Drawing No. EH–ME–010 // Mechanical Engineering
Shaft Critical Speed Calculator
Reviewed August 2026
Find a simply-supported shaft's first critical speed from a concentrated mass (such as a disk or coupling) and the shaft's own self-weight, combined using Dunkerley's approximation.
What problem does this solve?
Every rotating shaft has natural frequencies at which it will resonate — run a shaft at or near one of these critical speeds and vibration amplitude grows sharply, risking damage. This calculator estimates the first (lowest) critical speed for a simply-supported shaft carrying a concentrated mass, combined with the shaft's own distributed self-weight effect using Dunkerley's well-established approximation method.
Inputs
Shaft
Concentrated mass
Operation
Results
Background
For a point mass m at midspan of a simply-supported shaft, the shaft acts as a spring of stiffness k = 48EI/L³ (the same relationship as midspan beam deflection). The critical speed is ωcr = √(k/m), converted to rpm as Ncr = 60ωcr/(2π).
A uniform simply-supported shaft's own first-mode natural frequency is ω₁ = (π/L)² √(EI/m'), where m' is mass per unit length (ρ×cross-sectional area). This captures the shaft's own distributed inertia, separate from any added concentrated mass.
1/Ncr,combined² = 1/Ncr,mass² + 1/Ncr,self-weight². This classical approximation combines the effects of multiple mass contributions on a shaft's first critical speed, and is conservative (gives a combined critical speed at or below the true value), which is appropriate for a screening-level check.
Running a shaft too close to its critical speed causes rapidly growing vibration amplitude as speed approaches resonance. Common practice keeps continuous operating speed below roughly 70–80% of the first critical speed (for shafts operating below critical, sometimes called "stiff shaft" design) with adequate separation margin, though specific margin requirements vary by application and standard.
Frequently asked questions
Practical questions about inputs, assumptions and interpretation.
Operating a shaft at or near a critical speed causes resonance — vibration amplitude grows sharply, which can cause bearing damage, seal failure, fatigue cracking, or in severe cases catastrophic shaft failure. Rotating equipment is generally designed to operate with adequate margin away from critical speeds, either running well below the first critical speed ("stiff shaft") or, for some high-speed turbomachinery, safely passing through it during startup/shutdown and operating above it ("flexible shaft") with proper analysis.
Dunkerley's approximation systematically underestimates the true combined critical speed of a system with multiple mass contributions — meaning the real shaft's actual first critical speed is typically somewhat higher than this calculation suggests. This makes it a safe (conservative) screening tool: if Dunkerley's estimate already shows healthy margin, the real design is likely fine, but a low Dunkerley estimate always warrants closer analysis rather than dismissal.
No — this assumes ideal simply-supported (rigid, pinned) end conditions, common for a first-pass estimate. Real bearings have finite stiffness, which generally lowers the actual critical speed below a rigid-support prediction — for shafts running close to a critical speed margin, bearing stiffness should be included in a fuller rotordynamic analysis.