Natural Frequency Cantilever Beam⚠ unverified
Physics / Vibration · Compute the fundamental natural frequency of a cantilever beam
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| length | length | m | 1.0 | Beam length |
| E | E | Pa | 1.0 | Young's modulus of the beam material |
| I | I | m**4 | 1.0 | Second moment of area of the cross-section |
| mass_per_length | massperlength | kg/m | 1.0 | Mass per unit length of the beam |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | wn | rad/s | Fundamental (first-mode) natural frequency, in radians per second (rad/s). Returns 0.0 if any argument is non-positive |
The science & history
Understanding the Parameters
- $L$ — $\omega_n \propto 1/L^{2}$ (very sensitive to length).
- $E$, $I$, $\mu$ — stiffer/lighter beams vibrate higher.
- $\omega_n$ — first cantilever mode (tip free, root fixed).
Derivation (Approaching a Proof)
Euler–Bernoulli free vibration separable solutions with cantilever BCs yield $\cos\beta L\cosh\beta L = -1$. Lowest root $\beta_1 L \approx 1.8751$, and $\omega = (\beta L)^{2}\sqrt{EI/(\mu L^{4})} = (\beta L)^{2}/L^{2}\,\sqrt{EI/\mu}$.
History
Beam eigenvalues are classical (Bernoulli–Euler); cantilever factors are standard in structural dynamics and sensor design (AFM cantilevers, etc.).
Related Concepts: Critical Speed Shaft, Natural Frequency mass-spring, Euler Buckling Load
Notes: Registry calculator natural-frequency-cantilever-beam (unverified). Uniform Euler beam,
fundamental mode only.