An explicit damper between two inertias is stable on their reduced inertia, and a gear ratio shrinks it by the ratio squared

Fact

Coupling two rotating bodies with a spring-damper and integrating explicitly is the usual way to avoid writing a constraint solver. It is stable only while

c * dt / I_red < 2,      I_red = 1 / (1/I1 + 1/I2)

where I_red is the reduced inertia of the pair, not either body's own. A binary search on c puts the threshold at 2.001 for every combination of I1, I2 and dt tried — it is a property of the scheme, not of the numbers.

A gear ratio n between the two bodies is what makes this bite, because the far inertia referred to the near side is I2 / n². With a 0.26 kg·m² crank, a 2.4 kg·m² axle, c = 26 N·m·s/rad and dt = 1/400 s, the same coupling is comfortably stable in a 3.13:1 gear (c·dt/I_red = 0.52) and diverges within ten steps in a 13.5:1 one (5.19).

Why it matters

The symptom is not a NaN. It is a plausible-looking oscillation: torque alternating at its clamp, contact forces flipping sign every step, a body juddering. That reads as a friction, contact or controller problem, and it appears only in the configurations with the largest ratio — first gear, a winch's low range, a reduction drive — so it survives every test taken in the others.

Raising the substep rate helps, and halving dt only halves the number, so the fix looks like it "nearly worked" and invites more of it.

How to apply

• Compute c * dt / I_red at the worst ratio in the system and keep it under about 1. That is a two-line assertion worth having permanently.
• Where the ratio is large, drop the compliance: treat the coupling as either locked — one rigid body, the small inertia added to the large one — or slipping at a bounded torque, and choose per step from the torque the rigid solution would demand. That is unconditionally stable and needs no solver.
• Related: [[friction-must-scale-with-normal-impulse]].