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.

On regularity, and the irregularity at surfaces

01. The idea that regularity is a possibility derives directly from the foundations of Euclidean geometry. We allow by it such concepts as the straightness of a straight line, the flatness of a plane, the sphericality of a sphere. We can extend the notion to allow in the imagination such other things as, the exact ellipticality of a plane section through an ellipsoid, or the exact equi-angularity at the apices of some regular polyhedron. Extending the meaning of regularity into the realm of joints, we can envisage if we wish, and please refer to figure 6.01, that the convex conical surface at the joint element on link 1 and the flat plane surface at the joint element on link 2 can make, with one another, continuous line contact. We can similarly imagine that the convex surface of the accurately constructed spherical ball on 2 can make, with the concave surface of the matchingly constructed, equi-radial, spherical socket sunk in 3, continuous surface contact. These and other such imagined, continuous contacts between a pair of joint elements in mechanism – they are not single point contacts or even multiple point contacts but contacts involving ∞1or ∞2 of points – can exist in our imaginations only by virtue of the wholly idealised, but well established concepts due to Euclid.

02. But we know that no mould can ever exactly fit its pattern, that no casting can ever exactly fit its mould, that no welded construction ever matches its production drawing, and that no subsequent machining is ever done on any job accurately.

Some remarks connecting with chapter 6

01. Associated with each set of points of contact and its associated tangent planes and contact normals at a simple joint in mechanism, is a system of motion screws; this is explained in chapter 6. The so-called system of screws or the screw system is simply the set of all of the IS As which are available for relative motion at the joint. It is taken for granted in chapter 6 that the simple joint is considered alone (§ 1.28), and that contact is not being lost at any of the points of contact at the joint (§ 6.15, § 8.12).

02. Provided each one of the contact normals at a joint is projectively independent of the other contact normals at the joint (§ 3.32), the system of motion screws at the joint takes for its numerical name the number six minus the number of points of contact. This matter is thoroughly canvassed in chapter 6; with two points of contact there is discovered and explored the quasigeneral 4-system for example, with three, the quasigeneral 3-system, and so on.

03. It is important to see that in general the systems of screws of which I speak in chapter 6 – they are the systems of motion screws – contain no screws which lie along the contact normals at the points of contact at the joints. The contact normals at any one joint belong as screws of zero pitch to other systems of screws, which are, as we shall see, reciprocal to the mentioned systems.

Transition comment

01. The broad matters discussed in chapter 1, namely those of freedom, mobility and the need for sufficient constraint in mechanism, are of great importance in the wide realm of practical machine design. They are derived however from a somewhat narrow set of assumptions regarding (a) rigidity in the absence of elasticity, (b) crookedness in the absence of accuracy, and (c) direct contact at joints in the absence of lubricated clearances. In chapter 1 there is thus developed what might be seen to be a somewhat simple theory of constraint. It is a simple theory; it is directly applicable; but only in a very rough way is it able to predict, for much of our ordinary machinery, the actual mechanical behaviour of the moving parts.

02. We often see in ordinary working machinery the unhappy effects of friction due to eccentricities of loading (§ 17.24), the consequent continuous likelihood of jamming, chatter, and wear, and the necessity for lavish lubrication at badly affected joints. Such phenomena will be evident whenever the kinematic design for a smooth transmission of the wrenches has been poor (§ 17.26), or, alternatively, whenever the jamming phenomena themselves have been an intended activity of the machine (§ 10.61). Although such matters are important and need to be studied, they are in a curious way irrelevant here. They are not the matters being implied at § 2.01.

Some bases for an argument

01. For the reader already familiar with other discussions about the cylindroid (chapters 6, 15, 16 and elsewhere) it will be clear that a cylindroid is always lurking wherever there is a situation of two degrees of freedom – or wherever there are two forces or wrenches acting (§ 10.14). For the reader unaware as yet of the relevant generalities, the present chapter may be taken as a piece of purposely confusing, preliminary reading. It must be said in the kinematics (and the statics) of mechanism that the simpler a situation appears to be, the more baffling its deeper aspects often are. The science of the relative motion of rigid contacting bodies is bedevilled by degeneracies of its general geometry; these degeneracies, when taken in isolation, have a tendency to render the science a miscellany of unrelated facts and alleged separate theorems for which there appears to be no integrative binding. This chapter deals with some of that miscellany but with almost none of its binding; I shall be dealing here with the kinematics of only a few special cases of two degrees of freedom and I offer no conclusive results at the end of it.

02. However with no more than a primitive understanding of the relationship c + f= 6, where c is the number of constraints upon a body and/is its number of freedoms (§ 1.52), we can begin a useful argument here about the matter of two degrees of freedom.