VD-06Vibration & Structural Dynamics
Beam Natural Frequencies
Euler–Bernoulli bending modes for common boundary conditions, with animated mode shapes.
Beam
m
Cross-section
mm
mm
Mode 1
8.355Hz
Fundamental
8.355Hz
Mass per length
3.14kg/m
Second moment I
3,330mm⁴
Displayed mode
Mode 1 shape - Clamped–free (cantilever)no interior nodes
| Mode | βL | f (Hz) | f / f₁ |
|---|---|---|---|
| 1 | 1.8751 | 8.3552 | 1.00 |
| 2 | 4.6941 | 52.361 | 6.27 |
| 3 | 7.8548 | 146.61 | 17.55 |
| 4 | 10.9955 | 287.3 | 34.39 |
| 5 | 14.1372 | 474.93 | 56.84 |
§ Theory of operation
Euler–Bernoulli bending vibration
A slender beam (length much greater than depth) vibrating in bending obeys EI·∂⁴w/∂x⁴ + ρA·∂²w/∂t² = 0. Separating variables gives mode shapes built from sin, cos, sinh and cosh, and natural frequencies:
f_n = \dfrac{(\beta_n L)^2}{2\pi\,L^2}\,\sqrt{\dfrac{E\,I}{\rho\,A}}
The eigenvalues βₙL depend only on the boundary conditions:
| Boundary | β₁L | β₂L | β₃L |
|---|---|---|---|
| Pinned–pinned | π | 2π | 3π |
| Clamped–free | 1.8751 | 4.6941 | 7.8548 |
| Clamped–pinned | 3.9266 | 7.0686 | 10.2102 |
| Clamped–clamped / free–free | 4.7300 | 7.8532 | 10.9956 |
Observations
- Frequency scales with 1/L²: halving the span quadruples fₙ.
- The ratio √(E/ρ) is almost the same for steel, aluminium and titanium, changing metal hardly changes a beam's frequencies; changing the section does.
- Overtones are not harmonic (cantilever: 1 : 6.27 : 17.5), which is why struck bars sound clangorous unless shaped, as in marimba bars.
- Free–free beams share eigenvalues with clamped–clamped beams but have different shapes; they also have two rigid-body modes at 0 Hz (not shown).
Limitations
Shear deformation and rotary inertia are neglected. For stubby beams (L/h < 10) or higher modes, Timoshenko theory gives lower frequencies. Axial load, added masses and joint flexibility also shift the results.