INTRODUTION

Why do you suppose that Cadillac introduced first a V12 engine, and then a V16 engine, back in the early 1930s? They were closely followed by Marmon, with its own VI6, while Lincoln had a VI2. Was this simply a marketing question: large numbers of cylinders had a certain mystique that was attractive to the customers? Or was there more to it than that? VI2 engines have appeared elsewhere from time to time – Daimler in the UK had what they called a Double Six, and Jaguar introduced a VI2 for the E-Type.

Why did the stroke:bore ratio change progressively over time? Just fashion, or something more?

These are design questions. They have to do with the descisions that a designer must make when a new engine is planned. Certainly marketing considerations come into it. There is no question that manufacturers have recently decided on four valves per cylinder in large part because the customers expect it; a Porsche with two valves per cylinder would sound wimpy. There is no doubt that Bentley and William Lyons (Jaguar) felt that double overhead cams had great market appeal, quite aside from their technical advantages. However, there are good technical reasons for some of these choices. Some of these we have mentioned in passing. Here, we will look into this in greater detail.

SIMILARITY CONSIDERATIONS

It is possible to examine some of these questions by considering geometrically similar engines of different sizes, or geometrically similar cylinders of different sizes. Two objects are geometrically similar if all dimensions are in proportion.

Tensor analysis is the language in terms of which continuum mechanisms can be presented in the simplest and most physically meaningful fashion. For this reason, I suggest that those readers who are not already familiar with this subject should read at least a portion of this appendix before starting with the main text.

The degree to which tensor analysis must be mastered depends upon your aims. We have written this appendix with three types of people in mind.

Many first-year graduate students in engineering are anxious to get to interesting applications as quickly as possible. We suggest that they read only those sections marked with double asterisks. They should also understand those exercises marked with double asterisks. Not all of this need be done before embarking on Chapter 1. Sometimes it is helpful to alternate between Chapter 1 and this appendix.

Those students who are somewhat more curious about the foundations of continuum mechanics will want to read the unmarked sections as well as those marked with two asterisks. The unmarked sections not only allow you to be more critical in your reading but are required for a complete understanding of the transport theorem in Section 1.3.2.

The complete Appendix A is recommended for anyone who wishes to do serious research in any of the subareas of continuum mechanics. The single-asterisked sections are required to derive the forms of various results in curvilinear coordinate systems.

Classical lubrication theory predicts bearing performance on the assumption that the viscosity of the lubricant is uniform and constant over the whole film. As the bearing performance is strongly dependent on lubricant viscosity, and as the viscosity of common lubricants is a strong function of temperature (see Figure 9.1), the results of classical theory can be expected to apply only in cases where the lubricant temperature increase across the bearing pad is negligible.

Effective Viscosity

In many applications (small bearings and/or light running conditions) the temperature rise across the bearing pad, although not negligible, remains small. It is still possible in these cases to calculate bearing performance on the basis of classical theory, but in the calculations one must employ that specific value of the viscosity, called the effective viscosity, that is compatible with the average temperature rise in the bearing. This might be realized, for instance, by making an initial guess of the effective viscosity, followed by an iterative procedure, using Figure 9.1, for systematically refining the initial guess. Boswall (1928) calculated the effective viscosity on the basis of the following assumptions:

1. All the heat generated in the film by viscous action is carried out by the lubricant.

2. The lubricant that leaves the bearing by the sides has the uniform temperature Θ = Θi + ΔΘ/2, where ΔΘ = Θ0 – Θi is the temperature rise across the bearing.

Let Q and Qs represent the volumetric flow rate of the lubricant at the pad leading edge and at the two sides, respectively.

Fluid film bearings are machine elements which should be studied within the broader context of tribology, “the science and technology of interactive surfaces in relative motion and of the practices related thereto.”* The three subfields of tribology – friction, lubrication, and wear – are strongly interrelated. Fluid film bearings provide but one aspect of lubrication. If a bearing is not well designed, or is operated under other than the design conditions, other modes of lubrication, such as boundary lubrication, might result, and frictional heating and wear would also have to be considered.

Chapter 1 defines fluid film bearings within the context of the general field of tribology, and is intended as an introduction; numerous references are included, however, should a more detailed background be required. Chapters 2, 3, and 4 outline classical lubrication theory, which is based on isothermal, laminar operation between rigid bearing surfaces. These chapters can be used as text for an advanced undergraduate or first-year graduate course. They should, however, be augmented with selections from Chapter 8, to introduce the students to the all important rolling bearings, and from Chapter 9, to make the student realize that no bearing operation is truly isothermal. Otherwise, the book will be useful to the industrial practitioner and the researcher alike. Sections in small print may be omitted on first reading – they are intended for further amplification of topics. In writing this book, my intent was to put essential information into a rational framework for easier understanding.

The term tribology, meaning the science and technology of friction, lubrication, and wear, is of recent origin (Lubrication Engineering Working Group, 1966), but its practical aspects reach back to prehistoric times. The importance of tribology has greatly increased during its long history, and modern civilization is surprisingly dependent on sound tribological practices.

The field of tribology affects the performance and life of all mechanical systems and provides for reliability, accuracy, and precision of many. Tribology is frequently the pacing item in the design of new mechanical systems. Energy loss through friction in tribo-elements is a major factor in limits on energy efficiency. Strategic materials are used in many triboelements to obtain the required performance.

Experts estimate that in 1978 over 4.22 × 106 Tjoule (or four quadrillion Btu) of energy were lost in the United States due to simple friction and wear – enough energy to supply New York City for an entire year (Dake, Russell, and Debrodt, 1986). This translates to a $20 billion loss, based on oil prices of about $30 per barrel. Most frictional loss occurs in the chemical and the primary metal industries. The metalworking industry's share of tribological losses amount to 2.95 × 104 Tjoule in friction and 8.13 × 103 Tjoule in wear; it has been estimated that more than a quarter of this loss could be prevented by using surface modification technologies to reduce friction and wear in metal working machines. The unsurpassed leader in loss due to wear is mining, followed by agriculture.

The qualitative difference in performance between liquids and gases, in general, vanishes as M → 0, where the Mach number, M, is the ratio of the fluid velocity to the local velocity of sound. This general conclusion also holds for bearings, and at low speeds the behavior of gas film lubricated bearings is similar to liquid-lubricated bearings – in fact, many of the liquid film bearings could also be operated with a gas lubricant. This similarity between liquid and gas films no longer holds at high speeds, however, the main additional phenomenon for gas bearings being the compressibility of the lubricant.

Perhaps the earliest mention of air as a lubricant was made by Him in 1854. Kingsbury (1897) was the first to construct an air-lubricated journal bearing. But the scientific theory of gas lubrication can be considered as an extension of the Reynolds lubrication theory. This extension was made soon after Reynolds' pioneering work: Harrison in 1913 published solutions for “long” slider and journal bearings lubricated with a gas. Nevertheless, the study of gas lubrication remained dormant until the late 1950s, when impetus for the development of gas bearings came mainly from the precision instruments and the aerospace industries.

In self-acting bearings, whether lubricated by liquid or gas, lubrication action is produced in a converging narrow clearance space by virtue of the viscosity of the lubricant. As the viscosity of gases is orders of magnitude smaller than that of commonly used liquid lubricants, gas bearings generally must have smaller clearances and will produce smaller load capacities than their liquid-lubricated counterparts.

Fluid film lubrication naturally divides into two categories. Thin-film lubrication is usually met with in counter-formal contacts, principally in rolling bearings and in gears. The thickness of the film in these contacts is of order of 1µxm or less, and the conditions are such that the pressure dependence of viscosity and the elastic deformation of the bounding surfaces must both be taken into account.

Thick-film lubrication is encountered in externally pressurized bearings, also called hydrostatic bearings, and in self-acting bearings, called hydrodynamic bearings. Of the latter, there are two kinds: journal bearings and thrust bearings. The film thickness in these conformal-contact bearings is at least an order of magnitude larger than in counter-formal bearings. In consequence, the prevailing pressures are orders of magnitude smaller, so that neither the pressure dependence of viscosity nor the elastic deformation of the surfaces plays an important roles. If, in addition, the lubricant is linearly viscous and the reduced Reynolds number is small, the classical Reynolds theory, as derived in the previous chapter, will apply.

This chapter discusses isothermal processes only. It should be realized, however, that bearings never operate under truly isothermal conditions, and under near isothermal conditions only in exceptional cases. Viscous dissipation and consequent heating of the lubricant are always present, and the change in viscosity must be accounted for when analyzing thick-film lubrication problems. In restricted cases, where design and operating conditions are such as to suggest “uniform” temperature rise of the lubricant, the “effective viscosity” approach of Chapter 9 might be employed.

Fluid Mechanics

The equations employed to describe the flow of lubricants in bearings result from simplifications of the governing equations of fluid mechanics. It is appropriate, therefore, to devote a chapter to summarizing pertinent results from fluid mechanics. This discussion will not be limited only to concepts necessary to understand the classical theory of lubrication. A more than elementary discussion of fluid behavior is called for here, as various nonlinear effects will be studied in later chapters.

Our discussion begins with the mathematical description of motion, followed by the definition of stress. Cauchy's equations of motion will be obtained by substituting the rate of change of linear momentum of a fluid particle and the forces acting on it, into Newton's second law of motion. This will yield three equations, one in each of the three coordinate directions. But these three equations will contain twelve unknowns: three velocity components, (u, v, w) and nine stress components (Txx, Txy,…, Tzz). To render the problem well posed, i.e., to have the number of equations agree with the number of unknowns so that solutions might be obtained, we will need to find additional equations. A fourth equation is easy to come by, by way of the principle of conservation of mass. The situation further improves on recognizing that only six of the nine stress components are independent, due to symmetry of the stress matrix. However, on specifying incompressibility of the fluid (incompressible lubricants are the only type treated in this chapter) a tenth unknown, the fluid pressure, makes its debut.

Elastohydrodynamic lubrication (EHL) is the name given to hydrodynamic lubrication when it is applied to solid surfaces of low geometric conformity that are capable of, and are subject to, elastic deformation. In bearings relying on EHL principles, the pressure and film thickness are of order 1 GP and 1 µm, respectively – under such conditions, conventional lubricants exhibit material behavior distinctly different from their bulk properties at normal pressure. In fact, without taking into account the viscosity-pressure characteristics of the liquid lubricant and the elastic deformation of the bounding solids, hydrodynamic theory is incapable of explaining the existence of continuous lubricant films in highly loaded gears and rolling-contact bearings. This is illustrated in the next section, by applying isoviscous lubrication theory to a rigid cylinder rolling on a plane.

When two convex, elastic bodies come into contact under zero load, they touch along a line (e.g., a cylinder and a plane or two parallel cylinders) or in a point (e.g., two spheres or two crossed cylinders). On increasing the normal contact load from zero, the bodies deform in the neighborhood of their initial contact and yield small, though finite, areas of contact; this deformation ensures that the surface stresses remain finite. For a nominal line contact the shape of the finite contact zone is an infinite strip, for a nominal point contact it is an ellipse. Nominal line contacts possess only one spatial dimension and are, therefore, easier to characterize than the two-dimensional nominal point contacts.

It follows from the previous chapters that one of the key elements in the analysis of flow separation from a solid body surface is the investigation of the flow behavior in the region of boundary-layer interaction with the external inviscid flow. Although the interaction region is normally very small, it plays a major role in the separation phenomenon because of the mutual influence of the near-wall viscous flow and external inviscid flow in this region, with a sharp pressure rise prior to the separation point leading to very rapid deceleration of fluid particles near the wall and ultimately to the appearance of the reverse flow downstream of the separation. The complexity of the physical processes in the interaction region is accompanied, as might be expected, by mathematical difficulties in solving the equations that describe the flow in this region. While everywhere outside the interaction region the solution may often be obtained in analytical or at least self-similar form, the analysis of the interaction region requires that special numerical methods be used.

Numerical solution of the interaction problem serves not only to provide meaningful physical information on the development of events in the interaction region, but in many cases also appears to provide the only way of being sure that the solution for this problem really exists, and hence that the entire asymptotic structure of the flow, anticipated in the course of the asymptotic analysis of the Navier–Stokes equations, is self-consistent.