Nanomechanical Duffing Resonator

Drive & Damping

Drive frequency ω/ω₀ 0.800

Drive amplitude F₀ 1.00

Quality factor Q 80

Nonlinearity α +1.5

Frequency Sweep

Live Readouts

Amplitude |x|

0.00

ω/ω₀ now

0.800

Last jump detected

none yet — sweep ω slowly

How it works

A doubly-clamped nanobeam is not a perfect linear spring: as it bends, its neutral axis stretches, adding a cubic restoring term. Its motion x(t) obeys the driven, damped Duffing equation:

x'' + (ω₀/Q)·x' + ω₀²x + α·x³ = F₀·cos(ωt)

For α = 0 this is an ordinary resonance — a single symmetric peak at ω = ω₀. A nonzero α tilts the peak: α > 0 (stress-stiffening, typical of a tensioned nanobeam) bends the response curve toward higher frequency; α < 0 (softening) bends it toward lower frequency. The locus of peak amplitudes follows the weakly-nonlinear backbone curve (dashed on the chart):

ω(A) ≈ √(ω₀² + ¾·α·A²)

Past a critical drive amplitude, part of that bent curve becomes multivalued — three possible amplitudes exist at the same drive frequency, two stable and one unstable.
Sweeping ω up makes the amplitude ride the low branch until it suddenly jumps up to the high branch; sweeping down it jumps back at a different frequency — a hysteresis loop.
Use the sliders to set α, Q and F₀, then drag the frequency slider slowly (or use Auto sweep) through resonance in both directions and watch the amplitude trace on the chart below the beam.

Real-world relevance: this exact nonlinearity sets the dynamic range of every NEMS resonant mass sensor — drive it too hard for more signal and the response bends, becomes bistable and the readout frequency stops being a single-valued function of mass, corrupting the measurement.

Nanobeam (instanced segments) Fixed anchors

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