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Model checks
The science
What is modelled
A box chronometer sits in a gimbal ring so its dial stays nearly level while the ship rolls. In one axis the dial platform behaves like a pendulum hanging on a pivot with bearing friction: gravity pulls it back to true vertical, and friction damps its swing.
Equations
Tilt of the dial from true vertical ψ = roll + φ, with φ the gimbal angle relative to the ship: φ″ = −ωn²(roll + φ) − c φ′, ωn² = 44 s⁻² (pendulum length g/ωn² ≈ 22 cm), c = 2ζωn. For sinusoidal roll of frequency Ω the tilt amplitude is |H|·A with |H| = √(Ω⁴ + c²Ω²)/√((ωn² − Ω²)² + c²Ω²): waves much faster than ωn are filtered out, slow swells are followed.
Honest simplifications
One axis only (the 3D page has two), a point-mass pendulum with viscous bearing friction, a ship that rolls about a fixed axis through the gimbal pivot (no heave or sway accelerations), and no chronometer rate effects; an actual box chronometer also has balance-wheel and spring errors that are not modelled here.
Built-in checks
The forced-response amplitude against the closed-form transfer function, the logarithmic decrement of a free ring-down, and the removal of a constant heel.