Natural frequency and resonance, explained

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Mass and spring

Pull a mass on a spring and let go. It swings back and forth at one particular rate, whatever the size of the pull. That rate is the natural frequency: ω_n = √(k/m) in radians per second, or f_n = ω_n / 2π in hertz.

A stiffer spring raises it and a heavier mass lowers it, both under a square root. To double the natural frequency you need four times the stiffness, or a quarter of the mass. Real structures have many natural frequencies, one per mode shape, but each mode behaves like its own mass on a spring.

Damping

Damping takes energy out of the motion. It is measured by the damping ratio ζ, the actual damping divided by the critical damping 2√(km). Below ζ = 1 the system oscillates while the motion dies away, at the damped frequency ω_d = ω_n √(1 − ζ²). At ζ = 1 it returns to rest as fast as possible without overshooting. Above 1 it creeps back slowly.

Most metal structures have damping ratios of a few per cent, so in practice ω_d and ω_n are almost the same number.

Resonance

When a force pushes the system at its own natural frequency, every push arrives in step with the motion and the amplitude builds. Only damping limits it. At resonance the amplitude is 1/(2ζ) times the deflection the same force would cause statically, so a structure with 5 per cent damping moves ten times as far.

This is why designers keep running speeds away from natural frequencies, or pass through them quickly. Vibration isolation works the other way round: mounts soft enough that the natural frequency sits well below the forcing. Isolation only starts once the forcing frequency is above √2 times the natural frequency. Below that, the mounts make things worse.

A worked number

A 10 kg motor sits on mounts with a combined stiffness of 40 kN/m. ω_n = √(40,000 / 10) ≈ 63.2 rad/s, so f_n ≈ 10.1 Hz. If the motor runs at 600 rpm, any unbalance pushes it at 10 Hz: almost exactly at resonance. With ζ = 0.05 the vibration is about ten times the static deflection.

Softer mounts of 10 kN/m bring f_n down to about 5.0 Hz. Now 10 Hz is about twice the natural frequency, above the √2 threshold, and the mounts isolate instead of amplify.

Common mistakes

• Mixing hertz and radians per second. They differ by 2π, and a resonance check done in the wrong one misses by a factor of six.
• Using the weight in newtons, or the mass in grams, where the formula wants the mass in kilograms.
• Combining springs the wrong way: in parallel the stiffnesses add, in series their reciprocals add.