Natural Frequency Calculator

What is natural frequency?

Pull a mass on a spring and let go, and it swings at a rate of its own. That rate is the natural frequency, and it depends only on mass and stiffness: f = (1 / 2π) × √(k / m). A stiffer spring raises it and a heavier mass lowers it.

From static deflection

If you know how far the load compresses its spring under its own weight, you do not need mass and stiffness separately: f = (1 / 2π) × √(g / δ). With deflection in mm the practical form is f ≈ 15.76 / √δ. A deflection of 1 mm gives about 15.8 Hz, 4 mm about 7.9 Hz and 25 mm about 3.2 Hz. This is the relation used when choosing vibration mounts and spring isolators.

Example

A 10 kg mass sits on springs with a total stiffness of 4 N/mm. The angular frequency is √(4000 / 10) = 20 rad/s and the natural frequency is 20 / 6.283 = 3.18 Hz. That is 191 cycles per minute, so a machine turning near 191 rpm would drive the system into resonance.

Why does it matter?

When the forcing frequency approaches the natural frequency, the vibration amplitude grows sharply. This is resonance. Designs keep the running speed and the natural frequency well apart.

Limits

The calculator covers the undamped single degree of freedom model. Parts with distributed mass such as beams and shafts, systems with several masses and nonlinear springs are outside it. For the effect of damping and forcing, see the single degree of freedom vibration calculator.