Everything so far has been a still photograph: the ship upright, or listed, or trimmed, but not moving. This chapter lets her move. The GM that has been a number in a table becomes something a person on board can feel: it sets the beat of the roll, the lean in a turn, and the ship’s vulnerability to one particular kind of sea. The calculations are short; the judgement they support is not.
Push a ship over and let her go: she swings back through upright, out to the other side, and returns. The time for one complete swing, port to starboard and back to port, is her rolling period, T. Two facts about it matter. First, it belongs to the ship, not the sea: in still water she always rolls at the same beat. Second, the size of the roll makes no difference to the time it takes, at least for the moderate angles at which GZ = GM sin θ holds: a five degree roll and a fifteen degree roll both complete in the same T, like a pendulum. At large angles, where the GZ curve falls away from the straight line GM sin θ, the period lengthens.
K measures how far, on average, the ship’s weight sits from the rolling axis. It is not about how much weight there is, but where it is. Spread the same tonnes further from the axis, by winging out cargo or ballast, and K grows and the roll slows; concentrate them near the axis and the roll quickens. An ice skater does the identical trick: arms out, slow spin; arms in, fast spin. Moving weight sideways at the same height does not move G, so winging out changes K without touching GM: it is the one lever that slows the roll and costs no stability. For a merchant ship K comes out at between a third and two fifths of the beam; the IS Code estimate for MV Ninja at summer draught is K = CB with C = 0.373 + 0.023 B/d − 0.043 L/100 = 0.367, that is 8.89 m, which this chapter rounds to 8.9 m.
The formula puts GM under a square root in the bottom line: big GM, short period; small GM, long period. Seafarers have older words for the two ends of the scale. A stiff ship, loaded low with a large GM, snaps upright hard and rolls fast and jerkily: safe on paper, punishing on cargo lashings and crew. A tender ship, loaded high with a small GM, rolls slowly and lazily, hanging at the end of each swing: comfortable, and short of stability in reserve. The next two examples put numbers on both moods of the same ship.
MV Ninja at summer draught, loaded low: K = 8.9 m, GM = 2.24 m. Find her rolling period.
T = 2π × 8.9 ÷ √(9.81 × 2.24) = 55.92 ÷ 4.688 = 11.9 seconds.
A twelve second beat: brisk, positive, a stiff and safe ship that will be hard work in a seaway.
The same ship loaded to the same summer displacement with a tall deck cargo and her double bottom tanks empty, so that KG = 9.88 m: K = 9.4 m, GM = 0.45 m. Find the new period.
T = 2π × 9.4 ÷ √(9.81 × 0.45) = 59.06 ÷ 2.101 = 28.1 seconds.
The K barely moved; the GM collapsed, and the square root turned a twelve second roll into a twenty eight second one. A long lazy period is the feel of a small GM, and an officer who notices the ship hanging at the end of her roll is reading the GM without a calculator.
This condition would not be allowed to sail: the maximum KG table of the data booklet gives 9.64 m at 30456 t, and at KG 9.88 m the area under the GZ curve between 30 and 40 degrees is 0.017 m rad against the 0.030 m rad required. It is used here to show how a tender ship behaves.
The period formula runs backwards. Time the roll, assume a K, and out comes an estimate of GM. At sea this is done by timing several complete rolls and averaging, because one roll is hard to clock cleanly.
An officer times ten complete rolls in 126 seconds. Taking K = 8.9 m, estimate the GM.
One period T = 126 ÷ 10 = 12.6 seconds.
GM = (2π × 8.9 ÷ 12.6)² ÷ 9.81 = (55.92 ÷ 12.6)² ÷ 9.81 = 19.70 ÷ 9.81 = 2.01 m.
An estimate, not a survey: it stands on the assumed K. But it is honest enough to notice trouble, and close to the 2.24 m the ship actually had in Worked example 14.1.
Put the wheel over and the ship is forced round a circle. The turn pushes her outward through G; the water resists roughly through B. Those two forces, separated by the vertical distance BG, form a couple that heels her outward, and the GM is what resists it. Speed enters squared: half the speed means a quarter of the heel.
MV Ninja turns at 15 knots on a 300 m radius at summer draught (KB 5.041 m, KM 10.33 m, beam 24.2 m, draught 9.60 m), first loaded low (KG 8.09 m), then tender (KG 9.88 m, GM 0.45 m). Find the heel and the under keel clearance in 12.60 m of water for each condition.
Speed: 15 knots = 7.717 m/s, so v² = 59.55.
Stiff: BG = 8.09 − 5.041 = 3.049 m, GM = 2.24 m. Tan of the heel = 59.55 × 3.049 ÷ (9.81 × 300 × 2.24) = 0.0275, heel = 1.58°, call it 1.6° outward.
Tender: BG = 9.88 − 5.041 = 4.839 m, GM = 0.45 m. Tan of the heel = 59.55 × 4.839 ÷ (9.81 × 300 × 0.45) = 0.2176, heel = 12.28°, call it 12.3°. The same wheel heels this condition nearly eight times as far (the tangents are in the ratio 7.9): the lever BG grew by a factor of 1.59 and the resistance GM shrank by a factor of 4.98, and both changes point the same way.
Maximum draught when heeled = half beam × sin of the heel + draught × cos of the heel, using the heels to two decimal places. Stiff: 12.1 × sin 1.58° + 9.60 × cos 1.58° = 0.33 + 9.60 = 9.93 m, clearance 12.60 − 9.93 = 2.67 m. Tender: 12.1 × sin 12.28° + 9.60 × cos 12.28° = 2.57 + 9.38 = 11.95 m, clearance 0.65 m.
Upright, both conditions had 3.00 m under the keel. The tender turn spent 2.35 m of it. A tender ship is turned gently, and doubly so in shallow water.
A ship rolling among waves is a pushed swing. If the waves arrive at random compared with her beat, the pushes mostly cancel. But if the apparent period of the waves (the encounter period), the time between crests as the ship meets them, matches her natural rolling period, then every wave pushes just as she swings, and the roll grows cycle by cycle. This is synchronism, and it is the most dangerous rolling condition a ship can be in: cargo shift, structural damage, downflooding, and in the extreme, capsize.
MV Ninja is in the loaded condition of Worked example 14.1 (natural period 11.9 seconds) with a beam sea whose apparent period is 12 seconds. Assess the situation and state the measures available.
The apparent wave period matches her natural period within a tenth of a second: this is synchronism, and the rolling will build swing on swing until something changes.
The escapes, in order of practicality at sea: alter course, which changes the apparent wave period, and an alteration towards the sea shortens it fastest; alter speed, effective whenever the sea is anywhere but exactly abeam; and as harbour options rather than seagoing ones, change the GM or wing weights out to move the natural period itself.
The pattern to carry: short seas have periods near those of stiff ships and long swell has periods near those of tender ships, so a stiff ship is most at risk in short beam seas, and a tender ship in long ones. A sea on the quarter lengthens the apparent period, which is exactly the wrong direction for a tender, long period ship.
The rolling period T = 2πK ÷ √(g × GM): it belongs to the ship, and the size of the roll does not change it.
K is where the weight sits: winging out slows the roll without costing any GM.
Big GM rolls fast and hard (stiff); small GM rolls slow and lazy (tender): 11.9 against 28.1 seconds on the same hull.
Heel in a turn: tan = v² × BG ÷ (g × radius × GM), outward, with speed squared; a tender ship leans hardest and spends her own under keel clearance.
Synchronism is the matched swing: break it by altering course, towards the sea for the quickest change, or altering speed.