Beam Vibration Calculator

Study natural frequencies for common beam supports. Test geometry, elasticity, density, and mode effects instantly. Download reports with graphs, formulas, examples, and helpful notes.

Calculator Input

Use consistent SI units for reliable results.

End Condition

Mode Number

Beam Length L (m)

Elastic Modulus E (Pa)

Second Moment of Area I (m⁴)

Material Density ρ (kg/m³)

Cross-Sectional Area A (m²)

Damping Ratio ζ

Mass per Unit Length μ

Example Data Table

Example case: fixed-fixed steel beam with L = 2.0 m, E = 210 GPa, I = 0.0000075 m⁴, ρ = 7850 kg/m³, A = 0.0028 m², ζ = 0.02.

Mode	Support	Beta	Natural Frequency (Hz)	Damped Frequency (Hz)	Critical Speed (rpm)
1	Fixed-Fixed	4.730041	238.295799	238.248135	14,294.888094
2	Fixed-Fixed	7.853205	656.871538	656.740150	39,404.409007
3	Fixed-Fixed	10.995608	1,287.730768	1,287.473196	77,248.391767

Formula Used

Mass per unit length: μ = ρ × A

Natural angular frequency: ω_n = β_n² × √(EI / (μL⁴))

Natural frequency: f_n = ω_n / (2π)

Damped angular frequency: ω_d = ω_n × √(1 - ζ²)

Damped frequency: f_d = ω_d / (2π)

Time period: T = 1 / f_d

Critical speed: N = 60 × f_d

Support constants β: the calculator uses standard eigenvalues for pinned-pinned, fixed-free, fixed-pinned, and fixed-fixed beams.

How to Use This Calculator

Select the beam end condition that matches the real support case.
Choose the vibration mode from 1 to 5.
Enter beam length, elastic modulus, second moment of area, density, and cross-sectional area.
Add damping ratio if you want damped frequency and period.
Press Calculate to show results below the header and above the form.
Review the frequency graph and the mode summary table.
Use the CSV button for spreadsheet work.
Use the PDF button for a quick report download.

FAQs

1. What beam theory does this calculator use?

It uses the Euler-Bernoulli beam vibration model for slender, uniform beams. Shear deformation and rotary inertia are ignored, so thick or very short beams may need a more advanced model.

2. Which units should I enter?

Use a consistent SI set: meters, pascals, kilograms per cubic meter, and square or fourth-power meters. Mixing units will distort mass, stiffness, and frequency results.

3. Why does the end condition change the result so much?

Support restraint changes the beam’s effective stiffness and its eigenvalue. A fixed-fixed beam usually vibrates faster than a pinned-pinned or cantilever beam with the same material and geometry.

4. What does the damping ratio affect?

Damping reduces the oscillation frequency slightly and changes the time response. With small damping, the damped frequency stays close to the natural frequency, but it is still useful for engineering checks.

5. Can I use this for nonuniform beams?

This version assumes a uniform beam along its full length. For stepped, tapered, cracked, or heavily loaded members, use finite element analysis or a specialized vibration model.

6. What is the beta value in the output?

Beta is the characteristic eigenvalue constant for the selected support case and mode. It directly controls the natural frequency and comes from the beam vibration boundary-condition solution.

7. Why are five modes shown on the graph?

The first five modes give a practical view of how frequency rises with mode number. This helps compare resonance risk, machine excitation ranges, and modal spacing in one chart.

8. What is critical speed in rpm used for?

Critical speed is useful when rotating equipment, cyclic forcing, or repeated excitation can match a beam frequency. It helps engineers identify unsafe operating ranges and resonance zones.