Overview

This book begins with a brief historical introduction in which our aeronautical legacy is surveyed. The historical background illustrates the human quest to conquer the sky and is manifested in a system shaping society as it stands today: in commerce, travel, and defense. Its academic outcome is to prepare the next generation for the advancement of this cause.

Some of the discussion in this chapter is based on personal experience and is shared by many of my colleagues in several countries; I do not contest any differences of opinion. Aerospace is not only multidisciplinary but also multidimensional – it may look different from varying points of view. Only this chapter is written in the first person to retain personal comments as well as for easy reading.

Current trends indicate maturing technology of the classical aeronautical sciences with diminishing returns on investment, making the industry cost-conscious. To sustain the industry, newer avenues are being searched through better manufacturing philosophies. Future trends indicate “globalization,” with multinational efforts to advance technology to be better, faster, and less expensive beyond existing limits.

What Is to Be Learned?

This chapter covers the following topics:

• Section 1.2: A brief historical background

• Section 1.3: Current design trends for civil and military aircraft

• Section 1.4: Future design trends for civil and military aircraft

• Section 1.5: The classroom learning process

• Section 1.6: Units and dimensions

• Section 1.7: The importance of cost for aircraft designers

Coursework Content

There is no classroom work in this chapter, but I recommend reading it to motivate readers to learn about our inheritance.

Overview

This chapter is concerned with how aircraft design projects are managed in a company. It is recommended that newly initiated readers read through this chapter because it tackles an important part of the work – that is, to generate customer specifications so that an aircraft configuration has the potential to succeed. A small part of the course work starts in this chapter. The road to success has a formal step-by-step approach through phases of activities and must be managed.

The go-ahead for a program comes after careful assessment of the design with a finalized aircraft configuration having evolved during the conceptual study (i.e., Phase 1). The prediction accuracy at the end of Phase 1 must be within at least ±5%. In Phase 2 of the project, when more financing is available after obtaining the go-ahead, the aircraft design is fine-tuned through testing and more refined analysis. This is a time-and cost-consuming effort, with prediction accuracy now at less than ±2 to ±3%, offering guarantees to potential buyers. This book does not address project-definition activities (i.e., Phase 2); these are in-depth studies conducted by specialists and offered in specialized courses such as CFD, FEM, Simulink, and CAM.

This book is concerned with the task involved in the conceptual design phase but without rigorous optimization. Civil aircraft design lies within a verified design space; that is, it is a study within an achievable level of proven but leading-edge technology involving routine development efforts.

Overview

Chapter 6 proposes a methodology with worked-out examples to conceive a “firstcut” (i.e., preliminary) aircraft configuration, derived primarily from statistical information except for the fuselage, which is deterministic. A designer's past experience is vital in making the preliminary configuration. Weight estimation is conducted in Chapter 8 for the proposed first-cut aircraft configuration, revising the MTOM taken from statistics. Chapter 9 establishes the aircraft drag (i.e., drag polar), and Chapter 10 develops engine performance. From these building blocks, finally, the aircraft size can be fine-tuned to a “satisfactory” (see Section 4.1) configuration offering a family of variant designs. None may be the optimum but together they offer the best fit to satisfy many customers (i.e., operators) and to encompass a wide range of payload-range requirements, resulting in increased sales and profitability.

The two classic important sizing parameters – wing-loading (W/S) and thrust loading (TSLS/W) are instrumental in the methodology for aircraft sizing and engine matching. This chapter presents a formal methodology to obtain the sized W/S and TSLS/W for a baseline aircraft. These two loadings alone provide sufficient information to conceive of aircraft configuration in a preferred size. Empennage size is governed by wing size and location on the fuselage. This study is possibly the most important aspect in the development of an aircraft, finalizing the external geometry for management review in order to obtain a go-ahead decision for the project.

Appendix

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.

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.

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.

This appendix provides the stiffness coefficients of a shaft at a point along the shaft, which is typically a disk. These coefficients may be used in a simple two or four degrees of freedom model of a machine. The first column in Table A2.1 lists the boundary conditions for the shaft at the two bearing locations, which are at distances of a and b from the disk, for systems 1 through 5. A pinned-boundary condition forces the transverse displacement of the beam to be zero at the boundary but the beam is free to rotate. A clamped-boundary condition forces both the transverse displacement and the rotation of the beam to be zero at the boundary. If the boundary condition is free, then no constraints are applied. Generally, a beam is supported at the two ends and the boundary conditions are used to describe the beam system. For example, a clamped-free beam has one end clamped and the other free.

The stiffness matrices are obtained by a static reduction of the stiffness matrices (see Section 2.5.1), where the disk degrees of freedom are the master degrees of freedom. The stiffness matrix for each part of the shaft is equal to that obtained from the FEM, because a uniform beam with load applied only at the ends has its displacement defined by a cubic.

Table A2.2 lists the mass coefficients for systems 1 through 4 based on the displacement model given by the static deformation used to calculate the stiffness coefficients. These mass coefficients should be added to those of the disk to produce the equations of motion of the system.

Introduction

In this chapter, the stability of rotating machinery is considered. In stable systems, an initial disturbance decays to zero in the absence of excitation forces. By contrast, in an unstable system, the response grows, producing a large and undesirable response that may damage a machine. A simple example of instability is the motion of a pendulum. One equilibrium position is when the pendulum hangs vertically downward. This position is stable because if the pendulum is slightly displaced, it returns to the equilibrium position. In contrast, there is an equilibrium position when the pendulum is balanced vertically upward. This position is unstable because any slight disturbance from the vertical causes the pendulum to move away from the vertical and, in fact, rotate to the lower equilibrium position. For a linear system with constant coefficients, instability may be determined by considering the eigenvalues, computed in the usual way. Thus, the same or similar calculations used to determine eigenvalues of a system also provide a user with information about the stability of the system. As demonstrated in Chapter 2, the imaginary part of the eigenvalue gives the frequency of free oscillations, whereas the real part determines how rapidly the oscillations decay. The oscillations decay only if the real part of the eigenvalue is negative. A zero real part of the eigenvalue gives an undamped response in which the magnitude of the free oscillation remains constant and a positive real part causes the oscillation to grow.