Some first thoughts

01. If we agree (and we do) that no more than three parameters are needed to fix a single point in our Euclidean 3-space, we can argue that there are, all counted, an ∞3 of points. Next we can say (and we do) that, because the number of different radii available for the drawing of a sphere about a point is ∞1, there are, all counted, an ∞4 of spheres. Statements like that are typical of enumerative geometry, a method of mathematics much to be used in this book.

02. A parameter, in this metrical context, is a thing to which we can unambiguously ascribe a single number to correspond exactly with its size. Parameters turn out to be, either (a) a distance along some line either straight or curved which is measured from some origin point in the line, or (b) a magnitude of some angle across some surface either planar or ruled which is measured from some origin line in the surface. There appear to be no other kinds of parameter.

03. When we say that there exists for example an ∞n of some geometrical entity we mean, quite simply, that a minimum number n of parameters is needed to specify a nominated one of the entities from among its family. The family in question is then called an n-parameter family. The spheres in space, accordingly, are a 4-parameter family: any one of the spheres can be nominated by the use of only four parameters.

Introduction

01. I define the meanings of and occasionally use in this chapter two contrived adjectives: motional and actional. I use the two adjectives to distinguish two different kinds of straight line. Both of these I call, not because they are both straight but because they share another important characteristic, right lines. I speak about (a) motional right lines, and (b) actional right lines. The definition of the first kind of line comes from within the kinematics of its circumstance, while that of the second springs from the statics. The two kinds of line are similar and closely related with one another. At the beginning, to avoid confusion, I shall speak about them separately; I begin by discussing (a) the motional right line. I continue to do that until we reach § 3.43. For the sake of brevity throughout the passages leading up to § 3.43, however, I call the motional right line by its shorter name, which is common to both of the lines, right line. A right line in a moving body is any straight line which joins two points in the body whose linear velocities are perpendicular to the line. Along any such right line all points have linear velocities perpendicular to the line (§ 5.26, § 21.08); the tips of all of the attached velocity vectors along the line are in a straight line also (§ 5.25); there is an oo3 of right lines at an instant (§ 9.29); and, for a note about the terminology, please refer to § 10.50. I choose the simple, movable mechanism whose description follows as a vehicle for argument. The mechanism has four links and its mobility is unity (§ 1.34).

Some implications of a well-known idea

01. A widespread intuitive idea (and thus a useful definition of) the rigidity of a rigid body is as follows: having fixed any two points A and B in a moving body we argue that, if the body is rigid, the distance from A to B remains constant throughout a movement.

02. A number of important conclusions can be drawn from this fundamental statement. This chapter is devoted to drawing at least one of them, namely that, at any given instant in the motion of a rigid body, an axis with a pitch exists which is unique at the instant. The axis may be called the instantaneous screw axis for the body, or, more simply and often (§ 6.07), the screw in space about which the body is twisting (§ 5.50). Each of its aspects – orientation, location, and pitch – must be known before connections can be made between the so-called angular velocity of the body and the linear velocities of points such as A and B within it.

03. Because the points A and B in their line AB cannot move closer together or further apart, the component along the line of the velocity at B must be the same, at any instant, as the component along the line of the velocity at A. Applying this argument to other points along the line, we see that, for all points along a straight line in a rigid body, the velocity vectors at the points have equal components along the line. This is a general result. It holds for any line which may be drawn in a rigid body.

In April 1976 I began to see that this book was already begun. I found myself at that time drawing and writing a scatter of sketches and notes that gradually grew. During the years, however, I have never felt that finality was achieved. Even now, as I write no more and delete no more, the work is incomplete. It is no more than the limited remarks that I can make in the complicated circumstances of the wide field of TMM, the Theory of Mechanism and Machines.

I wish to acknowledge the primary influence of the International Federation for the Theory of Mechanisms and Machines. Without IFToMM, without its wide membership among the territories of the world, and without the magnificent opportunity this body has given me to consult with my colleagues in western and eastern Europe, the USSR, India, Asia, and in the Americas, this book would never have been attempted. Let me mention here with gratitude the encouragement and the general assistance I have had from the many individual scientific workers of IFToMM, and from their institutions. It should be needless to say that we (these colleagues and I) collectively encourage the life of IFToMM and trust that that life will continue to flourish.

At home I must mention my special colleague and sometimes collaborator Professor Kenneth Hunt of Monash University at Clayton in Australia. Without reserve – or so it has seemed to me – he and I have exchanged our views and openly expressed our occasional disagreements for a period now extending over twenty years.

Some observations

01. This book deals with mechanism. Mechanism is presented here as the geometric essence of machinery. The study of mechanism is important because the geometry of mechanical motion is often the crux of a real machine's design. The kind of motion I call mechanical motion is that which occurs between the rigid, contacting bodies or the material links of mechanism. This occurs in conjunction with two phenomena that happen together at the places of contact between the bodies. Firstly, transmission and other forces operate between the bodies as the motion occurs; and secondly, the shapes of the surfaces there guide the bodies as they move. A contact is a constraint upon a body's motion. Each movable body in machinery suffers its constraints accordingly. If we wish to penetrate the deeper issues at work in real machinery, if we wish to control (or at least not to be bamboozled by) the overwhelming complexity of the constraints upon the mechanical movements occurring constantly within machinery and thereby all about us, we must grapple with these constraints as directly as we can.

02. The digital computer demands on the part of its machine-designing users a ruthless competence in the algebraic processes needed for the manipulation of mechanical information and its numerical analysis. It is accordingly fashionable just now in the field of the theory of machines not so much to denigrate as simply to ignore the main bases in actual mechanical motion from which these algebraic processes grow. The main bases are essentially pictorial, geometrical. They arise from natural philosophy.

Opening remarks

01. In chapter 6 and in other parts of this book hitherto I have spoken about systems of ISAs or of screws, or more simply about screw systems. The origin in physical reality for most of these spoken remarks about screws was the capacity for instantaneous motion of some rigid body whose freedom to move at the instant was being restricted in some way. Relationships exist between these systems of screws about which small twists or rates of twisting of one body relative to another may occur, namely the systems of ISAs, and identical kinds of systems of screws about which wrenches and reaction wrenches between the same two bodies may act. An investigation of these two sets of systems of screws will reveal, at the end of this chapter 10, (a) an insight into the power expended in friction at working joints in mechanism, (b) an amplified meaning for the somewhat narrow term joint as defined for example at § 1.11, and (c) the beginnings of a method for calculating the forces at work at the joints of mechanism where mass and the consequent inertia of links is an important consideration.

02. With regard to (b) above I can mean by joint, as I shall show, the joint between for example the piston and the connecting rod of an engine designed for the transmission of power in the absence of loss, or the joint between for example a bulldozer blade and its one-off job where power is being releases in spurts, or the joint between a ploughing tool and its sod which is in a continuous, power-releasing action.