MD-01Machinery Diagnostics

Bearing Defect Frequencies

BPFO, BPFI, BSF and FTF of rolling-element bearings, plotted on a simulated spectrum.

Operating speed

Shaft speed

rpm

f_r = 24.67 Hz

Bearing geometry

Preset

Number of rolling elements n

Ball / roller diameter B_d

Pitch diameter P_d

Contact angle θ

BPFO · outer

88.42Hz

3.585 × f_r

BPFI · inner

133.6Hz

5.415 × f_r

BSF · ball spin

58.13Hz

2.357 × f_r

FTF · cage

9.825Hz

0.398 × f_r

Bearingcage at FTF

Simulate defect

In CPM:
BPFO 5305
BPFI 8015

Check:
BPFO+BPFI = 222.0
n·f_r = 222.0

Simulated envelope spectrum

Shaft 1×,2×,3×BPFOBPFI ± f_rBSFFTF

§ Theory of operation

Kinematics of a rolling-element bearing

When a rolling element passes over a localised defect, it produces a small impact. Those impacts repeat at frequencies fixed by the bearing geometry and shaft speed. Assuming pure rolling and a stationary outer ring:

\begin{aligned} \mathit{FTF} &= (f_r\,/\,2) \cdot (1 - (B_d/P_d)\,\cos\theta) \qquad \text{cage} \\ \mathit{BPFO} &= (n\,f_r\,/\,2) \cdot (1 - (B_d/P_d)\,\cos\theta) \qquad \text{outer race} \\ \mathit{BPFI} &= (n\,f_r\,/\,2) \cdot (1 + (B_d/P_d)\,\cos\theta) \qquad \text{inner race} \\ \mathit{BSF} &= (P_d\,/\,2B_d) \cdot f_r \cdot (1 - ((B_d/P_d)\,\cos\theta)^2) \qquad \text{ball spin} \end{aligned}

with $f_r$ the shaft frequency, n the number of rolling elements, $B_d$ their diameter, $P_d$ the pitch diameter and θ the contact angle. Useful checks: $\mathit{BPFO} + \mathit{BPFI} = n\cdot f_r$ , and $\mathit{BPFO} \approx 0.4\cdot n\cdot f_r$ , $\mathit{BPFI} \approx 0.6\cdot n\cdot f_r$ for most bearings.

Recognising the patterns

Outer-race defect: BPFO and harmonics. The defect is stationary in the load zone, so amplitudes are steady.
Inner-race defect: BPFI and harmonics with sidebands at ±1× shaft speed, because the defect rotates in and out of the load zone.
Rolling-element defect: often 2×BSF (the defect strikes both races per revolution), sidebanded by FTF.
Cage problems: FTF itself, or as sidebands; usually sub-synchronous.

In practice

Early-stage defects are best detected with envelope analysis (demodulation): the high-frequency structural resonances excited by the impacts are band-pass filtered and rectified, revealing the defect repetition rate. Real frequencies deviate by 1–2 % from these values due to slip, so look for families of peaks rather than exact matches. Always use the manufacturer's geometry data for the exact bearing designation.