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Whirling Speed Calculator

Enter the static deflection in millimetres to find the critical whirling speed.

How it is calculated

The tool uses the Rayleigh estimate in a short form: Nc = 946 / √δ rpm, where δ is the static deflection of the shaft in millimetres, meaning how much it sags under its own weight or an equivalent load when held horizontally. The constant 946 comes from (60 / 2π) × √g with g = 9810 mm/s², so it already includes gravity and works only when δ is in mm. The speed is then also given as rad/s = rpm × 2π / 60 and Hz = rpm / 60. The result is the first critical speed of a simple shaft. Deflection must be above zero, and rpm is rounded to a whole number. Real shafts with several discs or bearings need a fuller analysis.

Worked example

A shaft sags by 0.25 mm under its own weight. The square root of 0.25 is 0.5, so Nc = 946 / 0.5 = 1,892 rpm. Converting: 1,892 × 2π / 60 ≈ 198.13 rad/s, and 1,892 / 60 ≈ 31.53 Hz. Running the shaft anywhere near 1,892 rpm risks whirling, so a design would keep the operating speed well away from it. If the shaft were four times more flexible, with a 1 mm sag, the critical speed would halve to 946 rpm.

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Common mistakes

  • Entering the deflection in metres or inches. The constant 946 only works with millimetres, so 0.00025 (metres) gives a speed around 60,000 rpm instead of 1,892.
  • Entering the shaft's length or diameter in place of the static deflection. The deflection must be calculated or measured first from the beam formulas for your shaft and supports.
  • Operating at the result. The critical speed is a danger zone, and shafts are normally run well below or above it, not at it.
  • Expecting the deflection under the operating load. It is the sag when the shaft's weight (and any fixed masses on it) acts across its span while it is stationary.
  • Treating the value as exact. It is an estimate of the first critical speed only, and the real value depends on bearing stiffness, additional discs and the shaft's support.

About this calculator

Every rotating shaft has a natural bending frequency, and if it spins at that speed the small, unavoidable imbalance in the shaft gets amplified into a large, self-reinforcing bow called whirling or whipping — left unchecked, it can bend or snap the shaft. Rayleigh’s approximation puts the critical speed at Nc = (60/2π) × √(g/δ) rpm, where δ is the shaft’s static deflection under its own weight (or an equivalent load) and g is gravitational acceleration.

The formula shows that a stiffer, less-deflecting shaft has a higher, safer critical speed, while a longer or more flexible one whirls at a lower, more dangerous one. Mechanical and rotating-equipment engineers — designing turbine rotors, pump shafts, motor spindles and generator shafts — use this estimate to keep the normal operating speed safely away from the critical speed, or to plan for rapid pass-through if the machine must cross it during start-up.

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