Comparison of Righting Moment (R.M.) Values-Surface Monohull Ship vs. Submerged Submarine

Compare R.M. (righting moment) values of (a) surface monohull ship and (b) a submerged submarine with the help of diagrams showing positions of K, B, G and M.

The Fundamental Difference in Stability Mechanism

Before comparing R.M. values, it is essential to understand that a surface ship and a submerged submarine derive their transverse stability from entirely different geometric mechanisms, even though both are ultimately governed by the same basic principle — the righting moment is the product of the displacement (W) and the righting arm (GZ), i.e., R.M. = W × GZ. The difference lies in how GZ arises and behaves as the vessel heels, which fundamentally changes the shape and magnitude of the R.M. curve between the two cases.

(a) Surface Monohull Ship — Stability Mechanism

Positions of K, B, G, M:

• K (keel, baseline reference) is at the bottom.
• B (centre of buoyancy) lies below the waterline, at the centroid of the underwater hull volume.
• G (centre of gravity) is typically above B, at a height determined by the vessel’s loading and design.
• M (metacentre) is above G (for a stable ship) — the metacentre is the point about which the centre of buoyancy appears to pivot as the ship heels through small angles, and its height (KM) is a function of the underwater hull’s waterplane second moment of area (I): BM = I/∇ (where ∇ is the volume of displacement).

Mechanism of righting moment generation:

• As a surface ship heels, the underwater hull shape changes — one side emerges from the water while the other side becomes more deeply immersed, causing the centre of buoyancy B to shift horizontally toward the low (immersed) side.
• This horizontal shift of B, relative to the fixed position of G, creates the righting arm GZ, and hence the righting moment.
• Because this mechanism depends on the change of underwater wedge shape (emerged wedge/immersed wedge) at the waterplane, GM (metacentric height) and hence initial stability is strongly dependent on the ship’s waterplane area and beam — a wide waterplane gives a large I, hence a large BM, hence typically a stable, positive GM.
• The R.M. curve (GZ curve) for a surface ship typically rises from zero at zero heel, reaches a maximum GZ at a moderate angle of heel (often 30–50° depending on hull form and freeboard), and then declines, reaching zero (the angle of vanishing stability) at some larger angle — commonly 60–90°+ for a well-found ship — beyond which the ship would capsize since GZ becomes negative.

(b) Submerged Submarine — Stability Mechanism

Positions of K, B, G, M:

• For a submerged submarine, the entire hull is completely enclosed by water — there is no waterplane, no “emerged wedge/immersed wedge” effect possible, since the hull volume displaced does not change at all with heel angle (unlike a surface ship, whose displaced volume shape changes as it heels).
• Because of this, B (centre of buoyancy) is fixed relative to the hull — it does not shift as the submarine heels, since the underwater (fully submerged) volume and its shape relative to the hull remain constant regardless of heel angle.
• This is the single most important distinguishing point: for a submerged submarine, M and B effectively coincide (BM = 0, since there is no waterplane moment of inertia term to generate a metacentric radius) — i.e., the metacentre M is at the same point as the centre of buoyancy B.
• G, as with a surface ship, is positioned according to the vessel’s mass distribution, and for a submarine, G must be maintained below B for the vessel to be transversely stable when submerged — this is why submarines are specifically designed and ballasted (via the tank arrangement in Q.2) to keep a positive BG separation (G below B) once dived.

Mechanism of righting moment generation:

• Since M coincides with B for the submerged case, the righting arm mechanism becomes simply the direct “pendulum” effect of G being below B: as the submarine heels, gravity acting through G and buoyancy acting through B (which does not move relative to the hull) create a righting couple analogous to a weighted pendulum or a “keel-heavy” body — the righting arm GZ = BG × sin(heel angle), directly and simply, rather than through the more complex wedge-transfer mechanism of a surface ship.
• This is a fundamentally more robust and “large-angle stable” mechanism than the surface ship’s waterplane-dependent stability: because the righting arm is generated by a fixed BG separation acting like a pendulum, the submerged submarine’s GZ curve is close to a pure sine curve, remaining positive through very large heel angles — theoretically even to 90° heel and, in principle, only vanishing at 180° (fully inverted), as long as G remains below B (i.e., BG stays positive) at all heel angles, which it inherently does for a submerged body since B does not move relative to the hull as it rotates.
• Comparative R.M. magnitude: For a given displacement, the actual numerical value of BG for a submerged submarine is typically quite small compared to the GM of a well-designed surface ship, since submarine designers deliberately keep G close to B (a large BG would require unnecessarily heavy keel ballast and waste buoyancy/payload capacity) — submarines are designed with just enough positive BG for adequate stability, not the generous GM margins used in surface ship design. So while the range of positive stability (angle of vanishing stability) for a submerged submarine is far greater than for a surface ship, the magnitude of the righting moment at small angles may actually be comparable or even less than a surface ship’s, depending on the specific BG value chosen by the designer versus the ship’s GM.

Diagram Description (as would be drawn)

Surface ship (heeled condition):

• Waterline shown, hull cross-section heeled to one side.
• K at keel/baseline.
• B shown below the waterline, displaced horizontally toward the low side relative to the upright position (this horizontal shift is the key feature to illustrate).
• G shown on the vessel’s centreline (fixed, does not move with heel — this is important to show as a contrast to B’s movement).
• M shown above G on the vertical through the upright centre of buoyancy, essentially fixed for small angles (metacentre assumption valid up to ~10–15°).
• GZ shown as the horizontal distance between the vertical line through G and the vertical line through the (new) B — i.e., the perpendicular from G onto the line of action of the buoyancy force through B.

Submarine (heeled condition, submerged):

• Fully submerged hull cross-section, heeled to one side, with no waterline/waterplane shown at all (fully submerged — this is the key visual distinction from the surface ship diagram).
• B and M shown as coincident, at the geometric centre of the submerged volume, and this point does not move relative to the hull as the hull heels (contrast directly with the surface ship’s B which does move).
• G shown below B/M, fixed relative to the hull.
• As the hull heels, the line KG (through the hull) rotates with the hull, but the vertical line of buoyancy action through B/M remains vertical (through the fixed point B=M) — the righting arm GZ is the horizontal separation that develops between G and the vertical through B as the hull rotates, forming the BG·sin(θ) relationship directly.

Summary Table

Let me know when you’d like me to go back and cover Q.4 (which was skipped), or continue to Q.6.