Material resonance · driven damped oscillator
A = 0 · A/A_max = 100.0 % · Q = 125
Every solid has natural modes. Push it at exactly that frequency and each cycle adds energy faster than damping removes it, so the amplitude climbs to Q times the static deflection. In a wine glass the rim mode has almost no damping (Q ≈ 1000), so a sustained tone at ω₀ can stretch the rim past its fracture strain and it shatters. The same effect ruined the Tacoma Narrows bridge and is used deliberately in MRI, quartz clocks and ultrasonic cleaners.
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Sweep the driving frequency ω through the natural frequency ω₀. The response peaks at ω_r = ω₀√(1 − 2ζ²) with height Q = 1/(2ζ). Lower the damping ratio ζ (thin, annealed glass) and the peak grows until the rim amplitude passes the fracture strain.
strain 0 % of fracture
