What is the critical speed of a shaft?

The critical speed is the rotational speed (in RPM) at which a shaft's forcing frequency equals its own transverse natural frequency. At this speed, even a small residual imbalance causes large-amplitude lateral vibrations (whirling). Shafts should operate well below (subcritical) or well above (supercritical) this speed — typically with a ±20% separation margin.

What formula is used for the beam method?

For a uniform shaft the first critical speed equals the first bending natural frequency: N_c = (60/2π) × (β²/L²) × √(EI/ρA), where β² = π² for simply supported ends and β² ≈ 3.5160 for a cantilever. E is elastic modulus, I is second moment of area (πd⁴/64 for a solid round shaft), ρ is density, A is cross-sectional area, and L is shaft length.

How does Rayleigh's method work?

Rayleigh's method approximates the first natural frequency using static deflections caused by gravity. For each discrete mass W_i at position x_i, calculate the static deflection δ_i under that load. Then: ω² ≈ g × Σ(W_i·δ_i) / Σ(W_i·δ_i²). This always overestimates the true critical speed slightly, making it a safe conservative check.

What is a safe separation margin from critical speed?

API and ISO standards for rotating machinery typically require that the operating speed range stays at least 20% away from any critical speed. So if the critical speed is 3000 RPM, operate below 2400 RPM or above 3600 RPM. For lightly damped shafts or precision applications, a 25–30% margin is common.

Does shaft eccentricity affect the critical speed?

No. The critical speed is a property of the shaft stiffness and mass distribution alone. Eccentricity (imbalance) determines the amplitude of vibration at the critical speed, not the speed itself. A perfectly balanced shaft still has a critical speed — it just passes through it with minimal deflection.

Critical Speed of a Shaft Calculator

Name: Critical Speed of a Shaft Calculator
Availability: InStock
Author: Nham Vu

Enter shaft geometry, material, and support conditions to compute the first critical (whirling) speed in RPM and Hz using Rayleigh's method or beam formulas.

Inputs

Calculation Method

Support Condition

Shaft Length L (m)

Shaft Diameter d (m)

Elastic Modulus E (GPa)

Density ρ (kg/m³)

Fill in the inputs and click Calculate

Summary

Enter shaft geometry, material, and support conditions to compute the first critical (whirling) speed in RPM and Hz using Rayleigh's method or beam formulas.

How it works

Select the calculation method: Beam Formula (uniform shaft) or Rayleigh's Method (shaft with point masses).
For the Beam Formula method, enter shaft length, diameter, elastic modulus, and density; choose simply supported or cantilever support.
For Rayleigh's Method, enter the shaft stiffness (or let the tool derive it) and add each discrete mass with its static deflection under its own weight.
Click Calculate to see the first critical speed in RPM and Hz.
Compare the result against the operating speed to confirm an adequate separation margin (typically ±20%).

Use cases

Verifying that a motor shaft or pump shaft does not pass through resonance within the operating speed range.
Preliminary sizing of drive shafts in industrial machinery and turbines.
Checking overhung rotor configurations (cantilever) against whirling limits.
Teaching rotor dynamics and critical speed theory in mechanical engineering courses.
Estimating the effect of adding a disk or impeller mass on shaft critical speed.
Comparing solid versus hollow shaft cross-sections for resonance avoidance.

Frequently Asked Questions

Last updated: 2026-07-22 · Reviewed by Nham Vu