Shaft Whirling & Critical Speed

The Jeffcott rotor and the critical speed

The whole phenomenon collapses onto one idealization: a single rigid disc of mass m mounted at the midspan of a light, flexible shaft running in two bearings — the Jeffcott (or Laval) rotor. The shaft acts as a lateral spring of stiffness k; the disc is the mass. Strip away the rotation and you have a textbook spring-mass system whose lateral bending natural frequency is:

ω_n = √(k / m)        (rad/s)

N_c = (60 / 2π) · √(k / m)   (rev/min)

where:
  k = shaft lateral stiffness at the disc   (N/m)
  m = disc (rotor) mass                      (kg)

The critical speed ω_c is the spin speed at which the shaft rotates once per natural period — i.e. ω_c = ω_n. The deep insight is that for a symmetric, undamped Jeffcott rotor the bending critical speed is numerically identical to the non-rotating lateral natural frequency. You can find a turbine's first critical speed by gently rapping the parked rotor and reading the ring frequency, no spin test required.

The mid-span lateral stiffness of a simply-supported shaft carrying a central load is the familiar beam result, so the critical speed is set entirely by geometry and material:

k = 48 EI / L³        (simply supported, central disc)
k = 192 EI / L³       (fixed-fixed, central disc)

I = π d⁴ / 64         (solid round shaft)

⇒ ω_c = √(48 EI / (m L³))

Two levers jump out. Stiffness rises with the fourth power of diameter and falls with the cube of span, so a shaft's critical speed is exquisitely sensitive to how far apart its bearings sit. Halve the span and k goes up 8×, pushing ω_c up by √8 ≈ 2.83×.

Why it whirls: imbalance, resonance and the rotating force

No rotor is perfectly balanced. The center of mass sits a small distance e — the eccentricity — off the geometric spin axis. As the shaft turns at ω, that offset mass throws a rotating centrifugal force outward:

F_unbalance = m · e · ω²

This force sweeps around once per revolution: a synchronous (1×) forcing. The shaft responds like any damped oscillator driven at frequency ω. Define the speed ratio r = ω / ω_n and damping ratio ζ. The dynamic shaft deflection (whirl radius) of the disc center is:

          e · r²
δ = ───────────────────────────
     √[ (1 − r²)² + (2 ζ r)² ]

At low speed (r → 0) the deflection is tiny. As r → 1 the denominator collapses and δ blows up — that is resonance, and the bowed shaft orbits its bearing centerline once per turn, tracing the orbit we call whirling. In the undamped limit the amplitude is bounded only by yielding, bearing clearance, or the shaft striking its housing. Real damping (ζ) caps the peak; the maximum whirl amplitude at the critical speed is approximately e / (2ζ), so a rotor with 2% damping can whirl with 25× its own eccentricity.

The phase flip: self-centering above the critical speed

The remarkable, counter-intuitive payoff sits on the far side of the resonance peak. The phase angle φ between the rotating unbalance force and the shaft deflection swings from 0° well below critical, through 90° exactly at critical, to 180° well above it:

Regime	Speed ratio r = ω/ω_n	Phase φ	Whirl amplitude δ	Heavy spot sits…
Subcritical	r < 1 (≈ 0–0.8)	≈ 0°	grows with r², small	on the outside of the orbit
At critical	r ≈ 1	90°	peak ≈ e/(2ζ), large	90° ahead of deflection
Supercritical	r > 1 (≈ 1.3+)	→ 180°	decays toward e	on the inside, near spin axis
Far supercritical	r ≫ 1	180°	δ → e (self-centered)	the geometric axis becomes the inertial axis

Above the critical speed the shaft bends so that the heavy spot moves inward, toward the spin axis, and the rotor begins to spin about its own center of mass instead of its geometric center. The whirl radius shrinks back toward e. This is why high-speed machines — steam turbines, gas-turbine spools, cryogenic turbopumps, automotive turbochargers — are designed to operate supercritically, deliberately living above one or more criticals. The art is in the transit: you must accelerate through the critical speed fast enough that resonance never has time to build to a destructive amplitude.

Worked example: critical speed of a pump shaft

A 30 mm diameter steel shaft spans 0.6 m between bearings and carries a 12 kg impeller at midspan. Treat it as a simply-supported Jeffcott rotor. Find the first critical speed.

Geometry / material:
  d = 0.030 m,  L = 0.60 m,  m = 12 kg
  E = 200 GPa = 2 × 10¹¹ Pa

Second moment of area:
  I = π d⁴ / 64 = π (0.030)⁴ / 64
    = π × 8.1 × 10⁻⁷ / 64
    = 3.976 × 10⁻⁸ m⁴

Lateral stiffness (central load, simply supported):
  k = 48 EI / L³
    = 48 × (2 × 10¹¹) × (3.976 × 10⁻⁸) / (0.60)³
    = 48 × 7952 / 0.216
    = 1.767 × 10⁶ N/m

Critical speed:
  ω_c = √(k / m) = √(1.767 × 10⁶ / 12)
      = √(1.473 × 10⁵)
      = 383.8 rad/s

  N_c = ω_c × 60 / (2π) = 383.8 × 9.549
      ≈ 3,665 rev/min

So this pump must not be run continuously near 3,665 rpm. If the duty point is 2,950 rpm (a 2-pole 60 Hz induction motor), the rotor sits at r ≈ 0.80 — uncomfortably close. A designer would either fatten the shaft to 35 mm (k rises 1.85×, N_c rises to ≈ 4,990 rpm) or shorten the span to open the margin to the recommended ±20–25%.

Multiple critical speeds and higher modes

A real shaft is a continuous beam, not a single spring-mass, so it has an infinite series of bending modes — and therefore a series of critical speeds, one per mode. The first is the gentle single-bow shape; the second adds a node and looks like an S; the third has two interior nodes. For a uniform simply-supported shaft the mode shapes are half-sine waves and the natural frequencies scale as n² (1, 4, 9 …), so the criticals spread out rapidly.

• 1st critical (cylindrical / single bow): the whole rotor sags to one side. Lowest, most dangerous, almost always within the operating envelope of high-speed machines.
• 2nd critical (conical / S-bend): a node appears near midspan; the ends whirl out of phase. Common in long line shafts and multistage compressors.
• Higher modes: matter for very long, slender rotors and flexible-rotor balancing, where you must balance at speed in several planes simultaneously.
• Gyroscopic stiffening: for an overhung disc, the spin-rate-dependent gyroscopic moment splits each critical into a forward-whirl and backward-whirl branch — the rising forward-whirl line on a Campbell diagram is what synchronous (1×) unbalance actually intersects.

Design levers: moving the critical speed

• Diameter: the strongest lever. I ∝ d⁴, so k ∝ d⁴ and ω_c ∝ d². Going from 30 to 36 mm raises the critical speed 44%.
• Span: k ∝ 1/L³, so ω_c ∝ L^(−1.5). Add an intermediate bearing to a long line shaft and you split one long span into two stiff ones, jumping the critical well clear.
• Rotor mass: ω_c ∝ 1/√m. Hollow shafts and lighter discs help, but mass is usually fixed by function (an impeller has to move fluid).
• Bearing stiffness: the bearings are springs in series with the shaft. Soft hydrodynamic films lower the effective k and the critical speed — tilting-pad bearings let you tune it deliberately.
• Material: ω_c ∝ √E. Steel (E ≈ 200 GPa) versus titanium (≈ 115 GPa) versus aluminum (≈ 70 GPa) shifts the critical, though density partially offsets the gain.

Subcritical vs. supercritical design

	Rigid / subcritical rotor	Flexible / supercritical rotor
Operating speed	Below 1st critical (r < 0.75)	Above 1st (and often 2nd) critical
Typical machines	Slow pumps, fans, machine-tool spindles, electric-motor rotors	Steam/gas turbines, turbopumps, turbochargers, jet-engine spools
Whirl behavior	Heavy spot on outside; amplitude grows toward critical	Self-centering; amplitude → eccentricity e
Critical speed handling	Stay well below — never reach it	Accelerate quickly through it; use dampers
Damping requirement	Low	High during transit — squeeze-film dampers essential
Balancing	Single/two-plane rigid-rotor balance suffices	Modal / at-speed flexible-rotor balancing in multiple planes
Risk	Drifting upward into the critical (overspeed)	Lingering in the critical band (slow start, trip on a critical)

Failure modes and trade-offs

• Resonant whirl at the critical speed. The headline failure: an unbalanced rotor parked at or sweeping slowly through its critical builds whirl amplitude until the shaft contacts the casing, bearings overload, or the shaft yields. Mitigation: keep operating speed ±20–25% off any critical, or transit quickly with damping.
• Oil whirl and oil whip. A subsynchronous instability in plain journal bearings: the oil film drives a whirl at roughly 0.42–0.48× shaft speed. When that whirl frequency locks onto the first critical (oil whip), it self-excites and is destructive regardless of balance. Cure with tilting-pad or pressure-dam bearings.
• Bearing and seal wear. Even tolerable whirl applies a rotating 1× load to the bearings every revolution, driving fatigue spalling and seal rub. The unbalance force m·e·ω² grows with the square of speed, so a small eccentricity becomes a large force at high rpm.
• Mass-imbalance growth in service. A thrown turbine blade, fouling deposits on a fan, or a bent shaft suddenly increases e, shifting the rotor toward resonance and raising vibration. Condition-monitoring trip limits (e.g. ISO 20816 zones) catch this before failure.
• Internal (hysteretic) damping instability. Above the first critical, internal material damping can paradoxically drive a forward whirl unstable — a classic flexible-rotor trap. External (bearing) damping must dominate internal damping to stay stable.
• Thermal bow (Newkirk effect). A localized hot spot from a rub bows the shaft, increasing e, which worsens the rub — a runaway thermal-mechanical loop most dangerous near a critical speed.

The overarching trade-off mirrors beam design: stiffening the shaft (more diameter, shorter span, extra bearings) raises the critical speed and keeps operation subcritical, but adds weight, cost and bearing count. Letting the rotor be flexible and running supercritically buys lightness and high speed, but demands precise modal balancing, robust damping, and a control system that never lets the machine dwell in a critical band.