Every physical structure — a beam, a shaft, a building, a bolt — has a natural frequency at which it "wants" to vibrate when disturbed. Pluck it, and it oscillates at that frequency. Drive it externally at that same frequency, and amplitude grows without bound (limited only by damping). This is resonance, and it's the silent killer of mechanical systems.

For a simple spring-mass system, natural frequency is:

f_n = (1 / 2π) × √(k / m)

Where k is stiffness (N/m) and m is mass (kg). Stiffer = higher frequency. Heavier = lower frequency. That's it.

Why it matters in practice:

• Rotating machinery: A shaft has a "critical speed" equal to its natural frequency in bending. Run a pump or turbine at that RPM and the shaft whips violently. Designers either keep operating speed well below (rigid rotor) or accelerate quickly through it (flexible rotor).
• Buildings: Tall structures have natural frequencies around 0.1–1 Hz. Earthquakes and wind gusts contain energy in that band — hence tuned mass dampers like the 660-ton pendulum in Taipei 101.
• Electronics enclosures: A PCB mounted with screws has a natural frequency. Vibration testing (MIL-STD-810) sweeps frequencies to find it before the field does.

The classic failure: Tacoma Narrows Bridge, 1940. Wind didn't blow it down — it excited a torsional mode at ~0.2 Hz, and aeroelastic feedback pumped energy into that mode until the deck twisted itself apart. Not resonance with a periodic forcing, but the same principle: energy concentrating in one lightly damped mode.

Rule of thumb — the "factor of 2" rule: Keep operating frequencies at least 2× away from any natural frequency (either above or below). Inside that band, transmissibility exceeds 1, meaning vibration is amplified, not isolated. Above √2 × f_n, an isolator actually starts attenuating.

Quick calculation: A 5 kg motor on four rubber mounts, each with k = 20,000 N/m (total k = 80,000 N/m):

f_n = (1/2π) × √(80,000 / 5) = (1/2π) × √16,000 ≈ 20.1 Hz

If the motor runs at 3600 RPM = 60 Hz, you're at 3× f_n — comfortably in the isolation region. If it ran at 1200 RPM = 20 Hz, you'd be sitting on resonance and the mounts would amplify vibration instead of damping it.

Damping (energy dissipation per cycle) determines how bad resonance gets. Steel: very low damping, Q ≈ 1000. Rubber: high damping, Q ≈ 10. Concrete: moderate, Q ≈ 50. High-Q systems ring forever; low-Q systems die out fast — but also isolate less effectively above resonance.