Nonlinear Vibration of Multi-Load Annular and Circular Plates: A Parametric Study of Loading-Agnostic Frequency Behavior

MEMS resonators vibrate about equilibria deflected by whatever combination of temperature, pressure, and bias voltage is present—not about the flat, unloaded plate—raising a basic design question: does the resulting natural frequency depend on the specific loads that produced the deflection, or mainly on the deflection itself? We address this for annular and circular plates through a self-contained derivation of the governing multi-load equations and the linearized vibration eigenvalue problem about an arbitrary thermal, mechanical, and electrostatic equilibrium, validated against two classical benchmarks and applied to a parametric study spanning radius ratio, Poisson’s ratio, and five load combinations at matched deflection. The spread in squared frequency, ω12, across combinations shrinks monotonically from 29% at w/h=0.4 to 6% at w/h=2.4, a dynamic counterpart to a known static result in which the large-deflection boundary layer at a clamped edge depends only on the local membrane stress, not on which loads produced it. Going beyond the linear eigenfrequency, a single-mode Duffing-type reduction, compared against four independent classical benchmarks, reproduces the same asymmetry direction and hardening-to-softening crossover reported in the literature. For the one annular geometry and axisymmetric motion studied here, the results suggest that once deflection exceeds about twice the plate thickness, resonator frequency can be tabulated against deflection amplitude alone, rather than the full space of operating conditions.