Liapunov Stability Analysis of Elastic Shaft-Rigid Disk-Bearing Systems

A general method of analysis based on Liapunov’s direct method is presented for studying the dynamic stability of elastic shaft-rigid disk-bearing systems. A model comprised of a rigid disk rigidly attached at an arbitrary location along a flexible, rotating shaft which is mounted on two eight-component end bearings is used to develop stability criteria involving system stiffness and damping parameters. It is quantitatively shown by means of graphs for typical cases how the instability regions are reduced by (a) increasing the shaft dimensionless stiffness parameters, (b) increasing the bearing direct stiffness and damping parameters, (c) decreasing the bearing cross-coupling stiffness and damping parameters, (d) decreasing the mass ratio of the disk, and (e) increasing the disk’s radius ratio. These graphs present typical examples of the types of design information available to engineers through the equations provided in this paper. These graphs also verify that a two-modal term (N = 2) expansion is normally adequate to model the system deformations since the curves are not significantly altered by adding another term (N = 3) to the expansion. The critical value of the shaft dimensionless stiffness parameters is also studied.