Jeffcott [34] had intensively investigated the dynamics of simple rotors using two-degree-of-freedom model including damping effects. He further explained the meaning of critical speed and introduced the new terminology of “whirl instability”. Newkirk [35] made an intensive investigation on stability of rotor-bearing systems and concluded that the rotor dynamic behavior could not be attributed to a critical speed resonance and the reduction of unbalance had no effect upon rotor whirl amplitudes. Holzer [36] introduced the way to find natural frequencies of torsional systems. This method was later adapted by Myklestad [37, 38] to calculate the natural frequencies of airplane wings coupled in bending and torsion. At about the same time Prohl [39] showed how this method could be applied to rotor-bearing systems. It became one of the most powerful tools for solving problem in rotor dynamics. Later, Myklestad and Prohl had made a slight extension of this work to include other effects like shear deformation and gyroscopic moments. This matrix transfer method of Myklestad and Prohl has been fully extended into the analysis of critical speeds [40, 41], stability [42, 43], and forced response [44] of complex turbo rotors. Other methods such as the finite difference and finite element [45-47] approach were also used in study of rotor dynamics. The rotor system is simulated by using the combination of large numbers of small elastic shaft elements where rotor weights and inertia moment effects are lumped at the mass stations, interconnected between two node points of the shaft elements. By using the advanced high-speed computers, transient response motion of rotor-bearing systems were investigated through real time numerical integration of rotor acceleration. Shen [48] presented a formulation for flexible rotor analysis using the influence coefficient approach. Kirk [49] further discussed the transient motion of multi-mass rotor systems with effects of nonlinear bearing support. In his conclusion, the linear stability analysis of the system can be verified using transient response results. However, the use of transient response motion to predict nonlinear stability of the system was not pointed out. Hitching [47] outlined transient approach using finite element approach and also discussed the effects of random excitation functions and the application of Taylor series and curve fitting to the transient solution.

In the case for complex rotor system with a large number of mass stations, the use of modal method seems efficiently providing an alternative approach for solving large numbers of equations of motion with less time consuming because the modal method greatly reduces the number of degree of freedom of the system. This approach has been proven successful in computing transient response of rotor-bearing systems [50-57]. Childs [52, 53] performed a complete investigation in using undamped modes calculated from the averaged vertical and horizontal support stiffness of the system. Choy [54-56] developed a modal analysis for rotor systems including the effects of gyroscopic, unbalance, disk skew, nonlinear bearings,and rotor acceleration.

The primary goal of the vibration signature analysis for machine health monitoring is to aid fault detection and identification. Various signal processing techniques have been developed and applied for fault detection and diagnosis in rotating machinery. The signal processing methods for machine health monitoring may be classified into time domain analysis, frequency domain analysis and joint

time-frequency domain analysis.

The time domain methods analyze the amplitude and phase information of the vibration time signal to detect the fault of gear-rotor-bearing system [57-61]. One of the traditional techniques is FM0 developed by Stewart [62], where he used this fault detection parameter to investigate the changes in a gearbox vibration signal due to gear damage. Some statistical measurements and comparisons of the energy/amplitude of different components of the vibration signal are used in some later developed traditional fault detection techniques. The vibration signal is often averaged over a large number of cycles, and by removing some components at desired frequencies the residual signal is obtained. The statistical parameters are then calculated. Some of the techniques include Root mean Squared [63], Energy ratio [64], Crest Factor, Kurtosis [65], FM0 [62], FM4 [62], NA4 [66-69], NB4 [68, 70], M6 [71], M8A [71], etc.