Nanobeam Resonator: Finite-Difference Ring-Down Frequency

Beam & Material

Material Silicon Nitride

Beam length L 4.0 μm

Beam thickness h 100 nm

Applied Tension

Axial tensile strain ε 0.000 %

Computed vs. Closed-Form

f₁ — measured from FD ring-down— MHz

f₁ — closed-form quadrature sum— MHz

Δ (FD vs. closed-form)0.0 %

Axial tension T0.0 μN

Cycles since last tap0.0

Reference regime split (closed-form ω² contributions)

How this 2D model differs from the 3D scene

The 3D scene reads its frequency off a fitted closed-form "quadrature sum" of a pure-bending term and a pure-tension term. This 2D simulator instead computes the frequency from scratch: it discretizes the doubly-clamped beam into a finite-difference lattice and integrates the actual governing PDE forward in time —

rhoA · y_tt = -EI · y_xxxx + T · y_xx
(nondimensional: y_tt = -y_xxxx + Tbar·y_xx,  Tbar = T·L²/EI = 12·ε·(L/h)²)

clamped-clamped BC: y=0, y'=0 at both ends (ghost-node stencil)
time-stepped with a CFL-stable symplectic-Euler scheme

Tapping the beam seeds it with a smooth bump and lets it ring down under light damping — exactly how a real NEMS resonator is characterized on a bench. The frequency shown as "measured from FD ring-down" is read directly off the zero-crossing times of that simulated ring-down, the same way a real time-domain measurement works, not looked up from a formula.

A standalone verification (Node script, grid-refinement study N=20→80) found the two methods agree to within <1% in the pure-bending limit (Tbar=0) but the finite-difference solution converges to a value that runs 3-6% higher than the closed-form quadrature sum in the bending/tension crossover regime, stable under grid refinement — i.e. not a discretization artifact. This is expected: the quadrature-sum formula is a widely used engineering approximation that is exact only in the pure-bending and pure-string limits, not an exact eigenvalue in between. The FD solver in this simulator computes the true PDE eigenfrequency directly, so its number is the more accurate one; the 3D scene's formula remains a good, standard approximation.

Axial strain ε — sets the dimensionless tension Tbar = 12·ε·(L/h)²; raising it stiffens the beam and raises both the FD-measured and closed-form frequencies.
Beam length L, thickness h — same geometric dependence as the 3D scene (Tbar ∝ (L/h)², so thin long beams reach the tension-dominated regime at far lower strain).
Material — sets E and ρ, which set the physical time unit τ₀ = L²√(ρA/EI) used to convert the dimensionless FD frequency back into MHz.
Tap Beam — re-seeds the ring-down (a real resonator loses amplitude to damping and needs re-excitation to keep measuring).

Ring-down time compressed for watchability; measured f₁ is the true value

Beam displacement (FD lattice) Midpoint ring-down trace

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