Find the "critical speed" of a rotating shaft carrying a disc. Adjust the shaft diameter, span, disc mass and operating speed to see the bending stiffness, static deflection and bending natural frequency that set the whirl-resonance speed in real time, and design rotating machines that stay clear of resonance.
Parameters
Shaft diameter d
mm
Bearing span L
mm
Distance between the two bearings (span)
Mounted disc mass m
kg
Mass of the rotor at mid-span
Operating speed N
rpm
Steady-state running speed of the machine
Support condition
How firmly the shaft ends are held
Results
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Second moment of area I (m⁴)
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Bending stiffness k (N/m)
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Static deflection δ (mm)
—
Critical speed N_c (rpm)
—
Operating / critical speed ratio
—
Critical-speed verdict
—
Rotating-shaft whirl animation
A shaft on its two bearings with a disc at mid-span. The closer the operating speed is to the critical speed, the more the shaft bows out and whirls.
Critical speed N_c [rpm]. k: shaft bending stiffness [N/m], m: rotor mass [kg]. For fixed-fixed ends k=192EI/L³. A stiffer shaft (larger diameter, shorter span) or a lighter rotor raises the critical speed, and machines must avoid running at it.
Second moment of area of the round section I [m⁴] and the static deflection δ_st [m] under the rotor weight. E: Young's modulus (steel, 206 GPa), g: gravitational acceleration.
Lateral (bending) natural angular frequency ω_n [rad/s]. The smaller the static deflection δ_st, the higher the natural frequency and the critical speed.
What is the critical speed of a shaft?
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"Critical speed" sounds ominous. Does a rotating shaft really have a speed like that?
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It does. Every rotating shaft — in a motor, a pump, a turbine, a machine-tool spindle — is an elastic body, so it has natural frequencies of lateral (bending) vibration. When you spin the shaft at exactly one of those natural frequencies, it bows out and whirls violently. That rotational speed is called the critical speed.
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But you're only spinning it — why does it shake so hard when no one is pushing on it?
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Good question. The reason is an unavoidable fact: no real shaft-and-rotor is ever perfectly balanced. Manufacturing tolerances leave the centre of mass slightly off the axis of rotation. As the shaft spins, that tiny eccentricity throws a rotating centrifugal force, and the frequency of that force equals the rotational speed. When the speed rises to the critical speed, this rotating force resonates with the shaft's bending natural frequency and the deflection grows and grows. That growing motion is the "whirl".
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And what actually happens when it resonates?
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The shaft bows out dramatically and whirls, and the bearings see large loads. You get severe vibration and noise, and in the worst case fatigue failure or a rub where the rotor scrapes the casing and wrecks the machine. So designers strictly avoid the critical speed. Raise the operating speed on the left toward the critical speed and you'll see the shaft start to whirl wide on the canvas below.
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Then keeping the critical speed high should be safe. How do you make it higher?
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The critical speed is N_c ∝ √(k/m) — it depends only on the shaft's bending stiffness k and the rotor mass m. So a thicker shaft, a shorter span between supports, fixed end supports, or a lighter rotor all raise it. There's a neat shortcut too: the critical speed is closely related to the static sag of the shaft under its own weight. The less it sags, the higher the critical speed — so the self-weight deflection gives you a quick read.
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But I've heard big turbines run faster than their critical speed. Is that okay?
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Well spotted. There are two design schools. A "subcritical" (rigid-shaft) design runs comfortably below the critical speed; a "supercritical" (flexible-shaft) design runs comfortably above it. Large steam turbines are often made supercritical to save weight. The catch is they must accelerate quickly through the critical speed at every start-up and shut-down — racing past before the resonance can build. Subcritical or supercritical, the one thing you must never do is run continuously right at the critical speed (speed ratio 0.75 to 1.3).
Frequently Asked Questions
The critical speed is the rotational speed at which a shaft turns exactly at one of its lateral (bending) natural frequencies. No real shaft is perfectly balanced, so a tiny mass eccentricity throws a rotating centrifugal force whose frequency equals the rotational speed. As the speed approaches the critical speed, that rotating force resonates with the shaft's bending natural frequency, the shaft bows out and whirls violently, and bearing loads, vibration and noise rise sharply. This tool computes the critical speed from N_c = (60/2π)√(k/m).
The critical speed depends only on the shaft's bending stiffness k and the rotor mass m: N_c ∝ √(k/m). The stiffness k scales with the fourth power of the diameter and inversely with the cube of the span. So to raise the critical speed you can make the shaft thicker, shorten the span between supports, stiffen the end supports (fixed instead of simply supported), or use a lighter rotor. The tool plots how the critical speed responds as you vary the diameter.
A design that runs comfortably below the first critical speed is called subcritical (a rigid-shaft machine); one that runs comfortably above it is supercritical (a flexible-shaft machine). Most machines keep the critical speed at least 1.3 times the operating speed and run subcritical. Large steam turbines and high-speed rotors are made supercritical to save weight, and they accelerate quickly through the critical speed at start-up and shut-down before resonance can build. In every case, continuous running near the critical speed (speed ratio 0.75 to 1.3) is avoided.
A handy shortcut: the critical speed is closely tied to the static sag δ_st of the shaft under the rotor's own weight. N_c ≈ 946/√δ_st (with δ_st in mm and N_c in rpm, the basis of Dunkerley's formula) — the less it sags, the higher the critical speed. This follows because the natural angular frequency ω = √(k/m) and the static deflection δ_st = mg/k give ω = √(g/δ_st). Measuring the self-weight sag early in design gives a fast estimate of the critical speed.
Real-World Applications
Pumps, fans and electric motors: Centrifugal-pump shafts, fan and blower rotors and general-purpose motor armatures almost all run subcritical, comfortably below the first critical speed. Designers keep the critical speed at least 1.3 to 1.5 times the operating speed to avoid resonance. In overhung pumps where the impeller is mounted on a cantilevered shaft end, the cantilever deflection lowers the critical speed, so the shaft diameter and bearing span must be chosen carefully.
Steam and gas turbines: Large power-generation turbine rotors are long and heavy, so their first critical speed often falls below the operating speed and they are deliberately run supercritical. They pass through the critical speed at every start-up and shut-down, but accelerate fast enough to race past before resonance builds, and they avoid steady running near it. Real machines use damper bearings or flexible supports to limit the amplitude during the critical-speed passage.
Machine-tool spindles and high-speed spindles: Machining-centre and grinder spindles can turn at tens of thousands of rpm, and managing the critical speed directly affects accuracy. The basic recipe is a short, thick spindle on stiff bearings that keeps the critical speed well above the working range. A long tool overhang effectively lengthens the span and lowers the critical speed, so tool length and cutting conditions must be considered together.
Troubleshooting and field diagnostics of rotating machinery: Problems like "vibration suddenly increases at a certain speed" or "bearings wear out early" are often caused by the operating point sitting too close to a critical speed. A quick estimate from a tool like this points to the critical speed, and a vibration measurement that sweeps the speed confirms where the resonance peak sits. In practice bearing stiffness and gyroscopic effects also matter, so a detailed rotordynamics (FEM) analysis is used to back up the estimate.
Common Misconceptions and Pitfalls
The most common mistake is assuming there is only one critical speed. This tool uses the simplest model — a single disc at mid-span, a one-degree-of-freedom approximation — and finds the first critical speed. A real shaft is a continuous body, so it has first, second, third and higher critical speeds. In multistage pumps and turbines that carry several rotors, every one of those critical speeds must be kept out of the operating range. The value here is only a first-order estimate; higher modes and rotor interactions need a separate rotordynamics analysis.
Next, assuming the bearings can be treated as perfectly rigid. The simply-supported and fixed-fixed conditions here assume the bearings and support structure are infinitely stiff. Real rolling and journal bearings and support pedestals have finite stiffness, which acts in series with the shaft's own stiffness, so the critical speed of the actual machine comes out lower than the calculated value. In oil-film journal bearings the bearing stiffness and damping even change with speed, and can trigger an unstable whirl called "oil whip". Bearing stiffness is not a factor you can ignore.
Finally, overconfidence that "once past the critical speed you are safe". A supercritical machine passes through the critical speed at start-up, but gyroscopic effects (a restoring action from disc tilt), internal shaft damping and overhung masses split the motion into forward and backward whirl, and the behaviour is not simple. If the acceleration is slow, the rotor lingers near the critical speed and the resonant amplitude grows. Design the start-up and shut-down sequence of a supercritical machine to pass quickly through the critical speed, and always avoid continuous running or long dwell near it.
How to Use
Enter shaft diameter (mm) using dNum slider; range 10–100 mm for steel shafts.
Set span length (mm) between supports using lNum; typical 200–1000 mm for industrial rotors.
Input disc mass (kg) in mNum field; represents mounted impeller or flywheel mass.
Set operating speed (rpm) in nNum; simulator calculates critical speed and displays ratio N/N_c.
Review output: second moment of area I, bending stiffness k, static deflection δ, and critical-speed verdict (Safe/Warning/Critical).
Worked Example
Steel shaft: diameter 25 mm, span 600 mm, disc mass 8 kg, operating speed 3000 rpm. Calculated I = 38.35 × 10⁻⁹ m⁴, k = 3520 N/m (E=200 GPa for steel), static deflection δ = 2.27 mm, critical speed N_c = 3347 rpm. Operating ratio N/N_c = 0.896 (Safe verdict: operating below critical speed). If speed increased to 3500 rpm, ratio becomes 1.046 (Critical verdict—risk of resonance whirl).
Practical Notes
Whirl occurs near critical speed; maintain N/N_c < 0.75 for rigid-rotor machines or > 1.25 for flexible rotors to avoid resonance zone.
Increasing diameter to 32 mm raises I by 122% and critical speed significantly; small diameter changes dominate stiffness in shaft design.
Off-center disc mass or unbalance amplifies deflection near critical speed; balance tolerance per ISO 20816 critical.
Long spans (L > 800 mm) reduce critical speed; consider intermediate bearing placement for high-speed applications.