# The theory of machines - Mckay

THE THEORY OF MACHINES

BY ROBT. F. McKAY

LONDON, EDWARD ARNOLD, 1915

The theory of machines

PREFACE

This book attempts to deal in a comprehensive manner with the large amount of subject-matter which falls under the heading of the Theory of Machines. Although there are many text-books which cover adequately one or two special parts of the subject, there are none which deal systematically with the whole. It is hoped that the book will be found to meet the requirements of engineering students studying for University and kindred examinations in this subject, and also be of utility to engineers engaged on practical work.

The inclusion of the Elements of Mechanics in Part I. enables the author to elaborate those fundamental parts of the subject which, as his experience has shown him, give rise to grave misconceptions and difficulties on the part of the student. This initial survey of underlying principles obviates the disconcerting necessity of reverting to elementary matters when in the midst of a more advanced treatment of some sections of the subject proper.

In the treatment of Mass and Force, the so-called Engineer's unit of mass has been adopted. As both dynamical and statical forces have to be considered, no other course seems open if the book is to have any practical value. The unit of mass is then different to the standard mass, but the practical advantages of having one pound as the unit of force far outweighs other disadvantages.

In the Kinematics of Machines, Reuleaux's notation of lower and higher pairing has been used as a basis of classification of the subject. It is hoped that students will get a clear conception of kinematical problems by the author's classification given at the beginning of Chapter X. In the chapters devoted to the forms of wheel teeth, the important facts and conditions relating to wheel-teeth are given in the form of propositions, and it is thought that by this means the salient features of this difficult subject will be more readily grasped.

In a work of this description it is impossible to be entirely independent of previously-published works, and accordingly the author must express his indebtedness to many books of reference, to the Proceedings of the various Engineering Institutions, and also to the engineering periodicals. As far as possible, all assistance received has been acknowledged throughout. The author is indebted to the Crosby Steam Gauge and Valve Co., Messrs. Dobbie, Mclnnes and Co., and Messrs. Glenfield and Kennedy for the loan of blocks.

In preparing the book it has not been overlooked that, whilst there is a large number of students who can grasp book- work readily, there seems to be a general weakness in applying theoretical results to practical problems and difficulties. It is believed that this special faculty may be developed by solving numerous exercises, and a large number, culled from various sources, are given at the end of most chapters. The whole of the mathematical work involved in obtaining solutions has been undertaken by the author, and although every care has been exercised, it is perhaps too much to expect that there are no mistakes in the answers. The author will gratefully acknowledge any errors, clerical or otherwise, that may be discovered, whilst any suggestions for improving the book will be cordially appreciated.

CONTENTS

PART I - MECHANICS
I. INTRODUCTORY
II. PLANE MOTION OF A PARTICLE
III. FORCE AND TORQUE
IV. WORK AND ENERGY
V. PLANE MOTION OP A PARTICLE UNDER VARIABLE ACCELERATION - DISPLACEMENT VELOCITY AND ACCELERATION DIAGRAMS
VI. INTRODUCTION TO STATICS

PART II - KINEMATICS OF MACHINES
VII. ANALYSIS OF MOTION
VIII. THE SYNTHESIS OF A MACHINE
IX. LOWER PAIRING EXAMPLES OF MECHANISMS OBTAINED BY INVERSION
X. RELATIVE LINEAR VELOCITY AND ACCELERATION
XI. THE DETERMINATION OF THE ANGULAR MOTION OF LINKS
XII. THE RECIPROCATING MOTION OF THE SLIDER CRANK CHAIN
XIII. STRAIGHT LINE MOTIONS
XIV. HIGHER PAIRING TOOTHED GEARING
XV. GENERAL PROPOSITIONS ON THE FORMS OF WHEEL TEETH
XVI. INVOLUTE, CYCLOIDAL, AND PIN TEETH
XVII. HELICAL GEARING
XVIII. NON-CIRCULAR GEARS
XIX. TRAINS OF WHEELS AND EPICYCLIC GEARING
XX. BELTS AND BELTING
XXI. ROPE AND CHAIN DRIVES
XXII. CAMS
XXIII. SPHERIC MOTION HOOKE'S JOINT
XXIV. SCREW MOTION SCREW GEARS

PART III - DYNAMICS OF MACHINES
XXV. FRICTION
XXVI. STATIC EQUILIBRIUM OP MACHINES
XXVII. TURNING-MOMENT DIAGRAMS
XXVIII. GOVERNORS
XXIX. BALANCING
XXX. BRAKES AND DYNAMOMETERS

INTRODUCTORY

1. Like most of the ancient forms of human activity which have survived to the present time (cf. agriculture, navigation, etc., engineering now finds itself in the uncomfortable position of having to combine the qualities both of a science and an art. Original "rules of thumb" have, in the progress of modern civilization, become so numerous and so complex that it is no longer possible for any one craftsman to have an equally detailed facility in them all. Consequently the need arises for a classification of methods and material, and for organization of knowledge on a scientific basis, so that no one branch of the subject may suffer, as it assuredly would, through ignorance of the others. In all such cases, and particularly in engineering, the practical man will urge that the most phenomenal amount of book knowledge is, in the end, useless without his own specially developed faculty of rough-and-ready adaptation of means to an end. But the practical man must remember that it is just as certain in these days, that in the commencement and development of any new industry he can never guarantee to achieve his fullest possible results, i.e. attain his most complete efficiency, unless his practice is backed by a well-ordered and comprehensive theory.

In the study of every science, the first task is the classification or the adoption of some regular order in which to consider the, facts involved. This stage, completed in chemistry, physics, geology, etc., has not yet, in the branch of engineering with which we are directly concerned, been fully reached, owing partly to the shortness of tire period in which the subject has been scientifically considered, but more largely to the great diversity and multitudinous combinations of the mechanical contrivances which form its subject matter. In lieu, therefore, of this rigid classification, it is desired to point out in general terms at this initial stage, firstly, what must or what must not be included within the category of machines, and, secondly, the nature of the problems involved in the study of machines and the arrangement and classification of these problems. These two aims are best introduced by giving and examining the definition of a machine, which will be done in this chapter and Chapter VIII.

2. Definition of a Machine. A machine may be defined as an assemblage of resistant bodies whose relative motions are success- fully constrained so that some form of natural energy may be modified or transmitted to do some special kind of work.

This definition includes within its scope not only those contrivances and mechanical combinations which modify energy, as in the case of the steam engine, but also those which transmit energy, as in the case of a workshop machine. It will be advisable to give a detailed study of the phraseology of this definition in order to make clear exactly which combinations must or must not be considered as machines.

3. Simple Machines. In the first place, a machine has been denned as "an assemblage of resistant bodies." This last word would seem to exclude from our study the so-called simple machines, viz. the lever, wheel and axle, wedge, etc., which consist apparently of one body. However, this idea, if it exists, arises from too narrow a conception of the actual machine and is not due to a mistake in the definition itself. Both the acting and reacting bodies must be taken into account ; a lever, for example, is useless without a fulcrum ; the wheel and axle is useless with- out bearings ; likewise for each of the other simple machines. It will be found on analysis that there are no cases in which a machine consists of one body alone. The plural form of words in the definition is therefore necessary and correct.

4. Resistant Bodies. In the second place it will be advisable to give a wider interpretation to the term " resistant bodies " than that probably held by those newly introduced to the subject. In this term must be included all bodies by which either motion is constrained or through which force is transmitted. These bodies may be rigid, as in the case of the connecting rod of a steam engine; they may be elastic, as in the case of leather for belting; or they may be fluid, as in the case of water in the Brainah press, Whatever their form, each is called an element or link. As to the first of these types, it is well enough known that scientifically there is no such thing as a rigid body. But, for practical purposes, a machine part is called rigid when the deformations it undergoes when performing its customary functions are negligible, and the word is here used in this sense. Regarding the second type, elastic bodies, such as springs, belting, etc., have an important though limited utility, limited partly because of the deformations they undergo in practical employment and partly because they can in some cases only transmit forces in one direction. Belting or ropes, for example, can be used only in tension, but as long as no compressive force is put upon these materials they are more convenient for many purposes than a rigid connection. Fluid elements are less frequently used. They can only transmit such forces as are compressive.

In estimating the number of elements in a machine, care must be taken not to include redundant parts. In the example already considered, the wheel and axle, there are generally two bearings in which the axle of the pulley revolves. These two bearings in effect are rigidly connected together even if they do not form part of the same casting, and hence count only as one element. Similarly the piston, piston rod, and crosshead of a steam engine are rigidly connected together and have no motion relative to each other, so that they likewise form one element only in the steam engine. In general those parts of a machine which never have motion relative to another part are portions of he same element or link.

5. Differentiation between a Machine and a Structure. In the third place it will be advisable to draw attention to the statement that a machine modifies or transmits energy to do some special kind of work. It is this feature that differentiates a machine from a structure. The latter is subjected only to forces or straining actions; there is no relative motion between the component members, and hence energy cannot be usefully modified or transmitted. In a railway bridge, for example, the component parts are rigidly connected together, and the only relative motion is that due to the elasticity of the materials of construction. On the other hand, the useful purpose of a machine is only achieved by suitably directing the relative motion of its component parts. Of course, in the ultimate analysis of the machine, each moving piece, considered separately, must be treated as a structural part.

It may be pointed out that recording instruments must be included within the scope of our definition of a machine. These instruments, whether used for measuring purposes or otherwise, do not differ fundamentally from machines. The chief difference between the two is that the amount of energy utilized is very small in the case of an instrument. It is not necessary to emphasize this difference. Although the size of an instrument may be insignificant in comparison to a large machine, it is quite possible that from other standpoints an instrument may prove to be a more fruitful source of study than a machine.

6. Analysis of the Subject of Theory of Machines. In the modification or transmission of energy by machines, there are two possible variants, motion and force. The study of machines might very profitably follow the demarcation thus given. When the modification of motion is considered, neglecting the consideration of the forces producing or produced by that motion, the study is called the kinematics of machines. The general problem of this part of the subject is the determination of the comparative motions of the several parts of a machine. In this case, since considerations of force are not involved, the elements of a machine may be supposed to be skeleton links, and the relative motion of these links may be studied without further disturbing considerations. It will be found that the resulting problems can be solved to a large extent by purely geometrical means, though the study must be extended mathematically because of the introduction of the time- factor in dealing with velocity and acceleration.

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