MEC-32

Shaft critical speed

ωcr = √(g/δst). Whirl when rotation meets the bending natural frequency.

Reading speed

VibrationRayleigh

Governing equation

\omega_{\mathrm{cr}}=\sqrt{g/\delta_{\mathrm{st}}}

$\delta_{\mathrm{st}} \in \left[0.02, 8\right]$

where

\delta_{\mathrm{st}}: Static deflection (mm)
\omega_{\mathrm{cr}}: Critical ω (rad/s)
n_{\mathrm{cr}}: Critical speed (rpm)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

\omega_{\mathrm{cr}}=\sqrt{g/\delta_{\mathrm{st}}}

$\delta_{\mathrm{st}} \in \left[0.02, 8\right]$

$\delta_{\mathrm{st}} = 0.02$ $\omega_{\mathrm{cr}} = 6687.9$

Along the curve

②

Domain of the variable — blue

\delta_{\mathrm{st}}_{\min} = 0.02,\quad \delta_{\mathrm{st}}_{\max} = 8

$\delta_{\mathrm{st}} \in \left[0.02, 8\right]$

$\delta_{\mathrm{st}} = 0.02$ $\delta_{\mathrm{st}} = 0.02$

Along the curve

③

Response curve — green

\omega_{\mathrm{cr}}=\sqrt{g/\delta_{\mathrm{st}}}

$\delta_{\mathrm{st}} \in \left[0.02, 8\right]$

$\delta_{\mathrm{st}} = 0.02$ $\omega_{\mathrm{cr}} = 6687.9$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}\omega_{\mathrm{cr}}}{\mathrm{d}\delta_{\mathrm{st}}}

$\delta_{\mathrm{st}} \in \left[0.02, 8\right]$

$\delta_{\mathrm{st}} = 0.07$ $\omega_{\mathrm{cr}}^{\prime} = -39469.8$

Along the curve

Lecture brief

Historical brief

Machine design grew from Coulomb torsion and Hertz contact (1881) through Soderberg fatigue and the heat-engine cycle. The lab writes shaft, bearing, contact and thermodynamic limits in SI. This sheet (MEC-32 — Shaft critical speed) is the form associated with Rayleigh. Working symbols: $\delta_{\mathrm{st}}$ $\rightarrow$ $\omega_{\mathrm{cr}}$ , $n_{\mathrm{cr}}$ . Rayleigh: a shaft with static sag δst has ωn = √(g/δst). Crossing it is a critical speed.

Purpose

Purpose: compute

\omega_{\mathrm{cr}}

n_{\mathrm{cr}}

from

\delta_{\mathrm{st}}

in Mechanical via

\omega_{\mathrm{cr}}=\sqrt{g/\delta_{\mathrm{st}}}

ωcr = √(g/δst). Whirl when rotation meets the bending natural frequency. Use it when a real mechanical question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

\delta_{\mathrm{st}} = 0.400\,\mathrm{mm}

, the governing relation

\omega_{\mathrm{cr}}=\sqrt{g/\delta_{\mathrm{st}}}

yields

\omega_{\mathrm{cr}} = 156.6\,\mathrm{rad/s}

n_{\mathrm{cr}} = 1495.5\,\mathrm{rpm}

. A rotor, a sag, a whirl orbit. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Static deflection \delta_{\mathrm{st}}0.4 mm

Outputs

Critical ω \omega_{\mathrm{cr}}156.6 rad/s
Critical speed n_{\mathrm{cr}}1495.5 rpm

Reading speed

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Narration of this film

A rotor, a sag, a whirl orbit.

Rayleigh: a shaft with static sag δst has ωn = √(g/δst). Crossing it is a critical speed.

Reading speed

Watch on YouTube