In the early 1920s, a supercritical instability in built-up rotors was encountered. Thereafter, first shown by Newkirk (1924) and Kimball (1924) to be a manifestation of rotor internal damping (i.e., the friction damping between rotor components). Then, Newkirk and Taylor (1925) described an instability caused by the nonlinear action of the oil wedge in a journal bearing, which was named as the oil whip. Baker (1933) described self-excited vibrations due to contact between the rotor and the stator. The Soviet scientist Nikolai (1937) examined the stability of transverse and torsional vibrations in a shaft with a disc mounted in the center and the stability of a shaft with a disc attached to the free end. Kapitsa (1939) pointed out that a flexible shaft could become unstable due to friction conditions in its sliding bearings. In the middle of the twentieth century, Hori (1959) succeeded in explaining various fundamental characteristics of oil whip by investigating the stability of shaft motion and considering pressure forces due to oil films. The mechanism of vibrations due to the steam whirl in turbines was explained by Thomas (1958) and that in compressors was explained by Alford (1965). The vibration of hollow rotor containing the fluid was the problem of flow-induced vibrations. Instability due to liquids partially filling interior cavities of rotors was demonstrated by Kollmann (1962), and in 1967 Ehrich reported that fluid trapped in engine-shafts induced asynchronous vibration and also changed the shape of resonance curves. Kuipers (1964), and Wolf (1968) independently successed in explaining the appearance of an unstable speed range in a postcritical region of a rotor system containing inviscid fluid. In 1980s the rotor dynamic effects of seals in fluid handling machines received a great deal of attention. Rotor destabilization due to seals was predicted and demonstrated in an operational compressor by Jenny (1980).

As rotors became lighter and rotational speeds higher, the occurrence of nonlinear resonances such as subharmonics became a serious problem. Yamamoto (1955, 1957) studied various kinds of nonlinear resonances after he reported on the subharmonic resonance due to ball bearings in 1955. He also investigated combination resonances. Tondl (1965) studied nonlinear resonances due to oil films in journal bearings. Ehrich (1966) reported subharmonic resonances observed in an aircraft gas turbine due to strong nonlinearity produced by the radial clearance of squeeze-film dampers.

The non-stationary phenomena during passage through critical speeds have been studied since, Lewis (1932) reported his investigation on the Jeffcott rotor. Non-stationary phenomena that occur are one in a process with a constant acceleration (unlimited driving toque) and another with variable acceleration (limited driving toque). Natanzon (1952) studied shaft vibrations at critical speeds, and Grobov (1953, 1955) investigated in general form the shaft vibrations resulting from varying rotational speeds. The development of asymptotic method by Mitropol’skii (1965) for nonlinear problems considerably boosted the research on this subject.

Beginning in the early 1960s, most attention focused on hydrodynamic bearings, this was largely stimulated by Lund (1964). Gunter’s work (1966) related to rotor dynamic stability problems, combined with Ruhl and Booker’s (1972), and Lund’s (1974) methods for calculating damped critical speeds, stimulated a great deal of interest in rotor-bearing stability problems. Lund (1987) gave an overview of the field. In the mid 1970s, rotor dynamic instability experiences with various high-pressure compressors and the high-pressure fuel turbo-pump of the Space Shuttle main engine focused a great deal of attention on the influence of fluid-structure-interaction forces, particularly forces due to the liquid and gas seals, in pumps and turbines. Shaft seals have similar effect as fluid-film bearings. They influence the critical speeds, can provide damping or on the other hand cause instability. Shaft seals have acquired a significant role in their effect on rotor dynamics. Someya (1989), and Tiwari et al. (2004, 2005) complied extensive numerical and experimental results, and literatures related to identification of rotor dynamic parameters of bearings and seals.

Self-excited vibrations, which occurs due to non-conservative forces, in general lead to large vibration amplitudes which may ultimately damage or even destroy rotating machinery (Childs,  1993; Gasch et al., 2002; Tondl, 1965; Yamamoto and Ishida, 2001). Therefore, it is essential during the design stage of a new machine to consider the possibility of self-excitation and take measures against it. A strategy to suppress self-excited vibrations, which is based on an anti-resonance phenomenon (two neighbouring modes having opposite effects) that can occur in parametrically excited systems (Tondl, 1978, 1991, and 1998) was described by Ecker and Tondl (2004). The basic idea of parametric stabilization was adopted by introducing a time-dependent variable stiffness located at the bearing mounts. The non-conservative forces were generated through by the bearings of the rotor. They showed the cancellation of the self excited vibrations through the parametric excitation.

Recently, Shaw and Balachandran (2008) provided a comprehensive review of nonlinear dynamics of mechanical systems including the rotating machineries. For rotating machineries they considered both the ideal and non-ideal excitations. In ideal excitation case, it is assumed that the rotor speed is a specified function of time, which is a classical problem in the theory of non-stationary problem in dynamics, extensively covered in the book by Mitropolskii (1965). The problem of passage through resonance of non-ideal vibrating systems has obtained special attention of the engineering researchers in the last years but unfortunately little literature on this subject is available (Balthazar et al., 2003). Generally, non-ideal vibrating systems are those for which the power supply is limited. Probably, Laval was the first one to work with non-ideal problems via an experiment. He built, in 1889, a single-stage turbine and demonstrated that in the case of rapid passage through resonance with enough power, the maximum vibration amplitude may be reduced significantly compared with that obtained in the steady state resonant vibration. Simultaneously, it was also known that sometimes the passage through resonance required more input power than the excitation source had available. The consequence is the so-called the Sommerfeld effect that the vibrating system cannot pass the resonance or requires an intensive interaction between the dynamical system and the motor to do it. The worst case is that of a dynamical system constructed for an overcritical operation to become stuck just before resonance conditions are reached. A strong interaction results with fluctuating motor speed and fairly large vibration amplitudes. This phenomenon was studied intensively by (Sommerfeld, 1902). Balthazar et al., (2003) provided an excellent review on the topic of the limited power source, in which case the system is called the non-ideal vibrating system.

Instability from fluid-film bearings and shaft seals arises from the fact that, during radial displacement of a rotor, a restoring force is produced, which has a component at right angles to this displacement. The phenomenon of instability was described in detail by Newkirk (1924), whose interest was in turbo machineries. The cause of this instability, in fact, lay in the oil-film bearings. Notwithstanding, in the following years it was established that in a few cases, internal friction or damping could indeed be a cause of instability. The designer must thus be aware of these possibilities. Some of the important phenomena in rotor-bearing systems, its main causes, and investigators’ details are summarized in Table 1.1.