First critical speed
--Static deflection (max)
-Under shaft weight and discs, at - from the left bearingOperating speed ratio
-Enter an operating speedResults
| Critical speed, Rayleigh method | - |
| Critical speed, Dunkerley (lower bound) | - |
| Bare shaft only, no discs | - |
| Shaft mass | - |
| Bending stiffness EI | - |
Checks
How the calculator works
The shaft is modelled as a uniform beam on two rigid simple supports that carries the shaft's own weight and up to two point masses. The first lateral critical speed is the speed at which the rotation frequency equals the first bending natural frequency of the shaft, where unbalance is greatly amplified.
- Section. A = π(D2 − d2)/4 and I = π(D4 − d4)/64 for outside diameter D and bore d. Shaft mass = ρAL.
- Bare shaft (exact). ω1 = (π/L)2√(EI/ρA), and the speed is 30ω1/π rpm.
- Rayleigh method. The weight of the shaft (split into 200 equal pieces) and of each disc is applied as static loads. With the static deflection yi found from the standard simply supported beam influence formula, ω2 = g·Σ(miyi)/Σ(miyi2). This is a slight over-estimate of the true first frequency.
- Dunkerley. 1/ωc2 = 1/ω12 + Σmi/ki, with ki = 3EIL/(a2b2) the shaft stiffness at each disc. This is a slight under-estimate, so the two methods bracket the answer.
Material properties are typical room-temperature values (steel 207 GPa and 7850 kg/m³, stainless 193 GPa and 8000 kg/m³, aluminium 69 GPa and 2700 kg/m³, titanium alloy 114 GPa and 4430 kg/m³). Discs are treated as point masses without their own stiffness or rotary inertia.
Worked example: 50 mm steel shaft, 1 m span
A solid 50 mm steel shaft between simple supports 1 m apart, no discs. I = π·0.054/64 = 3.068×10-7 m4, A = 1.963×10-3 m2. ω1 = π2√(207×109·3.068×10-7/(7850·1.963×10-3)) = 633.5 rad/s, or 6,050 rpm (101 Hz). The calculator's Rayleigh result is within 0.1 percent of this. Adding a 50 kg disc at mid-span to a 10 mm shaft of the same length gives about 94 rpm from the classic single-mass formula √(48EI/mL3), which ignores the shaft's own mass; the calculator, which includes it, gives 94.0 rpm.
Questions
How do you calculate the critical speed of a shaft?
Treat the shaft as a beam on two simple supports and find its first natural frequency. For a bare uniform shaft, ωc = (π/L)²·√(EI/ρA), and the critical speed in rpm is 30·ωc/π. With discs, use Rayleigh's method: ωc² = g·Σ(m·y)/Σ(m·y²), where y is the static deflection under each mass including the shaft's own weight. Example: a 50 mm steel shaft 1 m long between simple supports has a critical speed of about 6,050 rpm.
How far should the operating speed be from the critical speed?
A common rule of thumb is to stay at least 20 to 25 percent away from the first critical speed, either below it (a stiff, subcritical shaft, usually kept under about 70 to 80 percent of the critical speed) or well above it for designs that are meant to run supercritical. Also check the second critical speed, which is higher; this calculator reports only the first.
Why is the real critical speed often lower than the calculated one?
Real bearings, housings and supports have finite stiffness and lower the critical speed compared with rigid simple supports. The result here assumes rigid, self-aligning simple supports, so it is an upper-bound estimate for the support condition; treat it as optimistic when the supports are soft.
What is the difference between the Rayleigh and Dunkerley methods?
Both estimate the first natural frequency from simple quantities. Rayleigh's method uses the static deflection curve and never underestimates the frequency (it is an upper bound), while Dunkerley's equation, 1/ωc² = 1/ω₁² + 1/ω₂² + …, is a lower bound. The two are normally within a few percent, and the calculator shows both.
Does the calculator include gyroscopic effects or damping?
No. It is a lateral, undamped first critical speed of a shaft on two simple supports. Gyroscopic stiffening of overhung or large discs, bearing stiffness, stepped diameters, whirl direction, unbalance response and damping are not included.
Engineering aid for guidance. First lateral critical speed of a straight uniform shaft on rigid simple supports with point masses; it does not include bearing and housing flexibility, stepped shafts, overhung masses, gyroscopic effects, higher modes, damping or torsional critical speeds. Verify critical designs with a full rotordynamic analysis, API 617 / ISO 21940 guidance where relevant, and testing; you remain responsible for the design. Nothing you enter leaves your browser except through a share link you copy yourself.