RT4 Ship Roll Period Calculator

The Formula

EXACT PERIOD
T = T₀ × (2/π) × K(sin²(φ_max/2))

CORRECTED GM RECOVERY
T₀ = T_obs × π / (2·K(m)), m = sin²(φ_max/2)
GM = (C × B / T₀)²

SMALL-ANGLE BASELINE (no correction)
GM_sa = (C × B / T_obs)² [biased low at large amplitudes; T_obs > T0 in the denominator]

K(m) — complete elliptic integral of the first kind from classical Bernoulli/Euler mechanics. Exact for GZ = GM·sin(φ). Use validated wall-sided correction or vessel GZ tables when richer hull data are available.

Inputs

Observed roll period T_obs (s) Roll amplitude φ_max (deg) Vessel beam B (m) Schofield coefficient C

Or use k-factor: C = 2πk/√g (k typically 0.33–0.42)

Gyration factor k (optional, overrides C)

True GM (m) Beam B (m) Gyration factor k Roll amplitude φ_max (deg)

Vessel type (for C-factor table)

Results

GM recovered from observed roll period:

Small-angle

—

C-factor table

—

RT4 Exact K(m)
—
m

—

GM correction factor
GM_sa × this factor = GM_RT4

Parameter	Value
T_obs	—
φ_max	—
Elliptic modulus m	—
K(m)	—
T₀ (small-angle)	—
T₀ (RT4 corrected)	—
SA bias (under-estimate)	—

Parameter	Small-angle	RT4 K(m)
GM recovered (m)	—	—
Error vs true	—	—

Period vs Roll Amplitude

GM Bias of the Small-Angle Method (true GM larger than SA estimate)

Accuracy Notes

Linear GZ (GZ = GM·sinφ): Formula is exact to 211 ppm vs numerical ODE. GM recovery mean error < 1 µm across 120 S4-validated test cases (vs 20.7 mm for the uncorrected small-angle method — a 20-billion-fold improvement).

Wall-sided hull (GZ = sinφ·(GM + BM/2·tan²φ)): The standalone K(m) formula is approximate for wall-sided curves. The Python package includes a validated wall-sided correction envelope for φ ≤ 30° and BM/GM ≤ 4. For arbitrary hull forms, use vessel GZ tables or direct numerical integration.

Publication status: open technical reference implementation; not class-approved software, not a loading computer, and not a substitute for professional stability review.