Math Bridge: Bearing Fault Frequencies

A field team faces an unresolved physical question: Why does this bearing fault land near 93.5 Hz? They must answer it before changing shaft speed on the real device. Predict the direction first.

See the relationship before changing it

The figure reads from left to right. The blue card is shaft speed. The middle card applies this page's relationship. The green card is shaft rate. Walk the arrows once: set the input, apply the rule, then read the result with its unit.

The retained audit below checks several chapter fixtures. This added model holds every other chapter fixture fixed, so the numeric fixture does not switch without explanation.

Walk the arrows. A slow dashboard trend is not a compressed vibration waveform. Without anti-alias filtering and retained features, it can show a plausible slow number that no longer identifies the bearing fault.

Derive the baseline in four named moves

1
Name the input. The chapter baseline for shaft speed is 1780.
2
Name the relationship. fr=1780/60=29.6667 Hz fcage=(29.6667/2)(1-0.30)=10.3833 Hz BPFO=9(10.3833)=93.45 Hz BPFI=(9/2)(29.6667)(1+0.30)=173.55 Hz Nyquistraw=10000/2=5000 Hz alias1Hz=|f-round(f/1)x1|=0.45 Hz
3
Substitute the chapter fixture. Set shaft speed to 1780. The page ledger gives shaft rate as 29.67 Hz.
4
Read the result. Keep Hz beside the value. Use it only inside the technical boundary on this page.

Predict, then change shaft speed

Try Predict the direction of shaft rate. Move one control, calculate, then check your prediction.

Observe A slow dashboard trend is not a compressed vibration waveform. Without anti-alias filtering and retained features, it can show a plausible slow number that no longer identifies the bearing fault. Reset the control to 1780 and compare shaft rate.

Explain Only shaft speed moves here. The other chapter fixtures remain fixed.

Check yourself

What should you do before trusting a moved-control result?

Answer: Predict its direction, apply the shown relationship, keep the units, and reset to the worked baseline.

What does this small model leave out?

Answer: Only shaft speed moves. Field effects named in the page's technical boundary stay fixed.

1. Start with the physical story

A shaft turns the inner race. Rolling elements pass a fixed defect at a geometry-dependent rate, so the useful vibration tone is not simply shaft speed. A sample clock must then be fast enough to distinguish that tone.

Motion Marley: First count how often the defect is struck; only then decide how often to measure.

2. Name every algebra move

Convert rotationDivide rpm by 60 to get shaft turns per second.

Project geometryMultiply the ball-to-pitch ratio by the cosine of contact angle.

Find cage rateHalve shaft rate and multiply by one minus the geometry term.

Count outer strikesMultiply cage rate by nine balls for BPFO.

Count inner strikesUse one plus the geometry term for BPFI.

Test the clockHalve sample rate for Nyquist, then fold frequencies around sample-rate multiples.

3. Reproduce the chapter case

fr=1780/60=29.6667 Hz
fcage=(29.6667/2)(1−0.30)=10.3833 Hz
BPFO=9(10.3833)=93.45 Hz
BPFI=(9/2)(29.6667)(1+0.30)=173.55 Hz
Nyquistraw=10000/2=5000 Hz
alias1Hz=|f−round(f/1)×1|=0.45 Hz

Keeping full precision shows both dashboard aliases at 0.45 Hz. The old rounded 93.5 and 173.6 values made them look like 0.5 and 0.4 Hz; that was a rounding artefact, not two different physics paths.

4. Try one real input

TryMove shaft speed and predict how both bearing tones and their 1 Hz aliases move.

ObserveFault tones scale with shaft speed, but the 1 Hz aliases jump and reverse direction as each tone crosses a folding boundary.

ExplainA slow dashboard trend is not a compressed vibration waveform. Without anti-alias filtering and retained features, it can show a plausible slow number that no longer identifies the bearing fault.

Technical boundaries.

This ledger uses ideal rolling kinematics and uniform sampling.

Bearing: Slip, load, contact angle, geometry tolerances, and multiple defects spread or shift peaks.
Signal path: Mounting, resonance, analogue filtering, noise, windowing, speed tracking, and sample jitter shape the spectrum.
Decision: A frequency match still needs baseline, operating state, severity evidence, and maintenance confirmation.

Correct, not complete: this calculation does not diagnose a bearing or set an alarm threshold.

5. Use the result in the design

Keep raw bandwidth or a validated edge feature around expected BPFO/BPFI bands. Send a slow trend only after preserving the feature definition, sample conditions, and uncertainty.

6. Record the evidence state

Store bearing geometry, shaft rpm, load, sensor and mount, analogue filter, sample rate, record length, window, feature version, raw trace reference, baseline, alert, and maintenance result.

7. Check yourself

Why is shaft rate not the outer-race fault rate?

Answer: Nine rolling elements pass the fixed defect at the cage rate, so ball count and geometry multiply the shaft motion.

Why can a 1 Hz trend show about 0.45 Hz?

Answer: Sampling makes a 93.45 Hz tone indistinguishable from its nearest 1 Hz multiple, leaving a 0.45 Hz alias.

Does a 93.45 Hz peak prove an outer-race defect?

Answer: No. Speed, load, mounting, other sources, baseline, and maintenance evidence must agree.

Honesty boundary.

The arithmetic reproduces the chapter's catalog-typical 1,780 rpm, nine-ball example without early rounding.

Bearing: Slip, load, contact angle, geometry tolerances, and multiple defects spread or shift peaks.
Signal path: Mounting, resonance, analogue filtering, noise, windowing, speed tracking, and sample jitter shape the spectrum.
Decision: A frequency match still needs baseline, operating state, severity evidence, and maintenance confirmation.

Correct, not complete: this calculation does not diagnose a bearing or set an alarm threshold.