Why looseness makes harmonics

A perfectly balanced, perfectly tight machine running at speed shows one clean peak in its spectrum: 1X, once per revolution. Loosen a bolt, crack a pedestal or let a bearing get sloppy in its housing, and suddenly the spectrum fills with peaks at 2X, 3X, 4X… sometimes even at 1.5X and 2.5X.

Nothing in the machine is turning at three times running speed. So where do those peaks come from?

The short answer: the force is still a clean sine wave. It's the response that stops being one. When a loose part hits a stop, its motion gets cut off, and any waveform that isn't a pure sine wave is, mathematically, a sum of sine waves at multiples of its base frequency.

Try it. The grey curve is the motion a tight machine would have. The teal curve is what happens when the motion is limited by a clearance, with the part hitting a stop. Drag the limits in and watch the spectrum.

Three things to notice

1. Clip both sides equally → odd harmonics only

Choose Both sides. You get 1X, 3X, 5X, but no 2X. A waveform that's symmetric top-to-bottom (the second half of each cycle is an upside-down copy of the first) can only contain odd harmonics. The even ones cancel out exactly.

2. Clip one side → even harmonics appear

Choose One side. Now the motion is limited only in one direction, like a foot lifting off and slapping back down on a loose baseplate. The symmetry is broken, and 2X shows up straight away, often as the biggest harmonic. This is why a strong 2X is one of the classic signs of structural looseness.

3. Hit every other revolution → half-orders

Choose Every other rev. If the impact only happens on alternate revolutions, the pattern now repeats every two revolutions. Its fundamental is therefore 0.5X, and the spectrum fills with 0.5X, 1.5X, 2.5X… This matches the field picture of advanced looseness (bearing loose in its housing, worn fits): a forest of harmonics plus half-orders and a raised noise floor.

What this means in the field

• Harmonics don't mean something is spinning faster. They mean the motion has a shape that isn't a pure sine, so check the time waveform as well as the spectrum.
• Look at which harmonics. Mostly odd suggests a symmetric limit; strong 2X suggests a one-sided limit; half-orders suggest something that rattles loose every other revolution.
• Other faults make harmonics too. Misalignment, rubs and bearing defects all distort the waveform. The harmonic pattern is a clue, not a diagnosis. Phase readings, the time waveform and the machine's history do the rest.

The maths, briefly

Any waveform that repeats every revolution can be written as a Fourier series, x(t) = Σ aₙ sin(nΩt + φₙ), with n = 1, 2, 3…. A pure sine has only a₁. Clip it, and energy moves into the higher aₙ. If the waveform satisfies x(t + T/2) = −x(t) (half-wave symmetry), every even aₙ is zero. And if the pattern only repeats every two revolutions, the series is built on Ω/2 instead, which is where the half-orders come from.

Downloads

• looseness_demo.py · the same three cases in Python (NumPy + Matplotlib)

Further reading

• Acoem: 3 types of mechanical looseness · structural, pedestal and fit-related looseness and their signatures
• Resources page · books and standards I recommend