Showing papers by "Arthur W. Lees published in 2010"

Dynamics of rotating machines

Abstract: This book equips the reader to understand every important aspect of the dynamics of rotating machines. Will the vibration be large? What influences machine stability? How can the vibration be reduced? Which sorts of rotor vibration are the worst? The book develops this understanding initially using extremely simple models for each phenomenon, in which (at most) four equations capture the behavior. More detailed models are then developed based on finite element analysis, to enable the accurate simulation of the relevant phenomena for real machines. Analysis software (in MATLAB) is associated with this book, and novices to rotordynamics can expect to make good predictions of critical speeds and rotating mode shapes within days. The book is structured more as a learning guide than as a reference tome and provides readers with more than 100 worked examples and more than 100 problems and solutions.

Dynamics of Rotating Machines: Free Lateral Response of Complex Systems

Abstract: Introduction In this chapter, we examine the characteristics of complex rotor–bearing systems in the absence of any applied forces. Chapter 3 shows how the characteristic dynamic behavior of simple rotors can be computed. As the rotor–bearing system becomes more complicated, the corresponding system model also becomes more complicated. The normal approach to handling this increased complexity is to discretize the system in a systematic way so that an approximate model can be created with a finite (but possibly large) number of coordinates. In the past, the transfer matrix method (or its predecessors: the Holtzer method for torsional analysis and the Myklestad-Prohl method for lateral analysis) were used for this purpose. With the increase in computing power in the last three decades, the FEM has become the de facto standard method for the static and dynamic analysis of structures as well as for the analysis of rotor–bearing systems. In this book, we choose to concentrate exclusively on the application of the FEM to model rotor–bearing systems. Example applications of the FEM to rotating machinery may be found in the works of Lalanne and Ferraris (1999), Nelson (1980), and Nelson and McVaugh (1976). The problem of creating an adequate model of a rotating machine is considered and the model is then analyzed to determine its dynamic characteristics. By “dynamic characteristics,” we mean the natural frequencies, the corresponding mode shapes, and the free response of the system. In Chapters 6, 7, and 8, we examine the effect of lateral forces acting on the rotor.

Dynamics of Rotating Machines: Finite Element Modeling

Abstract: Introduction The finite element method (FEM) has developed into a sophisticated method for the analysis of stress, vibration, heat flow, and many other phenomena. Although the method is powerful, its derivation is simple and logical. It is undoubtedly the combination of mathematical versatility with a simple geometric interpretation that led to the immense popularity of the method across wide areas of engineering and science. The texts by Bickford (1994), Cook et al. (2001), Fagan (1992), Irons and Ahmad (1980), and Zienkiewicz et al. (2005) provide details of the formulation of element matrices for various structural element types (e.g., beams, plates, shells, and continua). The National Agency for Finite Element Methods and Standards (NAFEMS, 1986) produced A Finite Element Primer , which is an excellent introduction to finite element (FE) methodology. This chapter explains the principles of FEA as they relate to vibrating structures. The same principles apply to the FE modeling and analysis of rotating machines. Two alternative methods produce the equations of motion of a system. The concept of generalized coordinates is explained in Section 4.2. The forces and moments produced by elastic deformation based on changes in these coordinates are calculated. For small deflections, these forces and moments, collectively called generalized forces , are linear functions of the generalized coordinates. Newton's second law is then used to equate the rate of change of momentum in the system to the forces on the system, both from the elastic deformation and externally applied forces, as in Chapter 2.

Dynamics of Rotating Machines: Introduction to Vibration Analysis

Abstract: Introduction In this chapter, we briefly examine the dynamic characteristics and properties of elastic systems composed of discrete components. In subsequent chapters, we extend and apply these ideas to the more complex problems arising from the dynamic analysis of continuous components, rotors, stators, and rotor–bearing–stator systems. Readers introduced to this material for the first time may need to consult other textbooks that develop these ideas in more detail and at a more measured pace (e.g., Inman, 2008; Meirovitch, 1986; Newland, 1989; Rao, 1990; Thompson, 1993). Here, the basic theory is reviewed in a manner suitable for those who already have some familiarity with it. For such readers, this chapter provides both a revision and a concise summary. The purpose of analyzing any elastic system or structure is to determine the static or dynamic displacements (or strains) and to find the internal forces (or stress) in the system or structure. To determine displacements, we require a frame of reference from which to measure them. Before we can begin the analysis, however, we must create a mathematical model of our system that may be very simple – perhaps devised intuitively – and easy to analyze but provides information of limited accuracy. Conversely, it may be a very complex model that requires significant computation but provides relatively accurate information.

Dynamics of Rotating Machines: Free Lateral Response of Simple Rotor Models

Abstract: 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.

Dynamics of Rotating Machines: More Complex Rotordynamic Models

Abstract: Introduction The purposes of this chapter are primarily to alert readers to the limitations of the analysis provided in previous chapters, to highlight the more complex behavior of certain types of rotor, and to indicate where detailed descriptions of the analysis of these systems can be found. Chapters 3, 5, and 6 consider the behavior of rotor–bearing systems when the rotor vibrates laterally; that is, the rotor whirls due to the actions of initial radial disturbances or, more important, radial forces. The rotor is modeled as an assembly of shaft elements and rigid disks. The shaft elements include the effects of inertia, bending and shear deflection, rotary inertia, and gyroscopic couples; the disks include the effects of inertia and gyroscopic couples. Bearings and foundations are modeled essentially as assemblies of axial springs and damping elements. Similarly, in Chapter 9, we examine the axial and torsional vibration of rotors and, in each case, we model the system as an assemblage of masses and inertias and axial or torsional springs. There are three basic assumptions in the aforementioned analysis: (1) the rotor– bearing–foundation system is linear; this assumption is present even in Chapter 7, in which instabilities of various types are examined; (2) although the rotor can deflect laterally, axially, and in torsion, it cannot otherwise deform; the shape of its cross section is fixed and plane cross sections remain plane; and (3) although the rotor can deflect laterally, axially, and in torsion, there is no coupling between these deflections; thus, an axial force can produce an axial deflection of the rotor, but it causes neither a lateral nor a torsional deflection.