9.4 A Brief Review on Application of FEM in Rotor-Bearing Systems

In the past several decades various methods have been developed to analyze the dynamic behaviour of multi-DOFs rotor-bearing systems, e.g., the influence coefficient, transfer matrix, dynamic stiffness, mechanical impedance, finite element methods, etc. Of several methods the finite element method (FEM) is one, which is particularly well suited for modelling the large scale and complex rotor-bearing-foundation systems. As compared to the torsional and axial vibrations, initially researchers attempted more complex transverse vibration analyses of rotor-bearing systems. Even the study on coupled axial, torsional and transverse vibrations of rotor systems have been performed with other complexities like the rotary inertia, shear deformation, gyroscopic couple, internal damping, etc.

The Euler-Bernoulli beam accounts for the effect of bending in beams, which is due to the pure bending . In this theory, any plane cross-section of the beam before bending is assumed to remain plane after bending, and remain normal to the elastic (neutral) axis. Therefore, a beam cross section has not only the translation but also the rotation. Raleigh's beam accounts for the energy arising out of this cross-sectional rotation, which he called the rotary inertia . Subsequently, Timoshenko accounted for the shear strain energy in the beam due to the shear force. The Timoshenko beam usually refers to a beam in which both the rotary inertia and shear deformation effects are taken into account. The effects of rotary inertia and shear deformation are predominant in the transverse vibration of the beam having large cross-section (i.e., for thick beams; for a circular shaft we have the condition for thick beam as where r is the radius of gyration of the shaft and L is the length of the shaft ). If Timoshenko beams spins also, then gyroscopic effects also have an important role along with the rotary inertia and shear effects.

Historically, Ruhl (1970), and Ruhl and Booker (1972) were amongst the first researchers to utilize the finite element method to study the stability and the unbalance response of turbo-rotor systems. In their finite element formulations, only the elastic bending and translational kinetic energies were included. However, many effects such as the rotary inertia, shear deformations, gyroscopic moments, and the internal and external damping were all neglected in their finite element analysis. These higher order effects can be a very important for some configurations, as discussed in books by Dimentberg (1961) and Tondl (1965). McVaugh and Nelson (1976) generalized Ruhl's work by utilizing the Rayleigh beam model to devise a finite element formulation including the effects of rotary inertia, gyroscopic moments and axial load to simulate a flexible rotor system supported on the linear stiffness and (viscous) damping bearings. In order to facilitate the computations of natural whirl frequencies and unbalance responses, element equations were transformed into a rotating frame of reference for the case of isotropic bearings (i.e., cross-coupled coefficients are equal and direct coefficients are equal for both the stiffness and the damping). Also to save the computational time the Guyan (static) reduction procedure (1965) was adopted to reduce the size of system matrices. Zorzi and Nelson (1977) extended the work of McVaugh and Nelson by the inclusion of both internal viscous and hysteretic damping in the same finite element model. At about same time, Rouch and Kao (1979), and Nelson (1980) utilized the Timoshenko beam theory for establishing shape functions. Based on these shape functions the finite element matrices of governing equations were derived. In these system finite element matrices, a shear parameter was included in shape functions to take into account the effect of transverse deformations. Comparisons were made of the finite element analysis with the classical closed form Timoshenko beam theory analysis for the non-rotating and rotating shafts.

Ozguven and Ozkan (1984), and Edney et al . (1990) presented combined effects of the shear deformation and the internal damping to analyze natural whirl frequencies and unbalance responses of rotor bearing systems. Ueghorn and Tabarrok (1992) developed a finite element model for the free lateral vibration analysis of linearly tapered Timoshenko beams. The derived mass matrix was approximate and the stiffness matrix was exact. Tseng and Ling (1996) developed a finite element model of the Timoshenko beam to analyse free vibrations of non-uniform beams mounted on variable two parameter foundations. The characteristic of this model was that the cross sectional area, the moment of inertia, and the shear foundation modulus were all assumed to vary in polynomial forms, implying that the beam element can deal with commonly seen non-uniform beams having different cross sections such as the rectangular, circular, tubular and even complex thin walled sections as well as foundations consists of beams which vary in a general way. This beam element model enabled user to handle the vibration analysis of more general beams like in structures. Chen and Peng (1997) studied the stability of the rotating shaft with dissimilar stiffness and discussed influences of the stiffness ratio and axial compressive loads. A finite element model of a Timoshenko beam was adopted to approximate the shaft, and effects of gyroscopic moments and torsional rigidities were taken into account. Results showed that with the existence of the dissimilar stiffness unstable zones occurred. Critical speeds decreased and instability regions enlarged when the stiffness ratio was increased. The increase of the stiffness ratio consequently made the rotating shaft unstable. When the axial compressive load increased, critical speeds decreased and zones of instability enlarged.