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Abstract-This paper reports a wafer-level technique for the systematic elimination of the modal frequency difference between a nominally degenerate pair of modes in an axisymmetric resonator design. A targeted etch process is developed in which masking resist and a conformal layer are ablated at specific sites on the resonator thereby exposing the underlying silicon and enabling site-specific mass removal by SiDRIE. A model of the perturbed resonator dynamics guides the selection of the ablation sites so that the subsequent timed etch reduces the modal frequency differences by a prescribed amount. This waferlevel process is demonstrated on seven resonators whose modal frequency differences are reduced below 100 mHz from initial splits as large as 15 Hz for a pair of modes with 13.5 kHz nominal frequencies.

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Perturbation expansions of solutions for a uniform, thin, linear elastic ring perturbed by point masses and radial massless springs are developed. The perturbation locations divide the ring into uniform segments so a variational formulation is used to determine the boundary conditions that must be satisfied between adjoining segments. The motion of each segment can be represented as a weighted sum of the eigenfunctions for the uniform thin ring so when the boundary conditions are enforced, the resulting algebraic relations are expanded as a function of the perturbation parameter (the perturbation mass normalized by the ring mass). A series of algebraic problems are sequentially solved to yield perturbation expansions for the modal frequencies and eigenmodes.Single-mass, dual-mass, and mass-spring case studies are considered. The perturbation results show excellent agreement with finite element analysis of a thin ring for mass perturbations up to 15% of the nominal ring mass. The results are also compared to Rayleigh-Ritz analysis.