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 Dunkerley's rule for the determination of l
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Approximate Evaluation of Eigenfrequencies
Abstract 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 Dunkerley’s 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.
1.1 Introduction Dynamics of rotating shafts has attracted attention a long time ago. Since the end of the nineteenth century, the theory of vibration was already extensively developed, and there was rapid progress in high-speed machinery construction, in particular to be used with locomotives and steam turbines. Whirling of shafts was anticipated by W. A. Rankine, who postulated that shaft operation above the critical speed was impossible. Extensive analytical investigations were performed by Dunkerley and Reynolds. De Laval observed and resolved experimentally most rotor dynamics problems, experimenting with steam turbines in the last quarter of the nineteenth century. The whirling problem was solved by A. Föpl, who
A. D. Dimarogonas et al., Analytical Methods in Rotor Dynamics, Mechanisms and Machine Science 9, DOI: 10.1007/978-94-007-5905-3_1, Springer Science+Business Media Dordrecht 2013
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1 Approximate Evaluation of Eigenfrequencies
explained analytically why operation above the critical speed is possible, as experimentally demonstrated by De Laval. His analysis is sometimes erroneously credited to Jeffcott and the De Laval rotor is sometimes misnamed the ‘‘Jeffcott rotor.’’ The early works of Rankine, Jeffcott and Stodola identified some of the fundamental aspects of the dynamics of rotating shafts [1–8]. In the 1920s the turbine industry designed machines to operate at substantially higher loads and at speeds above the lowest critical speed, and this introduced the modern rotor dynamics problems, which were treated by B. L. Newkirk and A. T. Kimball. Gyroscopic effects were introduced by A. Stodola. The influence of fluid bearings was investigated by Stodola and further quantified by B. L. Newkirk and H. D. Taylor and by A. Stodola. Vibration of shafts and beams of engineered shapes was first studied by Frahm, in particular, torsional vibration
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