Bearing Defect Frequencies BPFO, BPFI, BSF, FTF: Formulas and Calculation

Bearing defect frequencies are calculated with four formulas: BPFO (outer race), BPFI (inner race), BSF (rolling element) and FTF (cage). The inputs are the number of rolling elements N, the rolling element diameter d, the pitch diameter D, the contact angle θ and shaft speed. For an SKF 6205 ball bearing at 1500 rpm, BPFO is 89.6 Hz and BPFI is 135.4 Hz.

What BPFO, BPFI, BSF and FTF mean

Every rolling-element bearing has four defect frequencies, one for each component. If a component has a spall or pit, an impact occurs each time a rolling element passes over it, and the spectrum shows a peak at that component's frequency and its harmonics.

Symbol	Meaning	Bearing component
BPFO	Ball Pass Frequency, Outer race	Outer ring
BPFI	Ball Pass Frequency, Inner race	Inner ring
BSF	Ball Spin Frequency	Ball or roller
FTF	Fundamental Train Frequency	Cage

Formulas

Notation: N is the number of rolling elements, d the rolling element diameter, D the pitch diameter (centre to centre of opposite rolling elements), θ the contact angle, fr the shaft speed in Hz (rpm ÷ 60).

1. BPFO = (N / 2) × fr × (1 − (d / D) × cos θ)
2. BPFI = (N / 2) × fr × (1 + (d / D) × cos θ)
3. BSF = (D / 2d) × fr × (1 − (d / D)² × cos² θ)
4. FTF = (1 / 2) × fr × (1 − (d / D) × cos θ)

Two checks on the result:

• BPFO + BPFI = N × fr;
• BPFO = N × FTF.

Every formula contains the ratio d/D, so defect frequencies are not integer multiples of running speed. This separates them from unbalance (1x) and misalignment (2x). For radial deep groove ball bearings the contact angle is taken as 0°, so cos θ = 1.

Worked example: SKF 6205-2RS

The geometry comes from the open bearing data set of Case Western Reserve University: their test rig has an SKF 6205-2RS JEM at the motor drive end.

Parameter	Value
Number of balls N	9
Ball diameter d	0.3126 in (7.94 mm)
Pitch diameter D	1.537 in (39.04 mm)
Contact angle θ	0°
d / D	0.2034

Let the shaft run at 1500 rpm, so fr = 1500 / 60 = 25 Hz.

1. BPFO = 4.5 × 25 × (1 − 0.2034) = 4.5 × 25 × 0.7966 = 89.6 Hz
2. BPFI = 4.5 × 25 × (1 + 0.2034) = 4.5 × 25 × 1.2034 = 135.4 Hz
3. BSF = (39.04 / 15.88) × 25 × (1 − 0.0414) = 2.458 × 25 × 0.9586 = 58.9 Hz
4. FTF = 0.5 × 25 × 0.7966 = 10.0 Hz

Check: 89.6 + 135.4 = 225 Hz = 9 × 25 Hz. And 9 × 10.0 = 90 Hz, which matches BPFO within rounding.

Frequency	Multiple of running speed	At 1500 rpm	CWRU table
BPFO	3.585	89.6 Hz	3.5848
BPFI	5.415	135.4 Hz	5.4152
BSF	2.357	58.9 Hz	4.7135 (listed for the rolling element, this is 2 × BSF)
FTF	0.398	10.0 Hz	0.39828

The multiples are fixed for a given bearing. At another speed, multiply them by the new fr. At 3000 rpm (50 Hz), for example, BPFO of this bearing is 179.2 Hz.

Quick estimate without geometry

When the bearing type is unknown or catalogue data is missing, approximations are used:

Frequency	Estimate (fabrico.io)	Estimate (Ludeca)	For 6205 at 25 Hz
BPFO	0.4 × N × fr	0.43 × N × fr	90 Hz and 96.8 Hz
BPFI	0.6 × N × fr	0.57 × N × fr	135 Hz and 128.3 Hz
FTF	0.35–0.45 × fr	0.43 × fr	8.8–11.3 Hz and 10.8 Hz

Ludeca puts the error of these formulas at 10–15%. That is enough to know where in the spectrum to look, but not for a confident diagnosis. Exact values come from the bearing manufacturer or the bearing database in the analysis software.

How defect frequencies appear in the spectrum and envelope

1. Envelope first. Enveloping (demodulation) extracts low-amplitude, high-frequency repetitive impacts and suppresses non-repetitive machine vibration. Defect frequencies and their harmonics therefore show in the envelope spectrum before they appear in a normal velocity spectrum. According to SKF, an initial defect can produce harmonics up to 50 times the fundamental.
2. Harmonics. A peak at the defect frequency comes with a series of multiples: if BPFO = 3.585x, look for peaks at 7.17x, 10.75x and so on.
3. Outer race (BPFO). The outer ring is usually stationary, so the defect stays in the load zone. The pattern is stable, and this defect is relatively easy to detect.
4. Inner race (BPFI). The defect rotates with the shaft and passes through the load zone once per revolution. The signal is modulated at running speed, and sidebands spaced at 1x appear around BPFI.
5. Rolling element (BSF). Harder to detect. Often 2 × BSF is the strongest line, with sidebands spaced at FTF.
6. Cage (FTF). A low frequency, below running speed. It usually shows as sidebands spaced at FTF around other bearing frequencies.

SKF notes that sidebands at shaft speed near defect harmonics and fundamentals typically indicate severe bearing damage. Other signatures (unbalance, misalignment, looseness) are covered in How to read an FFT spectrum, and defect progression in Bearing failure stages and the P-F curve.

Why the calculated frequency may not match the peak

• Slip. The formulas assume pure rolling. Under light load the rolling elements skid, and the real peak sits slightly below the calculated value.
• Load and contact angle. Axial load changes the effective contact angle and with it d/D × cos θ.
• Inaccurate running speed. An induction motor runs slightly below synchronous speed. An error in fr carries into all four frequencies, so take speed from a tachometer or from the 1x peak in the spectrum.
• Wrong bearing type. If a bearing from another manufacturer or series was fitted during repair, the geometry may differ. Check the marking on site.

Allow a tolerance of a few percent when matching peaks. If a motor is noisy but bearing frequencies do not show in the envelope, see Electric motor bearing noise: causes.

Which instrument to choose

Bearing frequency analysis needs an instrument with spectrum, envelope and enough resolution to separate close peaks (for example BPFI and its 1x sidebands).

• Route-based diagnostics with enveloping — Bently Nevada vb8. Per the Bently Nevada datasheet: four simultaneous channels, up to 12,800 lines of FFT resolution, Fmax up to 80 kHz, demodulation for early detection of bearing faults, IEPE/ICP accelerometer input.

KEG TRK supplies Bently Nevada equipment in Kazakhstan. Price on request.

Sources

• Fabrico. Bearing Defect Frequencies: BPFO, BPFI, BSF, FTF formulas, checks, approximations, spectrum and envelope appearance.
• Case Western Reserve University Bearing Data Center. Bearing Information: SKF 6205-2RS JEM geometry and defect frequency multiples.
• SKF Evolution. Vibrations help find faulty components: defect frequencies, harmonics, sidebands, enveloping.
• Ludeca. Calculation of unknown bearing types in your equipment: approximation formulas and their error.
• Acoem. Understanding Bearing Fault Frequencies: how race, rolling element and cage defects appear.
• Bently Nevada. vb8 Datasheet 116M5343: vb8 analyzer specifications.