Dynamics of Rotating Systems

This book is the result of almost 30 years of work in the field of rotordynamics, which includes research, teaching, writing computer codes, and consulting. It is the outcome of an interdisciplinary research team that operated, and still operates, in the Mechanics Department and in the Interdepartmental Mechatronics Laboratory of Politecnico di Torino. The aim is mostly to write in a systematic way what has been the subject of a number of research papers, in such a way to give a consistent picture of the dynamic behavior of rotating machinery. As the title implies, this book is an attempt (only the reader can judge whether it is successful) to go beyond what is usually referred to as rotordynamics. The aim is that of dealing with the dynamic behavior of systems having in common the feature of rotating. This definition includes obviously those systems, like transmission shafts, turbine rotors, and gyroscopes, which are studied by rotordynamics, but also systems such as rotating blades (like in helicopter rotors) or flexible spinning spacecraft. Although rotordynamics usually deals only with the lateral behavior of rotors, some mention is made here also to torsional and axial vibration or to cases in which it is impossible to distinguish between them. However, the author imposed a limitation: No mention will be made of the dynamics of machines containing reciprocating parts, such as a crankshaft-connecting rod-piston mechanism. This arbitrary decision is based on the grounds that their vibration (mainly torsional vibration, but also axial and lateral vibration) is a very specialized topic, dealt with in many handbooks and textbooks and, above all, that to include it would have meant either to give a very insubstantial account or to double the size of the book. Another area in which a decision about where to stop was needed is controlled rotors. A thorough study of the dynamics of many controlled rotors, like those running on active magnetic bearings or supplied with active dampers, would have implied a detailed study of their control systems (hardware and, in case of digital systems, software) sensors and actuators (with the critical issue of the power amplifiers). As is typical of mechatronic systems, only an integrated and interdisciplinary approach allows us to exploit the advantages of the potentialities modern technology has opened. As this would have lead too far from the main topics of this book, these areas will be touched only marginally. The text is structured in two parts. The first one deals with what could be defined as classic or basic rotordynamics. The contents are basically well consolidated, although some incorrect statements can be found even in recent papers published on well-known journals. The basic assumptions are linearity, steady state operation, and at least some degree of axial symmetry. The second part, containing topics that are usually considered as specialized aspects of rotordynamics, could be titled advanced rotordynamics. The mentioned assumptions are dropped, and more detailed models are built for rotors departing from the classic configurations studied in rotordynamics. The contents of this part are more research topics than consolidated applications. The contents and the credits for the various chapters are the following: Chapter 1: Introduction. The basic concepts, graphical representation, and methods of rotordynamics are illustrated in a qualitative way. The expert reader, although familiar with these concepts, should not skip it altogether because the basic notation and the viewpoint that will be followed in the whole text are described. Part 1: Basic topics Chapter 2: Jeffcott rotor. The so-called Jeffcott rotor is the simplest rotor model that can be conceived. Although unable to account for some typical phenomena linked with rotordynamics, like gyroscopic effect or centrifugal stiffening, it allows us to gain a good insight into the peculiarities of rotating systems. In particular, it is essential for understanding the role of damping in rotordynamics. The topics dealt with are as a whole standard, but the part on nonsynchronous damping, is less common. Chapter 3: Model with four degrees of freedom: Gyroscopic effect. A simple model in which a rigid body is substituted for the point mass of the Jeffcott rotor is then studied, to allow the study of gyroscopic effects. This model is representative for the behavior of any rigid rotor on compliant bearings and allows us to define a modal gyroscopic system, on which modal decomposition of rotors can be based under some assumptions. Chapter 4: Discrete multi-degrees-of-freedom rotors. The lateral behavior of a flexible rotor modeled as a discrete parameter beamlike (1-D approach) system is then studied. Older approaches, like the transfer matrices methods, are dealt with together with more modern ones, like the finite element method (FEM). Chapter 5: Continuous systems: Transmission shafts. A short account on modeling simple rotors as continuous system is then included. This chapter can be considered more of academic rather than of practical relevance. Chapter 6: Anisotropy of rotors or supports. If either the rotor or the stator are not isotropic, it is still possible to obtain a closed-form solution for the linearized steady-state dynamics. Such systems are studied with particular reference to the backward whirling caused by unbalance in isotropic rotors on asymmetric supports and to the instability ranges of nonsymmetric rotors on isotropic supports. Chapter 7: Torsional and axial dynamics. The axial and torsional dynamics of rotors is briefly dealt with. Considering that the torsional and axial behavior is unaffected by the rotation of the system (at least if the basic assumptions of linearity and small displacements are made), just a brief account is reported. Chapter 8: Rotor-bearings interaction. The interaction between the behavior of the rotor and of the bearing is a complex subject, mainly because of the nonlinear behavior of the latter. The approach here followed is the classic one: The nonlinearity of the bearings is accounted for in computing their working conditions, and then the dynamic behavior is linearized assuming small displacements about the static equilibrium position (at speed). Rolling elements and lubricated and magnetic bearings are dealt with. Part 2: Advanced topics Chapter 9: Anisotropy of rotors and supports. The assumption that either the stator or the rotor is isotropic is dropped. No closed-form solution is any more possible, although a truncated series solution can be attempted. Chapter 10: Nonlinear rotordynamics. Here another assumption, that of linearity, is dropped. The phenomena typical of nonlinear systems, like jumps and even chaotic behavior are discussed. Chapter 11: Nonstationary rotordynamics. The spin speed is no more assumed to be constant, or other parameters, like unbalance, are allowed to change. In particular, the acceleration of the rotor through a critical speed and the occurrence of a blade loss are dealt with in detail. Chapter 12: Dynamic behavior of free rotors. Unconstrained rotating objects, like spinning celestial bodies or spacecraft, can be considered as rotors. The main aim of this section is to show that the assumption of constant angular momentum, typical of the dynamic study of free rotors, and that of constant angular velocity, typical of classic rotordynamics, coincide when the small displacement and rotations assumptions is made, so that the first can be approached with the methods of the latter. Chapter 13: Dynamics of rotating beams and blades. The effect of rotation, about an axis perpendicular to their longitudinal axis, on the dynamic behavior of beams and the blades-rotor interaction is studied using simple models. The well-known phenomena related to propeller and helicopter rotors' instability are dealt with, as well as other less-known phenomena regarding the effects of blade damping on the stability of a bladed rotor. Chapter 14: Dynamics of rotating discs and rings. Turbine and compressor discs are assumed, in classic rotordynamics, to behave as rigid bodies. In this chapter, this assumption is dropped and the effects of the flexibility of the discs are dealt with using simple models, starting from that introduced about 80 years ago by Southwell. Chapter 15: Three-dimensional modeling of rotors. This chapter deals with numerical modeling, mostly based on the FEM, of complex rotors. The topics dealt with in Chapters 13 and 14 using simplified models are here treated with the aim of building more accurate models, yielding precise quantitative results. Chapter 16: Dynamics of controlled rotors. Active vibration control is increasingly applied to rotors, either together with the use of active magnetic suspension or with techniques using active dampers or the control of more or less conventional bearings. As already stated, no attempt in modeling in detail the control, sensor or actuator dynamics is done, because it would lead too far from the central topics of this book. Appendix A: Vectors, matrices, and equations of motion. Some basic topics of system dynamics, particularly for the peculiar aspects linked with rotating systems, are summarized in this appendix. Appendix B: An outline on rotor balancing. As many very good books have been written on rotor balancing, only a short account on the basic topics are dealt with. Appendix E: Bibliography. Some of the books specifically devoted to rotordynamics are listed in chronological order. A CD-ROM comes with this book. It contains a simplified version of the DYNROT code and two short videos