Tap a glass bottle, a tuning fork or a brake disc – each rings differently. This "voice" of a part is a fingerprint of its geometry, mass and material. Knowing the fingerprint means recognising changes.
The simplest oscillator: mass-spring system
m · ẍ + c · ẋ + k · x = 0
Neglecting damping and using x(t) = X · cos(ωt):
ω0 = √(k / m) and f0 = (1 / 2π) · √(k / m)
- Stiffer parts vibrate higher.
- Heavier parts vibrate lower.
- Both enter via square root – changes are sub-proportional.
Real parts have many natural frequencies
A brake disc has infinitely many modes. The first few dominate the acoustic response:
- Mode 1: first axial bending (800–1,200 Hz)
- Mode 2: torsion (1,500–2,200 Hz)
- Mode 3: second bending (2,800–3,600 Hz)
- Mode 4: plate mode (4,000–5,500 Hz)
Why cracks shift natural frequencies
A crack locally reduces stiffness. If k drops by 1 %, the corresponding natural frequency drops ~0.5 % (square root). Sounds small – at 4 kHz that is 20 Hz, easily measurable.
Modal parameters
| Parameter | Symbol | Meaning |
|---|---|---|
| Natural frequency | f0 | stiffness / mass |
| Mode shape | Φ(x,y) | spatial vibration pattern |
| Modal damping | ζ | energy loss, often elevated by defects |
| Modal mass | mr | excitability of the mode |
Industrial use in four steps
- Capture 30–100 OK parts as masters.
- Statistical tolerance build per mode.
- Inline excitation, capture, comparison.
- ML refinement for subtle defect signatures.
Practical example: brake disc
A brake disc (Ø 320 mm, 6 kg) typically has Mode 4 at 4,320 Hz, ±18 Hz scatter (3σ). An 8 mm hairline crack reproducibly lowers this by 35–55 Hz. False negative rate < 10 ppm in series.
