Shaft critical speed
Calculate the bending critical speed of a stepped shaft on two simple supports carrying any number of discrete masses, with optional shaft self-weight. The calculator returns the deflection curve, the critical speed as a band between Dunkerley (lower bound) and Rayleigh (upper bound), and rates the safety margin to the operating speed.
Critical speed calculator
Model: stepped solid shaft on two rigid simple supports at the shaft ends; no damping, no gyroscopic effects. Real bearing and housing compliance lowers the critical speed. Only the bending critical speed (excited by unbalance) is computed, not the torsional one. When running up through resonance, pass through it quickly.
Results
Calculating …
Formulas and fundamentals
The model is a straight, stepped solid shaft on two rigid simple supports at the shaft ends, carrying any number of discrete masses (discs, hubs, couplings) in between. The deflection curve under the weight loads is computed numerically: the bending moment distribution M(x) yields the curvature M(x)/(E·I(x)), which is integrated twice; the boundary condition w = 0 at both supports fixes the integration constant. Jumps of the bending stiffness at shaft shoulders are captured exactly. The results are the maximum deflection, the slopes at the bearings and the values at every mass seat.
The Rayleigh quotient equates the maximum strain energy of bending with the maximum kinetic energy of the vibration. The static deflection curve under the weight forces of all masses and the self-weight serves as the trial function: ω² = g·(Σ m·y + ∫μ·y dx) / (Σ m·y² + ∫μ·y² dx). The result always lies above the true first natural frequency; with the gravity deflection curve as trial function the error typically stays below one to two percent. For a single mass on a massless shaft this collapses to the well-known engineering formula n_k ≈ 946/√f with the static deflection f in mm.
The Dunkerley formula sums the flexibilities of the subsystems: 1/ω² = 1/ω_W² + Σ 1/ω_i². Here ω_i is the critical angular frequency of the massless shaft carrying only mass i (from the influence coefficient δ_ii) and ω_W that of the bare shaft under self-weight. The result always lies below the true first natural frequency and is therefore conservative for subcritical design. The true value lies in the band between Dunkerley and Rayleigh; in addition the calculator solves the eigenvalue problem of the discrete masses exactly and, from two masses on, also reports the second critical speed. The traffic light rates the safety margin: subcritical green up to 0.8·n_k1 and yellow up to 0.9·n_k1 (both against the conservative Dunkerley bound), supercritical operation from 1.25·n_k1 based on Rayleigh.
Worked example
A steel shaft with a span of 800 mm consists of three segments: 200 mm at Ø 40 mm, 400 mm at Ø 60 mm and 200 mm at Ø 45 mm. A 20 kg disc sits at x = 300 mm and a 15 kg disc at x = 550 mm. The shaft self-weight is included and the operating speed is 3000 rpm.
The static analysis yields a maximum deflection of 45.6 µm. The first critical speed lies between 4533 rpm (Dunkerley) and 4650 rpm (Rayleigh); the exact solution of the discrete-mass system confirms 4649 rpm, showing that both bounds bracket the true value tightly. The operating speed of 3000 rpm stays below 0.8·4533 ≈ 3626 rpm, so the safety margin is met and the traffic light shows green.
Frequently asked questions
What is the bending critical speed?
The speed at which the shaft's rotational frequency coincides with its first bending natural frequency. The co-rotating centrifugal force from residual unbalance then excites the shaft in resonance and the amplitudes grow strongly. Operating speeds should keep sufficient margin from it.
How do Rayleigh and Dunkerley differ?
The Rayleigh quotient gives an upper bound of the first natural frequency, the Dunkerley formula a lower bound. Together the two methods bracket the true critical speed. For subcritical design the Dunkerley value governs because it is conservative.
How much margin to the critical speed is needed?
In practice ±20 to 25 percent is common, with about 15 percent as a minimum. Turbomachinery codes grade the required separation margin by damping up to roughly 26 percent. The calculator rates subcritical operation up to 0.8·n_k1 as green and up to 0.9·n_k1 as yellow; supercritical operation counts as sufficiently separated from 1.25·n_k1.
Does the calculation also apply to vertical shafts?
Yes. The critical speed is independent of the mounting orientation because the gravity deflection curve only serves as a trial function for the energy method. The common misconception that a vertical shaft has no deflection and therefore no critical speed is wrong.
Why is the real critical speed often below the calculated value?
The model assumes rigid bearings. Real rolling bearings and housings are compliant, which lowers the critical speed. The model also neglects the gyroscopic effect of the discs, which can matter significantly for large overhung discs. The calculated value should therefore never be exploited to the limit.
Does the tool also compute torsional critical speeds?
No. Only the bending critical speed excited by unbalance is computed here. Torsional critical speeds concern torsional vibrations from torque fluctuations (gear mesh, piston machines, inverters) and require a separate analysis with rotary inertias and torsional stiffnesses.