Ship Stability Simulator Online: Restoring Arms & Roll Physics

Q: How does the ship stability simulator evaluate initial metacentric height (GM)?

Initial metacentric height GM is evaluated via GM = KM - KG, confirming stable equilibrium when positive and governing initial resistance to heeling.

Q: Which equation governs the GZ righting arm for moderate heel inclinations?

GZ is evaluated using the wall-sided formula: GZ = GM * sin(theta) + 0.5 * BM * tan^2(theta) * sin(theta), simplifying to GZ = GM * sin(theta) for small inclinations.

Q: What defines the behavioral difference between a 'Tender' and a 'Stiff' ship?

A tender ship has low GM and long, sluggish roll cycles (18 to 25 seconds). A stiff ship has excessive GM, producing violent, rapid rolls (4 to 7 seconds) that induce dangerous accelerations on cargo securing systems.

Q: What is an Angle of Loll and how does the simulator demonstrate it?

An Angle of Loll results from negative GM. The upright position is unstable, causing the vessel to flop over to an angle of equilibrium theta_loll where form buoyancy stabilizes the list.

Q: How do Free Surface Effects (FSE) impair vessel stability?

Slack liquids shift outboard during rolling, creating a virtual rise in the center of gravity (GG1) that directly diminishes effective fluid metacentric height: GM_fluid = GM - GG1.

⚓ NAVLIB • NAVAL ARCHITECTURE

Ship Transverse Stability Simulator

Interactive naval architecture simulator of righting levers, centers, and forces. Click and drag the vessel to heel and watch dynamic restoration!

🎯 Scenarios:

Heights & Centers (m) 📐

KG (Center of Gravity): 7.2 m

Higher KG (deck cargo) degrades stability and reduces initial GM.

KM (Transverse Metacenter): 8.7 m

KM = KB + BM. Depends on hull geometry, beam, and draft.

KB (Center of Buoyancy): 4.4 m

Centroid of underwater submerged volume in upright condition.

Geometry & Loading 🚢

Draft (T): 8.0 m

Displacement (Δ): 12 000 t

Vessel mass. Determines magnitude of Weight P and Buoyancy Δ.

Heeling Moment (Wind): 0 t·m

Beam wind pressure or cargo shift causing steady list.

Heel Angle (θ) 🔄

Current Heel (θ): 0.0°

💡 Adjust slider OR drag vessel directly on canvas!

Heel Angle: 0.0°

GZ Lever: 0.000 m

M_righting: 0 t·m

👋 Click and drag the ship to heel

📈 Statical Stability Curve (GZ Curve) Heel: from -60° to +60°

Calculated Results 📊

GM (Metacentric Ht)
1.50m
KM - KG (initial)

GZ (Righting Arm)
0.00m
at current heel

Righting Moment

0t·m

M = Δ · GZ

Rolling Period (T)

12.4s

T = 2cB / √GM

BM (Radius M)

4.30m

KM - KB

Ship Beam (B)

20.0m

at waterline

Vectors & Layers 👁

Weight Vector (P = Δ) Buoyancy Vector (Δ) Righting Arm GZ Metacenter M & GM Immersed / Emerged Wedges Restoring Moment Arc Key Points K, B0, B', G, Z

• Center of Gravity (G) & KG

Point through which the ship's total weight P acts vertically downwards. KG is the height of G above keel baseline K. Higher KG (e.g. timber/containers on deck) decreases GM and degrades stability.

• Center of Buoyancy (B & B')

Centroid of displaced underwater volume. When vessel heels, one side immerses (immersed wedge) while the other rises (emerged wedge). This transverse volume transfer shifts B sideways to B'.

• Metacenter (M) & GM

Intersection of the vertical line of buoyant force with the ship's centerline. GM = KM - KG is the initial metacentric height. When GM > 0, vessel is stable; when GM < 0, vessel is unstable upright.

• Righting Lever (GZ)

Horizontal perpendicular distance between the vertical lines of action through G and B'. Produces the righting couple moment M = Δ · GZ, returning the vessel upright.

• Tender vs. Stiff Vessel

Tender (Low GM): long, sluggish roll period T, comfortable for crew but vulnerable to squalls.
Stiff (High GM): violent, rapid roll with high transverse accelerations, risking cargo lashing failure.

• Angle of Loll (Negative GM)

Occurs when GM < 0. The upright position (θ = 0) is unstable and the ship spontaneously lolls over to port or starboard until reaching the stable Angle of Loll θ_loll due to form stability.

Naval Architecture Hub STABILITY // REAL-TIME-PHYSICS

UKC Pro Calculator ↗ Squat & UKC Calculator ↗ GZ Curve & IMO Code ↗

Interactive Ship Stability Simulator: Restoring Levers, Metacentric Heights & Roll Physics

Mastering the principles of transverse stability is vital for navigating officers, Chief Mates, and Shipmasters during cargo loading, ballasting, and heavy weather passage planning. The NavLib Ship Stability Simulator provides an intuitive, real-time graphical environment to evaluate the equilibrium couple between downward gravitational force (weight P) acting through center of gravity G and buoyant lift acting through center of buoyancy B as the hull inclines.

The simulation engine is grounded in classical naval architecture: the wall-sided stability formula, damped periodic rolling equations, and Krylov's roll period determination. Users can adjust height of center of gravity (KG), height of metacenter (KM), draft (T), and displacement directly, or select operational presets—analyzing compliant loading conditions, tender vessel behavior, and hazardous negative metacentric heights (Angle of Loll).

Hydrostatic Vectors

GZ & Restoring Couples

Live rendering of key points K, B, G, M, and Z, calculating the horizontal righting arm GZ and net restoring moment M_rest = Displacement * GZ.

Volumetric Wedges

Submerged & Emerged Areas

Visual mapping of water transfer wedges, displaying how volume shifts drive center of buoyancy B outboard to B' as the vessel inclines.

Roll Dynamics Engine

Real-Time Motion & GZ Curve

Interactive periodic roll motion with hydrodynamic damping, beam wind moments, and concurrent static stability curve plotting from -60° to +60°.

Frequently Asked Questions: Ship Stability Simulator (FAQ)

How does the ship stability simulator evaluate initial metacentric height (GM)?

The initial metacentric height is calculated via the fundamental relation: GM = KM - KG, where KM is the height of transverse metacenter M above the keel (KM = KB + BM) and KG is the height of the center of gravity above the keel. A positive GM confirms stable upright equilibrium, while negative GM indicates instability.

Which equation governs the GZ righting arm for moderate heel inclinations?

For wall-sided vessels up to the deck edge immersion threshold, GZ is resolved using the wall-sided formula:
GZ = GM * sin(theta) + 0.5 * BM * tan^2(theta) * sin(theta)
For inclinations beyond deck edge immersion, the simulation engine dampens righting arm growth to reflect diminished reserve buoyancy.

What defines the behavioral difference between a "Tender" and a "Stiff" ship?

Motion characteristics are dictated by the natural roll period equation: T = (2 * c * B) / square_root(GM), where B is ship beam and c ≈ 0.38.
Tender Ship (small GM, under 0.20 m): generates low restoring moments with prolonged, sluggish roll cycles (18 to 25 seconds). Roll motions are gentle on crew but leave the ship vulnerable to wind squalls.
Stiff Ship (excessive GM, exceeding 2.5 to 3.5 m): creates rapid, jerky roll periods (4 to 7 seconds). The ship returns upright violently, exerting severe transverse forces on cargo lashings and container tiers.

What is an Angle of Loll and how does the simulator demonstrate it?

An Angle of Loll develops when initial GM is negative (GM < 0) because center of gravity G has risen above metacenter M. The upright conning position (theta = 0°) is unstable; the vessel flops over to an angle of equilibrium theta_loll where form buoyancy restores positive stability:
tan(theta_loll) = square_root( -2 * GM / BM ).
The simulator includes an "Angle of Loll" preset demonstrating that ballasting the low side can cause catastrophic dynamic rollover to the other side.

How do Free Surface Effects (FSE) impair vessel stability?

Liquids within slack tanks shift outboard during heeling, shifting the ship's aggregate center of gravity toward the low side. This creates a virtual rise in the center of gravity by a loss margin GG1: GG1 = (liquid_density * i) / Displacement, where i is the second moment of area of the liquid surface, decreasing effective fluid metacentric height: GM_fluid = GM_solid - GG1.

Regulations: IMO 2008 IS Code, SOLAS II-1, STCW Table A-II/1

Physics Engine: Wall-Sided Formula, Damped Roll Dynamics, Angle of Loll

Interaction: Interactive Drag Heel, Live GZ Chart, Preset States