Approximate closed-form solutions are developed for estimating the frequency separation caused by thickness, density, or radius azimuth variations in ring or hemispherical gyroscopes. For the nth flexural mode, frequency separation is caused by the 2n harmonic of density or thickness variation. These relations are combined with stochastic models to determine approximate closed-form solutions relating surface roughness to frequency separation. For 1-mm radius, 10-μm thick ring or sphere, with 0.5-μm 0-peak thickness variations that are 100-μm wide, the frequency separation is 0.05%.
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