Vibration and machine monitoring

A turning machine shakes in step with its shaft, and a crack in a bearing adds a faint, fast ringing. Watch the spectrum below: the shaft shows as tall lines, but the crack’s marked rate does not stand out from the noise.

Lines at the shaft’s orders

Put your hand on a running washing machine, a pump or a car’s engine, and you feel it shake. That shaking is a signal, and it tells a mechanic a surprising amount about what is going on inside.

Most machines turn. A shaft spins at a steady rate, and everything bolted to it goes round with it. So whatever shakes the machine comes back once every turn.

I call the shaft’s rate $f_\text{r}$, in turns per second, that is, in hertz. A shaft at 1500 revolutions per minute has $f_\text{r}=25$ Hz.

A spectrum in orders

A signal that repeats once per turn is periodic with period $1/f_\text{r}$. In “The line spectrum, and what a delay does” of Fourier series coefficients (7.2), a periodic signal had a spectrum of lines at whole multiples of its repeat rate. So a machine’s spectrum is a row of lines at $f_\text{r}$, $2f_\text{r}$, $3f_\text{r}$ and so on.

Mechanics read those lines in orders. A frequency $f$ is at order $f/f_\text{r}$. So $f_\text{r}$ itself is order 1, written 1×, and $2f_\text{r}$ is 2×.

Think of a car’s engine note as you press the pedal. It rises with the revs, and all its harmonics rise with it, in step. Read in orders, nothing moves at all.

Different faults shake the machine in different ways, and land on different orders:

• imbalance, a heavy spot on the rotor, pulls once per turn: it shows at 1×;
• misalignment, two shafts joined slightly off line, shows mostly at 2×;
• looseness, a part rattling in its mounting, gives lines at many orders.

So a spectrum read in orders is a first diagnosis.

A crack in a bearing

A shaft runs in rolling-element bearings. Between an inner race, a ring on the shaft, and an outer race, a ring held still in the housing, a set of balls rolls round.

Now suppose the outer race has a small crack. Each time a ball rolls over it, the ball strikes the edge. One strike is a tiny impact, like a tap with a hammer.

A tap makes the machine’s housing ring, the way a wine glass rings when you flick it. It rings at a structural resonance, a frequency set by the metal’s stiffness and mass, often kilohertz. Each ring dies away within milliseconds.

How often do the strikes come? With $N_\text{b}$ balls, the outer-race defect frequency is

Here $d_\text{b}$ is a ball’s diameter and $D_\text{p}$ the diameter of the circle through the balls’ centres. The contact angle $\theta$ is the angle at which the balls press on the races. BPFO stands for “ball pass frequency, outer race”. A crack on the inner race gives the same formula with a plus sign:

The machine on this page has $N_\text{b}=9$ balls, $d_\text{b}/D_\text{p}=0.2$ and $\theta=0$. Its outer-race defect frequency is 90.0 Hz, order 3.60, so a strike comes every 11.11 ms.

The page’s machine

Every signal on this page is synthetic: built from the formulas below and the site’s seeded random numbers, not recorded from a real machine.

The shaft part has three orders, 1×, 2× and 3×. To it I add the bearing’s strikes and some noise:

A strike at time $t_i$ adds a ringing at the 3 kHz resonance, which decays with the time constant $\tau=2$ ms. From $t=t_i$ on, it adds

By the next strike, 11 ms later, a ring has fallen below 0.4 % of its start.

The first strike is at 5 ms. Real strikes are not perfectly regular, so each gap is $1/f_\text{BPFO}$ times $1+0.02\,(\text{draw}-0.5)$. Each draw is a fresh number between 0 and 1 from the seeded generator (seed 3210), so each gap moves by at most 1 %.

In one second that makes 90 strikes, on average 11.108 ms apart. The noise is 0.5 times a seeded normal number (seed 321) at every sample.

I take one second at $f_s=16\,384$ Hz: 16 384 samples, so the DFT’s bins are exactly 1 Hz apart.

The strikes are weaker than the noise. Their RMS is 0.318, against 0.499 for the noise: 3.9 dB less power.

To get the spectrum I use a Hann window, as in Windowing & spectral leakage (15.1). Then I scale it as an amplitude spectrum, as in “One spectrum, four units” of Reading a spectrum: scaling and units (15.4):

shown as $20\log_{10}A_k$ in dB re 1. A sine of amplitude 1 then reads 0 dB. The picture at the top of the page shows this spectrum for the page’s machine.

Notice the defect frequency. At 90.0 Hz the plain spectrum reads −35.5 dB, 6.3 dB above the floor’s median. But between 80 and 200 Hz, six bins of plain noise stand taller still. Nothing marks 90 Hz as special.

Why the strikes hide

There are two reasons, and both come from where the strikes put their energy.

First, the strike train does repeat at 90 Hz, so it has lines at 90 Hz and its multiples. But each strike is a quick oscillation about 0, with almost no net push in one direction, so those lines are small. On its own, the strike train reads −37.0 dB at 90 Hz, 4.7 dB above the noise floor’s median and below its taller peaks.

Second, its real energy sits near 3 kHz. A decay with time constant $\tau$ falls by 3 dB at $1/(2\pi\tau)$ either side of its centre: here 80 Hz either side. So the resonance band rises in a hump, its median 3.9 dB above the floor. The hump says that something rings near 3 kHz, but not how often.

In “A weak tone beside a strong one” of Windowing & spectral leakage (15.1), a better window found a weak tone hiding in a strong one’s leakage. That does not help here. The strikes’ rate is not hiding in leakage; it is barely in the spectrum at all.

Envelope of the ringing: the defect’s rate

The strikes do have one thing the noise lacks: a rhythm. Every 11.1 ms, the 3 kHz ringing swells and dies. Envelope analysis turns that rhythm into lines in three steps.

1. Band-pass around the resonance, keeping 2.5 to 3.5 kHz. That throws away most of the noise, and the shaft’s lines with it.
2. Take the envelope: the length of the spinning arrow of “A spinning arrow’s length and speed” in The Hilbert transform and the analytic signal (27.2). Each ringing strike becomes one bump, and the 3 kHz oscillation is gone.
3. Take the envelope spectrum: the amplitude spectrum of the envelope, minus its mean, by the same recipe as before.

Think of a tap running into a sink while it also drips. The rush of water hides the drips’ sound in the mix, but you can still hear their rhythm.

The band-pass

The band-pass comes from “From the spec to the taps” of Window-method FIR design (19.1). It has $N_h=255$ taps, designed with SciPy’s firwin and its default Hamming window, cut off at 2.5 and 3.5 kHz.

Its gain is −3 dB at 2526 Hz and 3474 Hz. From 2.6 to 3.4 kHz its gain is flat to within about 0.05 dB. At the shaft’s 25, 50 and 75 Hz it is below −68 dB.

The filter delays its output by $\alpha=127$ samples, 7.75 ms. I shift the output back by that much, so that each ringing stays at its strike’s time.

The filter changes the balance. It keeps 90 % of the strikes’ power but only 12 % of the noise’s. In the band, the strikes have 4.7 dB more power than the noise, instead of 3.9 dB less.

The envelope and its spectrum

Call the band-passed signal $y[n]$ and its analytic signal $y_\text{an}[n]$. The envelope is $\lvert y_\text{an}[n]\rvert$. I subtract its mean, so that the spectrum’s bin at 0 Hz does not tower over the rest, and take the spectrum.

Why lines? The envelope is a row of bumps, one per strike, about 11.1 ms apart. A row of bumps that repeats is periodic, so by 7.2 its spectrum has lines at the repeat rate and its multiples. The 3 kHz is gone, because the envelope keeps only the arrow’s length.

Watch the envelope’s bumps line up with the strikes, then the envelope spectrum draw a line at the defect’s rate and at each of its multiples.

Notice the envelope spectrum’s peaks: 90 Hz, 180 Hz and 270 Hz, between 23.1 dB and 29.5 dB above its floor of −40.8 dB. The bearing that the plain spectrum hid now shows its rate.

At the shaft’s 25 Hz the envelope spectrum sits only 2.1 dB above its floor, well inside the noise. The band-pass removed the shaft’s lines before the envelope was taken.

What the picture shows

In the first 50 ms you can see each strike. The strikes come at 5.0, 16.2, 27.3, 38.3 and 49.3 ms. Band-passed, each one’s ringing peaks between 1.08 and 1.13, at least twice the noise’s largest swing of 0.53.

The envelope’s bumps top out 0.6 to 0.8 ms after their strikes, as the ringing builds up through the filter. The last one peaks at 50.2 ms, just past the panel’s edge.

Over the whole second, 88 of the 90 strikes make a bump within 1 ms of the strike and higher than everything else in its gap. The two exceptions, at 327 and 505 ms, are beaten by noise. The spectrum gathers all 90 anyway.

Noise makes peaks of its own in the envelope spectrum. The tallest one away from the defect’s multiples is at 231 Hz, 8.3 dB above the floor: 14.8 dB under the shortest defect line. The 4th and 5th lines, at 360 and 450 Hz, stand 19.5 and 16.6 dB above the floor.

An inner-race crack

A crack on the inner race would show at $f_\text{BPFI}=135.0$ Hz. That crack turns with the shaft, so its strikes grow and fade once per turn. As in Amplitude modulation (27.1), that puts sidebands at ±25 Hz around each line, at 110 and 160 Hz around the first.

Here the envelope spectrum at 135 Hz is 3.2 dB under its floor: this bearing’s inner race is fine.

Which band should you take? Here I knew the resonance was at 3 kHz. On a real machine it is found from the data, often with a chart called a kurtogram. I leave it out.

Worked example

1. The defect frequencies. With $N_\text{b}=9$, $f_\text{r}=25$ Hz, $d_\text{b}/D_\text{p}=0.2$ and $\theta=0$, so that $\cos\theta=1$, in hertz:

2. In orders. $90/25=3.60$ and $135/25=5.40$. Neither is a whole order, which is how a bearing line is told from a shaft harmonic.

3. The gaps. Without jitter, strikes come every $1/90$ s $=11.11$ ms. The jitter moves each gap by at most 1 %, about 0.11 ms.

4. Reading a line. The 2nd order has amplitude 0.4, and $20\log_{10}0.4=-7.96$ dB: the panel’s −8.0 dB. The 3rd has 0.15, which alone would read −16.5 dB. It reads −17.1 dB, because the noise in that bin, about 0.01, adds to it with its own phase.

5. The ringing’s width. With $\tau=2$ ms, the ring falls by 3 dB at $1/(2\pi\tau)=80$ Hz either side of 3 kHz, from about 2920 to 3080 Hz. That sits well inside the band-pass, which keeps 90 % of the strikes’ power.

Where you’ll meet this

Envelope analysis watches the bearings of wind turbines, rail axles, the pumps and fans of factories, and helicopter gearboxes. Predictive maintenance programmes track these lines over months, and plan a repair when a defect line starts to grow.

I kept the speed constant. When a machine speeds up, its lines move; order tracking samples the signal at fixed angles of the shaft instead of fixed times. That is resampling by a changing factor, as in Resampling by any factor (22.3), driven by a speed sensor.

I also left out gears, whose teeth add lines of their own. For more, see R. B. Randall, Vibration-based Condition Monitoring (2011), chapters 2 and 5, and R. B. Randall and J. Antoni, “Rolling element bearing diagnostics—a tutorial” (Mechanical Systems and Signal Processing, 2011).

The sensor in all of this is usually an accelerometer. Turning its raw readings into trustworthy numbers is the subject of Sensor signal conditioning (32.2).

Reference card