This document defines, reproducibly, the physical steering-trail model used by our calculator. It is not an explainer: it is the method record — the derivation, the formulas, the dynamic model with its sign, and how it was validated. Anyone can reproduce every figure.
The head angle θ is measured from the horizontal (the geometry-chart convention: 64° is slacker than 73°). 2D model in the central plane, rigid frame, point contact, trail at zero steering. It is the same convention manufacturers use to compute the trail in their charts, so results are directly comparable.
with R = wheel radius to the ground and offset = fork rake. Mechanical trail is the perpendicular distance to the steering axis —the one that generates the self-centering torque—; ground trail is what appears in charts. The wheel-flop factor follows Tony Foale's convention (Motorcycle Handling and Chassis Design). Reference case (enduro: R 370, offset 44, θ 64°): ground trail 131.5 mm, mechanical 118.2 mm, flop 51.8.
Radius takes two labeled modes. [MEASURED]: from the real rolled circumference, R = C/2π. [ESTIMATED]: R = BSD/2 + width, with the ISO bead-seat diameter (622 for 29"/700c, 584 for 27.5", 559 for 26") and tire height approximated by width. An estimate is never presented as a measurement.
The exact derivatives of ground trail (the citable asset; they yield reproducible magnet figures):
In the reference case: +1° slacker → +7.6 mm of trail; +7 mm of offset → −7.8 mm; +10 mm of diameter → +4.9 mm; going from 27.5" to 29" (Δdiameter 38 mm, same width) → +9.3 mm.
When the fork compresses the A2C (axle-to-crown) drops; the front falls, the head angle steepens (θ increases) and, since ∂trail/∂θ < 0, the trail drops. This is the opposite direction to overforking (a longer fork, larger A2C, makes the bike slacker).
| Scenario | θ | Trail | Reading |
|---|---|---|---|
| Front compression (braking) | 64° → 66.0° | 131.5 → 116.7 (Δ −14.8) | steering gets nervous when you grab the brakes |
| Balanced SAG (seated, full) | 64° → 63.8° | 131.5 → 133.2 (Δ +1.7) | ≈ no change: seated ≈ chart |
At balanced SAG the front (steepens) and the rear (slackens) nearly cancel: the net change is negligible. The dramatic number lives in the front-only transient (braking), and points to steeper / less trail.
The model was validated against an independent numerical oracle implemented from scratch. Two checks:
The linearization of the dynamic term (arctan ≈ x) is exact up to ~3° of change; the error versus a full trigonometric recompute is on the order of 0.01 mm of trail across the entire realistic SAG range.
License of the method and this document: CC BY 4.0 — use and cite it with attribution. © 2026 Carlos Eduardo Ravello Joo · BikeLab Studio.