How Machines Sing

Vibration is motion that repeats.

Hang a mass from a spring, pull it down and let go. It bobs up and down, again and again. Draw its position on a moving strip of paper and you get the shape at the heart of everything that follows: the sine wave.

Sound is vibration you can hear.

A vibrating surface pushes the air next to it, and the air pushes your eardrum. Machine vibration and sound are the same wave, measured in different places: in the air, or on the metal. A guitar string at 110 Hz and a bearing housing shaking 110 times a second draw the same curve.

It can be bigger…

The height of the wave is its amplitude. In a machine, amplitude means how far the metal actually moves. More amplitude, more energy, more damage.

…or faster.

How many times the motion repeats each second is its frequency, measured in hertz (Hz). One hertz is once per second. A higher frequency is a higher pitch.

Machines count in rpm. Waves count in hertz.

A motor turning 1800 revolutions per minute turns 30 times every second. Thirty times a second is 30 Hz. Divide rpm by 60 and you have the machine's fundamental note.

Machines are bass instruments.

Human hearing runs from about 20 Hz to 20 kHz. A piano spans 27.5 Hz to 4186 Hz. A machine's fundamental note sits at the very bottom of that range, or below it. Its bearings and gears sing much higher. The ruler below places them side by side.

Play with it.

Press and drag. Left and right changes the frequency, up and down the amplitude. Pure tones like this one are rare in machines, for reasons lesson 3 explains.

Where does the vibration come from?

No rotor is perfect. Somewhere on it there is a little extra mass: a heavy spot. As the shaft turns, the heavy spot pulls outward, around and around. A sensor on the bearing feels that pull once per revolution: a push when the spot passes under it, a tug when it passes the far side.

Once per revolution: 1X.

Vibration at exactly shaft speed is called 1X, "one times running speed". It is the fundamental note of the machine and the most common vibration there is. Every rotor has some. The question is how much.

Machines hum in B. Or in G.

Induction motors on a 60 Hz grid run just under 1800 or 3600 rpm: 30 and 60 Hz, a quarter-tone below the notes B0 and B1. On a 50 Hz grid they run near 1500 and 3000 rpm: 25 and 50 Hz, a shade above G0 and G1.

Hertz change. Ratios don't.

Slow a machine down and every frequency in it slides down together. The 1X note changes, and so does everything that depends on it. That is why analysts measure frequency in orders: multiples of running speed. In orders, 1X is always at 1, whatever the rpm.

A pure tone is rare.

The heavy spot makes a clean sine because the force it produces is smooth: one rotating pull. Most forces in a machine are rougher. A coupling pushed sideways or a bolt that lets a bearing lift also repeat once per revolution, but their shape is not a sine.

Below, the same 1X note twice: once as a pure sine, once with a different shape. Same pitch, different sound.

Any repeating shape is a stack of sines.

Every wave that repeats once per revolution can be built from sines at 1X, 2X, 3X, 4X…, each with its own height and timing. Those sines are the harmonics, and this is what makes vibration analysis possible. Build a few yourself.

Misalignment: twice per revolution.

When two coupled shafts do not share a centerline, the coupling pushes the shaft against one side of its bearing. The shaft can move freely one way and not the other, so its path flattens from a circle into a banana. A flattened circle, traced once per turn, has a strong second harmonic: 2X grows until it rivals 1X.

Looseness: the wave hits a wall.

Loosen the hold-down bolts of a bearing. When the 1X force pulls up, the housing lifts. When it pushes down, the base stops it. The sine is cut flat on one side, and a clipped wave is full of harmonics: 3X, 4X, 5X and on up. When the housing starts bouncing every other turn, half-orders appear as well: 0.5X, 1.5X, 2.5X.

The harmonics are a chord you already know.

In orders, 2X is an octave above 1X. 3X is an octave and a fifth. 4X is two octaves, 5X two octaves and a major third. This is the harmonic series, the set of notes a bugle can play with no valves at all, so a machine full of harmonics is playing a bugle call. Everything the shaft itself produces is in tune with itself. The next lesson is about the sources that are not.

Gears: a very high harmonic.

A pinion with 23 teeth meshes 23 times per revolution. Teeth meeting is a force, so there is a tone at 23X: the gear mesh frequency. At 1800 rpm that is 690 Hz, the whine you can hear across the plant. Change the number of teeth and listen to the pitch move.

Bearings: a bell struck by every ball.

A rolling-element bearing has an inner race, an outer race, and a set of balls or rollers held apart by a cage. When a flaw appears on a race, every ball that rolls over it strikes it, and each strike rings the bearing like a tiny bell. The strikes come at the rate balls pass the flaw: the ball pass frequency.

Bearing tones are out of key.

How often do balls pass a flaw? It depends on the geometry: how many balls, how big they are compared with the bearing, and the angle at which they touch the races. The answer is almost never a whole number of shaft turns. A typical outer race frequency is 3.6X: not 3X, not 4X. It falls between the harmonics, and that is how you tell it apart from everything the shaft itself produces.

Beats: two notes almost the same.

Two tones a hair apart do not sound like two tones. They sound like one tone that swells and fades, at a rate equal to their difference. A two-pole motor at 3580 rpm has its 2X at 119.3 Hz, right next to the 120 Hz electrical hum (twice the line frequency): the amplitude breathes every second and a half. Two identical fans side by side do the same thing through the floor.

Three families.

Every frequency in a machine's spectrum belongs to one of three families, named by where it sits relative to shaft speed.

The next lesson is about the sensors that pick all of this up.

One motion, three numbers.

Take a bearing housing shaking thirty times a second. You can describe that motion by how far it moves (displacement, in micrometres), how fast it moves (velocity, in mm/s), or how hard it is thrown back and forth (acceleration, in m/s² or g). They are three descriptions of one motion.

Same machine, three tone controls.

Because of that, the unit you choose changes what you see. In displacement the low notes dominate and the bearings vanish. In acceleration the bearings and gears shout and 1X disappears. Velocity sits in the middle, which is why the severity standards are written in velocity.

Three instruments.

Three families of sensors measure these three quantities. Each has a history, a sweet spot and a blind spot.

The proximity probe hears displacement.

A coil at the tip of the probe drives a radio-frequency magnetic field into the shaft. The field stirs up eddy currents in the metal, and the eddy currents drain energy from the coil: the closer the shaft, the more they drain. The probe's output voltage is therefore a measure of the gap, typically −7.87 volts per millimetre (200 mV per mil), and it never touches the shaft.

The velocity pickup hears velocity.

Inside a velocity transducer a magnet hangs on soft springs inside a coil fixed to the case. Shake the case fast enough and the magnet stays put while the coil moves around it. A coil moving through a magnetic field makes a voltage, and that voltage is proportional to how fast the coil moves: velocity, directly, with no power supply at all.

The accelerometer hears acceleration.

Squeeze a piezoelectric crystal and electric charge appears on its faces. Bolt a crystal to the machine with a small mass on top. Every time the machine accelerates upward, the mass's inertia presses the crystal harder; every time it accelerates downward, it presses less. The charge follows the acceleration, faithfully, up to tens of thousands of times a second.

Mounting is part of the sensor.

An accelerometer on a magnet is a mass on a spring, with a natural frequency of its own. Approaching that frequency its readings swell; above about a third of it they are no longer true. A stud-mounted sensor is good to about 10 kHz. The same sensor on a two-pole magnet is good to about 1 kHz. Held by hand on a probe tip, 500 Hz. The bearing frequencies live up there.

Which instrument for which machine.

Whatever the instrument, its output is a voltage that varies in time. The next lesson is about recording it.

The sensor gives a voltage. The analyser takes snapshots of it.

A sensor's output is a continuous voltage. A computer cannot hold a continuous anything, so it measures the voltage at regular instants and stores the numbers. Each number is a sample. How many it takes per second is the sample rate, Fs. The list of numbers is the time waveform.

Aliasing: the wagon wheel that turns backwards.

In old westerns, wagon wheels seem to turn slowly backwards: the camera's 24 frames per second sample the spokes too slowly. A strobe light does the same to a rotating shaft. An analyser sampling too slowly does the same to a vibration. The false frequency is called an alias, and once it is in the recording nothing can remove it.

Nyquist's rule.

You need more than two samples per cycle of the highest frequency present. This is why compact discs sample at 44.1 kHz: human hearing stops near 20 kHz, and 44.1 is a little more than twice that, with room for a filter. Machine analysers do the same sum.

Setting up a recording.

Two choices define a measurement: the highest frequency you want to see, Fmax, and how finely you want to see it, the number of lines. Everything else follows.

How big is it? Peak, peak-to-peak, RMS.

The size of a waveform is quoted three ways. Peak is the largest excursion from the centre. Peak-to-peak is the span from lowest to highest. RMS, root mean square, is the steady equivalent, the number that tracks energy. For a pure sine, peak = 1.414 × RMS. Add a few impacts and peak jumps while RMS barely moves. Their ratio, the crest factor, is one of the oldest bearing indicators there is.

A chord you cannot read by looking.

Below is a waveform made of three tones: 1X, 2X and a bearing tone at 3.6X. Added together they make a wiggle that is hard to read by eye. Your ear reads it at once: press play and you hear three notes. The spectrum gives your eyes what your ears already have.

The question Fourier asks.

How much of a given frequency is in this signal? To find out, multiply the signal, sample by sample, by a test sine of that frequency, and add up the products. Where the signal contains that frequency, signal and test sine rise and fall together: the products are mostly positive and the sum is large. Where it does not, the products come out positive as often as negative and the sum cancels to nothing. Ask the question for every frequency in turn and you have the spectrum. That is the discrete Fourier transform, the DFT.

The DFT is slow. The FFT is the same answer, fast.

Asking the question for N frequencies, each over N samples, costs N² multiplications. For a 4096-sample record that is nearly 17 million. In 1965 James Cooley and John Tukey published a way to split the job in half, and the halves in half again, until the whole thing costs about N·log₂N operations: 49,000 for the same record, 340 times fewer. That is the Fast Fourier Transform. It computes the same spectrum in a small fraction of the time.

How the split works.

Split the samples into evens and odds. The spectrum of the whole is the spectrum of the evens plus a twisted copy of the spectrum of the odds, so two half-size problems solve the full one. Do it again on each half. For eight samples that is three rounds of pairing, the "butterflies" below. For 4096 samples it is twelve rounds, and every round costs only N operations.

Resolution: you have to listen long enough.

Every line of the spectrum is Δf = 1/T wide, where T is the length of the recording. Two tones closer together than a line or two merge into one. The only cure is a longer recording. Here are the 2X of a two-pole motor at 3580 rpm, 119.3 Hz, and the 120 Hz electrical hum beside it.

Leakage, and the window that stops it.

The FFT treats the recording as if it repeated forever, end spliced to start. If a tone does not complete a whole number of cycles in the record, the splice is a jolt, and the jolt smears energy into the neighbouring lines: leakage. The cure is a window, a smooth shape that fades the record in and out so the ends always meet. The Hanning window is the everyday choice. It costs a slightly wider peak, and it reads amplitude up to 15% low when a tone falls between two lines, which is why peak readings jitter a little from one measurement to the next.

Averaging: let the noise settle.

Random noise draws a different spectrum every time; the machine's tones draw the same one. Average several spectra and the noise floor stops jumping around while the real peaks stay put. Averaging steadies random noise rather than lowering it, so that whatever stands above the floor can be seen.

The whole machine, on one page.

Here is a whole machine train, motor, gearbox and pump: 1X with its harmonics, the bearing tone with its harmonics, blade pass, gear mesh with its sidebands, and the hiss of a bearing beginning to wear. In the time waveform they are a tangle. In the spectrum each has its own line, and you can read them off one by one.

The spectrum says which frequencies are present and how large they are. It does not say when. The last lesson is about timing.

Two waves can agree, or disagree.

Two vibrations at the same frequency can still differ in one way: timing. The delay between them, measured as a fraction of a cycle, is their phase difference. We quote it in degrees: a whole cycle is 360°, half a cycle is 180°. In phase, they add. Half a cycle apart, they cancel.

Phase needs a reference. The tach provides it.

One sensor cannot tell you when a vibration peaks, only that it does. Add a once-per-turn pulse from the shaft, from a strip of reflective tape and an optical tachometer, or from a notch watched by a proximity probe (the keyphasor, a Bently Nevada trade name the whole field adopted), and every peak gets an address: so many degrees after the pulse. That number is the phase of the 1X vibration, and it separates several faults that the spectrum alone cannot.

Phase tells apart what the spectrum cannot.

Unbalance and misalignment can both show a large 1X. Phase separates them. Measure the 1X phase at every bearing, in the vertical, horizontal and axial directions, and compare. The patterns are simple.

Two probes make an orbit.

On a machine with proximity probes, two probes 90° apart see the shaft's horizontal and vertical motion at the same instant. Plot one against the other and you draw the path of the shaft centre inside its bearing: the orbit. The phase between the two probes sets its shape, the keyphasor puts a dot on it once per turn, and the result is a fault signature you can read at a glance.

Where to go next.

Frequency says what is vibrating, amplitude how much, phase where and when. Two modules put those to work.