A Nonclassical Vibration Analysis of a Multiple Rotating Disk and Spindle Assembly

Abstract

This paper studies natural frequencies and mode shapes of a spinning disk/spindle assembly consisting of multiple elastic circular plates mounted on a rigid spindle that undergoes infinitesimal rigid-body translation and rotation. Through use of Lagrangian mechanics, linearized equations of motion are derived in terms of Euler angles, rigid-body translation, and elastic vibration modes of each disk. Compared with a single rotating disk whose spindle is fixed in space, the free vibration of multiple disks with rigid-body motion is significantly different in the following ways. First of all, lateral translation of the spindle, rigid-body rotation (or rocking) of the spindle, and one-nodal diameter modes of each disk are coupled together. When all the disks (say N disks) are identical, the coupled disk/spindle vibration splits into N − 1 groups of “balanced modes” and a group of “unbalanced modes.” For each group of the balanced modes, two adjacent disks vibrate entirely out of phase, while other disks undergo no deformation. Because the out-of-phase vibration does not change the angular momentum, the natural frequencies of the balanced modes are identical to those of the one-nodal-diameter modes of each disk. For the group of the unbalanced modes, all disks undergo the same out-of-plane vibration resulting in a change of angular momentum and a steady precession of the spindle. As a result, the frequencies of the unbalanced modes are significantly lower than those of one-nodal-diameter modes of each disk. Secondly, axial translation of the spindle and the axisymmetric modes of each disk are couple together. Similarly, the coupled motion split into N − 1 groups of “balanced modes” and one group of “unbalanced modes,” where the frequencies of the balanced and unbalanced modes are identical to and smaller than those of the axisymmetric modes of each disk, respectively. Thirdly, the rigid-body motion of the spindle does not affect disk vibration modes with two or more nodal diameters. Response of those modes can be determined through the classical vibration analysis of rotating disks. Moreover, vibration response of the disk/spindle assembly from a ground-based observer is derived. Finally, a calibrated experiment is conducted to validate the theoretical predictions.

Free Vibration Analysis of a Spinning Flexible Disk–spindle System Supported by Ball Bearing and Flexible Shaft Using the Finite Element Method and Substructure Synthesis

“…In the modal testing, the natural frequency of unbalanced ;(0, 0) mode cannot be separated from the balanced B(0, 0) mode. Those modes are very close to each other as indicated by Shen and Ku [2]. In simulation, the frequency di!erence between two modes is around 13 Hz.…”

“…Natural frequencies except ;(0, 1) mode do not change due to the #exibility of the rotating spindle and stationary shaft. Compared with Figure 10 [2]. This research shows that shaft #exibility may be the critical parameter to predict the natural frequencies of the unbalanced ;(0, 1) mode accurately.…”

“…Simulation results agree very well with the experimental ones as shown in Figure 5. B(M, N) and ;(M, N) represent the balanced and unbalanced modes with M nodal circles and N nodal diameters [2]. Subscript b and f denote the backward and forward precession modes respectively.…”

“…Recently, many researchers have investigated the dynamics of the spinning #exible disk}spindle system in computer storage devices considering the modal interaction between a #exible spinning disk and spindle. Shen and Ku [2] analyzed the free vibrations of a spinning elastic disk and rigid spindle system by the assumed mode method, and they investigated the modal coupling of hard disk drives. They did not include the #exibility of the stationary shaft so that they did not fully explain the discrepancy of the rocking mode frequency between the theoretical prediction and modal testing.…”

Finite Element Analysis of Flexural Vibrations in Hard Disk Drive Spindle Systems

“…Also, the sixth mode is additionally sought similar to the third mode except for another nodal circle. According to the terms in reference [3], such modes as the second and fourth modes can be classi"ed as the balanced modes as opposed to the other four unbalanced modes.…”