The areas of machine tools, metal cutting, computer numerically controlled (CNC), computer-aided manufacturing (CAM), and sensor-assisted machining are quite wide, and each requires the academic and engineering experience to appreciate a manufacturing operation that uses all of them in an integrated fashion.

Although it is impossible to be an expert in all these subjects, a manufacturing engineer must be familiar with the engineering fundamentals for the precision and economical manufacturing of a part. This book emphasizes only the fundamentals of metal cutting mechanics, machine tool vibrations, feed drive design and control, CNC design principles, sensor-assisted machining, and the technology of programming CNC machines. The book is based on more than 120 journal articles and more than 60 research theses that reflect the engineering, research, and teaching experience of the author.

The book is organized as follows.

Chapter Two covers the fundamentals of metal cutting mechanics. The mechanics of two-dimensional orthogonal cutting is introduced first. The laws of fundamental chip formation and friction between the rake and flank faces of a tool during cutting are explained. The relationships among the workpiece material properties, tool geometry, and cutting conditions are presented. Identification of the shear angle, the average friction coefficient between the tool's rake face and moving chip, and the yield shear stress during machining is explained. The oblique geometry of practical cutting tools used in machining is introduced.

INTRODUCTION

Machine tools experience both forced and self-excited vibrations during machining operations. The cutting forces can be periodic, as in the case of milling. The nonsymmetric teeth in drilling, unbalance, or shaft runout in turning and boring can also produce periodically varying cutting forces. In all cases, the cutting forces can be periodic at tooth- or spindle-passing frequencies, which may have strong harmonics up to four to five times the tooth- or spindlepassing frequencies. If any of the harmonics coincide with one of the natural frequencies of the machine and/or workpiece structure, the system exhibits forced vibrations. The forced vibrations can simply be solved by applying the predicted cutting or disturbance forces on the transfer function of the structure by the use of the solution of ordinary differential equations in the time domain. However, self-excited, chatter vibrations are the most detrimental for the safety and quality of the machining operations, which are covered in this chapter.

Machine tool chatter vibrations result from a self-excitation mechanism in the generation of chip thickness duringmachining operations. One of the structural modes of the machine tool–workpiece system is initially excited by cutting forces. A wavy surface finish left during the previous revolution in turning, or by a previous tooth in milling, is removed during the succeeding revolution or tooth period, which also leaves a wavy surface owing to structural vibrations [112].

Geometry is one branch of mathematics that has an obvious relevance to the ‘real world’. Earlier, we studied some results in Euclidean geometry and we described the group of Euclidean transformations, the isometries. We saw that the Euclidean transformations preserve distances and angles, and have a definite physical significance.

In this chapter we study projective geometry, a very different type of geometry, that has important but less obvious applications. It was discovered through artists' attempts over many centuries to paint realistic-looking pictures of scenes composed of objects situated at differing distances from the eye. How can three-dimensional scenes be represented on a two-dimensional canvas? Projective geometry explains how an eye perceives ‘the real world’, and so explains how artists can achieve realism in their work.

In Section 3.1, we look at the development of perspective in Art and explain the concept of a perspectivity. We describe Desargues' Theorem, which concerns a curious property of two triangles whose vertices are in perspective from a single point, and so explain that perspective can play a key role in the statement and the proof of theorems in mathematics.

In Section 3.2, we define the term projective point (or Point) and call the set of all such Points the projective plane, which we denote by ℝℙ2. We also define a projective line (or Line). To enable us to tackle problems in projective geometry algebraically, we introduce homogeneous coordinates to specify the Points in ℝℙ2.

In Chapter 1 we studied conics in Euclidean geometry. In the rest of the book we prove a whole range of results about figures such as lines and conics, in geometries other than Euclidean geometry. In the process of doing this, we meet two particular features of our approach to geometry which may be new to you.

The first feature is the use of transformations in geometry to simplify problems and bring out their essential character. You may have met some of these transformations previously in courses on Group Theory or on Linear Algebra.

The second feature arises from the fact that the transformations we introduce form groups. Generally, we restrict our attention to geometry in the plane, ℝ2, but even in this familiar setting there may be more than one group of transformations at our disposal. This leads to the exciting new idea that there are many different geometries!

Each geometry consists of a space, some properties possessed by figures in that space, and a group of transformations of the space that preserve these properties. For example, Euclidean plane geometry uses the space ℝ2, and is concerned with those properties of figures that depend on the notion of distance. The group associated with Euclidean geometry is the group of isometries of the plane.

This idea, that geometry can be thought of in terms of a space and a group acting on it, is called the Kleinian view of geometry, after the 19th-century German mathematician Felix Klein who proposed it first.