Shaft Critical Speed (Rayleigh + Dunkerley)

dynamicsstress

Verified build 9 relations · 4 identities proven · 4 modeling steps · 3 parity samples

Spin any shaft with a wheel on it fast enough and it stops running true and starts to whirl — the shaft bows out and the bow chases the rotation, throwing the whole rotor into a widening orbit. The speed where this erupts is the critical speed ω_c, and it is nothing more exotic than resonance: the spinning rotor forcing itself at exactly the natural bending frequency of the shaft-and-disk it is made of. Turbine and pump rotors, motor armatures, machine-tool spindles, and long driveshafts are all designed by putting ω_c comfortably away from the running speed. This page is the simplest honest model of it — one disk at the middle of a shaft on two bearings.

The static sag already knows the answer

Rayleigh’s insight is that the whirl at ω_c has the same shape as the shaft’s own droop under gravity, so the critical speed follows from that one static number. Hang the disk’s weight $W = mg$ at midspan of a simply-supported shaft and it sags by

$\delta_{st} = \frac{W L^3}{48\,E I}$

— which is literally the headline result of the simply-supported beam page, the central-load deflection, reused here without change. Rayleigh’s method then gives the critical speed straight from that sag:

$\omega_c = \sqrt{\frac{g}{\delta_{st}}} = \sqrt{\frac{48\,E I}{m L^3}}$

The second form is the point. Substitute $\delta_{st}$ and $W = mg$ and the gravity cancels: what is left is $\omega_c = \sqrt{k/m}$ with the beam stiffness $k = 48EI/L^3$ — an ordinary mass-on-a-spring. Gravity sets the sag, but the same gravity sets the restoring force, so it drops out of the frequency entirely; a rotor would have the identical critical speed in orbit where there is no $g$ at all. Critical speed is pure stiffness over inertia.

g is a defined constant, not a knob

The $g$ that enters here is the standard acceleration of gravity, 9.80665 m/s². It is not a measurement to be dialled in — it is a conventional value fixed by definition (3rd CGPM, 1901), the same everywhere by agreement even though the real field runs from about 9.78 m/s² at the equator to 9.83 m/s² at the poles. So it appears on this page the way a material property does: a labeled, cited value with a source, never a slider. It is the site’s first such physical constant.

Two material axes pulling opposite ways

Rayleigh’s single mass ignores the shaft’s own weight. Dunkerley’s method puts it back by combining the disk’s critical speed with the bare shaft’s first bending mode $\omega_s = (\pi^2/L^2)\sqrt{EI/\rho A}$ through their reciprocal squares, and the result is provably never higher than Rayleigh’s — the true first critical speed is bracketed between them. Watching the two numbers move exposes the material story, and it is two stories, not one:

Stiffness $E$ raises the critical speed. A stiffer shaft is a stiffer spring; swap mild steel for a titanium alloy (lower $E$ ) and ω_c drops — the same counter-intuitive result the Euler column page keeps hammering: strength is not stiffness.
Density $\rho$ lowers the Dunkerley estimate. A heavier shaft carries more of its own distributed inertia, which drags $\omega_s$ — and with it $\omega_{cD}$ — down. Density and stiffness are independent axes, and here you can see them pull the two estimates in opposite directions on one screen.

The disk mass $m$ , by contrast, is a plain knob, not a material: it is a payload bolted to the shaft (an impeller, a flywheel, a grinding wheel), so you set it directly.

Reading the margin

The widget reports ω_c and the Dunkerley estimate as speeds (rpm by default), the frequency $f_c$ in Hz, and the ratio of running speed to critical speed. Drive the operating speed into the shaded band within 20% of ω_c and it warns: that is where an undamped rotor’s whirl amplitude runs away. As on the eccentric column, the danger is not where a first glance puts it — it is not about how strong the shaft is at all, but about a speed, and the margin has to be taken on that speed. Cross the band deliberately (many machines run supercritical, above ω_c) and you must pass through resonance on the way up.

Try it

Material

T3, bare flat sheet 0.010-0.128 in. thick, AMS 4037 / AMS-QQ-A-250/4 (MIL-HDBK-5J Table 3.2.3.0(b1), p. 3-71)

Bound properties of 2024-T3 aluminum sheet (bare)
E	10.5 Msi	typical	mil-hdbk-5j
rho	0.1 lb/inch**3	typical	mil-hdbk-5j

Constants

Cited physical constants used on this page
g	Standard gravity	9.80665 m/s²	nist

Materials modeled here: 2024-T3 aluminum sheet (bare) 304 stainless steel 6061-T6 aluminum 7075-T6 aluminum AISI 1045 medium-carbon steel AISI 4340 low-alloy steel (Ni-Cr-Mo) ASTM A36 structural steel (hot-rolled) C26000 Cartridge Brass (70/30) Douglas-fir (coast) Gray Cast Iron, ASTM A48 Class 30 Normal-weight concrete, f'c = 4000 psi class Nylon 6/6 (PA66), unfilled Ti-6Al-4V

Governing relations

I = \frac{\pi d^4}{64}

Assumes: solid circular shaft, lateral bending second moment of area

Source: Gere, J. M., & Goodno, B. J., Mechanics of Materials, 9th ed., Cengage, 2018 — App. E section properties (I = πd⁴/64, A = πd²/4 for a solid circular shaft) and the simply-supported beam small-deflection theory behind δ_st.

A = \frac{\pi d^2}{4}

Assumes: solid circular shaft cross-section

W = m\,g

Assumes: the disk is a rigid payload at midspan; its weight is what deflects the shaft; g is the cited standard gravity, not a knob

Source: Budynas, R. G., & Nisbett, J. K., Shigley's Mechanical Engineering Design, 10th ed., McGraw-Hill, 2015 — §7-6 (critical speeds of rotating shafts: the Rayleigh single-mass estimate ω = √(g/δ), Dunkerley's equation combining reciprocal squares, and the operating-speed margin) and Table A-9 (the simply-supported center-load deflection reused for δ_st).

\delta_{st} = \frac{W L^3}{48\,E I}

Assumes: simply supported (bearings as pin/roller), the disk weight applied at midspan; linear elastic, small deflections — the simply-supported center-load result reused verbatim from the beam page (its headline deflection)

\omega_c = \sqrt{\dfrac{g}{\delta_{st}}}

Assumes: single central mass on a massless shaft — Rayleigh's method; the whirl at ω_c resonates with the shaft's own static-deflection shape (physics enters here by citation, no ODE solved); undamped, small whirl; gyroscopic effects neglected · Valid while: The static sag exceeds L/100 — a very flexible rotor. The small-deflection beam theory behind δ_st is stretched, and the single-mode Rayleigh estimate loses accuracy. This is a short, stubby shaft (L < 10d): shear deflection, neglected by the Euler–Bernoulli theory behind δ_st and ω_s, is no longer small — the real shaft is more flexible than modeled, so the critical speeds shown here are OVERestimates.

f_c = \frac{\omega_c}{2\pi}

Assumes: the critical speed in cycles per second; ω_c is the same speed in radians per second

\omega_s = \frac{\pi^2}{L^2}\sqrt{\dfrac{E I}{\rho A}}

Assumes: the bare uniform shaft (no disk) as a simply-supported beam vibrating in its first bending mode, sin(πx/L) — the exact Euler–Bernoulli eigenvalue for that mode; uniform prismatic shaft, distributed mass ρA per length

\frac{1}{\omega_{cD}^2} = \frac{1}{\omega_c^2} + \frac{1}{\omega_s^2}

Assumes: Dunkerley's method — the reciprocal squares of the partial critical speeds add; a cited approximation that folds the shaft's own distributed mass into the disk estimate; always a LOWER bound on the true first critical speed (provably ω_cD ≤ ω_c here)

\mathrm{sr} = \frac{\omega_{op}}{\omega_c}

Assumes: how close the running speed sits to the Rayleigh critical speed; sr near 1 is resonance · Valid while: The operating speed is within 20% of the critical speed — the resonance band. Near ω_c an undamped rotor's whirl amplitude grows without bound, so running speeds are kept clear of the band (below about 0.8 ω_c or above about 1.2 ω_c). The operating speed is within 20% of the Dunkerley estimate ω_cD — the LOWER end of the bracket that contains the true first critical speed (ω_cD ≤ ω_true ≤ ω_c). The Rayleigh-based speed ratio sr can look safe here while the rotor runs near the real resonance; keep clear of the whole [0.8 ω_cD, 1.2 ω_c] band.

Derivation

Steps marked modeling step are where physics enters by citation — every other line is machine-proven to follow from them, and the cited models are independently re-derived in the test pipeline where possible. See what is and isn't machine-verified.

\delta_{st} = \frac{L^{3} W}{48 E I}

1. Start with the shaft as a spring. The disk sits at midspan of a simply-supported shaft, so its weight W deflects it by δ_st = WL³/48EI — the simply-supported center-load deflection, taken verbatim from the beam page (that page's headline result; the build does not re-integrate the beam here, so this line is imported by citation). — reuse the simply-supported center-load deflection modeling step

k_{beam} = \frac{48 E I}{L^{3}}

2. Invert it: the midspan stiffness — the force it takes to push the shaft down one metre — is k = W/δ_st = 48EI/L³. Fatter shaft (d⁴, through I) means a stiffer spring; longer span means a much softer one. — beam stiffness is the inverse of the deflection influence

\omega_{c}^{2} = \frac{g}{\delta_{st}}

3. The modeling step, where the physics enters by citation. At the critical speed the shaft whirls in resonance with its own static-deflection shape; Rayleigh's method treats the disk as a single mass on that beam-spring, so mÿ = −k y and ω_c² = k/m. With k = W/δ_st and W = mg this is ω_c² = g/δ_st. We do NOT integrate the whirl equation — the resonance result is cited from the rotordynamics literature and checked only for consistency. — Rayleigh single-mass critical speed (cited) modeling step

\omega_{c} = \frac{\sqrt{k_{beam}}}{\sqrt{m}}

4. Now substitute δ_st = WL³/48EI and W = mg. The gravity cancels: ω_c = √(g/δ_st) = √((W/δ_st)/m) = √(k/m) = √(48EI/mL³). The very weight that produced the sag also produces the restoring force, so g drops out entirely — the critical speed is a pure stiffness-over-inertia property. It is √(k/m), the same formula as any mass on a spring; you could measure ω_c on a rotor in orbit where there is no g at all. — substitute the beam-spring: g cancels

f_{c} = \frac{\omega_{c}}{2 \pi}

5. Cycles per second is radians per second over 2π. As on the oscillator page, f_c and ω_c are deliberately different quantity kinds despite sharing units of 1/s — the 2π is always explicit, so a 42 Hz whirl is never mistaken for a 42 rad/s spin. Critical speeds are usually quoted as N_c in rpm (the readout toggles), which is ω_c·60/2π. — definition of frequency

\omega_{s}^{2} = \frac{\pi^{4} E I}{A L^{4} \rho}

6. The single mass ignores the shaft's own weight. Model the bare uniform shaft as a simply-supported beam vibrating in its first bending mode, sin(πx/L); its natural frequency is ω_s = (π²/L²)√(EI/ρA). This is the exact first eigenvalue for that mode — the test pipeline re-derives it independently from the mode-shape energy (Rayleigh) quotient, constants and all. — uniform-shaft first bending mode (cited; re-derived in tests) modeling step

\frac{1}{\omega_{cD}^{2}} = \frac{1}{\omega_{s}^{2}} + \frac{1}{\omega_{c}^{2}}

7. Dunkerley's method combines the disk's critical speed and the shaft's own by adding their reciprocal squares. It brings the neglected shaft mass back in, giving a safe estimate that includes the distributed inertia the single-mass Rayleigh value leaves out. — Dunkerley superposition of reciprocal squares (cited) modeling step

\frac{1}{\omega_{cD}^{2}} - \frac{1}{\omega_{c}^{2}} = \frac{1}{\omega_{s}^{2}}

8. Rearranged, the gap between the two estimates is exactly 1/ω_s², which is positive, so ω_cD ≤ ω_c always — adding inertia can only lower a natural frequency. Dunkerley therefore underestimates and Rayleigh overestimates; the true first critical speed is bracketed between them. This inequality is machine-verified here and sampled numerically in the test pipeline. — Dunkerley ≤ Rayleigh, proven

How it fails

The failure at a critical speed is not a static overload — the shaft can be nowhere near its yield stress when it lets go. It is resonance. Every real rotor has a little residual imbalance, an eccentricity between its mass centre and its spin axis, and that imbalance is a rotating force that sweeps through every frequency as the machine runs up. When the running speed reaches ω_c it drives the shaft at exactly its own natural bending frequency, and the whirl amplitude climbs — in this undamped model, without bound. Real damping caps it, but the peak can still be tens of times the static sag, enough to close running clearances: the rotor rubs its seals or housing, the bearings see wildly reversing loads, and the reversing bending drives fatigue cracks. Machines have thrown blades and snapped shafts passing through a critical speed they were never meant to sit on.

Below, above, or straight through

Because the danger is a band and not a ceiling, there are two honest ways to live with it. Most machinery runs subcritical, below about 0.8 ω_c, which is why raising the critical speed (a fatter or shorter shaft, a stiffer material) buys margin. But many high-speed machines — turbines, turbochargers, some pumps — deliberately run supercritical, above ω_c, exploiting the fact that past the critical speed the rotor tends to spin about its mass centre and self-centres. The price is that they must accelerate through the resonance on the way up (and coast back through it on the way down) fast enough that the amplitude never has time to build. The one speed you may not choose is a steady one inside the band — which is exactly the state this widget shades and warns on.

Why the single number is a floor, not the truth

This page models one disk on a uniform shaft in rigid bearings, and every one of those words is a simplification that a real rotor breaks:

One mode, many criticals. A rotor with several wheels, or a long flexible shaft, has a whole sequence of critical speeds; this is only the first. The Rayleigh and Dunkerley estimates bracket that first one — Dunkerley from below, Rayleigh from above — so the true value sits between the two readouts, and quoting Rayleigh alone slightly overstates it.
The bearings are not rigid. Real bearing and pedestal flexibility adds series compliance, which can only lower the critical speed — so the rigid-support number here is optimistic, an upper bound on what a real machine achieves.
Gyroscopics and damping. An overhung or wide disk stiffens gyroscopically as it spins, so its critical speeds actually shift with rotational speed (splitting into forward and backward whirl); fluid-film bearings can even feed energy in and drive an unstable whirl that has nothing to do with imbalance at all. None of that lives in this static-sag model.

The companion vibration page, the torsional oscillator, is the twisting analogue of the same story: there the shaft rings in torsion rather than whirling in bending, but the lesson is identical — a natural frequency set by stiffness over inertia, and a resonance to be kept clear of. And as on the eccentric column, the margin that matters is not the one a stress calculation would hand you.

Rolling-Contact Bearing Life (Load–Life + Weibull Reliability)
A bearing's catalog rating C₁₀ is not a strength — it is the load at which 90% of a population survives one million revolutions. Halve the load and the fatigue life jumps eightfold (ball) or more (roller); ask for higher reliability than 90% and the life you can count on drops sharply. This page turns the two catalog numbers into a life in hours, and the reliability you demand into the life you actually get.
- dynamics
- reliability
Stepped Shaft — Shoulder-Fillet Stress Concentration
A shoulder where a shaft steps from a large diameter D to a small diameter d concentrates stress in the fillet of radius r. The peak stress is the nominal stress times a geometric factor K_t read from a cited chart — pure geometry: the material changes the safety factor but never K_t.
- stress
Bolted Joint with Gasket (External Tensile Load)
A preloaded bolt clamping a gasketed joint, then pulled by an external tensile load. The bolt and the members act as two springs in parallel, so the external load does NOT all go to the bolt — it splits by stiffness. The build solves the coupled bolt/member force system exactly and refuses the moment the members go slack.
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Helical Compression Spring
A torsion bar wound into a package: push on the coil and the wire twists. G sets the rate, σ_y sets the margin, and the geometry trades them against three envelopes — coil bind, buckling, and a spring index you can actually wind.
- stiffness
- stress
- mass-cost
Spur Gear Pair (Lewis Bending)
Two meshing spur gears carry power through one tangential tooth force. The Lewis equation turns that force into a root-bending stress, and a cited form-factor table sets how much a tooth of N teeth can take. Same load, same module — yet the pinion, with fewer teeth, always works harder.
- stress
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Axial Disk Clutch / Brake (Uniform Wear vs Uniform Pressure)
The torque an axial plate clutch can pass depends on an assumption you cannot see: how the contact pressure is distributed across the friction annulus. A new, rigid clutch presses uniformly; a worn-in one wears until pressure ∝ 1/r, concentrating load at the inner edge. This page shows both torque predictions side by side — never picking a winner — with the worn-in model always giving the smaller (safe) number, and the r_i = r_o/√3 that squeezes the most torque from a given lining.
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Chains with

Outputs whose SI dimension and quantity kind match another THING's input — the only wires the planner's connectionLegal accepts (invariant 2, computed at build time, not hand-listed). Wire these on the chaining demo.

Composite Bar (Core + Sleeve)
- A A_1
- A A_2
- delta_st L
- W P
Impact Loading (Falling Mass, Energy Method)
- delta_st b
- delta_st d
- delta_st h
- delta_st L
Symmetric Two-Bar Truss
- delta_st d
- delta_st L
- W P
Thermal Assembly (Two-Segment Bar Between Rigid Walls)
- A A_1
- A A_2
- delta_st L_1
- delta_st L_2
Cantilever Beam (End Load)
- delta_st b
- delta_st h
- delta_st L
- W P
Circular Plate under Uniform Pressure (Clamped vs Simply Supported)
- delta_st a
- delta_st t
Curved Beam in Bending (Winkler — Crane Hook, C-Clamp, Press Frame)
- delta_st r_i
- delta_st r_o
- delta_st w
- W P
Fixed-Fixed Beam (UDL)
- delta_st b
- delta_st h
- delta_st L

+ 28 more THINGs its outputs can legally feed (showing the first 8 in course order).

Sources

Budynas, R. G., & Nisbett, J. K., Shigley's Mechanical Engineering Design, 10th ed., McGraw-Hill, 2015 — §7-6 (critical speeds of rotating shafts: the Rayleigh single-mass estimate ω = √(g/δ), Dunkerley's equation combining reciprocal squares, and the operating-speed margin) and Table A-9 (the simply-supported center-load deflection reused for δ_st).
Gere, J. M., & Goodno, B. J., Mechanics of Materials, 9th ed., Cengage, 2018 — App. E section properties (I = πd⁴/64, A = πd²/4 for a solid circular shaft) and the simply-supported beam small-deflection theory behind δ_st.
Juvinall, R. C., & Marshek, K. M., Fundamentals of Machine Component Design, 5th ed., Wiley, 2011 — shaft design and critical (whirling) speed: the same Rayleigh √(g/δ) estimate and the Dunkerley combination, with operating speeds kept away from the resonance band.
BIPM, The International System of Units (SI Brochure), 9th ed., 2019, and NIST Special Publication 330 (2019): the standard acceleration of gravity gₙ = 9.80665 m/s², a conventional value fixed BY DEFINITION (3rd CGPM, 1901) — not a measurement.