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 m
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Mathematical Models for Rotor Dynamic Analysis
Abstract 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.
3.1 Introduction Almost 150 years ago Rankine published his paper on ‘Centrifugal whirling of shafts’. This marked the beginning of a special branch of applied mechanics dealing with the dynamics, and in particular with the stability, of rotating machinery. The ever increasing importance of the latter and the technical difficulties of extending its size and reliability have led to a considerable growth of the new discipline. In the 1920s there were very important milestones in rotor dynamics. It was the time when higher speeds and larger machines demanded supercritical machine operation. This introduced most of the rotor stability phenomena and inspired some fundamental analyses. Newkirk studied instability problems associated with the bearing effects [1] and he identified instabilities due to dry friction [2], while Kimball [3, 4] gave basic results for internal damping in the shafts.
A. D. Dimarogonas et al., Analytical Methods in Rotor Dynamics, Mechanisms and Machine Science 9, DOI: 10.1007/978-94-007-5905-3_3, Ó Springer Science+Business Media Dordrecht 2013
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3 Mathematical Models for Rotor Dynamic Analysis
In the last few decades work on rotor stability has been concentrated on improvements of rotor and bearing description. A vast number of publications, among which, several books on the subject are now available. In classical literature, Stodola [5], in his book on steam and gas turbines, devotes a considerable portion on rotor stability while the books by Dimentberg [6], Tondl [7], Ehrich [8], Childs [9], Lalanne and Ferraris [10], and by Yamamoto and Ishida [11] are exclusively on rotor dynamics. A total of 554 references on the subject are listed in Ref. [12].1 Internal or ‘‘rotating’’ damping is a well-known source of potential dynamic instability of shafts operating at supercritical speeds. This kind of destabilizing damping may be present due to energy dissipation in the shaft’s material or rubbing between rotating components. In some cases similar effects may also be the result of fluid flow in labyrinth seals, oil flow properties, journal bearings, etc. [13]. The Newkirk Effect, the vibration change due to thermal distortion of a rotor caused by rubbing on stationary components was stud
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