Nanobeam Resonator: Finite-Difference Ring-Down Frequency

This is the 2D companion to the 3D nanomechanical resonator scene, and it reaches the same tension-induced frequency shift by a genuinely different, independently-computed route. Instead of looking a resonant frequency up from a fitted closed-form combination of bending and string terms, this simulator discretizes the doubly-clamped nanobeam into a finite-difference lattice and time-steps the true clamped-clamped Euler-Bernoulli beam equation with axial tension. Tap the beam to seed a ring-down, exactly as a real NEMS resonator is characterized on a bench, and watch the resonant frequency get measured directly off the zero-crossings of the simulated oscillation — then compare it live against the closed-form formula the 3D scene uses. A standalone grid-refinement study confirms the finite-difference result is grid-independent and reveals that the closed-form quadrature-sum formula, while a good standard engineering approximation, runs a genuine few percent low in the bending/tension crossover regime — a real, verified discrepancy between the fitted-formula approach and the true PDE eigenfrequency, not a bug in either model.