Natural Frequency Calculator

Natural frequency of a mass on a spring, a pendulum, a cantilever or simply supported beam with a mass and a torsional system, with the static deflection check.

How it works

Any elastic system with mass vibrates at a natural frequency set by the ratio of its stiffness to its mass. If a periodic force acts near that frequency, resonance amplifies the motion.

ωn = √(k ÷ m), fn = ωn ÷ 2π

For a beam, the stiffness is that of the loading case (k = 3 E I ÷ L³ for a cantilever tip load, 48 E I ÷ L³ for a center load on a simply supported beam) and the beam’s own mass adds an effective share of 0.2357 (cantilever) or 0.4857 (simply supported) of its mass to the tip mass (Rayleigh’s method). A useful check is the static deflection: f = 0.4985 ÷ √δst with δst in meters. The theory of beams is in Structures and Strength of Materials.

Worked example

A steel strip 20 mm wide and 5 mm thick (E = 200 GPa, density 7 850 kg/m³) projects 300 mm as a cantilever and carries a 0.5 kg mass at its tip.

  • I = 0.020 × 0.005³ ÷ 12 = 208.3 mm⁴, so k = 3 × 200×10⁹ × 0.0000000002083 ÷ 0.3³ = 4 630 N/m.
  • The beam mass is 7 850 × 0.020 × 0.005 × 0.3 = 0.2355 kg, so the effective mass is 0.5 + 0.2357 × 0.2355 = 0.5555 kg.
  • ω = √(4 630 ÷ 0.5555) = 91.29 rad/s, so f = 14.53 Hz (872 cycles per minute).
  • Check: static deflection = 0.5555 × 9.80665 ÷ 4 630 = 1.18 mm and 0.4985 ÷ √0.00118 = 14.53 Hz.
  • A 0.5 m pendulum swings at f = √(9.80665 ÷ 0.5) ÷ 2π = 0.705 Hz.

Things to know

Damping lowers the frequency slightly and limits the amplitude at resonance, but for light damping (as in most structures) the natural frequency is a good estimate of where resonance occurs.

Keep the forcing frequency (motor speed, blade passing, wheel rotation) outside the band from 0.7 to 1.3 times the natural frequency, or add damping.

Real structures have many natural frequencies (modes). This calculator gives the first (lowest) mode, which is usually the most important.

The result assumes a rigid support. A flexible support or mounting lowers the frequency: add its compliance in series (1 ÷ k = 1 ÷ k₁ + 1 ÷ k₂).

For a rotating shaft, the critical speed in rpm equals the first bending natural frequency in cycles per minute.

Common questions

What is resonance?

A large vibration that builds up when a periodic force matches a natural frequency. The motion is limited only by damping and can break parts that carry the load easily when static.

How can I change the natural frequency?

Raise it by stiffening (a thicker or shorter beam, a stronger spring) and lower it by adding mass. Frequency goes as √(k ÷ m).

Why does the pendulum frequency not depend on the mass?

The restoring force and the inertia are both proportional to the mass, so it cancels: only the length (and gravity) matter.

Sources

  • Rao, S. S., Mechanical Vibrations, Pearson (Rayleigh’s method, effective mass of beams); Thomson, W. T., Theory of Vibration with Applications.
  • Blevins, R. D., Formulas for Natural Frequency and Mode Shape, Krieger.

Last reviewed: November 2, 2026