Math Bridge: Spindle Sampling and FFT Evidence

← Back to Manufacturing Applications
Math BridgeApplicationsStruggle-friendly runway

Why does a 10 kHz sample clock stop the FFT at 5 kHz?

Carry sample timing into the spectrum before trusting a bearing feature.

Motion Marley, the movement guideMotion Marley guides
The one targetTurn sample rate into trustworthy FFT coordinates.
The chapter case10 kHz, 1,024 samples, 1,800 Hz tone, ±5g 16-bit input.
What it buys youA spectrum whose limits and aliases are explicit.

A technician must decide whether nyquist frequency is safe before changing vibration sample rate on the real device. The result is unresolved until the rule and units are checked. Predict the direction first.

See the relationship before changing it

The figure reads from left to right. The blue card is vibration sample rate. The middle card applies this page's rule. The green card is nyquist frequency. 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 model keeps those stated values fixed and changes only vibration sample rate, so the numeric fixture does not switch without explanation.

Vibration sample rate changes nyquist frequency An input card leads through the rule FFT Nyquist = sample rate / 2 to the nyquist frequency result. INPUT PAGE INPUT APPLY THE RULE predict calculate check units OUTPUT RESULT
Walk the arrows. Higher sample rate raises the FFT frequency ceiling.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline is 10000 samples/s.

  2. 2

    Name the relationship. FFT Nyquist = sample rate / 2

  3. 3

    Substitute with units. 10,000 / 2 = 5,000 Hz

  4. 4

    Read the result. Keep the unit beside the value. Use it only inside the technical boundary on this page.

Predict, then change vibration sample rate

Try Predict the direction of FFT Nyquist = sample rate / 2. Test another vibration sample rate, then compare nyquist frequency.

10000 samples/s
Chapter baseline
Nyquist frequency

Observe Higher sample rate raises the FFT frequency ceiling. Reset vibration sample rate to 10000 and compare nyquist frequency.

Explain Higher sample rate raises the FFT frequency ceiling.

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 vibration sample rate moves here. Field effects named in the technical boundary stay fixed.

Honesty note. This teaching model changes only vibration sample rate. It is not a field prediction.

1. Start with the physical story

The ADC takes discrete snapshots of continuous vibration. Above half the sampling rate, different real frequencies produce the same sample pattern and fold into the trusted-looking FFT band.

Motion Marley: An FFT cannot repair information the sample clock never separated.

2. Name every algebra move

1

Set NyquistDivide sample rate by two.

2

Time the windowDivide 1,024 samples by samples per second.

3

Space the binsDivide sample rate by 1,024.

4

Locate the toneRound 1,800 Hz divided by bin width.

5

Fold the aliasSubtract the nearest sample-rate multiple from 6,200 Hz.

6

Check amplitude stepDivide the 10g span by 2 to the 16th power.

3. Reproduce the chapter case

fN=10000/2=5000 Hz
Twindow=1024/10000=102.4 ms
Δf=10000/1024=9.765625 Hz
bin=round(1800/9.765625)=184; centre=1796.875 Hz
falias=|6200−10000|=3800 Hz
q=10/2¹⁶=1.526×10⁻⁴ g

The chapter's 0–5,000 Hz range, roughly 100 ms cadence, and 1,800 Hz feature all follow from the same clock and window.

4. Try one real input

TryMove sample rate and predict Nyquist, window duration, bin width, and the 6.2 kHz alias.

Sample rate
Nyquist limit
FFT window
Bin width
Nearest tone bin
Bin centre
6.2 kHz apparent frequency
ADC step
Ideal RMS quantisation noise

ObserveHigher sample rate raises Nyquist and changes aliasing, but with a fixed 1,024 samples it shortens the window and widens each bin.

ExplainBandwidth and frequency resolution compete when record length is fixed. Anti-alias filtering protects the band before the FFT sees it.

Technical boundaries.

This is an ideal uniformly sampled, unwindowed FFT coordinate ledger.

Analogue path
Sensor resonance, mounting, gain, anti-alias filter, clipping, noise, clock jitter, and ADC ENOB shape the samples.
Spectrum
Window function, leakage, overlap, averaging, order tracking, and speed variation change feature estimates.
Diagnosis
A tone or amplitude shift needs baseline, operating state, labels, uncertainty, and maintenance confirmation.

Correct, not complete: this FFT ledger does not prove a bearing fault or predictive-maintenance diagnosis.

5. Use the result in the design

Select sensor and anti-alias bandwidth before sampling, then choose record length and window from the closest fault frequencies that operations must distinguish.

6. Record the evidence state

Keep sensor and mount, range, gain, filter, sample clock, window function, record length, overlap, machine speed and load, raw trace, feature version, baseline, label, and maintenance outcome.

7. Check yourself

Why is Nyquist exactly 5 kHz?
Answer: A real sampled signal needs at least two samples per cycle, so the unique band ends at half of 10 kHz.
Why does 6.2 kHz appear at 3.8 kHz?
Answer: Sampling makes it indistinguishable from |6200−10000|=3800 Hz unless the analogue filter removes it.
Does a clean 1,800 Hz bin prove a bearing fault?
Answer: No. Mounting, operating speed, load, leakage, baseline, labels, and maintenance evidence remain necessary.
Honesty boundary.

The arithmetic reproduces the chapter's 10 kHz, 1,024-sample, 1,800 Hz CNC screening case.

Analogue path
Sensor resonance, mounting, gain, anti-alias filter, clipping, noise, clock jitter, and ADC ENOB shape the samples.
Spectrum
Window function, leakage, overlap, averaging, order tracking, and speed variation change feature estimates.
Diagnosis
A tone or amplitude shift needs baseline, operating state, labels, uncertainty, and maintenance confirmation.

Correct, not complete: this FFT ledger does not prove a bearing fault or predictive-maintenance diagnosis.

Return to Manufacturing Applications with the maths unpacked →