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.

Introduction

The first two chapters provided the basics of small-rotation dynamics with applications to rigid bodies or near rigid bodies. The purpose of this chapter is to provide an introduction to the finite element method (FEM) of structural analysis to the limited extent necessary for the use of this textbook. That is, the present discussion of the FEM is limited mostly to one-dimensional structural elements.Specifically, all the example problems and exercises deal only with linear, beam finite elements and linear,spring finite elements. Neither of these elements require the finite series sophistication of two- orthree-dimensional elements such as plate or solid finite elements. A very brief introduction to multidimensional finite elements is presented in Endnote (). If the reader is already familiar with the FEM, then this chapter can be skipped, particularly because the next chapter provides ample further review of this topic. Since the finite element method is so extensively used for, and so particularly suited to, structural dynamics analyses, no other method of structural analysis is used for the calculations presented in this textbook. For the sake of instruction, the use of the FEM in this textbook is oriented to hand calculations rather than the use of one of the many available and routinely used commercial software programs that all do essentially the same things and differ only in style. Thus the reader should be able to gain insight into what all such FEM programs need to do.

Introduction

The purpose of this chapter is to introduce damping forces into the structural equations of motion. Simply speaking, damping forces are internal or external friction forces that dissipate the energy of the structural system. Although damping forces are usually much smaller than their companion inertia and elastic forces, they nevertheless can have a significant affect on a vibratory motion, especially after many periods of vibration, or when the system is vibrating at one of certain important frequencies called the system's natural frequencies. This chapter describes various ways of characterizing damping and explains how the damping properties of a vibratory system can be measured. Solutions for the motion of one-DOF systems are presented for force free and certain applied forces to better explain the role that damping plays in structural systems.

Descriptions of Damping Forces

When an actual, force free, structural system is set in motion by means of initial deflections or initial velocities, or both, any point within the system generally vibrates with amplitudes that are very little different over short time intervals; that is, time intervals lasting typically five or fewer periods of the vibration. Figure 5.1(a) shows the calculated amplitude–time trace of such a vibration where the period t of the vibration is 1 sec and the initial displacement has a unit value. As will soon be seen, the sinusoidal expression that describes the force free motion of a one-DOF undamped system, has to be modified, in this case by an exponential multiplier, when one representative form of system damping is present.