Math Bridge: Bearing Motion to MEMS Voltage

← Back to the Volkswagen Case Study
Math BridgeApplicationsStruggle-friendly runway

How does 5g become 88 millivolts?

Carry one bearing event through force, proof-mass motion, capacitance, and charge-amplifier voltage.

Motion Marley, the movement guideMotion Marley guides
The one targetInvert the MEMS sensing chain without skipping units.
The chapter case5g, 5 kHz, 0.1 ng, 0.4 mm², 2 µm, and 1 pF.
What it buys youA sanity-checkable voltage-to-acceleration path.

A field team faces an unresolved physical question: How does 5g become 88 millivolts? They must answer it before changing acceleration 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 acceleration. The middle card applies this page's relationship. The green card is base capacitance. 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.

Acceleration changes base capacitance An input card leads through the page relationship to the base capacitance result. SET INPUT ONE CONTROL APPLY RULE predict calculate check units READ RESULT
Walk the arrows. Every stage is linear only because displacement remains tiny compared with the 2 um gap and the amplifier is ideal.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline for acceleration is 5.

  2. 2

    Name the relationship. C0=ε0A/d=1.771 pF k=m(2πx5000)²=0.0987 N/m F=(1x10⁻¹⁰)(5x9.81)=4.905 nN x=F/k=49.70 nm ΔC=2C0x/d=88.02 fF Vout=(ΔC/1 pF)(1 V)=88.02 mV

  3. 3

    Substitute the chapter fixture. Set acceleration to 5. The page ledger gives base capacitance as 1.771 pF.

  4. 4

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

Predict, then change acceleration

Try Predict the direction of base capacitance. Move one control, calculate, then check your prediction.

5
Chapter baseline
Base capacitance

Observe Every stage is linear only because displacement remains tiny compared with the 2 um gap and the amplifier is ideal. Reset the control to 5 and compare base capacitance.

Explain Only acceleration 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 acceleration moves. Field effects named in the page's technical boundary stay fixed.

Honesty note. This teaching model changes only acceleration. It replays the chapter fixture; it is not a field prediction.

1. Start with the physical story

Bearing vibration accelerates a spring-mounted proof mass. Its nanometre displacement closes one capacitor gap and opens the other. A charge amplifier turns the differential capacitance into the voltage logged by the monitoring system.

Motion Marley: The sensor never sees “g” directly; it sees capacitance.

2. Name every algebra move

1

Find base capacitanceMultiply permittivity by plate area and divide by gap.

2

Find spring stiffnessMultiply proof mass by squared resonance angular frequency.

3

Find forceMultiply mass by acceleration.

4

Find displacementDivide force by spring stiffness.

5

Find differential CScale twice the base capacitance by displacement/gap.

6

Find voltageMultiply ΔC/Cf by excitation voltage.

3. Reproduce the chapter case

C0=ε0A/d=1.771 pF
k=m(2π×5000)²=0.0987 N/m
F=(1×10⁻¹⁰)(5×9.81)=4.905 nN
x=F/k=49.70 nm
ΔC=2C0x/d=88.02 fF
Vout=(ΔC/1 pF)(1 V)=88.02 mV

Those catalog-style assumptions produce 17.6 mV/g, a useful order-of-magnitude check rather than a Volkswagen sensor specification.

4. Try one real input

TryChange acceleration and predict force, displacement, capacitance, and output voltage.

Acceleration
Base capacitance
Spring constant
Proof-mass force (nN)
Displacement
Differential C (fF)
Output voltage
Sensitivity

ObserveIn this small-displacement model, doubling g doubles force, displacement, differential capacitance, and voltage.

ExplainEvery stage is linear only because displacement remains tiny compared with the 2 µm gap and the amplifier is ideal.

Technical boundaries.

This is a small-signal, single-axis, undamped MEMS model.

Geometry
The chapter names no Volkswagen die geometry; area, gap, mass, resonance, excitation, and feedback capacitance are catalog-style assumptions.
Dynamics
Damping, cross-axis response, bandwidth, resonance proximity, clipping, noise, temperature, and ageing are excluded.
Maintenance
One acceleration amplitude does not identify a bearing fault, remaining life, or safe intervention.

Correct, not complete: this ledger is a sensing-chain sanity check, not a machine-health diagnosis.

5. Use the result in the design

Calibrate voltage to acceleration on the installed axis, preserve spectra and operating state, then fuse vibration with temperature, acoustic, current, and maintenance evidence.

6. Record the evidence state

Keep sensor model and serial, axis, mounting, bandwidth, sample rate, range, calibration, temperature, RPM/load, raw waveform, spectrum, voltage, acceleration, model version, alert, inspection, and confirmed fault.

7. Check yourself

What electrical quantity does the MEMS element sense first?
Answer: A differential capacitance caused by proof-mass displacement.
Why does output voltage scale with g here?
Answer: Force, displacement, small-signal capacitance change, and ideal amplifier voltage are linear in this model.
Does 88 mV prove a bearing defect?
Answer: No. Geometry is assumed, and diagnosis also depends on frequency, operating state, mounting, history, and confirmed maintenance outcomes.
Honesty boundary.

The arithmetic preserves the chapter's illustrative 5g chain while labeling all catalog-style constants.

Geometry
The chapter names no Volkswagen die geometry; area, gap, mass, resonance, excitation, and feedback capacitance are catalog-style assumptions.
Dynamics
Damping, cross-axis response, bandwidth, resonance proximity, clipping, noise, temperature, and ageing are excluded.
Maintenance
One acceleration amplitude does not identify a bearing fault, remaining life, or safe intervention.

Correct, not complete: this ledger is a sensing-chain sanity check, not a machine-health diagnosis.

Return to the Volkswagen Case Study with the maths unpacked →