Stefanos A. Paipetis

Dynamics of Cracked Shafts

The problem of cracked rotor dynamics is discussed in Chap. 6. Analytical formulation for crack local flexibility in relation to crack depth yields a supervisory instrument which can give an early crack warning. Fracture mechanics methods provide stress intensity factors for the investigation of rotor’s dynamic performance for varying crack depth. Open cracks lead to linear systems, while closing cracks lead to non-linear ones. Analytical solutions are obtained, which can be used to monitor crack propagation or to identify cracks in service.

Variational Formulation of Consistent: Continuous Cracked Structural Members

In Chap. 9, modeling and formulation of the governing dynamic equations for cracked Euler-Bernoulli beams in flexural vibration are studied. The results of three independent evaluations of the lowest natural frequency of lateral vibrations of beams with single-edge cracks and various end conditions are investigated: continuous cracked beam vibration theory, lumped crack flexibility model vibration analysis, a finite element method, and experimental results. For the case of torsional vibration of a shaft with a peripheral crack, the Hu-Washizu-Barr variational formulation is adopted for obtaining the differential equation of motion, with plausible assumptions about displacements, momentum, strain and stress fields, along with the associated boundary conditions. For the experimental procedure crack propagation and formation of stationary cracks is achieved by a vibration technique. Continuous cracked beam theory agrees better with experimental results than lumped crack theory.

Thermal Effects Due to Vibration of Shafts

Chapter 8 deals with the inverse problem of the one encountered in Chap. 5, e.g. the heat generated by torsional vibration of rotating shafts. The corresponding mechanisms are associated with internal damping and plastic deformation. Practically all the energy of 476 plastic deformation is transformed into heat. For elastic deformation part of the strain energy is transformed into heat, depending on material loss factor. This phenomenon has been identified as the cause of large-scale failures of power equipment, with electrical disturbances being the cause of rotor torsional vibration. Maximum temperatures given in the form of design nomograms can assist in estimating the overheating of shaft of rotating machinery, where such phenomena are present. A typical turbo-generator shaft is analyzed for vibrations occurring during electrical transients.

Identification of Cracks in Rotors and Other Structures by Vibration Analysis

The question of crack detection from dynamic measurements is further extended and discussed in Chap. 7. A general stiffness matrix for cracked structural members is introduced, to model the respective dynamic system. This stiffness matrix can be further utilized for static, dynamic or stability analysis of a structure with cracked members of rectangular or circular cross-section. Off-diagonal terms indicate vibration coupling. The change in dynamic response is analytically evaluated for simple systems and by means of approximate methods for more complicated ones. The outlined procedure can be used for engineering analysis in two ways: (a) as a design tool, to assist in structural optimization with the objective of achieving certain specific dynamic characteristics; and (b) as a maintenance and inspection tool, to identify structural flaws, such as cracks, by linking the variations in service of the structures natural frequencies to structural changes due to the cracks.

Variable Elasticity Effects in Rotating Machinery

The effects of variable elasticity in rotating machinery occur with a large variety of mechanical, electrical, etc., systems, in the present case, geometrical and/or mechanical problems. Parameters affecting elastic behavior do not remain constant, but vary as functions of time. Systems with variable elasticity are governed by differential equations with periodic coefficients of the Mathieu-Hill type and exhibit important stability problems. In this chapter, analytical tools for the treatment of this kind of equations are given, including the classical Floquet theory, a matrix method of solution, solution by transition into an equivalent integral equation and the BWK procedure. The present analysis is useful for the solution of actual rotor problems, as, for example, in case of a transversely cracked rotor subjected to reciprocating axial forces. Axial forces can be used to control large-amplitude flexural vibrations. Flexural vibration problems can be encountered under similar formulation.

Mathematical Models for Rotor Dynamic Analysis

Chapter 3 presents the main mathematical models used in rotor dynamic analysis. The one disc-flexible rotor model, called Jeffcott or de Laval rotor, can be used to derive qualitative features, since it lends itself to analytical treatment. The transfer matrix is powerful to model very long and complex rotors but it is strictly limited to linear systems and has certain problems of numerical instability. Lumped mass systems lead to very tedious computations, compared with the transfer matrix method, but they can be used to describe nonlinear systems. For realistic rotor forms, a discrete finite element model is presented, applicable to very complicated rotor geometries, yet leading to a manageable system of equations for linear or non-linear analysis.

Approximate Evaluation of Eigenfrequencies

Approximate evaluation of rotors flexural eigenfrequencies is investigated in Chap. 1. However, the formulation is similar for torsional vibrations of shafts or even vibrations of elastic systems in general. The Dunkerleys rule for the determination of lowest eigenfrequency of a lumped-mass, multi-degree-of-freedom elastic shaft is applied along with its extension to higher modes. This procedure generally provides lower bounds for the eigenfrequencies, but its accuracy can be increased at will by means of the root-squaring process, as suggested by Graeffe and Lobachevsky, applicable both to undamped and damped systems. Extension to continuous systems is considered too, and an integral equation formulation of the eigenvalue problem, providing upper and lower bounds for the eigenvalues, which by means of an iterative process can be brought as close as desired. Those methods are useful for predicting bending and torsional fatigue life of rotors and shafts, and furthermore, for developing methodologies for damage detection, and the estimation of position and size of flaws and cracks in rotating machinery.