NEMS Multimode Mass Sensing — Time-Domain String Simulation (2D)

Beam Parameters

Axial tension T 1.00 µN

Beam length L 2000 nm

Particle mass δm 4.0 fg

No particle landed — reference string only

Mode Frequencies (time-domain measured)

Mode 1, f₁

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Mode 2, f₂

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Δf₁/f₁

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Δf₂/f₂

—

Inverted vs. True

Estimated x₀

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True x₀

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Estimated δm

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True δm

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Position error

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Mass error

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Unlike the 3D version, nothing here is plugged into the perturbation formula directly — both mode frequencies are clocked by counting real oscillation periods of a numerically time-stepped string with the particle's mass sitting on the mesh. The formula only appears afterwards, as the number the measurement is checked against.

How this 2D model differs from the 3D one

The 3D sim evaluates the standard mass-perturbation formula for a tensioned string directly and animates a matching sine curve. This 2D companion instead discretizes the beam into masses connected by tension and numerically integrates Newton's second law forward in time (velocity-Verlet leapfrog) with the particle's mass genuinely added to the mesh at its landing node:

mu(x) y_tt = T y_xx           (string wave equation, mu(x) = mu0 everywhere,
                                plus dm/dx smeared over one cell at x₀)
y_i'' = (T / mu(x_i)) (y_{i+1} - 2y_i + y_{i-1}) / dx²

Every rendered frame, the raw mesh displacement is projected onto the sin(nπx/L) basis — a discrete sine transform, exactly what a lock-in amplifier extracts from a real resonator's readout — giving each mode's own time-domain signal. Counting the time between that signal's zero-crossings gives the period, and hence the frequency, with no formula involved. A second, undisturbed copy of the same mesh runs alongside as the reference clock, so the fractional shift Δf₀/f₀ comes from two independently measured numbers, not from evaluating an equation.

R = (Δf₂/f₂) / (Δf₁/f₁) = 4cos²(πx₀/L)
x₀ = (L/π)·arccos(√(R/4))
δm = −M · (Δf₁/f₁) / sin²(πx₀/L)

A discrepancy worth flagging: deriving the shift from Rayleigh's method (a point mass dm raises mode n's effective modal mass by dm·sin²(nπx₀/L), and the modal mass of a string mode is M/2) gives Δf_n/f_n ≈ −(δm/M)·sin²(nπx₀/L). The time-domain measurement here reproduces exactly that relationship to within a few percent across every tested landing point and mass. The 3D version's theory panel states the shift as −(2δm/M)·sin²(...) — twice too large — which would recover a mass estimate exactly half the true value. This simulator uses the δm/M form, verified independently by the time-stepped mesh rather than assumed from either formula.

Tension — the engine's own internal clock is fixed for a watchable on-screen oscillation rate; tension instead scales how tall the visible string deflection sits and (through the standard unperturbed-string formula) the real MHz frequencies reported once a measurement locks.
Length — longer beams have lower real resonant frequencies for the same tension, exactly as in the 3D sim; the fractional shifts themselves depend only on x₀/L and δm/M, so they are unaffected.
Particle mass — the true δm applied to the mesh the next time you drop a particle.
Drop particle — restarts the loaded mesh from rest with the particle's mass on it and begins timing; watch the "measuring" status until enough oscillation cycles have been clocked to lock a frequency estimate, just as a real spectrum sweep needs integration time.

modal signal q₁(t), q₂(t)

Reference string (unloaded, running clock) Loaded string (with particle) Landed particle (true x₀)