you.
Self test questions
Work each question with pencil and paper first. Your score appears in the bar below.