Nonlinear Spring

Physics: Duffing equation: ẍ + δẋ + αx + βx³ = γcos(ωt). Hardening (β > 0): resonance peak bends to higher frequencies; softening (β < 0): bends lower. Jump phenomena occur at fold bifurcations in the frequency response — the system suddenly jumps to a different amplitude branch as ω is swept. Period-doubling and chaos occur for large γ. MEMS resonators exhibit Duffing behavior at nanometer amplitudes; Poincaré sections of chaotic Duffing reveal a strange attractor.