Closed-Form Forced Response of a Damped, Rotating, Multiple Disks/Spindle System

Abstract

This paper is to study forced vibration response of a rotating disk/spindle system consisting of multiple flexible circular disks clamped to a rigid spindle supported by two flexible bearings. In particular, the disk/spindle system is subjected to prescribed translational base excitations as well as externally applied loads. Because of the bearing flexibility, the rigid spindle undergoes infinitesimal rigid-body rotation and translation simultaneously. To model real vibration response that has finite resonance amplitudes, the disks and the bearings are assumed to be viscously damped. Equations of motion are then derived through use of Rayleigh dissipation function and Lagrange’s equation. The equations of motion include three sets of matrix differential equations: one for the rigid-body rocking of the spindle and one-nodal-diameter disk modes, one for the axial translation of the spindle and axisymmetric disk modes, and one for disk modes with two or more nodal diameters. Each matrix differential equation contains either a gyroscopic matrix or a damping matrix or both. The causal Green’s function of each matrix differential equation is determined explicitly in closed form through use of matrix inversion and inverse Laplace transforms. Closed-form forced response of the damped rotating disk/spindle system is then obtained from the causal Green’s function and the generalized forces through convolution integrals. Finally, responses of a disk/spindle system subjected to a concentrated sinusoidal load or an impulsive load are demonstrated numerically as an example.