Critical Speed of a Shaft

A rotating shaft is a beam that happens to be spinning, and like any beam it has a bending natural frequency. The critical speed is the rotational speed at which the shaft turns once per bending cycle — so the unbalance force, which rotates with the shaft, arrives in step with the shaft's own resonance and pumps it every revolution. The shaft bows out and whirls, and the deflection grows until damping or something breaking limits it.

The history is unusually instructive. Rankine analysed the problem in 1869 and concluded that a shaft simply could not be run above its critical speed — the deflection, he reasoned, would grow without bound. Dunkerley measured shafts through the 1890s and found otherwise, and Jeffcott explained why in 1919: above the critical, the rotor spins about its own centre of mass rather than about the bearing axis, and the whirl amplitude settles down instead of growing. That is why turbines, turbochargers and most high-speed centrifugal compressors run supercritical as a matter of course. The critical speed is a speed to pass through, not a ceiling — and confusing it with an operating limit is the classic error on this page.

The form here, $N_c = \frac{1}{2\pi}\sqrt{48EI/(mL^3)}$, is the simply supported shaft with a single central disc, because $48EI/L^3$ is the mid-span stiffness of exactly that beam. Change the support conditions and only that coefficient changes: a shaft built in at both ends gives $192EI/L^3$, four times stiffer and twice the critical speed; an overhung rotor is far softer. Notice the $L^3$ and the $d^4$ hiding inside $I = \pi d^4/64$ — shortening a span by 20% raises the critical speed about 40%, and that is nearly always the cheapest fix available.

What the equation leaves out matters as much as what it includes. It ignores the shaft's own distributed mass, which lowers the true critical; Dunkerley's method combines the shaft and the rotor contributions and is the standard hand correction. It assumes rigid bearings, and on soft pedestals or long housings the support stiffness can dominate the shaft entirely — machines have been rebuilt with stiffer shafts and shown no change at all for this reason. It ignores gyroscopic effects, which stiffen an overhung rotor and raise its critical with speed. And it finds only the FIRST critical: there are higher ones, and a machine climbing to a high running speed may pass through several. Common practice keeps the running speed below about 0.75 of a critical or above about 1.4 times it, and treats this calculation as the first estimate it is.