Shaft Critical Speed (Deflection Method)

The engineering

The elegance here is that gravity does your modal test for you: hang the rotor on its shaft, and the sag under its own weight already encodes the stiffness-to-mass ratio that sets the first critical. One millimeter of static deflection means roughly 946 rpm — memorize that anchor and you can scale any shaft by √δ in your head.

The number that bites is proximity, not the critical itself: run within ~25% of Nc and residual unbalance gets amplified into orbiting that eats bearings and seals. Big turbomachinery routinely operates *above* the first critical — the trick De Laval demonstrated — but only by accelerating through resonance briskly and with damping; lingering there is how shafts get bent.

Where this math comes from

William Rankine analyzed 'whirling' shafts in 1869 and concluded, wrongly, that operation above the critical speed was impossible — an error with a decent grip on the profession until Gustaf de Laval simply did it, running his 1889 cream-separator and steam-turbine shafts supercritically on deliberately flexible spindles that let the rotor spin about its own mass center.

Stanley Dunkerley's 1894 experiments gave engineers the summation formula for multi-mass shafts, and Henry Jeffcott's 1919 analysis of the whirling rotor finally explained *why* De Laval's trick works: above resonance the rotor self-centers. The Jeffcott rotor remains rotordynamics' hydrogen atom; this card is its ground state.

1. 1869William RankineWhirling speed identified — and supercritical running wrongly forbidden.
2. 1889Gustaf de LavalRuns turbine shafts above critical on flexible spindles.
3. 1894Stanley DunkerleyEmpirical formula for critical speeds of loaded shafts.
4. 1919Henry JeffcottWhirling rotor theory — self-centering above resonance explained.

See the full timeline of the math behind every calculator →

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