Formula Used
This calculator uses the classical uniform Euler–Bernoulli beam vibration model:
- ωₙ = (βₙ² / L²) · √(EI / μ)
- fₙ = ωₙ / (2π)
- μ = ρA (unless you override μ directly)
Here, E is elastic modulus, I is second moment of area, A is cross‑sectional area, L is beam length, and βₙ depends on boundary conditions and mode.
How to Use This Calculator
- Enter the beam length and choose its unit.
- Select the boundary condition and mode number.
- Provide elastic modulus and either density or mass per length.
- Choose a cross-section type and fill its dimensions.
- Press Calculate Natural Frequency to view results.
- Use the CSV or PDF buttons to export the mode table.
Example Data Table
| Case | Boundary | L (m) | Section | E (GPa) | ρ (kg/m³) | Mode | Approx. f (Hz) |
|---|---|---|---|---|---|---|---|
| Steel bar | Cantilever | 2.0 | Rectangular 50×100 mm | 200 | 7850 | 1 | ≈ 20.38 |
| Aluminum tube | Simply supported | 1.5 | Hollow circular 60/40 mm | 69 | 2700 | 2 | Varies by geometry and supports |
| Custom profile | Fixed-fixed | 3.0 | A=0.004 m², I=6e‑6 m⁴ | 210 | 7850 | 1 | Use calculator for exact value |
Tip: For complex shapes, use the Custom section and enter A and I from CAD.
Why Beam Natural Frequency Matters
Natural frequency is the vibration “sweet spot” of a beam. When excitation matches a mode, deflection and stress can rise sharply. Designers use frequency checks to avoid resonance from motors, rotating machinery, footsteps, wind, and seismic inputs. A quick estimate helps you decide whether stiffness, mass, or support changes are needed.
What the Calculator Assumes
The computation follows the Euler–Bernoulli model for a uniform beam. It assumes small deflections, linear elastic material behavior, and constant cross-section along the length. Shear deformation and rotary inertia are not included, so very deep beams or very short spans may need higher‑order theory. For many slender beams, this model is accurate and widely used.
Boundary Conditions Control β Values
Support conditions change the mode shapes and the eigenvalue term βL. A cantilever is more flexible than a fixed‑fixed beam at the same span, so it has a lower first frequency. Simply supported beams fall between these extremes. Free‑free beams have two rigid‑body modes at 0 Hz, and the listed modes start from the first flexible bending mode.
How Geometry Drives Stiffness
Stiffness enters through EI. Increasing E makes the beam stiffer and raises frequency. Increasing I is often more powerful than changing material. For a rectangular section, I scales with height cubed, so doubling height can raise I by about eight times. Tubes and I‑sections place material away from the neutral axis, boosting I efficiently.
Mass per Length Sets Inertia
The inertia term is μ, the mass per unit length. If density is used, μ is computed as ρA. Heavier materials or larger areas increase μ and reduce frequency. When you know the installed mass better than ρA, use the μ override. This is useful for beams with coatings, attachments, cable trays, or composite buildup.
Reading the Mode Table
The table lists the first five modes with βL, angular frequency ω, frequency f, and period. Higher modes rise rapidly because ω scales with β². If you are checking a forcing source at a known RPM, convert RPM to Hz by dividing by 60, then compare to the nearest mode. Keep a safety margin to account for tolerances and damping.
Practical Design Targets
Many vibration guidelines aim to separate operating frequencies from structural modes. For rotating equipment, designers often target the first bending mode to be well above the running speed. For floor beams, higher first frequencies generally reduce perceptible vibration. If the calculated first mode is too low, common fixes include shortening the span, adding bracing, increasing depth, or changing supports.
Limitations and When to Refine
Real structures may have taper, holes, joints, and distributed attachments. Damping reduces peak response but does not change the undamped natural frequency much. If your beam is thick, layered, or very short, consider shear‑deformable models or finite element analysis. Still, this calculator provides fast insight and is ideal for early sizing and comparison studies.
FAQs
1) What is “natural frequency” for a beam?
It is the frequency where a beam prefers to vibrate in a specific mode shape. If external forcing matches it, vibration amplitudes can grow significantly, especially near resonance.
2) Which inputs change frequency the most?
Frequency increases with stiffness (E and I) and decreases with mass per length (μ). Geometry often dominates because I can change dramatically with section depth and shape.
3) Should I use density or mass per length override?
Use density for clean, uniform beams. Use μ override when attachments, coatings, or built‑up assemblies add mass beyond ρA, or when you already know weight per unit length.
4) Why does support condition matter so much?
Supports change mode shapes and the eigenvalue β. A fixed‑fixed beam restrains rotation and deflection more than a cantilever, so it usually has a higher first frequency.
5) Are the first modes for a free‑free beam really zero?
Yes. Free‑free beams have two rigid‑body modes: translation and rotation, both at 0 Hz. The calculator lists the first five flexible bending modes instead.
6) Can I use this for very thick or short beams?
Use caution. The Euler–Bernoulli model ignores shear deformation and rotary inertia. For short, deep, or high‑frequency cases, a shear‑deformable model or FEA is more reliable.
7) How can I increase the first natural frequency?
Shorten the span, increase section depth, switch to a stiffer material, add bracing, or change the supports to provide more restraint. Reducing added mass also helps.