Shaft Critical Speed Calculator (Whirling)
First and second lateral critical speeds of a shaft carrying a lumped mass or its own distributed weight, for simply supported and cantilever arrangements. The exact beam eigenvalues are used for the distributed case rather than the approximate square root of g over delta rule, and the difference between the two is reported so the shortcut can be judged for what it is.
How to use this calculator
- Choose simply supported or cantilever, and whether the mass is lumped at one point or spread uniformly along the span as the shaft weight.
- Enter the span, the Young's modulus, and the second moment of area about the bending axis, taking the smallest value if the shaft steps.
- Enter the mass carried, and the speed the machine actually runs at.
- Read the first critical speed, the static deflection, and how far the operating speed sits from the critical, then keep at least 25 percent clear on one side.
Frequently asked questions
Why does a shaft whirl at all?
Any real rotor has a small residual unbalance, which produces a centrifugal force that grows with the square of the speed. The shaft bends in response, the bending moves the centre of mass further out, and the two effects reinforce each other. At the critical speed the bending stiffness is exactly balanced by the inertia term, so the amplitude is limited only by damping, which is small in a steel shaft.
Is the square root of g over delta rule accurate?
It is exact for a single lumped mass at the point where the deflection is measured, and it is a useful lower bound otherwise. For a uniform shaft carrying its own weight it comes out about eleven percent low, because the rule treats the distributed mass as if it were concentrated. It is a good sanity check and a poor final answer.
What clearance from the critical speed do I need?
The usual rule is at least 25 percent, and often more for a machine that will run for years. Below about 80 percent of the critical speed a shaft is comfortably subcritical, and above about 125 percent it is deliberately supercritical. Round numbers between those two are where machines fail, and they fail slowly, by shaking themselves apart rather than by snapping.
My rotor has several masses, can I add up the single mass answers?
No, and the answer will be wrong in the unsafe direction. Use the Dunkerley method, which adds the individual flexibilities, or a Rayleigh-Ritz calculation. As a rule the combined critical speed is lower than the lowest single mass value, so treating the largest mass alone as if it were the only one will overstate the critical speed and hide a problem.