On Unstable Vibrations of an Asymmetrical Shaft with Fluctuating Angular Velocity : 2nd Report, Approximate Solution and Instability for Lateral Vibrations of an Asymmetrical Shaft

Abstract

This paper deals with an approximate solution and instability for lateral vibrations of an asymmetrical shaft. When a rotating speed of the shaft fluctuates periodically with N times in a revolution, unstable vibrations occur at angular velocities near the major critical speed ω_c and near 2ω_c/|2±N|. In the cases that N=1, 3/2, 2 and 3, the approximation of the higher order than the first report is carried out for various values of an asymmetry of shaft stiffness Δ and an angular velocity fluctuation ε, and the effect of these parameters Δ, ε on the position and width of instability regions is made more clear. Then, the equations of motion are directly integrated, and loci of the shaft whirling are described in the x, y plane in order to know a dynamic behavior of a rotating shaft in instability regions. Furthermore, it is shown that very narrow instability regions occur at the angular velocities ω≒ω_c/N and 2ω_c/N.