# The theory of machines

THE THEORY OF MACHINES

PART I - THE PRINCIPLES OF MECHANISM

PART II - ELEMENTARY MECHANICS OF MACHINES

BY ROBERT W. ANGUS

McGRAW-HILL BOOK COMPANY, NEW YORK, 1917

The theory of machines

PREFACE

The present treatise dealing with the Principles of Mechanism and Mechanics of Machinery is the result of a number of years experience in teaching the subjects and in practising engineering, and endeavors to deal with problems of fairly common occurrence. It is intended to cover the needs of the beginner in the study of the science of machinery, and also to take up a number of the advanced problems in mechanics.

As the engineer uses the drafting board very freely in the solution of his problems, the author has devised graphical solutions throughout, and only in a very few instances has he used formulae involving anything more than elementary trigonometry and algebra. The two or three cases involving the calculus may be omitted without detracting much from the usefulness of the book.

The reader must remember that the book does not deal with machine design, and as the drawings have been made for the special purpose of illustrating the principles under discussion, the mechanical details have frequently been omitted, and in certain cases the proportions somewhat modified so as to make the constructions employed clearer.

The phorograph or motion diagram has been introduced in Chapter IV, and appeared in the first edition for the first time in print. It has been very freely used throughout, so that most of the solutions are new, and experience has shown that results are more easily obtained in this way than by the usual methods.

As the second part of the book is much more difficult than the first, it is recommended that in teaching the subject most of the first part be given to students in the sophomore year, all of the second part and possibly some of the first part being assigned in the junior year.

CONTENTS

PART I - THE PRINCIPLES OF MECHANISM
- The Nature of the Machine
- Motion in Machines
- Velocity Diagrams
- The Motion Diagram
- Toothed Gearing
- Bevel and Spiral Gearing
- Trains of Gearing
- Cams
- Forces Acting in Machines
- Crank Effort and Turning Moment Diagrams
- The Efficiency of Machines

PART II - MECHANICS OF MACHINERY
- Governors
- Speed Fluctuations in Machinery
- The Proper Weight of Flywheels
- Accelerations in Machinery and Their Effects
- Balancing of Machinery

CHAPTER I - THE NATURE OF THE MACHINE

1. General. - In discussing a subject it is important to know its distinguishing characteristics, and the features which it has in common with other, and in many cases, more fundamental matters. This is particularly necessary in the case of the machine, for the problems connected with the mechanics of machinery do not differ in many ways from similar problems in the mechanics of free bodies, both being governed by the same general laws, and yet there are certain special conditions existing in machinery which modify to some extent the forces acting, and these conditions must be studied and classified so that their effect may be understood.

Again, machinery has recently come into very frequent use, and is of such a great variety and number of forms, that it deserves special study and consideration, and with this in mind it will be well to deal with the subject specifically, applying the known laws to the solution of such problems as may arise.

2. Nature of the Machine. - In order that the special nature of the machine may be best understood, it will be most convenient to examine in detail one or two well-known machines and in this way to see what particular properties they possess. One of the most common and best known machines is the reciprocating engine, (whether driven by steam or gas is unimportant) which consists of the following essential, independent parts: (a) The part which is rigidly fixed to a foundation or the frame-work of a ship, and which carries the cylinder, the crosshead guides, if these are used, and at least one bearing for the crankshaft, these all forming parts of the one rigid piece, which is for brevity called the frame, and which is always fixed in position. (6) The piston, piston rod and crosshead, which are also parts of one rigid piece, either made up of several parts screwed together as in large steam and gas engines, or of a single casting as in automobile engines, where the piston rod is entirely omitted and the crosshead is combined with the piston. It will be convenient to refer to this part as the piston, and it is to be noticed that the piston always moves relatively to the frame with a motion of translation/ and further always contains the wristpin, a round pin to facilitate connection with other parts. The piston then moves relatively to the frame and is so constructed as to pair with other parts of the machine such as the frame and connecting rod now to be described, (c) The connecting rod is the third part, and its motion is peculiar in that one end of it describes a circle while the other end, which is paired with the wristpin, moves in a straight line, which latter motion is governed by the piston. All points on the rod move in parallel planes, however, and it is said to have plane motion, as has also the piston. The purpose of the rod is to transmit the motion of the piston, in a modified form, to the remaining part of the machine, and for this purpose one end of it is bored out to fit the wristpin while the other end is bored out to fit a pin on the crank, which two pins are thus kept a fixed distance apart and their axes are always kept parallel to one another, (d) The fourth and last essential part is the crank and crankshaft, or, as it may be briefly called, the crank. This part also pairs with two of the other parts already named, the frame and the connecting rod, the crankshaft fitting into the bearing arranged for it on the frame and the crankpin, which travels in a circle about the crankshaft, fitting into the bored hole in the connecting rod available for it. The stroke of the piston depends upon the radius of the crank or the diameter of the crankpin circle, and is equal to the latter diameter in all cases where the direction of motion of the piston passes through the center of the crankshaft. The flywheel forms part of the crank and crankshaft.

In many engines there are additional parts to those mentioned, steam engines having a valve and valve gear, as also do many internal-combustion engines, and yet a number of engines have no more than the four parts mentioned, so that these appear to be the only essential ones.

3. Parts of the Machine. - These two machines are typical of a very large number and from them the definition of the machine may be developed. Each of these machines contains more than one part, and in thinking of any other machine it will be seen that it contains at least two parts : thus a crowbar is not a machine, neither is a shaft nor a pulley; if they were, it would be difficult to conceive of anything which was not a machine. The so-called “simple machines,” the lever, the wheel and axle, and the wedge cause confusion along this line because the complete machine is not inferred from the name: thus the bar of iron cannot be called a lever, it serves such a purpose only when along with it is a fulcrum; the wheel and axle acts as a machine only when it is mounted in a frame with proper bearings; and so with the wedge. Thus a machine consists of a combination of parts.

5. Again, these parts must offer some resistance to change of shape to be of any value in this connection. Usually the parts of a machine are rigid, but very frequently belts and ropes are used, and it is well known that these serve their proper purpose only when they are in tension, because only when they are used in this way do they produce motion since they offer resistance to change of shape. No one ever puts a belt in a machine in a place where it is in compression. Springs are often used as in valve gears and governors, but they offer resistance wherever used. Thus the parts of a machine must be resistant.

6. Constrained Motion. - Now considering the nature of the motion, this also distinguishes the machine. When a body moves in space its direction, sense and velocity depend entirely upon the forces acting on it for the time being, the path of a rifle ball depends upon the force of the wind, the attraction of gravity, etc., and it is impossible to make two of them travel over exactly the same path, because the forces acting continually vary; a thrown ball may go in an approximately straight line until struck by the batter when its course suddenly changes, so also with a ship, that is, in general, the path of a free body varies with the external forces acting upon it. In the case of the machine, however, the matter is entirely different, for the path of each part is predetermined by the designer, and he arranges the whole machine so that each part shall act in conjunction with the others to produce in each a perfectly defined path.

Thus, in a steam engine the piston moves in a straight line back and forth without turning at all, the crankpin describes a true circle, each point on it remaining in a fixed plane, normal to the axis of the crankshaft during the rotation, while also the motion of the connecting rod, although not so simple is perfectly definite. In judging the quality of the workmanship in an engine one watches to see how exact each of these motions is and how nearly it approaches to what was intended ; for example, if a point on the crank does not describe a true circle in a fixed plane, or the crosshead does not move in a perfectly straight line the engine is not regarded as a good one.

The same general principle applies to a lathe; the carriage must slide along the frame in an exact straight line and the spindle must have a true rotary motion, etc., and the lathe in which these conditions are most exactly fulfilled brings the highest price.

These motions are fixed by the designer and the parts are arranged so as to constrain them absolutely, irrespective of the external forces acting; if one presses on the side of the crosshead its motion is unchanged, and if sufficient pressure is produced to change the motion the machine breaks and is useless. The carriage of the lathe can move only along the frame whether the tool which it carries is idle or subjected to considerable force due to the cutting of metal; should the carriage be pushed aside so that it would not slide on the frame, the lathe would be stopped and no work done with it till it was again properly adjusted. These illustrations might be multiplied indefinitely, but the reader will think out many others for himself.

This is, then, a distinct feature of the machine, that the relative motions of all parts are completely fixed and do not depend in any way upon the action of external forces. Or perhaps it is better to say that whatever external forces are applied, the relative paths of the parts are unaltered.

7. Purpose of the Machine. - There remains one other matter relative to the machine, and that is its purpose. Machines are always designed for the special purpose of doing work. In a steam engine energy is supplied to the cylinder by the steam from the boiler, the object of the engine is to convert this energy into some useful form of work, such as driving a dynamo or pumping water. Power is delivered to the spindle of a lathe through a belt, and the lathe in turn uses this energy in doing work on a bar by cutting a thread. Energy is supplied to the crank on a windlass, and this energy, in turn, is taken up by the work done in lifting a block of stone. Every machine is thus designed for the express purpose of doing work.

8. Definition of the Machine. - All these points may now be summed up in the form of a definition: A machine consists of resistant parts, which have a definitely known motion relative to each other, and are so arranged that a given form of available energy may be made to do a desired form of work.

9. Imperfect Machines. - Many machines approach a great state of perfection, as for example the cases quoted of the steam engine and the lathe, where all parts are carefully made and the motions are all as close to those desired as one could make them. But there are many others, which although commonly and correctly classed as machines, do not come strictly under the definition. Take the case of the block and tackle which will be assumed as attached to the ceiling and lifting a weight. In the ideal case the pulling chain would always remain in a given position and the weight should travel straight up in a vertical line, and in so far as this takes place the machine may be considered as serving its purpose, but if the weight swings, then motion is lost and the machine departs from the ideal conditions. Such imperfections are not uncommon in machines; the end long motion of a rotor of an electrical machine, the "flapping" of a loose belt or chain, etc., are familiar to all persons who have seen machinery running; and even the unskilled observer knows that conditions of this kind are not good and are to be avoided where possible, and the more these incorrect motions are avoided, the more perfect is the machine and the more nearly does it comply with the conditions for which it was designed.