“Introduction:

The first successful rotor model was proposed by August Foppl in 1895. It consisted of a single disk centrally located on a circular shaft, without damping. It demonstrated that supercritical operation was stable. Unfortunately, Foppl published his work in a German civil engineering journal, which was little read, if at all, by the rotordynamics community of his day.

In 1919 Henry Jeffcott conceived the same model, this time with damping, and published his work in a widely read English journal. As a result, in the UK and the USA a single disk rotor is called a Jeffcott rotor. Over time many variations of the Jeffcott rotor have been studied but its most frequent features are a single, rigid disk mounted on a circular, flexible shaft, which is supported by bearings at each end. This is the configuration that will be assumed in this paper, see Fig. 1.

The Jeffcott rotor model is obviously an oversimplification of real-world rotors but it nevertheless assists an understanding of many features of real-world rotor behavior, including critical speeds, gyroscopic action, and the destabilizing effect of internal damping. The full richness of the model is demonstrated in the book by Erwin Kramer [2].

Natural whirling (precessional) frequencies and modes:

The dynamics of a rotor are not exactly the same as the dynamics of a spring-mass system. The reference state of the latter is stationary while that of the former is rotational. Nevertheless, they share the existence of natural frequencies and natural mode shapes. Let the rotor have a spin speed of Ω rad/s (60Ω/2π rpm) and let Ω be such that the rotor’s reference state is stable. Then if an impact is applied in the same sense as Ω, the rotor will whirl in a bent shape around its reference state at a rate of ω rad/s (ω/2π Hz). This is called Forward Whirling (FW). If the impact is opposite to the sense of n, the rotor will undergo Backward Whirling (BW). The frequencies of these whirling motions are called natural whirling (precessional) frequencies and the associated shapes are called natural whirling (precessional) modes. For a Jeffcott rotor with a non-central disk the FW and BW natural whirling frequencies are different functions of Ω and will diverge when plotted against Ω. A typical variation is shown in Fig. 2(b).

The root-cause of this divergence is the wobbling of the non-central disk during whirling. The interaction of this wobbling motion with the disk angular momentum vector produces an effect that increases the FW frequency and decreases the BW frequency.”