Work Out What Your Bearing Will Sound Like Before It Fails

Most machine faults announce themselves only after they happen. A rolling-element bearing is the strange exception: months before a defect exists, you can already write down the exact rhythm it will drum out when it does. The reason is that a bearing is a tiny gearbox with no teeth — inner race, outer race, a train of rolling elements and the cage that spaces them, all rolling on each other at speed ratios fixed entirely by geometry. Put a spall on any one of those surfaces and it gets struck at a repetition rate the geometry decided the day the bearing was made.

That is why analysts talk about four famous frequencies: BPFO for a defect on the outer race, BPFI for the inner race, BSF for a damaged rolling element, and FTF for the cage itself. Know your bearing's geometry and shaft speed, and you can predict all four before the first measurement — then go looking for exactly those lines in the spectrum. This post derives them, shows why they are never neat multiples of running speed, and gives you a live calculator to try your own numbers.

Only four parameters matter: Z, the number of rolling elements; d, the rolling-element diameter; D, the pitch diameter (the circle through the ball centres); and θ, the contact angle between the load line and the radial plane — 0° for a deep-groove bearing, larger for angular-contact types. Everything scales with shaft speed fr, so the formulae are usually written in Hz with fr = RPM/60. The cage turns at FTF = (fr/2)·(1 − (d/D)·cosθ). Each of the Z elements passes a fixed point on the outer race once per cage turn, so BPFO = Z·FTF. The inner race spins past the ball train the other way, giving BPFI = (Z·fr/2)·(1 + (d/D)·cosθ). And each ball spins on its own axis at BSF = (fr·D/2d)·(1 − ((d/D)·cosθ)²).

Two sanity checks fall straight out of the algebra and are worth memorising. First, BPFO + BPFI = Z × fr, always — if your two ball-pass numbers do not sum to the element count times running speed, something is wrong. Second, a damaged rolling element strikes both races on each rotation, so ball defects usually show at 2×BSF rather than BSF itself. These checks catch typos faster than any lookup table.

The cage is the key. With the outer race fixed and no slip, each rolling element's centre moves at the average of the two surface speeds it touches — the moving inner race on one side, the stationary outer race on the other. But it is a weighted average, and the weighting is the geometry term (d/D)·cosθ: the fatter the ball relative to the pitch circle, the further the cage falls below half shaft speed. For a typical geometry that lands the cage at something like 0.397× running speed — close to a half, never exactly a half, and never a tidy fraction.

Every defect frequency inherits that awkward ratio, because each one is just the cage rate multiplied up by the element count or reflected through the inner race. This is a diagnostic gift. Unbalance, misalignment, looseness and blade pass all live at integer orders — 1×, 2×, 3× running speed. A confident peak at 3.57 orders can only be a bearing, and if 3.57 matches the BPFO you predicted for the drive-end bearing, you know which component and which surface before anyone touches a spanner.

The same frequency wears two units. In Hz it is what your analyser actually displays: a 6205-style geometry at 1480 RPM (fr = 24.67 Hz) puts BPFO at about 88.1 Hz, BPFI at 133.9 Hz, BSF at 57.3 Hz and FTF at 9.8 Hz. But those Hz values move every time the speed changes, which makes them useless for pattern memory. Divide by running speed and you get orders — 3.57, 5.43, 2.32 and 0.40 for that same geometry — and those numbers never move. Speed the machine up, slow it down, the comb slides along the Hz axis but stays nailed to the same orders.

So the working habit is: think in orders, verify in Hz. Orders tell you what kind of line you are looking at — integer means shaft-driven, a fraction like 3.57 means bearing. Hz tells you where to put the cursor on today's measurement at today's speed. The calculator below shows both side by side for exactly this reason.

The calculator below runs the four no-slip formulae live. Set shaft speed and the four geometry numbers, and it computes FTF, BSF, BPFO and BPFI in Hz and in orders, then draws the predicted comb on an order axis — grey peaks at the integer orders where shaft-driven energy lives, coloured lines where the bearing rates fall between them. The 2×BSF line is drawn too, since a damaged rolling element strikes both races per rotation, and the BPFI fundamental carries ±1-order sidebands because an inner-race defect rides in and out of the load zone once per revolution.

The presets are example geometries for illustration — a deep-groove set in the 6205 style and a steeper angular-contact set — not manufacturer's data. For a real programme, take Z, d, D and θ from the manufacturer's datasheet or a bearing database rather than measuring by hand: these formulae assume pure rolling, and real bearings slip a little under load, so measured rates typically sit a percent or two below the kinematic prediction. Treat the calculator as a search window, not a verdict.

A representative example, not a specific customer: a maintenance team inherits a fan with no bearing records beyond the number on the housing. From the bearing database they pull Z, d, D and θ, run the formulae at the fan's 1480 RPM, and write four Hz values on the route sheet before the first measurement. The next reading shows a small comb whose spacing sits about one percent below the predicted outer-race rate — exactly the shortfall slip produces — and the peak trends upward over the following months. Because the number was predicted in advance, nobody debates what the line is; the argument is only about when to schedule the replacement, which is the argument you want to be having.

The habit generalises. Compute the expected rates for every monitored bearing once, store them with the measurement point, and every future spectrum arrives pre-annotated with where trouble would show. If you want the full theory behind these four formulae — including slip, load-zone modulation and envelope analysis — the free Bearing & Gear Analysis 101 primer at 101.tieraonline.in walks through it step by step; it is a free primer, not an accredited ISO certification. When you need the formal analyst credential, TIERA's TCAT programme (see /services) runs structured training with proctored examinations at exams.tieraonline.in.

A predicted frequency is only useful if your measurement chain can resolve it. PhonoVibe DAQs capture vibration at 24-bit with IEPE sensor power across 2- to 16-channel models, and TVIB's TSAP201 analyser puts harmonic and sideband cursors directly on the spectrum — so the BPFO you computed above becomes a cursor you drop on real data and check line by line.

And because the best way to learn these signatures is to make them happen, the TMFSS simulator seeds real outer-race, inner-race, cage and rolling-element defects on a benchtop machine — you predict the comb with the formulae, then measure it and watch the two agree.