Critical Speed Shaft⚠ unverified
Physics / Vibration · Compute the approximate critical whirling speed of a simply supported shaft
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| length | length | m | 1.0 | Shaft length between supports |
| E | E | Pa | 1.0 | Young's modulus of the shaft material |
| I | I | m**4 | 1.0 | Second moment of area of the cross-section |
| mass | mass | kg | 1.0 | Mass of the shaft |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | wc | rad/s | Critical whirling speed, in radians per second (rad/s). Returns 0.0 if any argument is non-positive |
The science & history
Understanding the Parameters
- $E$, $I$, $L$ — stiffer/shorter shafts raise critical speed.
- $m$ — confirm whether ToolBox treats this as total mass or as $\mu$ (per length).
- $\omega_c$ — convert to rpm via $N = 60\omega_c/(2\pi)$ for machinery ratings.
Derivation (Approaching a Proof)
Euler–Bernoulli free vibration eigenvalues for a pinned–pinned beam: $\omega_n = (n\pi/L)^{2}\sqrt{EI/\mu}$. The $n=1$ mode is the usual first critical speed for a simple rotor idealisation. The registry form should be cross-checked against that standard result.
History
Critical speeds (Rankine, Dunkerley, Rayleigh) dominate rotor design and balancing practice.
Related Concepts: Natural Frequency mass-spring, Natural Frequency Cantilever Beam, Euler Buckling Load
Notes: Registry calculator critical-speed-shaft (unverified). Confirm mass parameter definition
in ToolBox.