Mechanical Engineering Calculators: Shaft Critical Speed / Whirling Calculator

Static-deflection critical-speed equation

Critical speed occurs when a rotating excitation coincides with a lateral natural frequency of the rotor system. This reference calculator uses a deliberately narrow screening model: the first lateral natural frequency is estimated from the static deflection produced by the rotating mass weight.

The comparison outputs are:

Positive signed separation means the operating point is below the estimate. Negative signed separation means it is above. The absolute separation output removes that direction but does not convert the result into an acceptance decision.

Worked example

Use the defaults: gravity static deflection δₛ = 1 mm, operating speed n = 600 rpm, and a user-selected review band of ±20%.

1. Convert deflection to metres: δₛ = 0.001 m.
2. Calculate angular frequency: ω₁ = √(9.80665 / 0.001) = 99.028531 rad/s.
3. Convert to frequency: f₁ = 15.760880 Hz.
4. Convert to rotational speed: n₁ = 945.652815 rpm.
5. Calculate speed ratio: r = 600 / 945.652815 = 0.634482.
6. The operating point is 36.5518% below the estimate.
7. The user’s ±20% review range is 756.522252 to 1,134.783378 rpm, so the default operating point lies outside it.

Automated tests verify this result directly from √(g/δₛ), metric and inch equivalence, inverse-square-root scaling, review-band bounds, stationary, near-critical and supercritical states, URL serialization, warnings, and invalid inputs.

The required deflection is a gravity-load result

Do not enter just any shaft deflection. The static-deflection method requires displacement caused by the weight of the rotating mass in the same lateral direction being evaluated. A displacement calculated from transmitted gear force, belt pull, chain tension, torque, process force, or an arbitrary design load does not satisfy that requirement.

Calculator #024 can supply a candidate value only if its point loads represent rotating component weights under gravity and its uniform-shaft, ideal-simple-support assumptions are suitable. Even then, using only maximum deflection is a simplified first-mode approximation. The complete rotor may contain several concentrated masses, distributed shaft weight, bearings with finite stiffness, seals, couplings, and a flexible support structure.

Understanding the review band

The orange SVG band is generated from the percentage entered by the user. It is not a protected operating zone, prohibited zone, standard-mandated margin, or statement about vibration amplitude. It exists to make proximity visible and to produce a separate engineering warning when operating RPM falls inside the selected range.

When the operating speed is above the estimate, the calculator adds a supercritical-operation warning. This is not an automatic failure. Some rotor systems are designed to pass through a critical speed and operate above it, but the run-up rate, unbalance response, damping, stress, bearing loads, seals, clearances, vibration limits, controls, coast-down behavior, and all relevant modes must be evaluated by an applicable method.

Why the actual critical speed can differ

The simplified equation compresses rotor mass and lateral stiffness into one measured or calculated gravity deflection. Actual behavior can differ because:

• bearing, seal, housing, pedestal, and foundation stiffness may be comparable with shaft stiffness;
• damping determines resonance amplitude and response during acceleration or coast-down;
• shaft and rotor mass are distributed rather than concentrated at one coordinate;
• several disks, impellers, gears, couplings, or other components create multiple degrees of freedom;
• bearing coefficients, fluid forces, fits, clearances, and thermal state can change with speed and load;
• gyroscopic effects, shaft asymmetry, cracks, anisotropic supports, and cross-coupled forces can shift modes or create instability;
• nonsynchronous excitation and harmonics can produce important responses away from a simple 1× crossing.

Use an appropriate Rayleigh, Dunkerley, transfer-matrix, finite-element, or full rotordynamic analysis when the rotor cannot be represented by this single-deflection estimate. Verification may also require modal testing, run-up/coast-down data, shaft displacement probes, bearing vibration measurements, balancing records, and manufacturer criteria.

Engineering scope and limitations

This calculator provides a first synchronous lateral critical-speed screening estimate from one positive gravity static deflection. It excludes:

• derivation of mass, weight, stiffness, shaft geometry, static deflection, or bearing coefficients;
• multiple concentrated masses, distributed shaft mass, overhung rotors, stepped shafts, flexible disks, and coupled shafts;
• higher bending modes, torsional and axial modes, gyroscopic splitting, anisotropy, cracks, rubs, nonlinear clearances, and fluid-induced instability;
• bearing and seal stiffness or damping, cross-coupled coefficients, housing, foundation, piping, and structural dynamics;
• unbalance-response amplitude, resonance stress, bearing load, orbit, phase, amplification factor, damping ratio, transient run-up or coast-down, and vibration limits;
• a universal separation margin, permissible operating range, pass/fail classification, standards compliance, machinery protection settings, or engineering approval.

Establish the complete rotor-bearing-support model, operating range, applicable excitation orders, verification method, acceptance criteria, and qualified engineering review before releasing a design or operating through a predicted critical speed.