# Shaft Critical Speed Calculator

Calculate the critical (resonance) speed of a rotating shaft using the single-DOF rotor model. ω_n = √(k/m), n_cr = (60/2π)·√(k/m) rpm. Also returns the natural frequency f_n in Hz.

> Canonical page: https://elysiatools.com/en/tools/shaft-critical-speed

- **Category:** Math & Numbers

- **Keywords:** critical speed, resonance speed, shaft, rotor dynamics, natural frequency, whirling, n_cr, mechanical engineering

## Overview

The Shaft Critical Speed Calculator helps mechanical engineers and rotor dynamics specialists determine the critical (resonance) speed of a rotating shaft using a single-degree-of-freedom (single-DOF) rotor model. By entering the shaft stiffness (k) and rotor mass (m), you can instantly calculate the angular frequency (ω_n), natural frequency (f_n), and critical whirling speed (n_cr) in RPM to prevent destructive mechanical resonance.

## Inputs

- **Stiffness k (N/m)** (number): e.g. 200000
- **Mass m (kg)** (number): e.g. 5
- **Decimal Places** (number)

## When to use

- When designing rotating machinery, such as turbines, pumps, or drive shafts, to ensure operating speeds do not match the shaft's natural frequency.
- During rotor dynamics analysis to identify the whirling threshold and avoid destructive resonance.
- When troubleshooting vibration issues in existing industrial rotating equipment by verifying the system's stiffness-to-mass ratio.

## How it works

- Enter the equivalent stiffness (k) of the shaft in Newtons per meter (N/m).
- Input the concentrated mass (m) of the rotor or disc in kilograms (kg).
- Adjust the decimal places for precision and click calculate to compute the angular frequency, natural frequency, and critical speed in RPM.

## Use cases

- Designing industrial electric motor shafts to operate safely below or above the first critical whirling speed.
- Verifying the structural safety of centrifugal pump shafts under varying rotor mass configurations.
- Teaching mechanical engineering students the fundamentals of rotor dynamics and single-DOF vibration models.

## Frequently asked questions

### What is critical speed in a rotating shaft?

Critical speed is the rotational speed at which a shaft becomes unstable and begins to vibrate violently (whirl) due to resonance with its natural frequency.

### What formula does this calculator use?

It uses the single-DOF rotor model where angular frequency is ω_n = √(k/m) rad/s, natural frequency is f_n = ω_n / 2π Hz, and critical speed is n_cr = (60/2π)·ω_n RPM.

### Can I calculate multiple critical speeds for multi-mass systems?

No, this calculator uses a single-degree-of-freedom model, which calculates the primary (first) critical speed for a single concentrated mass.

### What units should I use for stiffness and mass?

Use Newtons per meter (N/m) for stiffness (k) and kilograms (kg) for mass (m).

### How do I prevent a shaft from operating at critical speed?

You can change the operating speed, increase shaft stiffness (e.g., by increasing diameter or shortening span), or reduce the rotor mass.

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