Executive Summary

Geothermal energy has the potential to provide long-term, secure base-load energy and greenhouse gas (GHG) emissions reductions. Accessible geothermal energy from the Earth's interior supplies heat for direct use and to generate electric energy. Climate change is not expected to have any major impacts on the effectiveness of geothermal energy utilization, but the widespread deployment of geothermal energy could play a meaningful role in mitigating climate change. In electricity applications, the commercialization and use of engineered (or enhanced) geothermal systems (EGS) may play a central role in establishing the size of the contribution of geothermal energy to long-term GHG emissions reductions.

The natural replenishment of heat from earth processes and modern reservoir management techniques enable the sustainable use of geothermal energy as a low-emission, renewable resource. With appropriate resource management, the tapped heat from an active reservoir is continuously restored by natural heat production, conduction and convection from surrounding hotter regions, and the extracted geothermal fluids are replenished by natural recharge and by injection of the depleted (cooled) fluids.

Global geothermal technical potential is comparable to global primary energy supply in 2008. For electricity generation, the technical potential of geothermal energy is estimated to be between 118 EJ/yr (to 3 km depth) and 1,109 EJ/yr (to 10 km depth). For direct thermal uses, the technical potential is estimated to range from 10 to 312 EJ/yr. The heat extracted to achieve these technical potentials can be fully or partially replenished over the long term by the continental terrestrial heat flow of 315 EJ/yr at an average flux of 65 mW/m2.

Introduction

In Chapter 1, it is noted that the dynamic behavior of many rotors can be divided into three different classes: lateral, axial, and torsional. This chapter addresses both the axial and torsional behaviors of rotors. Generally, these two categories of behavior do not interact with one another, except in worm drives and bevel gears. Treating axial and torsional behavior together is justifiable because, mathematically, they are close analogies of one another. For cyclically symmetrical rotors, the analysis of both axial and torsional behavior is relatively simple. Thus, it is possible to provide an overview treatment of both in the space of a single chapter.

The degree to which a single rotor can be analyzed in isolation is different for the three classes of vibration. Lateral vibrations of a rotor are usually strongly coupled to the vibrations of the supporting stator and structure. The only significant exception to this is where very flexible bearings are in use. Passive magnetic bearings often provide this condition. By contrast, there is usually little coupling between torsional or axial vibrations of a rotor and any motion of the stator, except in the case of a geared system. Although this fact is helpful in analyzing the rotor, it often has the undesirable effect that even very severe torsional or axial vibrations in a rotor easily may go undetected by vibration probes on the stator. The analysis of torsional behavior is often carried out for a complete shaft train, whereas it is sometimes possible to analyze separately the axial and lateral behavior of the individual rotors in a shaft train.

Introduction

In all rotating machinery, some degree of mass unbalance is always present. Chapter 6 shows that small deviations in mass symmetry about the axis of rotation can lead to significant unbalance forces being exerted on the bearings. It is imperative to minimize these forces because they can lead to damaging vibration levels in the rotor, bearings, supporting structure, and ancillary equipment. Mass unbalance also can cause large shaft responses, leading to rotor–stator rubs or high stresses in the rotor. This unbalance is controlled primarily by attention to tolerances in rotor manufacture, but this alone is rarely sufficient and some means of further reducing vibration levels on the complete machine is necessary.

In this chapter, we discuss methods to achieve a satisfactory state of balance for a machine by adjusting the distribution of mass on the rotor, by either adding or removing correction masses at specific locations. Here, we concentrate on how to determine which unbalance corrections are appropriate; we do not focus on how, practically, these corrections are achieved. There are several approaches to the balancing of a rotor and all are based on the assumption of system linearity. Occasionally, iteration is required in the case of machines the bearings of which exhibit some degree of nonlinearity. The fundamental concepts of balancing are to monitor the effect on the synchronous (i.e., 1X) response of the machine due to one or more small masses attached to the rotor and then to scale those influences to determine the extent of the unbalance present in the rotor.

Introduction

In this chapter, we consider the process of creating adequate models of simple rotor systems and examine their lateral vibration in the absence of any applied forces. By “simple,” we mean a rotor system that can be modeled in terms of a small number of degrees of freedom. These simple models consist of either a rigid rotor on flexible bearings and foundation or a flexible rotor on rigid bearings and foundation. Obviously, rotating machines are not designed specifically with these properties; the reality is that rotating machines are designed for a purpose. Shaft dimensions and inertias and the type and dimensions of the bearings are chosen appropriately for the machine function. It may be that the rotor is short with a large diameter, resulting in a shaft that is much stiffer than the bearing and foundations on which it is supported. In such circumstances, it might well be appropriate to model the system as a rigid rotor on flexible bearings and foundations. In these simple models, we assume that both the bearing and foundation can be represented by simple linear springs in the x and y directions. Therefore, the stiffness of the bearing and foundation can be combined and considered as a single entity, using the formula for the stiffness of springs in series, Equation (2.9). Conversely, a machine design might demand a long shaft supported on relatively stiff rolling-element bearings and a stiff foundation. In this case, the bearing and foundation stiffness relative to the shaft stiffness is very high and it may be acceptable to model the system as a flexible rotor on rigid supports. Both models are studied in this chapter.

Overview

The aim of this book is to introduce readers to modern methods of modeling and analyzing rotating machines to determine their dynamic behavior. This is usually referred to as rotordynamics. The text is suitable for final-year undergraduates, postgraduates, and practicing engineers who require both an understanding of modern techniques used to model and analyze rotating systems and an ability to interpret the results of such analyses.

Before presenting a text on the dynamics of rotating machines, it is appropriate to consider why one would wish to study this subject. Apart from academic interest, it is an important practical subject in industry, despite the forbidding appearance of some of the mathematics used. There are two important application areas for the techniques found in the following pages. First, when designing the rotating parts of a machine, it is clearly necessary to consider their dynamic characteristics. It is crucial that the design of a machine is such that while running up to and functioning at its operating speed(s), vibration does not exceed safe and acceptable levels. An unacceptably high level of rotor vibration can cause excessive wear on bearings and may cause seals to fail. Blades on a rotor may come into contact with the stationary housing with disastrous results. An unacceptable level of vibration might be transmitted to the supporting structure and high levels of vibration could generate an excessive noise level. The second aspect of importance is the understanding of a machine's behavior when circumstances change, implying that a fault has occurred in the rotating parts of the machine.

This book addresses the dynamics of rotating machines, and its purpose may be considered threefold: (1) to inform readers of the various dynamic phenomena that may occur during the operation of machines; (2) to provide an intuitive understanding of these phenomena at the most basic level using the simplest possible mathematical models; and (3) to elucidate how detailed modeling may be achieved. This is an engineering textbook written for engineers and students studying engineering at undergraduate and postgraduate levels. Its aim is to allow readers to learn and gain a comprehensive understanding of the dynamics of rotating machines by reading, problem solving, and experimenting with rotor models in software.

The book deliberately eschews any detailed historical accounts of the development of thinking within the dynamic analysis of rotating machines, focusing exclusively on modern matrix-based methods of numerical modeling and analysis. The structure of the book (described in Chapter 1) is driven largely by the desire to introduce the subject in terms of matrix formulations, beginning with the exposition of the necessary matrix algebra. All of the authors are avid devotees of matrix-based approaches to dynamics problems and all are constantly inspired by the intricacy and detail that emerge from even relatively simple numerical models. The emergence of software packages such as MATLAB that enable what would once have been considered large matrix computations to be conducted easily on a personal computer is one of the most exciting and important innovations in dynamics in the past two decades. With such a package, sophisticated models of machines can be assembled “from scratch” using only a few prewritten functions, which are available from the Web site associated with this book.