Mechanical Engineering Calculators: Dunkerley Multi-Mass Shaft Critical Speed Calculator

Dunkerley multi-mass critical-speed equation

Dunkerley’s method combines partial natural frequencies obtained by considering each lumped rotor mass separately. For case i, determine the gravity static deflection at that mass location with only that mass weight acting. The partial frequency and combined estimate are:

The published relationship is often described as an underestimate of the fundamental frequency for the ideal linear multi-degree-of-freedom model. That mathematical tendency does not turn the output into a guaranteed safe lower bound for a real machine with uncertain bearings, supports, damping, geometry, and operating conditions.

The isolated-case requirement

Each field represents a separate structural load case. For δ₁, apply only rotor weight 1 and measure or calculate deflection at its own station. Repeat independently for every other rotor weight. Preserve the same shaft geometry, elastic properties, bearing and support boundary conditions, and lateral direction.

Do not enter:

• total deflection with all rotor weights acting together;
• deflection from transmitted gear force, belt pull, chain tension, process force, or torque;
• maximum shaft deflection at a coordinate different from the mass station;
• a mixture of deflections obtained from different support models;
• arbitrary values selected to reach a target speed.

Calculator #026 uses the total multi-weight deflection field required by Rayleigh’s energy method. Those yᵢ inputs are deliberately not transferred to #027 because their physical definitions are different.

Worked example

Use three active isolated cases and one disabled case:

1. Sum the isolated deflections: Σδᵢ = 0.25 + 0.60 + 0.15 = 1.00 mm.
2. Calculate the combined angular frequency: ωD = √(9,806.65 / 1.00) = 99.028531 rad/s.
3. Convert to frequency: fD = 15.760880 Hz.
4. Convert to critical speed: nD = 945.652815 rpm.
5. The combined estimate is 22.5403% below the lowest active partial speed of 1,220.832535 rpm.
6. At 600 rpm, the speed ratio is 0.634482 and the operating point is 36.5518% below the estimate.
7. The default ±20% review range is 756.522252 to 1,134.783378 rpm, so 600 rpm lies outside it.

The contribution bars visualize δᵢ / Σδᵢ. A larger bar identifies a larger reciprocal-frequency contribution and therefore a stronger reduction of the combined estimate. It does not represent physical rotor position or vibration amplitude.

Cross-check with the single-deflection relationship

Because substituting ωᵢ = √(g/δᵢ) reduces the reciprocal-frequency sum to ωD = √(g/Σδᵢ), Calculator #025 can reproduce the #027 numerical result using the summed isolated deflection. The related-action link transfers this sum together with operating RPM and the review band.

That cross-check confirms the algebra and unit conversion only. It does not validate whether the individual deflections were generated correctly or whether Dunkerley’s assumptions fit the rotor-bearing-support system.

Engineering scope and limitations

This calculator provides four input slots and combines up to four positive isolated gravity-deflection cases. It excludes:

• derivation or verification of the isolated deflections and influence coefficients;
• direct input of weights, masses, stiffnesses, shaft geometry, or partial frequencies;
• distributed shaft mass and a separate bare-shaft critical-speed term;
• overhung, stepped, branched, coupled, anisotropic, cracked, or nonlinear rotor behavior unless independently represented by a valid method;
• higher bending modes, gyroscopic splitting, speed-dependent bearing or seal coefficients, cross-coupling, fluid-induced instability, rubs, and nonlinear clearances;
• unbalance-response amplitude, resonance stress, orbit, phase, amplification, damping ratio, run-up or coast-down response, and protection settings;
• universal separation criteria, permissible operating ranges, pass/fail classification, standards compliance, or engineering approval.

Use an applicable transfer-matrix, finite-element, modal, or full rotordynamic analysis when the machine cannot be represented by consistent isolated single-mass cases. Confirm predicted critical speeds against qualified engineering review and suitable test or operational evidence before releasing a design or operating through a critical-speed region.