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Sea, gimbal and compass
The bowl behaves like a damped pendulum with natural frequency 5.2 rad/s (period 1.21 s): waves much slower than that are filtered out, waves near it are amplified unless damping is high. The needle is a separate damped dipole with natural frequency 3.7 rad/s.
Gimbal
Model checks
The science
What is modelled
The hull rolls and pitches sinusoidally with a slowly modulated amplitude. The gimballed bowl hangs from pivots and is restored toward the vertical by gravity, with viscous damping. The magnetic needle is a damped dipole that is pulled toward the magnetic bearing by a restoring torque proportional to the sine of its misalignment.
Equations
- Gimbal:
θ'' = -ωn² sin(θ + roll) - 2ζωn θ', withωn = 5.2 rad/s. - Needle:
α'' = -ωm² sin(α - target) - 2ζmωm α'. - Isolation ratio (small angles):
r √(r² + 4ζ²) / √((1 - r²)² + (2ζr)²), withr = ωwave/ωn.
Checks run in the page
- Free gimbal period 2 pi / wn.
- Forced response against the closed-form isolation ratio.
- Damped needle period.
- Energy of the undamped finite-amplitude pendulum.
Honest simplifications
Two uncoupled pendulum axes, a prescribed roll motion that ignores the ship's own dynamics, no deviation or variation, no acceleration errors and no magnetic dip. It teaches the mechanics of the gimbal; it is not a navigation instrument model.
What to try
- Press Resonant swell: with the wave period near 1.2 s the bowl tilt exceeds the hull roll.
- Press Heavy sea and compare the bowl tilt with the hull roll, then lock the gimbal.
- Press Over-damped and watch the bowl follow the hull slowly.