Oscillations of a Rotating Shaft with Symmetrical Nonlinear Spring Characteristics

Abstract

A rotating shaft system composed of a circular disc, an elastic shaft and two bearings is usually subject to some gyroscopic effect and has four natural frequencies p1p4 which correspond to four modes of whirling motion. Of these four, the first two are of forward precession (p1>p2>0) and the other two are of backward precession (0>p3>p4). We consider a vertical shaft system, in which a shaft is supported by a double-row self-aligning ball bearing at the upper end and by a single-row deep groove ball bearing at the lower end. If the two center lines of the upper and lower bearings are well aligned and if the shaft is situated at the middle of the angular clearance of the lower bearing, the elastic restoring force of the shaft has symmetrical nonlinear spring characteristics. For this rotating shaft system, summed-and-differential harmonic oscillations of the types [2p2p3], [2p2-p4], [p2-2p3], [p2-2p4], and [p2-p3-p4], and a subharmonic oscillation of order 1/3 of the type [3p2] can occur. In this paper, these nonlinear forced whirling oscillations are studied experimentally. Furthermore, it is pointed out that the polar coordinates are suitable for representing the nonlinear restoring force characteristics in these whirling oscillations.