Inputs
Rotor mass m and operating speed n are always required; additionally either the combined shaft/bearing stiffness k or the static deflection δ under the rotor's own weight.
Shafts & hubs · Rotor dynamics
Critical speed of a shaft – basic model
Use the classical single-degree-of-freedom (Jeffcott/Laval rotor) model to calculate the natural angular frequency and critical speed of a rotor on an elastic shaft/bearing support, and compare it against the planned operating speed.
This covers a single rotor as a concentrated mass on a massless, undamped elastic support. This is not a DIN 743 strength verification and not a complete rotor-dynamics model.
ωnn_krit
02
Critical speed
Set the inputs and calculate.
Inputs and method
Critical speed of a shaft – basic model
Estimate a single-mass rotor's natural frequency and critical speed on an elastic shaft/bearing support, and compare it against the operating speed.
Calculation
From k and m follows the natural angular frequency ωn=√(k/m). Alternatively, Rayleigh's approximation ωn=√(g/δ) from the static deflection gives the same result, since δ=m·g/k. The critical speed is ncrit=60·ωn/(2π); the ratio n/ncrit shows how close operation is to resonance.
Example
m=50 kg, k=2,000 N/mm give ωn=200 rad/s and ncrit≈1,910 min⁻¹; at n=1,000 min⁻¹ the ratio is 0.52.
Sources and limits: Classical vibration theory, single-degree-of-freedom model (Jeffcott/Laval rotor); explicitly not a DIN 743 strength verification and not a complete rotor-dynamics model.
Critical speed of a shaft – basic model in detail
Estimate a single-mass rotor's natural frequency and critical speed on an elastic shaft/bearing support, and compare it against the operating speed.
What is a rotor's critical speed?
A rotor on an elastic shaft/bearing support can, in the simplest case, be idealized as a single-degree-of-freedom system (Jeffcott/Laval rotor): a concentrated mass on a massless, elastic spring. The critical speed is the speed at which the unbalance excitation frequency coincides with this system's natural frequency.
Formula and variables
ωn = √(k/m)
ωn = √(g/δ) (δ = m·g/k)ncrit = 60·ωn/(2π)ratio = n / ncrit
| Symbol / input | Meaning |
|---|---|
| m | Rotor mass, assumed as a concentrated point mass, in kg. |
| k | Combined stiffness of shaft and bearings at the rotor location, in N/mm. |
| δ | Static deflection under the rotor's weight alone, in mm. |
| ωn | Natural angular frequency of the single-degree-of-freedom system, in rad/s. |
| ncrit, n | Critical speed and chosen operating speed, in min⁻¹. |
Choose the inputs correctly
Rotor mass m and operating speed n are always required; additionally either the combined shaft/bearing stiffness k or the static deflection δ under the rotor's own weight.
How to use the calculator
Enter the rotor mass m and the operating speed n. Then choose whether the combined shaft/bearing stiffness k is directly known, or whether the static deflection δ that occurs under the rotor's weight alone should be used instead.
Worked example
m=50 kg, k=2,000 N/mm give ωn=200 rad/s and ncrit≈1,910 min⁻¹; at n=1,000 min⁻¹ the ratio is 0.52.
How should the ratio n/ncrit be interpreted?
The ratio n/ncrit shows how close the chosen operating speed is to resonance. Values well below or well above 1 are generally considered less critical than values near 1; a specific figure for the required safety margin depends on damping, unbalance quality and application, and is deliberately not hard-coded here as a fixed limit.
Mass in kg, speed in min⁻¹, stiffness in N/mm, deflection in mm; results in rad/s, min⁻¹ and dimensionless (ratio).
Typical applications
A first estimate of whether a planned operating speed lies dangerously close to a rotor's first critical speed, e.g. when pre-sizing pump, fan or turbine shafts, before a full rotor-dynamics analysis is carried out.
Assumptions, limits and common mistakes
Classical vibration theory, single-degree-of-freedom model (Jeffcott/Laval rotor); explicitly not a DIN 743 strength verification and not a complete rotor-dynamics model.
Common mistake: Don't confuse the critical speed with an allowable operating speed in the sense of a strength verification -- this calculator checks only proximity to resonance, not material strength, notch effects or fatigue per DIN 743. For the static deflection, don't enter the deflection under an arbitrary operating load -- specifically use the one under the rotor's weight alone.
Frequently asked questions
What happens when the operating speed equals the critical speed?
The excitation frequency from rotor unbalance coincides with the natural frequency of the rotor-support system (resonance); without sufficient damping, vibration amplitudes grow strongly, which can cause excessive deformation or damage.
Why do stiffness and static deflection give the same result?
Both inputs describe the same relationship: the static deflection of a single-degree-of-freedom system under its own weight is δ=m·g/k. Substituting that into ωn=√(k/m) gives ωn=√(g/δ) -- mathematically the same natural frequency.
Why does varying the mass change nothing in deflection mode?
Because the mass is already contained in the deflection: δ=m·g/k includes m. From δ alone follows ωn=√(g/δ), so m does not enter a second time. A heavier mass produces a larger deflection at the same stiffness -- and that larger deflection is the input that lowers the critical speed. If you know the stiffness itself, use the stiffness mode.
Is this a DIN 743 strength verification?
No. This calculator checks only rotor dynamics (proximity to resonance) via a single-mass model. A strength or fatigue verification of the shaft itself is a separate step, e.g. with the existing DIN 743 shaft calculator.
What effects does the model not account for?
The shaft's own mass, multiple bearing locations or spans, gyroscopic effects, damping and higher critical speeds are not represented in this basic model.