1.1 From the Rakine to Jeffcott Rotor Models: Rotor dynamics has a remarkable history of developments, largely due to the interplay between its theory and its practice (Nelson, 2003). Rotor dynamics has been driven more by its practice than by its theory. This statement is particularly relevant to the early history of rotor dynamics. Research on rotor dynamics spans at least 14 decades of history.

Rankine (1869) performed the first analysis of a spinning shaft (see Fig. 1.3a). He predicted that beyond a certain spin speed ". . . the shaft is considerably bent and whirls around in this bent form." He defined this certain speed as the whirling speed of the shaft. In fact, it can be shown that beyond this whirling speed the radial deflection of Rankine's model increases without limit, which is not true in actaul case. However, Rankine did add the term whirling to the rotor dynamics vocabulary. Whirling refers to the movement of the centre of mass of the deflected disc (or discs) in a plane perpendicular to the bearing axis (see Fig. 1.3b). In general, the frequency of whirl, v, depends on the stiffness and damping of the rotor (except for the synchornous whirl in which case it is equal to the unbalnce excitaion force frequency, ω, i.e., the spin speed of the rotor), and the amplitude is a function of the excitation force’s frequency, ω, and magnitude. A critical speed, ω_cr, occurs when the excitation frequency coincides with a natural frequency, ω_nf; and can lead to excessive vibration amplitudes. Rankine’s neglect of the Coriolis acceleration in his analysis, which led to erroneous conclusions that confused engineers for one-half century.

The turbine built by Parsons in 1884 (Parsons, 1948) operated at speeds of around 18 000 rpm, which was fifty times faster than the existing reciprocating engine at that time. In 1883 Swedish engineer de Laval developed a single-stage steam impulse turbine (Fig. 1.3b) (named after him) for marine applications and succeeded in its operation at 42 000 rpm. He aimed at the self-centering of the disc above the critical speed, a phenomenon which he instinctively recognized. He first used a rigid rotor, but later used a flexible rotor and showed that it was possible to operate above critical speed by operating at a rotational speed about seven times the critical speed (Stodola, 1924).