Symmetric gyroscope frequency separation by geometry and roughness

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 n^th 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%.