Timber Raft: Draught and Stability

What is modelled

A raft of touching round logs floats where the weight of wood and load equals the weight of the displaced water. How well it resists rolling depends on the metacentric height, the sum of the centre of buoyancy and the water-plane stiffness minus the height of the centre of gravity; a high load can make a wide raft unstable. Wind pushes the raft sideways against water drag, and a sweep blade steers it by pushing against the flow.

Equations

• Archimedes: submerged circular-segment area per log A = (ρw Alog + load/(nL)) / ρ, depth found by bisection.
• Stability GM = KB + BM − KG, BM = I / V with I = Σ (L c³/12 + L c x²).
• Drift v = √(2F / (ρ Cd A)); sweep force F = ½ ρ Cl As v².

Checks run in the page

• Displacement by numerically integrating the submerged logs against the raft mass.
• A heeled raft: the exact righting arm by slicing the logs against GM sin(angle).
• The settled wind slip from the stepped equation of motion.
• Sweep motion from a stepped simulation against the closed form.

Honest simplifications

Logs are straight, touching and identical, the raft is one layer, drag coefficients are single round numbers and the roll period uses a rough radius of gyration. The wind and sweep forces are steady and the river is treated as still relative to the raft. Real rafts are lashed in several sections with a flexible joint pattern, so this is a physics sketch, not a river-safety guide.

• Press Overloaded cargo raft and watch the freeboard shrink until the raft is waterlogged.
• Press Narrow raft, tall load to see the metacentric height fall and the righting arm curve go negative.
• Press Strong gale to see the sideways slip grow with the square of the wind speed.