Does a twin with a rigid drivetrain miss the hunting cycle caused by gear backlash?

research/backlash-twin

A rigid-drivetrain twin says the PI loop converges but gear backlash makes the real loop hunt at 1.3–2.9× the gap (exactly ∝ h); describing-function prediction 7–10% high, and bistable starts can hide it.

8 tests

What it found

On a discrete PI loop the rigid twin has spectral radius 0.71–0.89 and converges, yet the real loop with a gap of half-width h settles on a limit cycle of motor amplitude 1.30–2.94h that scales exactly with h (1.302317 for h=0.01, 1, 100). The backlash describing function is an excitation-dependent gain (0.21 at A/h=1.2, 0.99 at 30) with a phase lag, and harmonic balance predicts the cycle but overestimates its amplitude by 6.7–10.1%. Negative result: the loop is bistable in some settings (hunting 1.165h from small starts, converging from large steps), so a large-step validation can pass a configuration that hunts. Follow-up to the deadband twin. Stylised: scalar loop, symmetric known gap, noiseless.

A figure from its own output

Figure produced by research/backlash-twin: describing_function — backlash-twin/experiments/describing_function.svg, written by the project's own experiment code.

Paper, code and results

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Produced with AI assistance (Claude). Results are from the code in this repository and are simulated or stylised unless the abstract says otherwise; nothing here implies affiliation with or endorsement by any lab, researcher or company cited. MIT licensed.