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

A doubly-clamped nanobeam under strong axial tension behaves like a vibrating string, and a nanoparticle landing on it shifts each flexural mode's frequency by an amount that depends on both the particle's mass and exactly where on the string it landed — a single mode's shift can't separate the two, but the ratio of two modes' shifts can. This 2D companion to the 3D NEMS sensor demonstrates that inversion without ever evaluating the perturbation formula to get its own numbers: it discretizes the beam into a mesh of point masses joined by tension, drops the particle's mass onto one mesh node, and integrates Newton's second law forward in time with a velocity-Verlet leapfrog scheme. Each mode's frequency is then read off the simulated motion itself, by projecting the raw mesh displacement onto its sine mode-shape every frame and timing the zero-crossings of that signal against an identical unloaded reference mesh running alongside. The two independently measured fractional shifts are inverted with the same closed-form ratio the 3D sim uses to recover the particle's landing position and mass — and, checked against the numerically time-stepped mesh, the correct proportionality constant turns out to be half what the 3D sim's own theory panel states, a discrepancy this simulator corrects and reports openly.