Vibration and Coupling Phenomena in Asymmetric Disk-Spindle Systems

This paper analytically treats the free vibration of coupled, asymmetric disk-spindle systems in which both the disk and spindle are continuous and flexible. The disk and spindle are coupled by a rigid clamping collar. The asymmetries derive from geometric shape imperfections and nonuniform clamping stiffness at the disk boundaries. They appear as small perturbations in the disk boundary conditions. The coupled system eigenvalue problem is cast in terms of “extended” eigenfunctions that are vectors of the disk, spindle, and clamp displacements. With this formulation, the eigenvalue problem is self-adjoint and the eigenfunctions are orthogonal. The conciseness and clarity of this formulation are exploited in an eigensolution perturbation analysis. The amplitude of the disk boundary condition asymmetry is the perturbation parameter. Exact eigensolution perturbations are derived through second order. For general boundary asymmetry distributions, simple rules emerge showing how asymmetry couples the eigenfunctions of the axisymmetric system and how the degenerate pairs of axisymmetric system eigenvalues split into distinct eigenvalues. Additionally, properties of the formulation are ideal for use in modal analyses, Ritz-Galerkin discretizations, and extensions to gyroscopic or nonlinear analyses.