Theory of Imperfections of Three-Dimensional Tuning Plate Gyroscope. Part II: Geometric Imperfections

Imperfections in a 3D (three-dimensional) disc/plate gyroscopes are major factors that limit the gyroscope’s sensitivity. Sources of imperfections may include nonuniform mass and stiffness distributions, non-homogeneity of damping distributions, axial misalignments, variations in radial distances, tilt-axis misalignment, et cetera. Geometric imperfections of a plate gyroscope will be covered in this paper. Firstly, we considered geometric imperfections based on the hypothesis of plane mid – surface of the plate in the form of outer rim imperfections of the plate. This type of imperfection arises due to imperfections in the plate’s shape or mounting. Imperfections in the outer rim of the plate can cause fluctuations in the gyroscope’s output, affecting its precession. This study also reveals that the outer rim radius imperfection causes the effect of “frequency splitting” for some harmonics. Secondly, we considered the effects of geometric imperfections of non-planar mid-surface on the plate/disc based on the theory of Mikhlin–Reissner shallow shells by introducing additional terms in the expression of the shear strain in comparison to the Novozhilov’s theory of shells. Hence, the main effects of non-planar mid-surface of the plate are concentrated in the potential energy of elongations and shear strains. This type of imperfection may arise from assembly errors or external forces acting on the gyroscope and it may affect the gyroscope’s output by impacting on its accuracy and stability. This paper is dedicated to modelling geometric imperfections that affect the accuracy of a 3D tuning plate gyroscope. Consequently, its main purpose is to derive equations of the motion and the associated forces that involve a geometrically imperfect plate gyroscope which vibrates both in-plane and out-of-plane with circumferential wave numbers m and n = m + 1, respectively.