What formula is used for the natural frequency?

The fundamental natural frequency is f = (β²/2π) × √(EI / (mL⁴)), where β is a mode shape constant that depends on boundary conditions. For a simply supported beam β² = π²; for a cantilever β² ≈ 3.5160; for a fixed-fixed beam β² ≈ 22.3733. These are the lowest roots of each beam's characteristic equation derived from Euler-Bernoulli beam theory.

What is the linear mass density m?

Linear mass density m (kg/m) is the mass of the beam per unit length. It equals the material density ρ (kg/m³) multiplied by the cross-sectional area A (m²): m = ρA. For example, a steel beam (ρ = 7850 kg/m³) with a cross-section area of 0.005 m² has m = 39.25 kg/m.

What is the second moment of area I?

The second moment of area I (m⁴) quantifies a cross-section's resistance to bending. For a solid rectangle of width b and height h: I = bh³/12. For a solid circular section of diameter d: I = πd⁴/64. Section tables list I for standard steel profiles. Consistent SI units (m⁴) are required here.

What units should I use for the elastic modulus?

Enter E in Pascals (Pa). Common values: structural steel ≈ 200×10⁹ Pa (200 GPa), aluminum ≈ 69×10⁹ Pa, concrete ≈ 25–35×10⁹ Pa, timber ≈ 10–14×10⁹ Pa. If you have E in GPa, multiply by 1×10⁹ to convert to Pa.

What does Euler-Bernoulli beam theory assume?

Euler-Bernoulli theory assumes the beam is slender (length much greater than depth), cross-sections remain plane and perpendicular to the neutral axis after bending, and the beam is uniform along its length. It neglects shear deformation and rotary inertia. For deep beams or higher modes, Timoshenko beam theory gives more accurate results.

How do I avoid resonance?

Keep the beam natural frequency at least 20–30% away from any excitation frequency. If resonance cannot be avoided by geometry, consider adding damping (rubber mounts, tuned mass dampers) or changing the cross-section (higher I) or material (higher E) to shift the natural frequency.

Natural Frequency of a Beam Calculator

Name: Natural Frequency of a Beam Calculator
Availability: InStock
Author: Nham Vu

Enter beam length, elastic modulus, second moment of area, and linear mass density to compute the fundamental natural frequency for simply supported, cantilever, or fixed-fixed boundary conditions.

Beam Parameters

Boundary Condition

Simply Supported (Pinned-Pinned)

β² = π² ≈ 9.8696

Cantilever (Fixed-Free)

β² ≈ 3.5160

Fixed-Fixed (Clamped-Clamped)

β² ≈ 22.3733

Beam Length (L)

Elastic Modulus (E)

Second Moment of Area (I)

m⁴

Rectangle b×h: I = bh³/12 | Circle d: I = πd⁴/64

Linear Mass Density (m = ρA)

kg/m

Fill in the beam parameters and click Calculate

Summary

Enter beam length, elastic modulus, second moment of area, and linear mass density to compute the fundamental natural frequency for simply supported, cantilever, or fixed-fixed boundary conditions.

How it works

Select the boundary condition that matches your beam: simply supported, cantilever, or fixed-fixed.
Enter the beam length L in meters (m).
Enter the elastic modulus E in Pascals (Pa) — e.g., 200e9 for structural steel.
Enter the second moment of area I in m⁴ — e.g., from section tables or your own calculation.
Enter the linear mass density m (mass per unit length) in kg/m.
Click Calculate to see the fundamental natural frequency in Hz and the angular natural frequency in rad/s.

Use cases

Checking whether a floor beam natural frequency exceeds the 8 Hz serviceability threshold.
Designing machine support frames to avoid resonance with equipment operating frequencies.
Estimating bridge girder natural frequencies for pedestrian loading assessment.
Validating finite element modal analysis results against closed-form solutions.
Teaching structural dynamics and Euler-Bernoulli beam vibration in engineering courses.
Preliminary sizing of cantilever brackets to avoid resonance with pump or motor speeds.
Comparing natural frequencies of different cross-sections or materials for beam selection.
Checking that antenna masts or signpost poles will not resonate at wind vortex-shedding frequencies.

Frequently Asked Questions

Last updated: 2026-07-04 · Reviewed by Nham Vu