Bearing Defect Frequency Calculator

BPFO, BPFI, BSF and FTF derived from the bearing’s own rolling kinematics, in hertz and in orders — with the BPFO + BPFI = Z × shaft speed identity printed live, the 0.4Z rule’s exact error, and the slip band you actually search in.

Last updated 24 Sep 2026

Bearing defect frequencies

Geometry and speed → BPFO, BPFI, BSF, FTF

Shaft speed (rev/min)Every one of the four frequencies is proportional to this, so they are usually quoted as ORDERS of shaft speed and converted at the end.

Number of rolling elements ZCount them, or get them from the maker. Getting Z wrong is the commonest input error and the identity BPFO + BPFI = Z × shaft speed is how you catch it: the sum of the two race frequencies in orders must be the element count exactly.

Rolling element diameter B_d (mm)Ball or roller diameter. Only the RATIO B_d/P_d matters, so consistent units are enough.

Pitch diameter P_d (mm)The diameter of the circle the element centres lie on, which is very close to (d+D)/2 for most bearings — 38.5 mm for a 6205, against a (25+52)/2 = 38.5. Not the bore and not the outside diameter.

Contact angle β (degrees)Zero for a deep groove ball bearing under radial load. 15° to 40° for an angular contact bearing, and around 10° to 30° for a tapered roller. It enters as cos β on the diameter ratio, so a 40° angle changes the frequencies by about a quarter of the ratio.

Assumed slip (%)Real bearings skid a little, so the measured tones sit BELOW the calculated ones — typically 1 to 2 per cent. This sets the search band the page prints.

Not a circuit: one axis in orders of shaft speed, with the four defect frequencies on it as pointers and the whole numbers ticked underneath. The ticks are the point of the figure. A spectrum from a rotating machine is dense with content at whole multiples of shaft speed — imbalance at 1×, misalignment at 2×, looseness, blade passing, gear mesh — and a bearing tone is identifiable precisely because it lands BETWEEN those lines. Watch the four pointers as you change the ball and pitch diameters: they move continuously and generally sit nowhere near a tick. If one of them lands on a tick, that frequency will be hidden in a real measurement and you should look at its harmonics and sidebands instead. The line along the top is the free self-check drawn: because BPFO and BPFI always add to exactly the rolling element count, whatever the geometry, the two race pointers always straddle Z/2 symmetrically — and the bar reaches from zero to Z/2 so you can see them do it. If they do not, an input is wrong. The residual printed below should be zero to the last decimal place.

107.16HzExample

A 6205 deep groove ball bearing — nine balls of 7.94 mm on a 38.5 mm pitch diameter, zero contact angle — at 1,800 rev/min

Four frequencies from rolling without slip, and the identity that checks them

FTF = ½f_r(1 − r·cos β) · BPFO = Z·FTF · BPFI = ½Z·f_r(1 + r·cos β) · BSF = f_r(1 − r²cos²β)/(2r) · r = B_d/P_d · BPFO + BPFI = Z·f_r exactly

f_r: shaft rotational frequency in Hz, which is rev/min divided by 60. Everything scales with it, which is why these are quoted as orders
Z: the number of rolling elements. The identity BPFO + BPFI = Z·f_r is how you check you have it right
r = B_d/P_d: the ball or roller diameter over the pitch diameter — the diameter of the circle the element centres lie on. Only the ratio matters. About 0.18 to 0.32 for real deep groove ball bearings
β: contact angle. Zero for a deep groove bearing under radial load; it enters only as cos β on r, so a 40° angle reduces the effective ratio by about a quarter
BSF: the element’s spin rate about its own axis, measured in the CAGE’s frame. Usually observed at 2×BSF, because one defect on a ball hits both races per ball revolution
1 to 2 per cent: how far BELOW these figures the real tones sit, because real bearings slip. Search a band, not a line

Worked example

A 6205 deep groove ball bearing — nine balls of 7.94 mm on a 38.5 mm pitch diameter, zero contact angle — at 1,800 rev/min
Shaft frequency first, because everything scales with it: f_r = 1,800/60 = 30 Hz. And the geometry ratio, which is the only geometry that matters: r = B_d/P_d = 7.94/38.5 = 0.2062. Note how close that is to 0.2, which will matter in a moment
The cage rate comes first in the derivation, because the other three follow from it. FTF = ½f_r(1 − r·cos β) = ½ × 30 × (1 − 0.2062) = 11.91 Hz, which is 0.3969 of shaft speed. A cage always turns at less than half shaft speed, and that is a useful sanity check in itself
BPFO is then simply Z times the cage rate, because a fixed point on the outer race is passed once by each of the nine balls per cage revolution: 9 × 11.91 = 107.2 Hz, or 3.5719 orders
BPFI counts the inner race's rate RELATIVE to the cage: ½Z·f_r(1 + r·cos β) = 162.8 Hz, 5.4281 orders. And BSF, the ball's own spin in the cage's frame, is f_r(1 − r²)/(2r) = 69.6 Hz — usually looked for at 2×BSF = 139.3 Hz, because one defect on a ball strikes both races once per ball revolution
NOW THE FREE CHECK, and use it every time. BPFO + BPFI = 107.2 + 162.8 = 270.0 Hz, and Z × f_r = 9 × 30 = 270 Hz. Exactly equal, and it must be: adding the two formulas cancels the geometry entirely, leaving Z·f_r whatever the ball diameter, pitch diameter and contact angle are. So if the two orders you compute do not add up to the element count, you have an input wrong — and if you can read both race frequencies off a spectrum but do not know how many balls the bearing has, their sum over shaft speed tells you
The rules of thumb, checked. 0.4Z = 3.6 orders against the exact 3.5719, which is 0.79 per cent out; 0.6Z = 5.4 against 5.4281. Very close indeed, and the reason is the ratio: 0.4Z and 0.6Z ARE the exact formulas at r = 0.2, and this bearing's r is 0.2062. On a bearing with r = 0.32 the same rule would be nearly twenty per cent out
AND THE PARAGRAPH THAT MAKES ANY OF THIS USEFUL. These are theoretical frequencies computed for pure rolling, and real bearings slip — the elements skid slightly entering and leaving the load zone and the cage lags. Measured tones sit one to two per cent BELOW the calculated ones, so at 1.5 per cent slip you search for BPFO between 105.6 and 107.2 Hz rather than at a single line. The slip is also why these frequencies are NON-INTEGER multiples of shaft speed: 3.572 orders sits between the third and fourth harmonics of running speed rather than on top of either, and that is exactly what makes a bearing fault identifiable in a spectrum otherwise full of shaft-order content from imbalance, misalignment and looseness
Finally, what to look for beside each peak. A BPFI peak should carry SIDEBANDS spaced at shaft speed (132.8 and 192.8 Hz), because the defect rotates with the shaft and passes in and out of the load zone once per revolution. A 2×BSF peak should carry sidebands at the CAGE frequency (127.4 and 151.2 Hz). A BPFO peak normally does not have sidebands, because the defect sits still in the load zone — but it does have strong harmonics. The presence or absence of sidebands is often a better discriminator than the peak frequency itself

The four frequencies, and what each one means

All four follow from the kinematics of rolling without slip and none of them is a lookup: in the cage’s frame the rolling element’s surface speed must match the inner race at one contact and the outer race at the other, which is two equations in the cage rate and the element’s spin rate. Solve them and these four fall out. The orders column is the one to work in, because everything scales with shaft speed. Two structural facts worth keeping: BPFO is exactly Z times FTF, because a fixed point on the outer race is passed once by each element per cage revolution; and BPFI is Z times the inner race’s rate RELATIVE to the cage, which is why the two race frequencies add to Z·f_r exactly. This is a first-pass calculation on an idealised part — uniform section, static load, no stress raisers beyond those stated, and room-temperature material properties. Real parts have fillets, keyways, surface finish and duty cycles that a closed-form answer cannot see.
	Formula	Orders (this page’s example)	Hz	What it indicates, and what to look for beside it
BPFO	½·Z·f_r·(1 − (B_d/P_d)cos β)	3.572×	107.2 Hz	A defect on the OUTER race. The commonest bearing fault frequency seen in practice, because the outer race is where the load zone is fixed. Harmonics of it are usual; sidebands are not, because the defect does not move in and out of the load zone
BPFI	½·Z·f_r·(1 + (B_d/P_d)cos β)	5.428×	162.8 Hz	A defect on the INNER race. Expect SIDEBANDS spaced at shaft speed, because the defect rotates with the shaft and passes in and out of the load zone once per revolution, amplitude-modulating the tone
BSF	(P_d/2B_d)·f_r·(1 − ((B_d/P_d)cos β)²)	2.321×	69.6 Hz	A defect on a ROLLING ELEMENT. Usually seen at 2×BSF rather than BSF, because a single defect on a ball strikes both races once per ball revolution. Expect sidebands at the CAGE frequency, because the defective ball passes through the load zone once per cage revolution
FTF	½·f_r·(1 − (B_d/P_d)cos β)	0.397×	11.9 Hz	The cage rate, always below half shaft speed. Seen on its own only when the CAGE itself is damaged, which is a late-stage and dangerous fault; more often it appears as the sideband spacing around a rolling element frequency

The free self-check: BPFO + BPFI = Z × shaft speed, exactly

Add the two formulas and the geometry cancels: ½Z(1 − r·cos β) + ½Z(1 + r·cos β) = Z, whatever the ball diameter, the pitch diameter and the contact angle are. So the sum of the two race frequencies in orders of shaft speed is the ELEMENT COUNT, exactly, and that is a free check on every calculation and on every input. Use it two ways. Forwards, to verify a calculation — if the two orders do not add to a whole number equal to the element count, something is wrong with the geometry you typed. And BACKWARDS, to find the element count from a spectrum: if you can identify both race frequencies but do not know how many balls the bearing has, their sum divided by shaft speed tells you. This table is at B_d/P_d = 0.2 precisely, which is why the last two columns agree with the first two exactly — the 0.4Z and 0.6Z rules of thumb ARE the formulas at that ratio. This is a first-pass calculation on an idealised part — uniform section, static load, no stress raisers beyond those stated, and room-temperature material properties. Real parts have fillets, keyways, surface finish and duty cycles that a closed-form answer cannot see.
Elements Z	BPFO (orders)	BPFI (orders)	Sum	0.4Z	0.6Z
7	2.800×	4.200×	7.000×	2.80×	4.20×
8	3.200×	4.800×	8.000×	3.20×	4.80×
9	3.600×	5.400×	9.000×	3.60×	5.40×
10	4.000×	6.000×	10.000×	4.00×	6.00×
11	4.400×	6.600×	11.000×	4.40×	6.60×
12	4.800×	7.200×	12.000×	4.80×	7.20×
13	5.200×	7.800×	13.000×	5.20×	7.80×

The 0.4Z rule’s error, and the slip band you actually search in

The 0.4Z and 0.6Z rules are the exact formulas at B_d/P_d = 0.2, so their error is exactly 0.4/(0.5(1 − r)) − 1 and it is a known quantity rather than a vague one. Real deep groove ball bearings run from about 0.18 to 0.32, giving errors from two per cent low to twenty per cent high, and the rule is an OVERestimate above r = 0.2 and an underestimate below. AND HERE IS THE PARAGRAPH THAT MAKES THE PAGE USEFUL. These are theoretical frequencies computed on the assumption of pure rolling, and real bearings slip: the rolling elements skid a little as they enter and leave the load zone, and the cage does not quite keep up. Measured tones therefore sit one to two per cent BELOW the calculated ones — which is why the last column is a BAND and not a line, and why matching a spectrum peak to within a per cent is a match rather than a near miss. The slip is also why these frequencies are NON-INTEGER multiples of shaft speed: they sit between the harmonics of running speed rather than on top of them, and that is precisely what makes them identifiable in a spectrum that is otherwise full of shaft-order content. This is a first-pass calculation on an idealised part — uniform section, static load, no stress raisers beyond those stated, and room-temperature material properties. Real parts have fillets, keyways, surface finish and duty cycles that a closed-form answer cannot see. This page sizes a part; it does not certify one. Where the answer carries a consequence — a load path, a lifting duty, a pressure boundary, a fastener holding something that can fall — confirm it against the design code that governs the application, and against the manufacturer’s own rating, before relying on it.
B_d/P_d	BPFO, exact (orders)	0.4Z, for Z = 9	Error of the rule	BPFO at 1,800 rev/min	Search band with 1.5 % slip
0.15	3.825×	3.60×	-5.9%	114.7 Hz	113.0–114.7 Hz
0.18	3.690×	3.60×	-2.4%	110.7 Hz	109.0–110.7 Hz
0.20	3.600×	3.60×	0.0%	108.0 Hz	106.4–108.0 Hz
0.22	3.510×	3.60×	2.6%	105.3 Hz	103.7–105.3 Hz
0.25	3.375×	3.60×	6.7%	101.3 Hz	99.7–101.3 Hz
0.30	3.150×	3.60×	14.3%	94.5 Hz	93.1–94.5 Hz
0.35	2.925×	3.60×	23.1%	87.8 Hz	86.4–87.8 Hz

Four frequencies from the kinematics, the identity that checks them, and why the slip matters

All four frequencies follow from rolling without slip, and none of them is a lookup. In the cage’s own frame the rolling element’s surface speed has to match the inner race at one contact and the outer race at the other. That is two equations in two unknowns — the cage rate and the element’s spin rate — and solving them gives the cage frequency FTF = ½f_r(1 − r·cos β) and the element spin BSF = f_r(1 − r²cos²β)/(2r), where r is the element diameter over the pitch diameter. The two race frequencies then follow by counting: BPFO is Z times the cage rate, because a fixed point on the outer race is passed once by each element per cage revolution, and BPFI is Z times the inner race’s rate relative to the cage.

BPFO + BPFI = Z × shaft speed, exactly, and it is free. Add the two formulas and the geometry cancels: ½Z(1 − r·cos β) + ½Z(1 + r·cos β) = Z, whatever the element diameter, the pitch diameter and the contact angle. So the sum of the two race frequencies in orders of shaft speed is the ELEMENT COUNT. Use it forwards, as a check that the geometry you typed is self-consistent — this page prints the residual and it should be zero to the last decimal. And use it backwards, which is more useful: if you can identify both race frequencies in a measured spectrum but do not know how many balls the bearing has, their sum divided by shaft speed tells you. That is worth more than a data sheet, because the same designation is made with different element counts by different makers.

The 0.4Z and 0.6Z rules of thumb are the exact formulas at r = 0.2. Not rough averages of anything: ½Z(1 − 0.2) is 0.4Z identically, and ½Z(1 + 0.2) is 0.6Z. That makes their error a known quantity rather than a vague one — exactly 0.4/(0.5(1 − r)) − 1 for BPFO — and it tells you which direction it goes: the rule OVERestimates BPFO for a bearing with a larger ball relative to its pitch diameter and underestimates it for a smaller one. Real deep groove ball bearings run from about 0.18 to 0.32, so the rule ranges from two per cent low to twenty per cent high. Useful when you have no geometry; not good enough when you do, because five per cent is wider than the slip band you would otherwise search in.

These are theoretical frequencies, real bearings slip, and that is what makes them identifiable. The calculation assumes pure rolling. Real elements skid a little as they enter and leave the load zone and the cage lags, so measured tones sit one to two per cent BELOW the calculated ones. Two consequences, and the second is the important one. You search a BAND, not a line — and matching a peak to within a per cent is a match, not a near miss. And these frequencies come out as NON-INTEGER multiples of shaft speed: 3.57 orders, 5.43 orders, 0.40 orders. A spectrum from a rotating machine is full of content at whole numbers of shaft speed — imbalance at 1×, misalignment at 2×, looseness at 0.5× and its harmonics, blade and vane passing, gear mesh — and a bearing tone stands out precisely because it lands between those lines. If one of your frequencies happens to land ON an integer order, it will be hidden, and the answer is to look at the harmonics and the sidebands instead.

Sidebands tell you more than the peak does. A BPFI defect rotates with the shaft, so it passes in and out of the load zone once per revolution and the tone is amplitude-modulated at shaft speed — expect sidebands at BPFI ± f_r. A rolling element defect passes through the load zone once per cage revolution, so expect sidebands at 2×BSF ± FTF. A BPFO defect sits still in the load zone and is not modulated, so it normally has strong harmonics and NO sidebands. That pattern is often a better discriminator than the frequency itself, especially when two candidate frequencies are close together. And FTF appearing on its own, rather than as a sideband spacing, points at the cage — which is a late-stage fault and the one that fails suddenly.

What a defect frequency does and does not tell you. It tells you WHERE: outer race, inner race, element, cage. It does not tell you why, and the why is what decides whether to change the bearing or change something else. ISO 15243 names the failure modes and they have different remedies: subsurface fatigue is the bearing reaching the end of its rating life and the answer is a bigger one; surface-initiated fatigue is contamination or a thin film and the answer is sealing or lubrication; smearing is too little load or too little film; false brinelling is vibration while stationary; fretting corrosion at the seats is a fit that is too loose; and an electrically induced pattern is a bearing carrying current it should not. A spectrum that says BPFO says none of that. It says look at the outer race.

Frequently asked questions

What are BPFO, BPFI, BSF and FTF?

The four frequencies at which a bearing defect makes itself heard. BPFO is the ball pass frequency of the outer race — how often a rolling element passes a fixed point on it. BPFI is the same for the inner race. BSF is the rate a rolling element spins about its own axis, usually observed at twice that because one defect on a ball strikes both races per ball revolution. FTF is the cage rate, always below half shaft speed. All four come from the kinematics of rolling without slip, not from a table.

What is the check BPFO + BPFI = Z × shaft speed for?

It is free and exact, and it works two ways. Adding the two formulas cancels the geometry entirely, leaving Z times shaft speed whatever the ball diameter, pitch diameter and contact angle are. Forwards, that verifies the geometry you typed: if the two orders do not add to the element count exactly, something is wrong. Backwards, it finds the element count from a measured spectrum: identify both race frequencies, add them, divide by shaft speed. That is more reliable than a data sheet, because the same bearing designation is made with different element counts by different makers.

How accurate are the 0.4n and 0.6n approximations?

Exact at a ball-to-pitch-diameter ratio of 0.2, and their error off that is computable: 0.4/(0.5(1 − r)) − 1 for BPFO. That is what those rules ARE — the formulas at r = 0.2, not averages of anything. Real deep groove ball bearings run from about 0.18 to 0.32, so the rule ranges from two per cent low to twenty per cent high, and it overestimates for a larger ball. Useful when you have no geometry at all; not good enough when you do, because five per cent is wider than the one to two per cent slip band you would otherwise be searching in.

Why do the measured frequencies not match the calculated ones exactly?

Because the calculation assumes pure rolling and real bearings slip. The elements skid slightly as they enter and leave the load zone, and the cage lags behind the pure-rolling rate, so measured tones sit one to two per cent BELOW the calculated ones. Search a band rather than a line, and treat a peak within a per cent as a match. The slip also varies with load, speed and lubricant film, so a bearing tone can drift by a per cent between measurements on the same machine with nothing wrong.

Why are bearing frequencies non-integer multiples of shaft speed?

Because of the ball-to-pitch-diameter ratio: BPFO is ½Z(1 − r·cos β) orders, which for a real bearing lands on something like 3.57 or 4.11 rather than on a whole number. And that is the single most useful thing about them. A spectrum from a rotating machine is dense with content at whole multiples of shaft speed — imbalance, misalignment, looseness, blade passing, gear mesh — and a bearing tone is identifiable precisely because it sits BETWEEN those lines. If one of them happens to land on an integer order, look at the harmonics and the sidebands instead.

What do sidebands around a bearing frequency mean?

That the defect is moving in and out of the load zone, which amplitude-modulates the tone — and the spacing tells you what is moving. Sidebands at shaft speed around BPFI mean the defect is on the inner race, rotating with the shaft. Sidebands at the cage frequency around 2×BSF mean the defect is on a rolling element, passing through the load zone once per cage revolution. A BPFO defect sits still in the load zone and so is not modulated: it shows strong harmonics and no sidebands. That pattern is often a better discriminator than the peak frequency itself.

Does a defect frequency tell me why the bearing failed?

No — it tells you where, and the why decides what to do. ISO 15243 names the failure modes and their remedies differ completely: subsurface fatigue means the bearing reached the end of its rating life and needs to be bigger; surface-initiated fatigue means contamination or too thin a lubricant film; smearing means too little load or too little film; false brinelling means vibration while stationary; fretting corrosion at the seat means the fit is too loose. A spectrum showing BPFO is consistent with several of those. Take the bearing out and look at it.

Related calculators

References

Industry Digits, Bearing defect frequencies — field reference. Used as the PUBLISHED WORKED EXAMPLE this page’s four formulas are checked against: a 6205 with nine 7.94 mm balls on a 38.5 mm pitch diameter at 1,800 rev/min, giving BPFO 107.2, BPFI 162.8, BSF 69.7 and FTF 11.9 Hz. This page reproduces all four to four significant figures. It is also the source for the statement that measured tones “typically sit 1–2 % below calculated values” because “real bearings slip slightly”.
Fabrico, “Bearing relubrication intervals” and “Bearing defect frequencies”. The source for the n·d_m speed-factor bands (below 200,000 a year or more, 200,000 to 400,000 five to twelve months, above 400,000 weeks), for the 30 to 50 per cent housing fill, and for the statement that grease life halves for every 10 to 15 °C — a range where SKF gives a single 15, so both are printed.
ISO 15243:2017, Rolling bearings — Damage and failures — Terms, characteristics and causes. Cited by number. It is the document that gives the failure modes this batch keeps naming — surface-initiated and subsurface fatigue, smearing, fretting corrosion, false brinelling — their agreed names, which matters because a defect frequency identifies a LOCATION and not a cause.
ISO 5753-1, Rolling bearings — Internal clearance — Part 1: Radial internal clearance for radial bearings. Cited by number. The C2/CN/C3/C4/C5 clearance groups used here were read from SKF’s published table 3 for deep groove ball bearings and from NTN’s table 8.8 in its technical section on bearing internal clearance and preload; the two agreed exactly at every bore step from 6 to 120 mm, which is why the numbers are used.