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70

MODERN

MACHINE-SHOP

PRACTICE

VOL. I.— I.

Modern Machine-Shop Practice

OPERATION, CONSTRUCTION. AND PRINCIPLES OF SHOP MACHINERY, STEAM ENGINES, AND ELECTRICAL MACHINERY

BY

JOSHUA ROSE, M.E.

ILLUSTRATED WITH NEARLY 4000 ENGRAVINGS

THIRD EDITION REVISED AND ENLARGED

VOLUME L

NEW YORK

CHARLES SCRIBNER'S SONS 1 90 1

Who

\

Copyright, 1887, 1891, 1899 By CHARLES SCRIBNER'S SONS

PREFACE TO FIRST EDITION

ODERN MACHINE SHOP PRACTICE is presented to American mechanics as a complete guide to the operations of the best equipped and best managed workshops, and to the care and management of engines and boilers.

M'

The materials have been gathered in part from the author's experience of thirty-one years as a practical mechanic; and in part from the many skilled workmen and eminent mechanics and engineers who have generously aided in its preparation. Grateful acknowledgement is here made to all who have contributed information about improved machines and details of new methods.

The object of the work is practical instruction, and it has been written throughout from the point of view, not of theory, but of approved practice. The language is that of the workshop. The mathematical problems and tables are in simple arithmetical terms, and involve no algebra or higher mathematics. The method of treatment is strictly progressive, following the successive steps necessary to becoming an intelligent and skilled mechanic.

The work is designed to form a complete manual of reference for all who handle tools or operate machinery of any kind, and treats exhaustively of the following general topics: I. The construction and use of machinery for making machines and tools; II. The construction and use of work-holding appliances and tools used in machines for working metal; III. The construction and use of hand tools for working metal; IV. The con- struction and management of steam engines and boilers. The reader is referred to the Table of Contents for a view of the multitude of special topics considered.

The work will also be found to give numerous details of practice never before in print, and known hitherto only to their originators, and aims to be useful as well to master-workmen as well as to apprentices, and to owners and managers of manufacturing establishments equally with their employees, whether machinists, draughtsmen, engineers, or operators of special machinery.

viii PREFACE.

The illustrations, over three thousand in number, are taken from modern practice; they represent the machines, tools, appliances and methods now used in the leading manufactories of the world, and the typical steam engines and boilers of American manufacture.

The new Pronouncing and Defining Dictionary at the end of the work, aims to include all the technical words and phrases of the machine shop, both those of recent origin and many old terms that have never before appeared in a vocabulary of this kind.

The wide range of subjects treated, their convenient arrangement and thorough illustration, with the exhaustive Table of Contents of each volume and the full Analytical Index to both, will, the author hopes, make the work serve as a fairly complete ready reference library and manual of self-instruction for all practical mechanics, and will enlighten, while making more profitable, the labor of his fellow-workmen.

JOSHUA ROSE.

New York, 1887.

J

T

PREFACE TO THE THIRD EDITION

HE vast strides made in mechanics during recent years, make it an urgent necessity to provide a thorough revision of Modern Machine Shop Practice.

American mechanics and engineers excel those of any other country in their efforts to keep abreast of the progress of the age.

The latest machines and machine tools are greatly superior to those preceding them; and recent improvements make the engines of only a few years ago seem almost useless.

Realizing this, the author has carefully examined the many claims of mechani- cal improvement, and embodied in this revision those having the highest practical advantages.

As a result of all the changes and additions to the subjects of Lathe Work, Vice Work, Planing, Gear Cutting Machines, Grinding Machines, Milling Machines, and Steam Engines, the size of the work has been greatly enlarged and the number of plates and illustrations increased.

Since the First Edition was published the use of electricity has become so universal and the machinery for generating and applying it so varied and elaborate as to take a very important place in the world of mechanics. In revising this subject it has been found expedient to entirely replace the old text and engravings with new material; and to embody a complete description of the most modern electrical machines and appliances. The aim has been to follow as closely as possible the original intent of the work, and in pursuance of this plan the rules to be observed in operating electrical machines, the details regarding their construction, the use that each part in the mechanism is intended to serve, together with the principles that regulate the movements of the electric current so far as they are known positively, have been explained from the standpoint of the practical electrician and stated in plain work-shop language. The author is deeply indebted to the leading electricians in this

VOL. I.— 2.

X PREFACE TO THE THIRD EDITION.

country who have so kindly aided him in this division of the work; also to engineers having charge of the most important electric plants both at home and abroad. Placing himself in the position of a learner, he has sought from one and another the separate facts that go to make up the complete chain of information needed by practical men. To aid in making so complex a subject clear to the ordinary reader, a free use of illustrations has been made which it is believed will remove all difficulties; and a valuable series of questions and answers introduced, so that any person of average ability, even without any previous knowledge of the subject, can become an intelligent and practical electrician.

Refrigeration and Ice-making, another rapidly expanding industry, is also wholly rewritten and newly illustrated, embodying the latest improvements in machinery and practice.

•

The author believes that the large expense and labor incurred in this revision are amply justified by the enhanced value of the work in its new form.

JOSHUA ROSE.

New York, November, 1898.

CONTENTS.

Volume 1.

CHAPTER I. THE TEETH OF GEAR-WHEELS.

PAGE

Gear-Wheels. Spur-wheels, bevel-wheels, mitre-wheels, crown- wheels, annular or internal wheels i

Trundle-wheels, rack and pinion-wheel and tangent screw, or worm and worm-wheel i

The diameter of the pitch circle of i

Gear-Wheel Teeth. The face, the flank, the depth or height. . i

The space, the pitch line, the point, the arc pitch, the chord pitch, the line of centres 2

Rules for finding the chord pitch from the arc pitch ; table of natural sines ; diametral pitch ; finding the arc from the diametral pitch ; table of arc and diametral pitches. . . 3 Gear-Wheels. The driver and follower, a train of gears 3

Intermediate gears 3

The velocity of compounded wheels 4

Finding the diameters of the pitch circles of 4

Considered as revolving levers • 5

Calculating the revolutions of, and power transmitted by. . . 5

The angular velocity of 6

Gear-Wheels. Hunting tooth in, stop motion of 7

Gear- Wheel Teeth. The requirements and nature of the teeth

curves 7

Cycloidal curves for the faces of ; epicycloidal and involute curves ; the hypocycloidal curve ; method of forming or generating the e|)icycloidal and hypocycloidal curves for the faces and flanks of gear teeth 8

Applications of the epicycloidal and hypocycloidal curves in the formation of gear teeth 9

The diameter of the circle for generating the epicycloidal and hypocycloidal curves ; graphical demonstration that the flank curves are correctly formed to work with the face curves of the other wheel lo

Graphical demonstration that the curves are correct inde- pendent of either the respective sizes of the wheels, or of

the curve generating circles n

Gear-Wheels. Hand applications of the rolling or generating

circle to mark the tooth curves for a pair of wheels 12

Gear-Wheel Teeth. The variation of curve due to different di- ameters of wheels or of rolling circles 12

Tracing the path of contact of tooth upon tooth in a pair of gear-wheels ; definition of the " arc of approach ; " defini- tion of the " arc of recess ; " demonstration that the flanks of the teeth on the driver or driving-wheel have contact with the faces of the driven wheel during the arc of ap- proach, and with the flanks of the driven wheel during the arc of recess 13

Confining the action of the teeth to one side only of the line of centres, when motion rather than power is 10 be con- veyed... 13

Demonstration that the appearance or symmetry of a tooth has no significance with regard to its action 14

Finding how many teeth will be in constant action, the di- ameter of the wheels, the pitch of the teeth, and the diam- eter of the rolling circle being given 15

Example of the variation of tooth form due to variation of

wheel diameter 15

Gear Teeth. Variation of shape from using different diameters

of rolling circles 16

Thrust on the wheel shafts caused by different shapes of

teeth 16

Gear-Wheels. Willis' system of one size of rolling circle for

trains of interchangeable gearing 16

Conditions necessary to obtain a uniform velocity of i6

Gear Teeth. The amount of rolling and of sliding motion of. .. 16

The path of the point of contact of 16

The arcs of approaching and of receding contact 16

Lengths of the arcs of approach and of recess 16

The influence of the sizes of the wheels upon the arcs of con- tMt 17

PAGB

Influence of the size of the rolling circle upon the amount of flank contact 18

Demonstration that incorrectly formed teeth cannot correct themselves by wear 18

The smaller the diameter of the rolling circle, the less the sliding motion 18

Influence of the size of the rolling upon the number of teeth in contact in a given pair of wheels 19

Demonstration that the degrees of angle the teeth move through exceed those of the path of contact, unless the tooth faces meet in a point 19

Influence of the height of the teeth upon the number of teeth in contact 20

Increasing the arc of recess without increasing the arc of ap- proach 20

Wheels for transmitting motion rather than power 21

Clock wheels 21

Forms of teeth having generating or rolling circles, as large

or nearly as large as the diameters of the wheels 2i

Gear-Wheels. Bevel 21

The principles governing the formation of the teeth of bevel- wheels 22

Demonstration that the faces of the wheels must be in line with the point of intersection of the axis of the two shafts 22 Gear Teeth. Method of finding the curves of, for bevel gear. . . 22 Gear-Wheels. Internal or annular 23 to 27

Demonstration that the teeth of annular wheels correspond

to the sT)aces of spur-wheels 23

Gear-Wheels Internal. Increase in the length of the path of contact on spur-wheels of the same diameter, and having the same diameter of generating or rolling circle 23

Demonstration that the teeth of internal wheels may inter- fere when spur- wheels would not do so 23

Methods of avoiding the above interference 23

Comparison of, with spur-wheels 23

The teeth of : demonstration that it is practicable to so form the teeth faces that they will have contact together as well as with the flanks of the other wheel 24

Intermediate rolling circle for accomplishing the above re- sult 24

The application of two rolling circles for accomplishing the above result 24

Demonstration that the result reached by the employment of two rolling circles of proper diameter is theoretically and practically perfect 24

Limits of the diameters of the two rolling circles 25

Increase in the arc of contact obtained by using two rolling circles 25

Demonstration that the above increase is on the arc of recess or receding contact, and therefore gives a smooth action. 25

Demonstration that by using two rolling circles each tooth has for a certain period two points of contact 25

The laws governing the diameters of the two rolling circles. 25

Practical application of two rolling circles 26

Demonstration that by using two rolling circles the pinion may contain but one tooth less than the wheel 26

The sliding and rolling motion of the teeth of 27

CHAPTER II.

THE TEETH OF GEAR-WHEELS (Continued).

Worm and Worm- Wheel, or wheel and tangent screw ... .28 to 31

General description of 28

Qualifications of 28

The wear of 28

Worm- Wheel Teeth, the sliding motion of 28

When straight have contact on the centres only of the tooth

sides 28

That envelop a part of the worm circumference 28

The location of the pitch line of the worm 23

Xll

CONTENTS.

PAGE

Cutting curved worm wheel teeth with a hob 28

The proper number of teeth in the worm-wheel. 29

Locating the pitch line of the worm so as to insure dura- bility 29

Rule for finding the best location for the pitch line of the

worm : • • 29

Increasing the face of tlie worm to obtain a smoother action 29

Worms, to work with a square thread 29

Worm-Wheels, applications of 3°

End Thrust of Worm-Shaft, prevented by right and left hand

worms, driving gears of different diameters 31

Gear- Wheels with involute teeth 31*034

Gear Teeth. Generating the involute curve 31

Templates for marking the involute curve 32

Involute Teeth, the advantages of 34

Gear Teeth, Pratt and Whitney's machine for cutting templates

for 33

CHAPTER III.

THE TEETH OF GEAR-WHEELS (Continued).

New gear generating and cutting engine 36

Rack transformed into a cutting tool 36

Bisected gear teeth cutters '. 36

Prof. McCord's new gearing 36

The Willis theory 36

Gear Teeth, revolving cutters for 37

Pantagraph engine for dressing the cutters for 38

Numbers of cutters used for a train of wheels 39

Gear-Wheel Teeth. Table of equidistant value of cutters 41

Depth of, in the Brown and Sharpe systen 42

Cutting the teeth of worm-wheels 42

Finding the angle of the cutter for cutting worm-wheels. . . 43

The construction of templates for rolling the tooth curves. . 43

Rolling the curves for gear teeth 43

Forms of templates for gear teeth 44

Pivoted arms for tooth templates 44

Marking the curves by hand 45

Former or Template of the Corliss level gear-wheel engine

or cutting machine 45

The use of extra circles in marking the curves with com- passes ■ 46

Finding the face curves by geometrical constructions 47

The Willis odontograph for finding the radius for striking

the curves by hand 47

The method of using the Willis odontograph 48

Professor Robinson's odontograph 49

Method of using Professor Robinson's odontograph 49

Application of Professor Robinson's odontograph for trains

of gearing 51

Tabular valufes and setting numbers for Professor Robinson's

odontograph 51

Walker's patent wheel scale for marking the curves of cast

teeth 51

Scale for relations of pitch, number of teeth, and radius of

gear-wheels 52

Two parts known to find the third 52

Gear to clear a shaft or other obstruction ... 53

The amount of side clearance in cast teeth 53

Filleting the roots of epicycloidal teeth with radial flanks. . 53

Scale of tooth proportions given by Professor Willis 54

The construction of a pattern for a spur-wheel that is to be

cast with the teeth on 54

Template for planing the tooth to shape 54

Method of marking the curves on teeth that are to be glued on 55

Method of getting out the teeth of 5*^

Spacing the teeth on the wheel rim 56

Methods of accurately spacing the pattern when it has an

even number of teeth 58

Method of spacing the wheel rim when it has an odd num- ber of teeth 58

Gear-Wheels, Bevel-Pinion, drawings for 59

Getting out the body for a bevel-wheel 59

Template for marking the division lines on the face of the

wheel 59

Marking the lines of the division on the wheel 60

Gear-Wheels, Pinion, with dovetail teeth 60

Testing the angel of bevel-wheels while in the lathe 60

Gear-Wheels, Skew Bevel. Finding the line of contact 61

Marking the inclination of the teeth 61

Gear-Wheels, Bevel, drawing for built up 61

Gear-Wheels, Worm, or endless screw 62

Constructing a pattern from which the worm is to be cast. . 62

Tools for cutting the worm in a lathe 62

Cutting the teeth by hand 62

Gear-Wheels, Mortise or cogged 63

Methods of fastening cogs 63

Methods of getting out cogs for 63

Gear-Wheel Teeth, calculating the strength of epicycloidal. ... 64

Factors of safety for 64

Tredgold's rule for calculating the strength of 65

Cut, calculating the strength of 65

Gear-Wheel Teeth. The strength of cogs 66

The thickness of cogs 66

The durability of cogs 66

Table for calculating the strength of different kinds of 67

The contact of cast teeth 67

Table for determining the relation between pitch diameter,

pitch, and number of teeth in gear-wheels 68

Examples of the use of the above table 68

With stepped teeth 69

Angular or helical teeth 69

End thrust of angular teeth 6g

Herring-bone angular teeth 69

For transmitting motion at a right angle by means of angu- lar or helical teeth 69

Cutting helical teeth in the lathe 69

For wheels whose shaft axes are neither parallel nor meet- ing 70

Elliptical 70

Elliptical, marking the pitch lines of 70

Elliptical, drawing the teeth curves of 73

For variable motion 74

Form of worm to give a period of rest 74

Various applications of. 74

Gear-Wheels, arrangement of, for periodically reversing the di- rection of motion 75

Watt's sun and planet motion 75

Arrangements for the rapid multiplication of motion 75

Arrangement of, for the steering gear of steam fire engines. 75

Various forms of mangle gearing 79

Gear-Wheel and Rack, for reciprocating motion 77

Friction Wheels 77

The material for 77

Paper 78

For the feed motion of machines 78

The unequal wear upon grooved 79

Form of, for relieving the journals of strain 79

Cams, for irregular motion 80

Finding the pitch line of 80

Finding the working face of 80

The effect the diameter roller has upon the motion produced

by a cam 80

Demonstration of the different motion produced by different

diameters of rollers upon the same cam 80

Diagram of motion produced from the same cam with differ- ent diameters of rollers 81

Return or backing 82

Methods of finding the shape of return or backing 82

Cam Motion, for an engine slide valve without steam lap 83

For a slide valve with steam lap 83

Groove Cams, proper construction of 84

The wear of 84

Brady's improved groove cam with rolling motion and ad- justment for wear 84

CHAPTER IV.

SCREW-THREADS.

Screw-Threads, the various forms of 85

The pitch of 85

Self-locking 85

The Whitworth 86

The United States standard 86

The Common V 86

The requirements of 86

Tools for cutting 87

Variation of pitch from hardening 87

The wear of thread cutting tools 88

Methods of producing 88

Alteration of shape of, from the wear of the tools they are

cut by 8g

Screw Thread Cutting Tools. The wear of the tap and the

die 89

Improved form of chaser to equalize the wear 90

Form of, to eliminate the effects of the wear in altering the

fit 90

Originating standard angles for gi

Standard micrometer gauge for the United States standard

screw thread 91

Standard plug and collar gauges for 91

Producing gauges for . . 92

Table of United States standard for bolts and nuts 93

Table of standard for the V thread 95

United States standard for gas and steam pipes 95

Taper for standard pipe threads 95

Methods of measuring thread 95

Shape of caliper nibs 95. 96

Rules for computing depths of threads 95. 96

Rule for computing depth of United Slates standard

thread g6

Examples of measuring screws by the sides of their threads. 96

CONTENTS.

xiu

Machine for measuring^ the pitch of threads g6

Routine for measuring a sixty degree screw thread 96

Measuring odd and even number of threads 96

Screw-cutting hand tools 96

Thread-Cutting Tools. American and English forms of stocks

and dies q/

Adjustable or jamb dies 98

The friction of jamb dies 98

The sizes of hobs that should be used on jamb dies 99

Cutting right or left hand thread with either single, double,

or treble threads with tlie same dies gg

Hobbs for hobbing or threading dies 100

Improved thread cutting dies loi

Various forms of stocks with dies adjustable to take up the

wear lot

Dies for gas aiul steam pipes loi

Thread-Cutting Tool Taps. The general forms of taps 102

Reducing the friction of 102

Giving clearance to 102

The friction of taper 103

Improved forms of 103

Professor J . E. Sweet's form of tap 104

Adjustable standard 104

The various shapes of flutes employed on taps 105

The number of flutes a tap should have 105

Demonstration that a tap should have four cutting edges

rather than three 106

The position of the square or driving end. with relation to

the cutting edges 106

Taper taps for blacksmiths 106

Collapsing taps for use in tapping machines 107

Collapsing lap for use in a screw machine 107

The alteration of pitch that occurs in hardening 108

Gauging the pitch after the hardening loS

Correcting the errors of pitch caused by the hardening 109

For lead 109

Elliptical in cross section loi)

For very straight holes log

Tap wrenches solid and adjustable no

Thread-Cutting. Tapping no

Appliances for tapping standard work in

CHAPTER V. FASTENING DEVICES.

Bolts, classification of, from the shapes of their heads 112

Classification of, from the shapes of their bodies 112

Countersunk "2

Holes for, classification of 112

For foundations, various forms of 113

Hook bolls 113

The United States standard for finished bolts and nuts .. 113 The United Stales standard for rough bolts and nuts, or

black bolts H4

The Whitworth standard for bolts and nuts 1 14

Screws ii4

Keys and Keyways 1 15

Various kinds of I '5

Hearing surface of "5

With set screws 1 1 5

Application of keys to a square shaft '15

Proportions of "5

Taper keys "5

Application of set screws to hubs or bosses 115

Bushing used as keys nf'

P'eathers and their application n6

Through keys and keyways nf'

Draft "^

Removal of keys ""

Reverse keys ""

Pins and their application "7

Screws, Studs, and Bolts "7

Set screws "7

Bolt heads, application of "7

Nuts, application of ^'7

Check nuts "7

Differential screws "°

Slackening of nuts ''*

Fine end adjustments ''°

Split nuts I"

CHAPTER VI.

THE LATHE.

Lathe, the importance and advantages of 1 19

Classification of lathes "0

Foot "9

Methods of designating the sizes of 120

. Bench 120

Power 120

Hand 120

Slide Rest for 121

American form of, advantages and disadvantages ... 122

English forms of 122

Improved features of slide rests 122

For spherical work 123

Methods of turning true spheres 123

Engine Lathe, general construction of 123

F. E. Reed lathe 124

Elevating rest 124

Details of construction 124

Feed motions 126

Taper turning attachments 126

Flathers Machine Tool Co. 's 126

Dwighl Slate patent • • • 127

Hazlelt & Lord patent 127

Specifications of 127

Le Blond engine lathes • • ■ 128

Automatic feeds 128

Taper turning 129

Putnam Tool Co.'s lathe i33

The construction of the shears of 134

Construction of the headstock 134

Construction of the bearings 134

Construction of the back gear . I35

Means of giving motion to the feed spindle 135

Construction of the tailstock 135

Method of rapidly securing and releasing the tailstock 136

Lathe Tailstock, setting over for turning tapers 136

Engine Lathe, construction of carriage 137

Feed motion for carriage or saddle I37

Lathe Apron. Construction of the feed traverse 13S

Construction of the cross-feed motion 138

Engine Lathe, lead screw and change wheels of 139

Feed spindle and lead screw hearings I3g

Swing frame for lead screw 139

Lead screw nuts 14°

With compound slide rest 140

Construction of compound slide rest 14'

Advantages of compound slide rest 14'

For taper turning 142

Taper-turning attachments 142

Slide rest employing two cutting tools 143

With compound duplex slide rest '43

Detachable slide rest 143

Three-tool slide rest for turning shafting 143

With flat saddle for chucking work on 143

The Sellers Lathe 143

Construction of the headstock and treble gear 144

Construction of the tailstock and method of keeping it in line 145

Construction of the carriage and slide rest 145

Methods of engaging and disengaging the feed motions. . . . 146 Car Axle Lathe, with central driving' motion and two slide rests. 147

The feed motions of '4^

Self-Acting Lathe, English form of 148

Pattern Maker's Lathe 148

Brake for cone pulley 149

With wooden bed I49

Slide rest for 149

Chucking Lathe, English 149

Feed motions of 15°

Pulley Lathe 150

Gap or Break Lathe 151

Extension Lathe 151

Wheel Lathe 151

Chucking Lathe for boring purposes 151

Hand and drilling lathe 151

Set-over lathe .. . 15'

Improved Foot Block 151

Dressing Lathe for hexagon globe valves 152

Double-head lathe for turning keys of cocks 152

Hendey-Norton lathe '52

Screw-cutting features 152

Forming Monitor lathe with improvements 152

The forming tool slide and improved tool slide 152

Increased capacity '53

Automatic chuck I53

Pieces inserted and removed without stopping the lathe 153

Improved turret lathe '53

Ring friction clutch '53

Taking up the wear of the clutch 153

Screw machine with automatic feed '53

Spring collar 153

Sixty-inch forge lathe , i53

Footstock rollers '53

Pit lathe 153

Triple geared Putnam lathe '53

Surfacing Lathe with Cutter Face Plate i53

Position of cutting tools 153

XIV

CONTENTS.

Chucking lathes i54

Double axle lathe i54

Improved gap lathes i54

Horizontal chucking machine, improved 154

Combination turret lathe 154

Hollow spindle with chuck at each end 154

Automatic turret revolution 1 54

Regular lathe carriage instead of block with cut-off slide . . . 154

Provision for turning tapers 154

Chasing bar 155

Flat turret lathe (2 x 24) 155

Cut begins at the chuck 155

Large circular plate in place of a turret 155

Backward rest on turning tools 155

Automatic chuck 155

Toggle joint connection for sleeve lever 155

Spindle box caps on hollow posts 1 56

Turning and locking the turret 1 56

Power feed for turret carriage 1 56

Three point bearing 156

Irregular and taper turning tool 157

Chuck for irregular pieces 157

Screw-cutting machine 157

riartness patent dies 157

Construction of chasers, dies, and holders 157

Chaser produced by milling process 158

J. E. Reinecker lathe i5g

Gisholt's lathes 159

Hexagonal turrets 159

Finishing cone pulleys 160

New Haven Manufacturing Co.'s lathe 160

Pattern makers' lathes 160

Lodge and Shipley engine lathe 160

Taper turning attachment 160

Method of obtaining exact diameter 160

Screw-cutting gears 160

CHAPTER VIL DETAILS IN LATHE CONSTRUCTION.

Live Spindle of a lathe, the fit of 162

Taking up the wear of the back bearing 162

The wear of the front bearing of 162

Driving Cone, arranging the steps of 162

Requirements of proportioning the steps of 162

Rules for proportioning diameters of steps of, when two pul- leys are exactly alike and connected by an open belt. . . 162-164

When the two pulleys are unlike 164-167

Back Gear, methods of throwing in and out 168

Conveying motion to the lead screw 168

Attaching the swing frame 169

Feed Motion, to run quickly for polishing 170

On small lathes, in one direction only 170

Carriage, to discontinue its traverse, or feed 170

To lock in a fixed position 1 70

Slide Rest, weighted elevated 170

Double too! holder for 171

Gibbed elevating 171

Examples of feed motions 171

Feed Regulators for screw cutting 172

The star feed 172

Rajtchet Feeds 173

Tool Holding devices, the various kinds of 173

Tool Rest swiveling 174

Tool Holder for compound slide rests 174

For octagon boring tools 175

Lathe Lead and Feed Screws 175

Lead screws, supporting, long 176

Position of the feed nut 177

Form of threads of lead screws 177

The effect the form of thread has in causing the nut to lock

properly or improperly 177

Example of a lead screw with a pitch of three threads per inch. 177

Example of a lead screw with five threads per inch 178

Example with a lead screw of five threads per inch 179

Device for correcting the errors of pitch of 179

Table for finding the change wheels for screw cutting when the

teeth in the change wheels advance by four 180

For finding the change wheels when the teeth in the wheels

advance by six 180

Constructing a table to cut fractional threads on any lathe . 181 Finding the change wheels necessary to enable the lathe to

cut threads of any given pitches. 181

Finding the change wheels necessary to cut fractional pitches. 181

Determining the pitches of the teeth for change wheels 182

Lathe Shears or beds 182

Advantages and disadvantages of, with raised V-guide-ways. 1S2

Examples of various forms of 183

Lathe Shears with one V and one flat side 183

Methods of ribbing 184

PACE

The arrangement of the legs of 184

Lathe Tailblock 185

With rapid spindle motion 185

With rapid fastening and releasing devices 185

The wear of the spindles of 185

Spindles, the various methods of locking 186

Testing, various methods of i8j

CHAPTER VIII. SPECIAL FORMS OF THE LATHE.

Watchmaker's Lathes 188

Construction of the headstock 188

Construction of chucks for 188

Expanding chucks for 188

Contracting chucks for 188

Construction of the tailblock 189

Open spindle tailstocks for 189

Filing fixture for 189

Fixture for wheel and pinion cutting 189

Jewelers' rest for 189

Watch Manufacturer's Lathe 190

Special chucks for 190

Pump centre rest 190

Lathe, hand 191

Screw slotting 191

With variable speed for facing purposes 191

Cutting-off machine 192

Grinding Lathes 192

Landis' Universal grinding lathe 192

Swiveling platen for taper grinding 193

Traversing mechanism 193

Change of speed without stopping machine 193

Capacity 193

Change of tension on friction traverse 193

Change of point of traverse reverse 193

Grinding light shafts 193

The centre rest 194

The headstock spindle 194

Footstock spring tension 194

To do very accurate grinding 194

Adjustment to grind one two-thousandth of an inch 194

Chuck for grinding thin saws 194

Improved No. 2 Universal grinding machine 194

Internal grinding fixture 195

Care and use of machines 195

Truing and mounting emery wheels 195

Need of various sizes, shapes, and grades of wheels. 195

Speed of work on grinding machines 195

Effects of change of temperature in work 196

Use of water in grinding 196

Back, follow, and spring rests 196, 197

.Shoes for rests 197

Special chucks for grinding lathes 197

The Morton Poole calender roll grinding lathe 198

The construction of the bed and carriages 198

Principles of action of the carriages 198

Construction of the emery-wheel arbors and the driv- ing motion 198, 199

The advantages of 199

The method of driving the roll 200

Construction of the headstock 200

The transverse motion 200

The Brown and Sharpe Screw Machine, or screw-making lathe. 200

Threading tools for 203

Examples of the use of 203

The Secor Screw Machine, construction of the headstock 204

The chuck 205

The feed gear 205

The turret 205

The cross slide 205

The stop motions 206

Pratt and Whitney's Screw Machine 206

Parkhurst's wire feed, construction of the headstock, chuck

and feed motion 207

Box tools for 208

Applications of box tools 208

Threading tool for 208

Cutting-off tool for 208

Special Lathe for wood working 208

The construction of the carriage and reducing knife 209

Construction of the various feed motions 209

Construction of the tailstock 209

Lathes for irregular forms 210

Axe-handle 210

Back knife gauge 210

SpecLal, for pulley turning 211

Boring and Turning mill or lathe 2H

Construction of the feed motions 213

Construction of the framing and means of grinding the lathe 214

CONTENTS.

XV

Construction of the verticaV feed motions. . . . i 215

The Morton Poole roll turning lathe • 215

Construction of the slide rest 2l6

The tools for 2l6

Special Lathes for brass work 216,217

Boring Lathe with traversing spindle 218

Lathe for facing and boring cylinders 219

Quartering Machine 2ig

Heavy upright boring machine 221

Vertical boring and turning machine 221

CHAPTER IX.

DRIVING WORK IN THE LATHE.

Drivers, carriers, dogs, or clamps, an<l their defects 222

Lathe clamps 222

Equalizing drivers 223

The Clements driver 223

Driver and face plate for screw cutting 223

Shartle's self-tightening dog (Fig. 756) 223

Work driver for turning bolts (Fig. 762) 224

Forms of, for bolt heads 224

Adjustable, for bolt heads 224

For threaded work 225

For steady rest work 225

For cored work 225

For wood ^^5

Centres for hollow work 226

For taper work , 226

Lathe Mandrels, or arbors 227

Drivers for *27

For tubular work 2^7

Expanding mandrels 227

With expanding cones 228

With expanding pieces 228

Expanding, for L-irge work 228

For threaded work 228

For nuts, various forms of 22g

For eccentric work 229

Centring devices for crank axles • • • 230

The Steady Rest or back rest 231

Steady rest, improved form of 232

Cone chuck 232

Steady rest for square and taper work 233

The cat head 233

Clamps for *33

Follower rests ^34

Chucks and Chucking 234

Simple forms of chucks 234

Adjustable chucks for true work 235

Geared scroll chuck 235

Little Giant drill chuck 235

Two-jawed chucks 236

Box body chucks ^37

Reversible jawed chucks 237

Three and four-jawed chucks 237

Combination chucks 237

The wear of scroll chuck threads 237

Universal chucks 23°

The wear of chucks 240

Special forms of chucks ; 241

Cushman chuck with independent and connected jaws 241

Cushman chuck with reversible jaws 241

Horton combination independent chuck 241

Locking joint ^4'

Graduations ^4'

Expanding chucks for ring-work 241

Cement chuck ^42

Chucks for wood-working lathes 242

Lathe Face Plates ^43

Face plates, errors in, and their effects 243

Work-holding straps ^44

Face plate, clamping work on 245

Forms of clamps for • • 245

Examples of chucking work on 246, 247

For wood work ^47

Special Lathe Chuck for cranks 248

Face Plate Work, examples of 249

Errors in chucking ^5°

Movable dogs for ^5°

The angle plate ^5 1

Applications of

Angle plate chucking, examples of ■

Cross-head chucking ■

CHAPTER X.

CUTTING TOOLS FOR LATHES.

Principles governing the shapes of lathe tools 254

pJiraond-pointed, or front tool ^54

... 251 ... 251 251-253

FAGB

Principles governing use of tools 254

Front rake and clearance of front tools 254

Influence of the height of a tool upon its clearance and keen- ness 255

Tools with side rake in various directions 256

The effect of side rake 256

The angle of clearance in lathe tools 257

Variation of clearance from different rates of feed and dia- meters of work 257

Round-nosed tools 258

Utmost Duty of cutting tools 258

Judging the quality of the tool from the shape of its cut-

ting 259

Square-nosed tools 260

The height of lathe tools 260

Si 1e tools for lathe work 261

Cutting-off or grooving tools 262

Facing tools or knife tools 262

Spring tools 263

Brass Work, front tools for 264

Side tools for 264

Threading tools 264

Internal threading tools ■ 264

The length of threading tools 265

The level of threading tools 265

Gauges for threading tools 266

Setting threading tools 266

Circular threading tools 267

Threading tool holders 267

Chasers ". 2^8

Chaser holders 268

Setting chasers 268

Square Threads, clearance of tools for 269

Diameter at the roots of threads 269

Cutting coarse pitch square threads 269

Dies for finishing square threads 269

Tool Holders for outside work 270

For circular cutters 272

Swiveled ^73

Combined tool holders and cutting- off tools 273

Power required to drive cutting tools 273

CHAPTER XI. DRILLING AND BORING IN THE LATHE.

The Twist Drill ^74

Twist drill holders 274

The diametral clearance of twist drills 274

The front rake of twist drills 275

The variable clearance on twist drills as usually ground. . . . 275 Demonstration of the common error in grinding twist drills. 276

The effects of improper grinding upon twist drills 276

Table of speeds and feeds for twist drills 277

Special drill for boring small holes very deep 278

Grinding twist drills by hand 279

Twist drills for wood work 279

Tailstock Chucks for drilled work 279

Flat Drills for lathe work 280

Holders for lathe work 281

Half-round bit or pod auger 281

With front rake for wrought iron or steel 281

With adjustable cutter 231

For very true work 281

Chucking Reamer 281

The number of teeth for reamers 2»2

Spacing the teeth of reamers 282

Spiral teeth for reamers 282

Grinding the teeth of reamers 2S2

Various positions of emery-wheel in grinding reamers 282

Chucking reamers for true work 283

Shell reamers 2°3

Arbor for shell reamers 283

Rose-bit or rose reamers 283

Shell rose reamers 284

Adjustable reamers 204

Stepped reamers for taper work 285

Half-round reamers 285

Reamers for rifle barrels 285

Boring Tools for lathe work 285

Countersinks 285

Shapes of lathe boring tools 285

Boring tools for brass work 2b6

The spring of boring tool;; 286

Boring tools for small work 287

Boring tool holders 287

Boring Devices for Lathes 2»»

Boring Heads fJ"

Boring Bars ^ag

Boring bar cutters ■'°^

Three versus four cutters for boring bars 290

XVI

CONTENTS.

PAGE

Boring bars with fixed heads 290

With sliding heads 290

Bar cutters, the shapes of 291

Boring head with nut feed 291

Boring bars for taper work, various forms of 292

Boring double-coned work 293

Boring bar, centres for. 293

Cutting Speeds and feeds for wrought iron 294

Examples of speeds taken from practice 295

CHAPTER XII.

EXAMPLES IN LATHE WORK.

Technical Terms used in the work 296

Lathe Centres. .. . 296

Devices for truing 297

Tools for testing the truth of, for fine work 298

Shapes of, for light and heavy work 299

Centre Drilling, attachment for lathes 300

The error induced by straightening work after 300

Machine 300

Combined centre-drill and countersink 300

Countersink with adjustable drill 300

Centring square 300

Centre-punch 300

Centre-punch guide 301

Centring work with the scribing block 301

Finding the centre of very rough work 3or

Centre-drill chuck 302

The proper form of countersink for lathe work 302

Countersinks for lathe work 302

Various forms of square centres 303

The advantage of the square centre for countersinking 303

Novel form of countersink for hardened work 303

Chucks for centre-drilling and countersinking 303

Recentriiig turned work 304

Straightening work in the lathe 304

Centring machine 304

Chucks connected by gearing 304

The centring chuck 305

Positive stop for regulating drill action 305

Temperature of work 305

Straighteniirg Work. Straightening machine for bar iron 305

Hand device for straightening lathe work 305

Chuck for straightening wire 305

Cutting Rods into small pieces of exact length, tools for 305

Roughing cuts, the change of shape of work that occurs from re- moving the surface by 306

Feeds for 306

Rates of feed for 307

Finishing Work, the position of the tool for 307

Finishing cast-iron with water 307

Specks in finished cast-iron work 307

Scrapers for finishing cast-iron work 307

Method of polishing lathe work 308

Filing lathe work 308

The use of emery paper on lathe work 308

The direction of tool feed in finishing long work 309

Forms of laps for finishing gauges or other cylindrical lathe

work 310

Forms of laps for finishing internal work 311

Grinding and polishing clamps for lathe work 311

Burnishing lathe work 311

Taper Work, turning 312

The wear of the centres of 312

Setting over the tailstock to turn 312

Gauge for setting over 313

Fitting 313

Grinding 313

The order of procedure in turning 313

The influence of the height of the tool in producing true. . . 314

Special Forms. Curved work. 314. 315

Standard gauges for taper work 316

Methods of turning an eccentric 317

Turning a cylinder cover 318

Turning pulleys 318

Chucking device for pulleys. 318

Cutting Screws in the lathe 319

The arrangement of the change gears 319

The intermediate wheels 319

The compounded gears 320

Finding the change wheels to cut a given thread 320

Finding the change wheels for a lathe whose gears are com- pounded 321

Finding the change gears for cutting fractional pitches 321

To find what pitch of thread the wheels already on the lathe

will cut 322

Cutting left-hand threads 322

Cutting double threads 322

fAGH

Cutting screws whose pitches are given in the terms of the

metric system 322

Cutting threads on taper work 323

Errors in cutting threads on taper work 324

CHAPTER XIII. EXAMPLES IN LATHE WORK (Continued).

Ball Turning with tubular saw 325 •

With a single tooth on the end of a revolving tube 325

With a removable tool on an arbor 325

Tool holder with worm feed 325

By hand 325

Cams, cutting in the lathe 326

Improved method of originating cams in the lathe 326

Motions for turning cams in the lathe 326, 327

Application of cam motions to special work 327

Cam chuck for irregular work 328

Milling or knurling tool 328

Improved forms of 328

Winding Spiral Springs in the lathe 329

Hand Turning 330

The heel tool 330

The graver and its applications 330, 331

Hand side tools 33 1

Hand round-nosed tools for iron 331

Hand finishing tool 331

Hand Tools, for roughing out brass work 332

Various forms and applications of scrapers 332, 333

Clockmakers' hand tool for special or standard work 334

Centring needle-bars, etc 335

Round finish 335

Finishing small caps 335

Screw cutting with hand tools 335

Outside and inside chasers 335

Hobs and their uses. 335

The application of chasers, and errors that may arise from the position in which they are presented to

the work ■ 336

Errors commonly made in cutting up inside chasers. 337

V-tool for starting outside threads ... 337

Starting outside threads 338

Cutting taper threads 338

Wood turning hand tools 338

The gauge and how to use it 338

The chisel and its use 339

The skew chisel and how to use it 339

Wood turners' boring tools for lathe work 340

CHAPTER XIV.

MEASURING MACHINES, TOOLS AND DEVICES.

Standards of Measurements, in various countries 341

Use of, by sight and by the sense of feeling 341

Variations in standard gauges 341

The necessity for accurate standards 341

The Rogers Bond standard measuring machine 342

Details of construction of 343, 344

The principle of construction of 344

The methods of using 345

The Whitworth measuring machine 345

The Betts Machine Company's measuring machine 346

Professor Sweet's measuring machine 347

Measuring machine for sheet metal 348

Measuring machine, microscopic 348

Measurements finer than one fifty-thousandth of an inch. . . 348

Circle, division of the 348

Throughton's method of dividing the circle 348, 349

Ramsden's dividing engine 349

The construction of 350, 351

Pratt and Whitney's dividing device 352

Practical application of 353

Index wheel, method of originating, by R. Hoe & Co 353

Application of the index wheel (Hoe & Co's system). 353

Micrometer Caliper and its principle of construction 354, 355

Use close to a shoulder 354

Readings for ten-thousandths of an inch 354

Gauges. Standard plug and collar gauges 356

Methods of comparing standard plug and collar gauges. . . . 356 The effects of variations of temperature upon standard gauges 356

Plug and collar gauges for taper work 357

The Baldwin standards for taper bolts 359

Workmen's gauges for lathe work 359

Calipers, outside, the various forms of 360

Inside calipers 360

Calipers with locking devices 360

Spring calipers 360

The methods of holding and using 361, 36!>

Keyway calipers -'^3

CONTENTS.

xvii

PAGE

The advantages of calipers 363

Fitting. The four kinds of fit in machine work 363

Ihe inlluence of the diameter of the work in limiting the

application of standard gauges 363

The wear of tools and its inlluence upon the application

of the standard gauge system 364

The influence of the smoothness of the surface upon the

allowance to be made for drilling or hydraulic fits 365

Examples of allowance for hydraulic fits 365

Parallel holes and taper plugs for hydraulic fits 365

Practicable methods of testing the fit of axle brasses

forced in by hydraulic pressure 366

Shrinkage or contraction fits Tfib

Allowances for 3^6

Gauge for 3^7

The shrinkage system at the Royal Gun Factory at

Woolwich • • • y>l

Experiments by Thomas Wrightson upon the shrink- age of iron under repeated heatings and cool- ings 368-374

Shrinking work, to refit it 374. 375

CHAPTER XV.

MEASURING TOOLS.

End Measurements of large lathe work 376

Template gauges for 376

Trammels or Trains 377

Adjustable gauges for 377

Compasses — Dividers 377

Compass calipers 378

Key Seating rule 378

Surface Gauge 378

Pattern makers' pipe gauge 379

Squares. The try square 379

The T square 379

Various methods of testing squares 379' 38°

Bevel squares 380

Bevel Protractors 380

Hexagon Gauge • • • 381

Straight Edge and its applications 381, 382

Winding strips and their application 382

Surface Plate or planimeter 383

Templates for curves 384

Wire Gauges, notch 384

Standard gauges for wire, etc 384. 386

Gauge for music wire ■■ 386

Brown and Sharpe wire gauge 387

Birmingham wire gauge for rolled shell silver and gold ... 387

Sheet iron gauge, Russian 387

Galvanized iron 387

Belgian sheet zinc 3=7

American sheet zinc 387

Rifle Bore gauge • • • 38?

Strength of Wire, Kirkaldy's experiments 387. 388

CHAPTER XVI. SHAPING AND PLANING MACHINES.

Shaping machines 389

Work table moved to feed the work to the cut 389

Quick return shaper 389

Construction of swivel head 389

Slide 390

Vice chuck 39°

Feed motion 39°

Hand shaping machine 39^

Quick Return Motion, Whitworth's ...... 391

Vice Chucks, tlie principles of construction of plain, for planing

macliine 39^

Morton shaping machine 39^

Detailed description of 392

Reversing mechanism 393

Tool holders 394

Setting work in Shaper Vices 394

Holding taper work in 394

Holding cylindrical work in 394

Swiveling 395

Rapid motion • 39

For vice work 39

Centres for shaping machines 397

Traveling Head in shaping machine :••■•• 397

Planer Shapers or shaping machines, having a tappet motion lor

reversing the direction of motion. 398,399

Quick Return Motion shaping machines, link 399

The Whitworth • ''■°°

Comparisons of the link motion and Whitworth 401

Simple Crank, investigating the motion of 401

Hendry Machine Co.'s shaper 402

Friction clutch 40

VOL. I.— 3-

TM.y.

Uniform speed 402

Instantaneous reverse 402

Planing Machines, or phiner 402

Tlie various motions of 402, 403

The table driving gear 404

Planing machine with double heads 404

Rotary planing machine 405

CHAPTER XVII.

PLANING MACHINERY.

The Sellers planing machine

The belt shifting mechanism

The automatic feed motions 406,

Sellers planer, improved

Increase of speed .

Driving arrangement

Friction clutches

Abutment cones

Adjustment for wear

Countershafting

Feed mechanism

Shifting mechanism ■

Device for transmitting and arresting motion

Post planing machine

V^aried cutting strokes

Tool head on radius bar

Open side planer

Gray Co.'s Planers

Spiral gear driving mechanism

Vertical motion of the feed rack

Throwing out the feed of all heads at once.

Independent adjustment of feed device of side heads

Four cutting tools, starling the cut simultaneously

Fitchburg Machine Works' Planer

Extra head with automatic vertical feed

Sliding Head

Cross Bar

Slides of Planers, the various forms of construction of

Wear of the Slides of planer heads, various methods of taking

up the

Swivel heads

Tool Aprons

Swivel Tool-Holding devices for planers

Planer Heads, graduations of

Safety devices for

Feed motions for

V-guideways for

Flat guideways for

Oiling devices for

Planing Machine Tables

Slots and holes in planing machine tables

Forms of bolts for planer tables

Supplenientary tables for planer tables

Angle plates for planer tables

Chucking devices for planer tables

Planer Centres

Planer Chucks

For spiral grooved work

For curved work

Chucking machine beds on planer tables

For large planing machines

Chucking the halves of large pulleys on a planer

Gauges for planing V-guideways in machine beds

Planing guideways in machine beds

Gauge for planer tools

Planer Tools, the shapes of

For broad-finishing feeds

The clearance of

For slotted work

Planer Tool Holder, with swivel tool post . .■

Various applications of

406 406 407 407 407 408 409 409

409 409 409 410 410 411 411 411 411 411 411 411 411 411 412 412 412 412 412 412

412 412 413 4>3 413 413 414 414 415 415 415 416

417 417

418 418 418

419 419 420 420 422

423 421 422 424 424 424 424 424 425 425

CHAPTER XVIII. DRILLING MACHINES. Drilling Machines. General description of a power drilling

machine 426

Turret drilling machine 420

Tapping device 427

Reversing feature 427

Lever feed 429

Barnes' sensitive drill with friction disk 429

Regulating drill speed while running 429

Foot or hand feed drill 429

For deep drilling 429

Bickford Company's drills 429

Counterbalanced spindle 429

Boiler maker's drill y 429

With automatic and quick return feed motions 43°

XVlll

CONTENTS.

Improved, with simple belt and uniform motion two series of rates of automatic feed, and guide for bormg bar 430

Gould & Eberhardt's improved drill press 430

Feed speed by friction disk 4 J"

Tapping attachment 43

Lodge Davis Co.'s drilling machine 43'

Automatic stop motion

Radial , • ;

Cincinnati Radial Drill Co. s machines

Swiveling spindle head

Universal beam drill

Double column machine

Post drill

Portable drilling head

Drilling and Boring machine

Feed motion of

Combined Drilling Machine and lathe

Boring Machine, horizontal

Drilling Machine, three spindle

Four spi ndle

Quartering Machine •

Drilling and Turning Machine for boiler makers 435

Feed motions of 43

For boiler shells "I^

Three or four holes at once 43°

For car wheels ,\.,:,\. ;; ' V : I^a

Universal Boring, Drilling, and Milling Machine 43»

CHAPTER XIX. DRILLS AND CUTTERS FOR DRILLING MACHINES.

431 431 431 431 431 431 431 432 432 432 433 433 434 434 434

.440

Jigs or Fixtures for drilling machines

Limits of error in

Examples of, for simple work, as for links, etc

Considerations in designing

For drilling engine cylinders

For cutting out steam ports

Drills and Cutters for drilling machines

Table of sizes of twist drills, and their shanks

Flat drills for drilling machines

Errors in grinding flat drills

The tit-drill

The lip drill

Cotter or keyway drills ■

Drilling holes true to location with flat drills

Drilling hard metal

Table of sizes of tapping holes

Drill Shanks and sockets

Improved form of drill shank

Square shanked drills and their disadvantages. . .

Drill Chucks

Stocks and Cutters for drilling machines

Tube plate cutters

Stocks and Cutters. Adjustable stock and cutter

Facing tool with reamer pin

Counlerbores for drilling machines

Drill and counterbore for wood work.

Facing and countersink cutters

Device for drilling square holes ■

Device for drilling taper holes in a drilling machine

CHAPTER XX.

HAND-DRILLING AND BORING TOOLS, AND DEVICES.

439 439 440 440

441 441 442 442 442

443 443 443 446

444 444 445 445 446 446 446

447 448 448 449 449 449 449 450 451

The auger bit

Cook's auger bit • • ■ • •

Principles governing the shapes of the cutting edges of auger bits

Auger bit for boring end grain wood

The centre bit

The expanding bit

Drills. Drill for stone

The fiddle drill

The fiddle drill with feeding device

Drill with cord and spring motion

Drill stock with spiral grooves

Drill brace

Drill brace with ratchet motion

Universal joint for drill brace ■

Drill brace with multiplying gear and ratchet motion

Breast drill with double gear

Drilling levers for blacksmiths

Drill cranks

Ratchet brace

Flexible shaft for driving drills

Drilling device for lock work

Hand drilling machine

Slotting Machine

Sectional view of

Key-Seating machines

Slotting or paring machine

Relief motion for the tool

Adjustable nut

Friction brake

Tool holders • •

Tools 461

CHAPTER XXI.

The Brad-awl

Bits. The gimlet bit. .

The German bit.

The nail bit. . . .

The spoon bit. . .

The nose bit. . . .

453 453

453 453 454 454 454 455 455 455 455 455 456

456 456 456

457 457 457

458

459 459 459 460 460 460 460 461 461 461 , 462

452

452 452 452 452 453

THREAD-CUTTING MACHINERY AND BROACHING

PRESS.

Pipe Threading, die stock for, by hand 4^3

Die stock for, by power 4^3

Pipe threading machines, general construction of 463

Bolt Threading hand machine 4^4

With revolving head 4^5

Power threading machine 4^5

With automatic stop motion 4^6

Construction of the head 466

Construction of the chasers 4^6

Bolt threading machine with back gear 467

Single rapid bolt threading machine 4f>7

Double rapid bolt threading machine 4^7

Construction of the heads of the rapid machines 468

Bolt cutting machines 4^

Bolt threading machinery, the Acme 4*^8

Construction of the head of 468-470

Capacity of 470

Cutting Edges for taps, the number of 47'

Examples when three and when four cutting edges are used,

and the results upon bolts that are not round 471, 472

Demonstration that four cutting edges are correct for bar iron 472 Positions of Dies, or chasers, in the heads of bolt cutting machine 473

Dies, methods of bobbing to avoid undue friction. 473

The construction of, for bolt threading machines 473

Method of avoiding friction in thread cutting 474

Hob for threading 474

Cutting speeds for threading 474

Nut Tapping machine 475

Automatic socket for 475

Rotary 475

Three-spindle 475

Pipe Threading Machine 475-477

Tapping Machine for steam pipe fittings 478

Broaching Press 478

Principles of broaching 478

Examples in the construction of broaches 479

FULL-PAGE PLATES.

Volume I.

Frontispiece

Plate

I.

-

II.

I'-

III.

ll.

IV.

<<

V.

c<

VI.

((

VI.— A.

1<

VII.

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VII.— A.

((

VII.— B.

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VII.— C.

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VII.— D.

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VIII.

It

IX.

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XIII.

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XIV.

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XV.

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XVI.

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XVI.— A.

iX

XVII.

Facing

PORTRAIT OF THE AUTHOR Title Page

EXAMPLE OF LATHE OPERATION i

WORM AND WORM WHEEL 28

TEMPLATE CUTTING MACHINES FOR GEAR TEETH .... 34

GEAR CUTTING AND GENERATING MACHINE 35

FORMS OF SCREW THREADS 85

MEASURING AND GAUGING SCREW THREADS 93

MEASURING SCREW THREADS 95

SCREWS, STUDS, AND BOLTS n6

END ADJUSTMENT AND LOCKING DEVICES 117

THE F. E. REED LATHE 124

THE F. E. REED LATHE 125

THE F. E. REED LATHE 126

THE F. E. REED LATHE 127

HAZLETT AND LORD'S PATENT TAPER TURNING ATTACHMENT . 128

EXAMPLES IN LATHE CONSTRUCTION 148

CHUCKING LATHES 150

EXAMPLES OF LATHES 151

EXAMPLES OF SMALL LATHES 150

DETAILS OF SET OVER LATHE 151

LATHES FOR BRASS WORKERS 152

DETAILS OF KEY LATHE 153

EXAMPLES OF LATHES 152

EXAMPLES OF LATHES 153

EXAMPLES OF LATHES 152

FORMING MONITOR LATHE 153

XX

\TE

XVIII.

t«

XIX.

X

XX.

(<

XXI.

<(

XXII.

II

XXIII.

<i

XXIV.

(C

XXV.

(<

XXVI.

11

XXVII.

IC

XXVIII.

11

XXVIII.— A.

((

XXIX

l<

XXX.

«

XXXI.

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XXXI.— A.

<<

XXXII.

<l

XXXIII.

<l

XXXIV.

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XXXV.

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XXXVI.

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XXXVI.— A.

11

XXXVI.— B.

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XXXVI.— c.

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XXXVI.— D.

II

XXXVI.— E.

It

XXXVI.— F.

II

XXXVI.— G.

II

XXXVI.— H.

<i

XXXVI.— I.

II

XXXVI.— J.

It

XXXVI.— K.

11

XXXVII.

<t

XXXVIII.

FULL-PAGE PLATES.

IMPROVED FORMING LATHE TOOL REST .

MONITOR LATHES

FRICTION HEAD OF MONITOR LATHE . WIRE FEED OF SCREW MACHINE . EXAMPLES IN HEAVY LATHES

Facing

SURFACING LATHE WITH CU'ITER FACE PLATE 153

EXAMPLES OF CHUCKING LATHES iS4

DOUBLE AXLE LATHE iS5

EXAMPLES IN GAP LATHES i54

FRICTION CLUTCH MOTION FOR LATHES, ETC iS5

COMBINATION TURRET LATHE iS4

THE FLAT TURRET LATHE AND ITS PRODUCT 155

THE FLAT TURRET LATHE 154

WORK DONE BY THE FLAT TURRET LATHE 155

DETAILS AND WORK OF THE FLAT TURRET LATHE .... 156

TURRET AND HEAD DETAILS OF FLAT TURRET LATHE ... 157

DETAILS OF FLAT TURRET LATHE 156

DETAILS OF FLAT TURRET LATHE 157

END ELEVATION OF FLAT TURRET LATHE 156

FLAT TURRET LATHE TOOL HOLDERS 157

SCREW CUTTING MACHINE 158

J. E. REINECKER LATHE 159

DETAILS OF REINECKER LATHE 158

DETAILS OF REINECKER LATHE . . . 159

GISHOLT LATHE 160

GISHOLT LATHE 161

NEW HAVEN MANUFACTURING CO.'S LATHE 160

PATTERN MAKERS' LATHES 161

PATTERN MAKERS' LATHES 160

ENGINE LATHE 161

DETAILS IN LATHE CONSTRUCTION 171

DETAILS IN LATHE CONSTRUCTION . ,172

TOOL-HOLDING AND ADJUSTING APPLIANCES 173

WATCHMAKER'S LATHE 188

FULL-PAGE PLATES. xxi

facing

Plate XXXIX. DETAILS OF WATCHMAKER'S LATHE 189

" XL. SCREW SLOTTING AND HAND LATHES 190

" XLI. LATHE WITH DIFFERENTIAL SPEED MOTION 191

•' XLII. CUTTING OFF MACHINE AND GRINDING LATHE 192

'» XLIIL GRINDING LATHE 193

" XLIV. DETAILS OF GRINDING LATHE 192

XLV. DETAILS OF GRINDING LATHE 193

XLVL IMPROVED UNIVERSAL GRINDING MACHINE 194

XLVIL IMPROVED UNIVERSAL GRINDING MACHINE 195

XLVIII. IMPROVED UNIVERSAL GRINDING MACHINE 196

XLIX. IMPROVED UNIVERSAL GRINDING MACHINE 197

L, IMPROVED UNIVERSAL GRINDING MACHINE 196

'• LL IMPROVED UNIVERSAL GRINDING MACHINE 197

LIL SPECIAL CHUCK— CALENDER ROLL GRINDING LATHE .... 198

LIII. DETAILS OF CALENDER ROLL GRINDING LATHE 199

LIV. EXAMPLES OF SCREW MACHINES 200

'« LV. ROLL TURNING LATHE 215

LV.— A. HEAVY UPRIGHT BORING MACHINE 220

LV.— B. VERTICAL BORING AND TURNING MACHINE 221

*' LVI. LATHE CHUCKS 235

LVII, LATHE CHUCKS 241

«' LVIII. EXAMPLES IN ANGLE CHUCKING 252

LIX. CENTRING MACHINE 304

LX. STRAIGHTENING LATHE WORK 305

LXI. METHODS OF BALL TURNING 325

" LXIL HAND LATHE AND APPLIANCES 334

«« LXIII. CENTRING AND FINISHING DUPLICATE PARTS 335

LXIV. THE ROGERS-BOND UNIVERSAL COMPARATOR 342

LXV. MEASURING MACHINE 348

LXVI. MEASURING MACHINE 348

" LXVII. DIVIDING ENGINE AND MICROMETER . . . . . . . -354

" LXVIII. MICROMETER CALIPERS 355

'« LXIX. SHAPING MACHINE 39°

LXIX.— A. SHAPING MACHINES 39^

xxu

Plate LXIX.— B.

it

LXIX.— C.

n

LXIX.— D.

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LXX.

a

LXXI.

it

LXXII.

tt

LXXIII.

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LXXIV.

ti

LXXV.

ti

LXXVI.

ti

LXXVII.

it

LXXVIII.

tt

LXXIX.

tt

LXXX.

ti

LXXXI.

tt

LXXXII.

n

LXXXIII.

it

LXXXIII.— A.

it

LXXXIII.— B.

tt

LXXXIII.— C.

it

LXXXIII.— D.

a

LXXXIII.— E.

it

LXXXIV.

it

LXXXV.

it

LXXXVI.

tt

LXXXVII.

it

LXXXVIII.

tt

LXXXIX.

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XC.

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XCI.

it

XCI.— A.

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XCI.— B.

a

XCI.— C.

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XCII.

FULL-PAGE PLATES.

Facing

MORTON SHAPING MACHINE 392

MORTON SHAPING MACHINE 393

MORTON SHAPING MACHINE 394

SHAPING MACHINES AND TABLE-SWIVELLING DEVICES , . .398

SHAPING MACHINE 402

DETAILS OF SHAPING MACHINE 403

EXAMPLES OF PLANING MACHINES 404

PLANING MACHINERY 406

DETAILS OF PLANING MACHINE 407

DETAILS OF PLANING MACHINE 406

DETAILS OF PLANING MACHINE .407

DETAILS OF PLANING MACHINE 406

DETAILS OF PLANING MACHINE 407

DETAILS OF PLANING MACHINE 406

DETAILS OF PLANING MACHINE 407

DETAILS OF PLANING MACHINE 406

DETAILS OF PLANING MACHINE 407

DETAILS OF PLANING MACHINE 408

DETAILS OF PLANING MACHINE 409

DETAILS OF PLANING MACHINE 409

POST PLANING MACHINE 410

OPEN SIDE PLANER 411

PLANING MACHINE 410

DETAILS OF PLANING MACHINE 411

PLANING MACHINE 412

PLANING MACHINE 413

DETAILS OF PLANER HEADS 4,2

DETAILS OF PLANER HEADS 413

DETAILS OF PLANER HEADS 413

EXAMPLES IN PLANING WORK 422

TOOL HOLDERS FOR PLANING MACHINES 425

TURRET DRILLING MACHINE 426

TURRET DRILLING MACHINE 427

LIGHT DRILLING MACHINES 428

FULL-PAGE PLATES, xxiii

Facing

Plate XCIII. SMALL DRILLING MACHINES 429

XCIV. DRILLING MACHINES 428

XCV. DRILLING MACHINES 429

XCVL HEAVY DRILLING MACHINE 428

XCVII. DRILLING MACHINES WITH FRICTION SPEED 429

XCVIIL DRILLING MACHINE WITH STOP MOTION 43°

XCIX. DETAILS OF DRILLING MACHINE 431

C. RADIAL DRILLING MACHINE 43°

CL DETAILS OF HEAD OF RADIAL DRILLING MACHINE . . . .43'

CIL DOUBLE COLUMN RADIAL DRILL 43°

CIIL DOUBLE COLUMN RADIAL DRILL WITH TWO HEADS .... 431

CIV. POST AND PORTABLE DRILLING HEADS 43°

CIV.— A. DRILLING MACHINES 431

CV. EXAMPLES IN BORING MACHINERY 434

CVI. BOILER-DRILLING MACHINERY 436

CVII. SLOTTING MACHINE 46°

CVIII. DETAILS OF SLOTTING MACHINE 461

CIX. BOLT-CUTTING MACHINE 468

ex. BOLT-CUTTING MACHINE 469

CXI. NUT-TAPPING MACHINERY 475

VOL. I.

MODERN MACHINE SHOP PRACTICE.

FRONTISPIECE.

FINISHING CONE PULLEYS.

MODERN

MACHINE SHOP PRACTICE.

Chapter I.— THE TEETH OF GEAR-WHEELS.

AWHEEL that is provided with teeth to mesh, engage, or gear with similar teelh upon another wheel, so that the motion of one may be imparted to the other, is called, in general terms, a gear-wheel.

When the teeth are arranged to be parallel to the wheel-axis, as in Fig. i, the wheel is termed a spur-wheel. In the figure, A

Fig. I.

represents the axial line or axis of the wheel or of its shaft, to which the teeth are parallel while spaced equidistant around the rim, or face, as it is termed, of the wheel.

When the wheel has its teeth arranged at an angle to the shaft, as in Fig. 2, it is termed a bevel-wheel, or bevel gear; but when

this angle is one of 45°, as in Fig. 3, as it must be if the pair of wheels are of the same diameter, so as to make the revolutions of

Fig. 3.

their shafts equal, then the wheel is called a mitre-wheel. When the teeth are arranged upon the radial or side face of the wheel,

piMAJUL/LnJUAJ\AAA/U|lJ^^

Fig. 4.

as in Fig. 4, it is termed a crown-wheel. The smallest wheel of a pair, or of a train or set of gear-wheels, is termed the pinion ; VOL. 1.— 4.

and when the teeth are composed of rungs, as in Fig. 5, it is termed a lantern, trundle, or wallower ; and each cylindrical piece serving as a tooth is termed a stave, sj>indle, or round, and by some a leaf.

An annular or internal gear-wheel is one in which the faces of the teeth are within and. the flanks without, or outside the pitch- circle, as in Fig. 6 ; hence the pinion P operates within the wheel.

When the teeth of a wheel are inserted in mortises or slots

fig-5.

provided in the wheel-rim, it is termed a mortised-wheel, or a cogged-wheel, and the teeth are termed cogs.

When the teeth are arranged along a plane surface or straight line, as in Fig. 7, the toothed plane is termed a rack, and the wheel is termed a pinion.

A wheel that is driven by a revolving screw, or worm as it is termed, is called a worm-wheel, the arrangement of a worm and worm-wheel being shown in Fig. 8. The screw or worm is some- times also called an endless screw, because its action upon the wheel does not come to an end as it does when it is revolved in one continuous direction and actuates a nut. So also, since the worm is tangent to the wheel, the arrangement is sometimes called a wheel and tangent screw.

The diameter ot a gear-wheel is always taken at the pitch circle, unless otherwise specially stated as "diameter over all," "diameter of addendum," or "diameter at root of teeth," &c., &c.

Fig. 7.

When the teeth of wheels engage to the proper distance, which is when the pitch circles meet, they are raid tc be in gear, or geared together. It is obvious that if two wheels are to be geared together their teeth must be the same distance apart, or the same fitch, as it is called.

The designations of the various parts or surfaces of a tooth of a gear-wheel are represented in Fig. g, in which the surface A is the face of the tooth, while the dimension F is the width of face of the wheel, when its size is referred to. B is the flank or dis- tance from the pitch line to the root of the tooth, and C the

MODERN MACHINE SHOP PRACTICE.

point. H is the space, or the distance from the side of one tooth to the nearest side of the next tooth, the width of space being measured on the pitch circle PP. E is the depth of the tooth, and G its thickness, the latter also being measured on the pitch circle P P. When spoken of with reference to a tooth,

Fig. 8.

P P is called the pitch line, but when the whole wheel is referred to it becomes the pitch circle.

The points C and the surface H are true to the wheel axis.

The teeth are designated for measurement by the pitch ; the height or depth above and below pitch line ; and the thickness.

The pitch, however, may be measured in two ways, to wit, around the pitch circle A, in Fig. lo, which is called the arc or circular pitch, and across B, which is termed the chord pitch.

In proportion as the diameter of a wheel (having a given pitch) is increased, or as the pitch of the teeth is made finer (on a wheel

Fig. 9.

of a given diameter) the arc and chord pitches more nearly coincide in length. In the practical operations of marking out the teeth, however, the arc pitch is not necessarily referred to, for if the diameter of the pitch circle be made correct for the required number of teeth having the necessary arc pitch, and the wheel be accurately divided off into the requisite number of

divisions with compasses set to the chord pitch, or by means of an index plate, then the arc pitch must necessarily be correct, although not referred to, save in determining the diameter of the wheel at the pitch circle.

The difference between the width of a space and the thickness of the tooth (both being measured on the pitch circle or pitch line) is termed the clearance or side clearance, which is necessary to prevent the teeth of one wheel fiom becoming locked in the spaces of the other. The amount of clearance is, when the teeth are cut to shape in a machine, made just sufficient to prevent contact on one side of the teeth when they are in proper gear (the pitch circles meeting in the line of centres). But when the teeth are cast upon the wheel the clearance is increased to allow for the slight inequalities of tooth shape that is incidental to casting them. The amount of clearance given is varied to suit the method employed to mould the wheels, as will be ex- plained hereafter.

The line of centres is an imaginary line from the centre or axis of one wheel to the axis of the other when the two are in gear ; hence each tooth is most deeply engaged, in the space of the other wheel, when it is on the line of centres.

There are three methods of designating the sizes of gear-wheels. First, by their diameters at the pitch circle or pitch diameter and the number of teeth they contain ; second, by the number of teeth in the wheel and the pitch of the teeth ; and third, by a system known as diametral pitch.

The first is objectionable because it involves a calculation to find the pitch of the teeth ; furthermore, if this calculation be

Fig. 10.

made by dividing the circumference of the pitch circle by the number of teeth in the wheel, the result gives the arc pitch, which cannot be measured correctly by a lineal measuring rule, especially if the wheel be a small one having but few teeth, or of coarse pitch, as, in that case, the arc pitch very sensibly differs from the chord pitch, and a second calculation may become necessary to find the chord pitch from the arc pitch.

The second method (the number and pitch of the teeth) possesses the disadvantage that it is necessary to state whether the pitch is the arc or the chord pitch.

If the arc pitch is given it is difficult to measure as before, while if the chord pitch is given it possesses the disadvantage that the diameters of the wheels will not be exactly proportional to the numbers of teeth in the respective wheels. For instance, a wheel with 20 teeth of 2 inch chord pitch is not exactly half the diameter of one of 40 teeth and 2 inch chord pitch.

To find the chord pitch of a wheel take 180 (= half the degrees in a circle) and divide it by the number of teeth in the wheel. In a table of natural sines find the sine for the number so found, which multiply by 2, and then by the radius of the wheel in inches.

Example. — What is the chord pitch of a wheel having 12 teeth and a diameter (at pitch circle) of 8 inches ? Here 180 -^ 12 = 15 ;

THE TEETH OF GEAR-WHEELS.

(sine of 15 is -25881). Then -25881 x 2 = -51762 x 4 (= radius of wheel) = 2-07048 inches = chord pitch.

TABLE OF NATURAL SINES.

Degrees.

Sine.

Degrees.

Sine.

DeKrees.

Sine.

I

•o'74S

16

•27563

31

•51503

2

•03489

17

•29237

32

•52991

3

-05233

18

•30901

33

•54463

4

-06975

'9

-32556

34

■55919

5

•08715

20

•34202

35

•57357

6

•10452

21

■358.^6

36

•58778

7

•I2I86

22

-37460

37

•6oi8t

8

-13917

23

•3907?

38

•61566

9

•15643

24

•40673

39

•62932

10

-17364

25

•42261

40

•64278

It

-19080

26

•43837

41

•65605

12

-20791

27

•45399

42

•66913

13

•2J49S

28

•46947

43

•68199

'4

•24192

29

•48480

44

-69465

«S

•25881

30

•50000

45

•70710

The principle upon which diametral pitch is based is as follows : —

The diameter of the wheel at the pitch circle is supposed to be divided into as many equal parts or divisions as there are teeth in the wheel, and the length of one of these parts is the diametral pitch. The relationship which the diametral bears to the arc pitch is the same as the diameter to the circumference, hence a diametral pitch which measures i inch will accord with an arc pitch of 3^i4i6 ; and it becomes evident that, for all arc pitches of less than 3-1416 inches, the corresponding diametral pitch must be expressed in fractions of an inch, as \, \, \ and so on, increasing the denominator until the fraction becomes so small that an arc with which it accords is too fine to be of practical service. The numerators of these fractions being 1, in each case, they are in practice discarded, the denominators only being used, so that, instead of saying diametral pitches of \, \, or \, we say diametral pitches of 2, 3, or 4, meaning that there are 2, 3, or 4 teeth on the wheel for every inch in the diameter of the pitch circle.

Suppose now we are given a diametral pitch of 2. To obtain the corresponding arc pitch we divide 3-1416 (the relation of the circumference to the diameter) by 2 (the diametral pitch), and 3"i4i6 4- 2 = 1-57 = the arc pitch in inches and decimal parts of an inch. The reason of this is plain, because, an arc pitch of 3'i4i6 inches being represented by a diametral pitch of i, a diametral pitch o\\ (or 2 as it is called) will be one half of 3-1416. The advantage of discarding the numerator is, then, that we avoid the use of fractions and are readily enabled to find any arc pitch from a given diametral pitch.

Examples. — Given a 5 diametral pitch ; what is the arc pitch ? First (using the full fraction J) we have \ x 3-1416 = -628 = the arc pitch. Second (discarding the numerator), we have 3*i4i6 ■- 5 = ^628 = arc pitch. If we are given an arc pitch to find a corresponding diametral pitch we again simply divide 3- 14 16 by the given arc pitch.

Example. — What is the diametral pitch of a wheel whose arc pitch is \\ inches ? Here 3-1416 ^ 1-5 = 2-og = diametral pitch. The reason of this is also plain, for since the arc pitch is to the diametral pitch as the circumference is to the diameter we have : as 3'i4i6 is to i, so is f5 to the required diametral pitch ; then 3-1416 X I -=- 1-5 =2-09 = the required diametral pitch.

To find the number of teeth contained in a wheel when the diameter and diametral pitch is given, multiply the diameter in inches by the diametral pitch. The product is the answer. Thus, how many teeth in a wheel 36 inches diameter and of 3 diametral pitch ? Here 36 x 3 = 108 = the number of teeth sought. Or, per contra, a wheel of 36 inches diameter has 108 teeth. What is the diametral pitch ? 108 -^ 36 = 3 = the diametral pitch. Thus it will be seen that, for determining the relative sizes of wheels, this system is excellent from its simplicity. It also possesses the advantage that, by adding two parts of the diametral pitch to the pitch diameter, the outside diameter of the wheel or the diameter

of the addendum is obtained. For instance, a wheel containing 30 teeth of 10 pitch would be 3 inches diameter on the pitch circle and 3]^ outside or total diameter.

Again, a wheel having 40 teeth of 8 diametral pitch would have a pitch circle diameter of 5 inches, because 40 -j- 8 = 5, and its full diameter would be 5J inches, because the diametral pitch is J, and this multiplied by 2 gives J, which added to the pitch circle diameter of 5 inches makes 5^ inches, which is therefore the* diameter of the addendum, or, in other words, the full diameter of the wheel.

Suppose now that a pair of wheels require to have pitch circles of 5 and 8 inches diameter respectively, and that the arc pitch requires to be, say, as near as may be ^ inch ; to find a suitable pitch and the number of teeth by the diametral pitch system we proceed as follows :

In the following table are given various arc pitches, and the corresponding diametral pitch.

Diametral Pitch.

Arc Pitch.

Arc Pitch.

Diametral Pitch.

Inch.

2

••57

1-75

1-79

2-25

1-39

'•5

2^09

25

1-25

'•4375

2^l8

2-75

PI4

'•375

228

3

1^04

'•3125

2^39

3^5

•890

1-25

2-51

4

•785

PI875

2-65

5

•628

II2S

2-79

6

•523

I^0625

2-96

7

•448

I 0000

3^i4

8

•392

09375

3^35

9

•350

0-875

3'§?

10

•3'4

©•8125

3^86

II

•280

o^75

4-19

• 12

•261

0-6875

4^57

14

•224

0625

5^03

16

•196

0-5625

5^58

18

•'74

0^5

6-28

20

•157

0-437S

7^.8

22

•'43

0^375

8-38

w

-130

0-3125

10-00

•120

0-25

12-56

From this table we find that the nearest diametral pitch that will correspond to an arc pitch of ^g inch is a diametral pitch of 8, which equals an arc pitch of -392, hence we multiply the pitch circles (5 and 8,) by 8, and obtain 40 and 64 as the number of teeth, the arc pitch being -392 of an inch. To find the number of teeth and pitch by the arc pitch and circumference of the pitch circle, we should require tofind the circumference of the pitch circle, and divide this by the nearest arc pitch that would divide the cir- cumference without leaving a remainder, which would entail more calculating than by the diametral pitch system.

The designation of pitch by the diametral pitch system is, how- ever, not applied in practice to coarse pitches, nor to gears in which the teeth are cast upon the wheels, pattern makers generally preferring to make the pitch to some measurement that accords with the divisions of the ordinary measuring rule.

Of two gear-wheels that which impels the other is termed the driver, and that which receives motion from the other is termed the driven wheel or follower ; hence in a single pair of wheels in gear together, one is the driver and the other the driven wheel or follower. But if there are three wheels in gear together, the middle one will be the follower when spoken of with reference to the first or prime mover, and the driver, when mentioned with reference to the third wheel, which will be a follower. A series of more than two wheels in gear together is termed a train of wheels or of gearing. When the wheels in a train are in gear continuously, so that each wheel, save the first and last, both receives and imparts motion, it is a simple train, the first wheel being the driver, and the last the follower, the others being termed intermediate wheels. Each of these intermediates is a follower with reference to the wheel that drives it, and a driver to the one that it drives. But the velocity of all the wheels in the train is the same in fact per second (or in a given space of time), although the revolutions in

MODERN MACHINE SHOP PRACTICE.

that space of time may vary ; hence a simple train of wheels transmits motion without influencing its velocity. To alter the velocity (which is always taken at a point on the pitch circle) the gearing must be compounded, as in Fig. ii, in which A, B, C, E are four wheels in gear, B and C being com- pounded, that is, so held together on the shaft D that both make an equal number of revolutions in a given time. Hence the velocity of C will be less than that of B in proportion as the diameter, circumference, radius, or number of teeth in C, varies from the diameter, radius, circumference, or number of teeth (all the wheels being supposed to have teeth of the same pitch) in B, although the rotations of B and C are equal. It is most convenient, and therefore usual, to take the number of teeth, but if the teeth on C (and therefore those on E also) were of different pitch from those on B, the radius or diameters of the wheels must be taken instead of the pitch, when the velocities of the various wheels are to be computed. It is obvious that the compounded pair of wheels will diminish the velocity when the driver of the compounded pair (as C in the figure) is of less radius than the follower B, and con- versely that the velocity will be increased when the driver is of greater radius than the follower of the compound pair.

The diameter of the addendum or outer circle of a wheel has no influence upon the velocity of the wheel. Suppose, for example, that we have a pair of wheels of 3 inch arc or circular

Fig. II

pitch, and containing 20 teeth, the driver of the two making one revolution per minute. Suppose the driven wheel to have fast upon its shaft a pulley whose diameter is one foot, and that a weight is suspended from a line or cord wound around this pulley, then (not taking the thickness of the line into account) each rotation of the driven wheel would raise the weight 31416 feet (that being the circumference of the pulley). Now suppose that the addendum circle of either of the wheels were cut otf down to the pitch circle, and that they were again set in motion, then each rotation of the driven wheel would still raise the weight 3 '14 1 6 feet as before.

It is obvious, however, that the addendum circle must be suffi- ciently larger than the pitch circle to enable at least one pair of teeth to be in continuous contact ; that is to say, it is obvious that contact between any two teeth must not cease before contact between the next two has taken place, for otherwise the motion would not be conveyed continuously. The diameter of the pitch circle cannot be obtained from that of the addendum circle unless the pitch of the teeth and the proportion of the pitch allowed for the addendum be known. But if these be known the diameter of the pitch circle may be obtained by subtracting from that of the addendum circle twice the amount allowed for the addendum of the tooth.

Example. — A wheel has 19 teeth of 3 inch arc pitch ; the ad- denduni of the tooth or teeth equals ^ of the pitch, and its

addendum circle measures i9'943 inches; what is the diameter of the pitch circle ? Here the addendum on each side of the wheel equals (j^j of 3 inches) = -9 inches, hence the -9 must be multiplied by 2 for the two sides of the wheel, thus, -9x2= i'8. Then, diameter of addendum circle 19943 inches less i'8 inches = i8'i43 inches, which is the diameter of the pitch circle.

Proof. — Number of teeth = 19, arc pitch 3, hence 19 x 3 = 57 inches, which, divided by 3'i4i6 (the proportion of the circum- ference to the diameter) = 18143 inches.

If the distance between the centres of a pair of wheels that are in gear be divided into two parts whose lengths are in the same proportion one to the other as are the numbers of teeth in the wheels, then these two parts will represent the radius of the pitch circles of the respective wheels. Thus, suppose one wheel to contain 100 and the other 50 teeth, and that the distance between their centres is 18 inches, then the pitch radius or pitch diameter of one will be twice that of the other, because one contains twice as many teeth as the other. In this case the radius of pitch circle for the large wheel will be 12 inches, and that for the small one 6 inches, because 12 added to 6 makes 18, which is the distance between the wheel centres, and 12 is in the same pro- portion to 6 that 100 is to 50.

A simple rule whereby to find the radius of the pitch circles of a pair of wheels is as follows : —

Rule. — Divide number of teeth in the large wheel by the number in the small one, and to the sum so obtained add i. Take this amount and divide it into the distance between the centres of the wheels, and the result will be the radius of the smallest wheel. To obtain the radius of the largest wheel subtract the radius of the smallest wheel from the distance between the wheel centres.

Example. — Of a pair of wheels, one has 100 and ttie other 50 teeth, the distance between their centres is 18 inches; what is the pitch radius of each wheel ?

Here 100 -H 50 = 2, and 2 -t- i = 3. Then 18 -e- 3 = 6, hence the pitch radius of the small wheel is 6 inches. Then 18 — 6 = 12 = pitch radius of large wheel.

Example 2. — Of a pair of wheels one has 40 and the other 90 teeth. The distance between the wheel centres is 32J inches ; what are the radii of tlie respective pitch circles ? 90 -^ 40 = 2-25 and 2-25 + I = 3*25. Then 32-5 -;- 3-25 = 10 = pitch radius of small wheel, and 32-5 — 10 = 22-5, which is the pitch radius of the large wheel.

To prove this we may show that the pitch radii of the two wheels are in the same proportion as their numbers of teeth, thus : —

Proof. — Radiusof small wheel =10 x 4 = 40 radius of large wheel = 22'5 x 4 = go'O

Suppose now that a pair of wheels are constructed, having respectively 50 and 100 teeth, and that the radii of their true pitch circles are 12 and 6 respectively, but that from wear in their journals or journal bearings this 18 inches (12 -H 6 = 18) between centres (or line of centres, as it is termed) has become 18J inches. Then the acting effective or operative radii of the pitch circles will bear the same proportion to the i8§ as the numbers of teeth in the respective wheels, and will be I2'25 for the large, and 6'i25 for the small wheel, instead of 12 and 6, as would be the case were the wheels 18 inches apart. Working this out under the rule given we have 100 -;- 50 = 2, and 2 + i — 3, Then 1 8*375 -4-3 = 6" 125 = pitch radius of small wheel, and i8*37S — 6"I2S = i2'25 = pitch radius of the large wheel.

The true pitch line of a tooth is the line or point where the face curve joins the flank curve, and it is essential to the transmission of uniform motion that the pitch circles of epicycloidal wheels exactly coincide on the line of centres, but if they do not coincide (as by not meeting or by overlapping each other), then a false pitch circle becomes operative instead of the true one, and the motion of the driven wheel will be unequal at different instants of time, although the revolutions of the wheels will of course be in proportion to the respective numbers of their teeth.

If the pitch circle is not marked on a single wheel and its arc pitch is not known, it is practically a difficult matter to obtain either the arc pitch or diameter of the pitch circle. If the wheel

THE TEETH OF GEAR WHEELS.

is a new one, and its teeth are of the proper curves, the pitch circle will be shown by the junction of the curves forming the faces with those forming the flanks of the teeth, because that is the location of the pitch circle ; but in worn wheels, where from play or looseness between the journals and their bearings, this point of junction becomes rounded, it cannot be defined with certainty.

In wheels of large diameter the arc pitch so nearly coincides with the chord pitch, that if the pitch circle is not marked on the wheel and the arc pitch is not known, the chord pitch is in practice often assumed to represent the arc pitch, and the diameter of the wheel is obtained by multiplying the number of teeth by the chord pitch. This induces no error in wheels of coarse pitches, because those pitches advance by J or J inch at a step, and a pitch measuring about, say, i^ inch chord pitch, would be known to be \\ arc pitch, because the difference between the arc and chord pitch would be too minute to cause sensible error. Thus the next coarsest pitch to i inch would be ij, or more often \\ inch, and the difference between the arc and chord pitch of the smallest wheel would not amount to anything near \ inch, hence there would be no liability to mistake a pitch of \\ for I inch or vice versa. The diameter of wheel that will be large enough to transmit continuous motion is diminished in proportion as the pitch is decreased ; in proportion, also, as the wheel dia- meter is reduced, the difference between the arc and chord pitch increases, and further the steps by which fine pitches advance are more minute (as \, -^i, A" &c.). From these facts there is much more liability to err in estimating the arc from the measured chord pitch in fine pitches, hence the employment of diametral pitch for small wheels of fine pitches is on this account also very advantageous. In marking out a wheel the chord pitch will be correct if the pitch circle be of correct diameter and be divided off into as many points of equal division (with compasses) as there are to be teeth in the wheel. We may then mark from these points others giving the thickness of the teeth, which will make the spaces also correct. But when the wheel teeth are to be cut in a machine out of solid metal, the mechanism of the machine enables the marking out to be dispensed with, and all that is necessary is to turn the wheel to the required addendum diameter, and mark the pitch circle. The following are rules for the purposes they indicate.

The circumference of a circle is obtained by multiplying its diameter by j'i4i6, and the diameter may be obtained by dividing the circumference by 3'i4i6.

The circumference of the pitch circle divided by the arc pitch gives the number of teeth in the wheel.

The arc pitch multiplied by the number of teeth in the wheel gives the circumference of the pitch circle.

Gear-wheels are simply rotating levers transmitting the power they receive, less the amount of friction necessary to rotate them under the given conditions. All that is accomplished by a simple train of gearing is, as has been said, to vary the number of revolu- tions, the speed or velocity measured in feet moved through per minute remaining the same for every wheel in the train. But in a compound train of gears the speed in feet per minute, as well as the revolutions, may be varied by means of the com- pounded pairs of wheels. In either a simple or a compound train of gearing the power remains the same in amount for every wheel in the train, because what is in a compound train lost in velocity is gained in force, or what is gained in velocity is lost in force, the word force being used to convey the idea of strain, pressure, or pull.

In Fig. 12, let A, B, and C represent the pitch circles of three gears of which A and B are in gear, while c is compounded with B ; let E be the shaft of A, and G that for B and C. Let A be 60 inches, b = 30 inches, and 0 = 40 inches in diameter. Now suppose that shaft E suspends from its perimeter a weight of 50 lbs., the shaft being 4 inches in diameter. Then this weight will be at a leverage of 2 inches from the centre of E and the 50 must be multiplied by 2, = 100 lbs. at one inch from centre of E. At the perimeter of A this 100 will become one-thirtieth of one hundred, because from the centre to the perimeter of A is 30. One-thirtieth of 100 is 3 Aft) lbs., which will be the force exerted by A on the

perimeter of H. From the perimeter of 11 to its centre (that is, its radius) is 15 inches, hence the },^i^, lbs. at its perimeter will be- come fifteen times as much at one inch from the centre G of B, and 3^^^ x 15=49,2^,, lbs. From the centre G to the perimeter of C being 20 inches, the 491*,^ lbs. will be only one-twentieth of that amount at the perimeter of C, hence 49V'o*i-J-20=2|iu''i lbs., which is the amount of force at the perimeter of C.

Here we have treated the wheels as simple levers, dividing the weight by the length of the levers in all cases where it is trans- mitted from the shaft to the perimeter, and multiplying it by the length of the lever when it is transmitted from the perimeter of the wheel to the centre of the shaft. The precise same result will be reached if we take the diameter of the wheels or the number of the teeth, providing the pitch of the teeth on all the wheels is alike.

Suppose, for example, that A has 60 teeth, B has 30 teeth, and C has 40 teeth, all being of the same pitch. Suppose the 50 lb. weight be suspended as before, and that the circumference of the shaft be equal to that of a pinion having 4 teeth of the same pitch as the wheels. Then the 50 multiplied by the 4 becomes 200, which divided by 60 (the number of teeth on A) becomes 3r''f)%, which multiplied by 30 (the number of teeth on b) becomes 99^(5,

Fig. 12.

which divided by 40 (the number of teeth on C) becomes 2^ lbs. as before.

It may now be explained why the shaft was taken as equal to a pinion having 4 teeth. Its diameter was taken as 4 inches and the wheel diameter was taken as being 60 inches, and it was supposed to contain 60 teeth, hence there was 1 tooth to each inch of diameter, and the 4 inches diameter of shaft was there- fore equal to a pinion having 4 teeth. From this we may perceive the philosophy of the rule that to obtain the revolutions of wheels we multiply the given revolutions by the teeth in the driving wheels and divide by the teeth in the driven wheels.

Suppose that A (Fig. 13) makes i revolution per minute, how miny will c make, A having 60 teeth, B 30 teeth, and C 40 teeth .•> In this case we have but one drivingwheel A, and one driven wheel b, the driver having 60 teeth, the driven 30, hence 60 -h 30 = 2, equals revolutions of B and also of c, the two latter being on the same shaft.

It will be observed then that the revolutions are in the same proportion as the numbers of the teeth or the radii of the wheels, or what is the same thing, in the same proportion as their diameters. The number of teeth, however, is usually taken as being easier obtained than the diameter of the pitch circles, and easier to calculate, because the teeth will be represented by a whole number, whereas the diameter, radius, or circumference, will generally contain fractions.

MODERN MACHINE SHOP PRACTICE.

Suppose that the 4 wheels in Fig. 14 have the respective numbers of teeth marlced beside them, and that the upper one having 40 teeth makes 60 revolutions per minute, then we may obtain the revolutions of the others as follows : —

devolu- tions.

Teeth in first driver.

I'eeth in Teeth in first driven, second driver.

Teeth in second driven.

60

X 40 -T

60 X 20 4

120 =

6,^

and a remainder of the reciprocating decimals. We may now prove this by reversing the question, thus. Suppose the 120 wheel to make 6^5 revolutions per minute, how many will the 40 wheel make ?

Revolu- tions.

6-66

Teeth in first driver.

120

Teeth tn first driven.

Teeth in Teeth in

second driver, second driven.

60

40 =

?1%5

revolutions of the 40 wheel, the discrepancy of j J^ being due to the 6-66 leaving a remainder and not therefore being absolutely correct.

That the amount of power transmitted by gearing, whether compounded or not, is equal throughout every wheel in the train, may be shown as follows : —

Referring again to Fig. 10, it has been shown that with a 50 lb.

Fig- 13-

weight suspended from a 4 inch shaft E, there would be y>f§^ lbs. at the perimeter of A. Now suppose a rotation be made, then the 50 lb. weight would fall a distance equal to the circum- ference of the shaft, which is (3.1416 x 4 = I2i^) 12,'^ inches. Now the circumference of the wheel is (60 dia. x 3'i4i6 = i88j^ cir.) i88f\^ inches, which is the distance through which the j/d^ lbs. would move during one rotation of A. Now yT^T, lbs. moving through i88'49 inches represents the same amount of power as does 50 lbs. moving through a distance of 12-56 inches, as may be found by converting the two into inch lbs. (that is to say, into the number of inches moved by i lb.), bearing in mind that there will be a slight discrepancy due to the fact that the fractions -t^i in the one case, and '56 in the other are not quite correct. Thus :

= 627-67 inch lbs., and = 628

i88'49 inches x 3-33 lbs. 12-56 „ X 50 ,,

Taking the next wheels in Fig. 12, it has been shown that the 3'33 lbs. delivered from A to the perimeter of B, becomes 2-49 lbs. at the perimeter of C, and it has also been shown that C makes two revolutions to one of A, and its diameter being 40 inches, the distance this 2-49 lbs. will move through in one revolution of A

will therefore be equal to twice its circumference, which is (40 dia. X 3-1416 = 125-666 cir., and 125-666 x 2 = 251-332) 251-332 inches. Now 2-49 lbs. moving through 251-332 gives when brought to inch lbs. 627.67 inch lbs., thus 251-332 x 2-49 = 627-67. Hence the amount of power remains constant, but is altered in form, merely being converted from a heavy weight moving a short distance, into a lighter one moving a distance exactly as much greater as the weight or force is lessened or lighter.

Gear-wheels therefore form a convenient method of either simply transmitting motion or power, as when the wheels are

Fig. 14.

all of equal diameter, or of transmitting it and simultaneously varying its velocity of motion, as when the wheels are compounded either to reduce or increase the speed or velocity in feet per second of the prime mover or first driver of the train or pair, as the case may be.

In considering the action of gear-teeth, however, it sometimes is more convenient to denote their motion by the number of degrees of angle they move through during a certain portion of a revolution, and to refer to their relative velocities in terms of the ratio or proportion existing between their velocities. The first of

Fig. IS-

these is termed the angular velocity, or the number of degrees of angle the wheel moves through during a given period, while the second is termed the velocity ratio of the pair of wheels. Let it be supposed that two wheels of equal diameter have contact at their perimeters so that one drives the other by friction without any slip, then the velocity of a point on the perimeter of one will equal that of a point on the other. Thus in Fig. 15 let A and B represent the pitch circles of two wheels, and C an imaginary line joining the axes of the two wheels and termed the line of centres. Now the point of contact of the two wheels will be on the line of

THE TEETH OF GEAR-WHEELS.

centres as at D, and if a point or dot be marked at d and motion be imparted from A to B, then when each wheel has made a quarter revolution the dot on A will have arrived at E while that on n will have arrived at F. As each wheel has moved through one quarter revolution, it has moved through 90° of angle, because in the whole circle there is 360°, one quarter of which is 90°, hence instead of saying that the wheels have each moved through one quarter of a revolution we may say they have moved through an angle of 90", or, in other words, their angular velocity has, during this period, been 90". And as both wheels have moved through an equal number of degrees of angle their velocity ratio or pro- portion of velocity has been equal.

Obviously then the angular velocity of a wheel represents a portion of a revolution irrespective of the diameter of the wheel, while the velocity ratio represents the diameter of one in propor- tion to that of the other irrespective of the actual diameter of either of them.

Now suppose that in Fig. 16 A is a wheel of twice the diameter of B ; that the two are free to revolve about their fixed centres, but that there is frictional contact between their perimeters at the line of centres sufficient to cause the motion of one to be imparted to the other without slip or lost motion, and that a point be marked on both wheels at the point of contact D. Now let motion be communicated to A until the mark that was made at D has moved one-eighth of a revolution and it will have moved through an eighth of a circle, or 45°. But during this motion the mark on B will have moved a quarter of a revolution, or through an angle of

Fig. 16.

90° (which is one quarter of the 360° that there are in the whole circle). The angular velocities of the two are, therefore, in the same ratio as their diameters, or two to one, and the velocity ratio is also two to one. The angular velocity of each is therefore the number of degrees of angle that it moves through in a certain portion of a revolution, or during the period that the other wheel of the pair makes a certain portion of a revolution, while the velocity ratio is the proportion existing between the velocity of one wheel and that of the other ; hence if the diameter of one only of the wheels be changed, its angular velocity will be changed and the velocity ratio of the pair will be changed. The velocity ratio may be obtained by dividing either the radius, pitch, diameter, or number of teeth of one wheel into that of the other.

Conversely, if a given velocity ratio is to be obtained, the radius, diameter, or number of teeth of the driver must bear the same relation to the radius, diameter, or number of teeth of the follower, as the velocity of the follower is desired to bear to that of the driver.

If a pair of wheels have an equal number of teeth, the same pairs of teeth will come into action at every revolution ; but if of two wheels one is twice as large as the other, each tooth on the small wheel will come into action twice during each revolution of the large one, and will work during each successive revolution with the same two teeth on the large wheel ; and an application of the principle of the hunting tooth is sometimes employed in clocks to prevent the overwinding of their springs, the device being shown in Fig. 17, which is from " Willis' Principles of Mechanism."

For this purpose the winding arbor C has a pinion A of 19 teeth

fixed to it close to the front plate. A pinion B of 18 teeth is mounted on a stud so as to be in gear with the former. A radial plate c D is fixed to the face of the upper wheel A, and a similar plate F E to the lower wheel B. These plates terminate outward in semicircular noses D, E, so proportioned as to cause their extremities to abut against each other, as shown in the figure, when the motion given to the upper arbor by the winding has brought them into the position of contact. The clock being now wound up, the winding arbor and wheel A will begin to turn in the opposite direction. When its first complete rotation is effected the wheel B will have gained one tooth distance from the line of centres, so as to place the stop D in advance of E and thus avoid a contact with E, which would stop the motion. As each turn of the upper wheel increases the distance of the stops, it follows from the principle of the hunting cog, that after eighteen revolutions of A and nineteen of B the stops will come together again and the clock be prevented from running down too far. The winding key being applied, the upper wheel A will be rotated in the opposite direction, and the winding repeated as above.

Thus the teeth on one wheel will wear to imbed one upon the other. On the other hand the teeth of the two wheels may be of such numbers that those on one wheel will not fall into gear ■ with the same teeth on the other except at intervals, and thus an inequality on any one tooth is subjected to correction by all the teeth in the other wheel. When a tooth is added to the

Fig. 17.

number of teeth on a wheel to effect this purpose it is termed a hunting cog, or hunting tooth, because if one wheel have a tooth less, then any two teeth which meet in the first revolution are distant, one tooth in the second, two teeth in the third, three in the fourth, and so on. The odd tooth is on this account termed a hunting tooth.

It is obvious then that the shape or form to be given to the teeth must, to obtain correct results, be such that the motion of the driver will be communicated to the follower with the velocity due to the relative diameters of the wheels at the pitch circles, and since the teeth move in the arc of a circle it is also obvious that the sides of the teeth, which are the only parts that come into contact, must be of same curve. The nature of this curve must be such that the teeth shall possess the strength necessary to transmit the required amount of power, shall possess ample wearing surface, shall be as easily produced as possible for all the varying conditions, shall give as many teeth in constant contact as possible, and shall, as far as possible, exert a pressure in a direction to rotate the wheels without inducing under wear upon the journals of the shafts upon which the wheels rotate. In cases, however, in which some of these requirements must be partly sacrificed to increase the value of the others, or of some of the others, to suit the special circumstances under which the wheels are to operate, the selection is left to the judgment of the designer, and the considerations which should influence his determinations will appear hereafter.

8

MODERN MACHINE SHOP PRACTICE.

Modern practice has accepted the curve known in general terms as the cycloid, as that best filling all the requirements of wheel teeth, and this curve is employed to produce two distinct forms of teeth, epicycloidal and involute. In epicycloidal teeth the curve forming the face of the tooth is designated an epicy- cloid, and that forming the flank an hypocycloid. An epicycloid may be traced or generated, as it is termed, by a point in the circumference of a circle that rolls without slip upon the circum- ference of anotlier circle. Thus, in Fig. 18, A and B represent two wooden wheels, A having a pencil at P, to serve as a tracing or marking point. Now, if the wheels are laid upon a sheet of

Fig. 18.

paper and while holding B in a fixed position, roll A in contact with B and let the tracing point touch the paper, the point p will trace the curve C C. Suppose now the diameter of the base circle B to be infinitely large, a portion of its circumference may be represented by a straight line, and the curve traced by a point on the circumference of the generating circle as it rolls along the base line B is termed a cycloid. Thus, in Fig. 19, b is the base line, A the rolling wheel or generating circle, and C C the cycloidal curve traced or marked by the point D when A is rolled along B. If now we suppose the base line B to represent the pitch line of a rack, it will be obvious that part of the cycloid at

Fig. 19.

one end is suitable for the face on one side of the tooth, and a part at the other end is suitable for the face of the other side of the tooth.

A hypocycloid is a curve traced or generated by a point on the circumference of a circle rollirvg within and in contact (without slip) with another circle. Thus, in Fig. 20, A represents a wheel in contact with the internal circumference of B, and a point on Us circumference will trace the two curves, C C, both curves startmg from the same point, the upper having been traced by rolhng the generating circle or wheel A in one direction and the lower curve by rolling it in the opposite direction.

To demonstrate that by the epicycloidal and hypocycloidal curves, forming the faces and flanks of what are known as epicy- cloidal teeth, motion may be communicated from one wheel to another with as much uniformity as by frictional contact of their circumferential surfaces, let A,B, in Fig. 21, represent two plain

wheel disks at liberty to revolve about their fixed centres, and let C C represent a margin of stiff white paper attached to the face of B so as to revolve with it. Now suppose that A and B are in close contact at their perimeters at the point G, and that there is no slip, and that rotary motion commenced when the point E (where as tracing point a pencil is attached), in conjunction with the point F, formed the point of contact of the two wheels, and

Fig. 20.

continued until the points E and F had arrived at their respective positions as shown in the figure ; the pencil at E will have traced upon the margin of white paper the portion of an epicycloid denoted by the curve E F ; and as the movement of the two wheels A, B, took place Dy reason of the contact of their circumferences, it is evident that the length of the arc E G must be equal to that

Fig. 21.

of the arc G F, and that the motion of A (supposing it to be the driver) would be communicated uniformly to B.

Now suppose that the wheels had been rotated in the opposite direction and the same form of curve would be produced, but it would run in the opposite direction, and these two curves may be utilized to form teeth, as in Fig. 22, the points on the wheel A working against the curved sides of the teeth on B.

To render such a pair of wheels useful in practice, all that is necessary is to diminish the teeth on B without altering the

THE TEETH OF GEAR-WHEELS.

nature of the curves, and increase the diameter of the points on A, making them into rungs or pins, thus forming the wheels into what is termed a wheel and lantern, which are illustrated in Fig. 23.

A represents the pinion (or lantern), and B the wheel, and C,c, the primitive teeth reduced in thickness to receive the pins on

Fig. 22.

A. This reduction we may make by setting a pair of compasses to the radius of the rung and describing half-circles at the bottom of the spaces in B. We may then set a pair of compasses to the curve of c, and mark off the faces of the teeth of B to meet the

half-circles at the pitch line, and reduce the teeth heights so as to leave the points of the proper thickness ; having in this opera- tion maintained the same epicycloidal curves, but brought them closer together and made them shorter. It is obvious, however.

Fig. 24-

that such a method of communicating rotary motion is unsuiled

to the transmission of much power ; because of the weakness of,

and small amount of wearing surface on, the points or rungs in A.

In place of points or rungs we may have radial lines, these

VOL. I.— 5.

lines, representing the surfaces of ribs, set equidistant on the radial face of the pinion, as in Fig. 24. To determine the epicycloidal curves for the faces of teeth to work with these radial lines, we may take a generating circle C, of half the diameter of A, and cause it to roll in contact with the internal circumference of A, and a tracing point fixed in the circumference of C will

Fig. 25.

draw the radial lines shown upon A. The circumstances will not be altered if we suppose the three circles, A, B, C, to be movable about their fixed centres, and let their centres be in a straight line ; and if, under these circumstances, we suppose rotation to be imparted to the three circles, through frictional contact of their perimeters, a tracing point on the circumference of C would trace the epicycloids shown upon B and the radial lines shown upon A, evidencing the capability of one to impart uniform rotary motion to the other.

To render the radial lines capable of use we must let them be the surfaces of lugs or projections on the face of the wheel, as shown in Fig. 25 at D, E, &c., or the faces of notches cut in the wheel as at F, G, H, &c., the metal between F and G forming a tooth J, having flanks only. The wheel B has the curves of each

Fig. 26.

tooth brought closer together to give room for the reception of the teeth upon A. We have here a pair of gears that possess suffi- cient strength and are capable of working correctly in either direction.

But the form of tooth on one wheel is conformed simply to suit those on the other, hence, neither two of the wheels A, nor would two of B, work correctly together.

They may be qualified to do so, however, by simply adding to

lO

the tops of the teeth on A, teeth of the form of those on B, and adding to those on B, and within the pitch circle, teeth corres- ponding to those on A, as in Fig. 26, where at k' and j' teeth are provided on B corresponding to J and K on A, while on A there are added teeth 0',n', corresponding toO.N.on B, with the result that two wheels such as A or two such as B would work correctly together, either being the driver or either the follower, and rotation may occur in either direction. In this operation we have simply added faces to the teeth on A, and flanks to those on B, the curves being generated or obtained by rolling the gene- rating, or curve marking, circle C upon the pitch circles P and p'. Thus, for the flanks of the teeth of A, C is rolled upon, and within the pitch circle P of A ; while for the face curves of the same teeth C is rolled upon, but without or outside of P. Similarly for the teeth of wheel B the generating circle C is rolled within P' for the flanks and without for the faces. With the curves rolled or produced with the same diameter of generating circle the wheels will work correctly together, no matter what their relative diameter may be, as will be shown hereafter.

MODERN MACHINE SHOP PRACTICE.

to wheel Q a pencil whose point is at n. If then rotation be given to (Z a in the direction of the arrow s, all three wheels will rotate in that direction as denoted by their respective arrows j.

Assume, then, that rotation of the three has occurred until the pencil point at n has arrived at the point m, and during this period of rotation the point n will recede from the line of centres A B, and will also recede from the arcs or lines of the two pitch circles a a, b b. The pencil point being capable of marking its path, it will be found on reaching m to have marked inside the pitch circle b b the curve denoted by the full line m x, and simultaneously with this curve it has marked another curve out- side of a a, as denoted by the dotted line y m. These two curves being marked by the pencil point at the same time and extending from y to m, and x also to m. They are pro- longed respectively to / and to K for clearness of illustration only.

The rotation of the three wheels being continued, when the pencil point has arrived at O it will have continued the same curves as shown at oy, and O g, curve O / being the same as

Fig. 27.

In this demonstration, however, the curves for the faces of the teeth being produced by an operation distinct from that employed to produce the flank curves, it is not clearly seen that the curves for the flanks of one wheel are the proper curves to insure a uniform velocity to the other. This, however, may be made clear as follows : —

In Fig. 27 let a a and b b represent the pitch circles of two wheels of equal diameters, and therefore having the same number of teeth. On the left, the wheels are shown with the teeth in, while on the right-hand side of the line of centres A B, the wheels are shown blank ; a a\s the pitch line of one wheel, and b b that for the other. Now suppose that both wheels are capable of being rotated on their shafts, whose centres will of course be on the line A B, and suppose a third disk, q, be also capable of rotation upon its centre, c, which is also on the Hne A B. Let these three wheels have sufficient contact at their perimeters at the point n, that if one be rotated it will rotate both the others (by friction) without any slip or lost motion, and of course all three will rotate at an equal velocity. Suppose that there is fixed

m X placed in a new position, and O g being the same as my, but placed in a new position. Now since both these curves (o J and O g) were marked by the one pencil point, and at the same time, it follows that at every point in its course that point must have touched both curves at once. Now the pencil point having moved around the arc of the circle Q from « to m, it is obvious that the two curves must always be in contact, or coincide with each other, at some point in the path of the pencil or describing point, or, in other words, the curves will always touch each other at some point on the curve of Q, and between n and O. Thus when the pencil has arrived at m, curve m y touches curve K x at the point m, while when the pencil had arrived at point O, the curves oy and O^will touch at O. Now the pitch circles a a and b b, and the describing circle Q, having had constant and uniform velocity while the traced curves had constant contact a( some point in their lengths, it is evident that if instead of being mere lines, my was the face of a tooth on a a, and m x was the flank ot a tooth on (5 3, the same uniform motion may be trans- mitted from a a,\a b b, by pressing the tooth face wz y against

THE TEETH OF GEAR-WHEELS.

II

the tooth flank m x. Let it now be noted that the curve y m corresponds to the face of a tooth, as say the face E of a tooth on a a, and that curve x m corresponds to the flank of a tooth on b b, as say to the flank F, short portions only of the curves being used for those flanks. If the direction of rotation of the three wheels was reversed, the same shape of curves would be produced, but they would lie in an opposite direction, and would, therefore, be suitable for the other sides of the teeth. In this case, the contact of tooth upon tooth will be on the other side of the line of centres, as at some point between n and Q.

In this illustration the diameter of the rolling or describing circle Q, being less than the radius of the wheels a a ox b b, the flanks of the teeth are curves, and the two wheels being of the same diameter, the teeth on the two are of the same shape. But the principles governing the proper formation of the curve remain the same whatever be the conditions. Thus in Fig. 28 are segments of a pair of wheels of equal diameter, but the de- scribing, rolling, or curve-generating circle is equal in diameter

Fig. 28.

to the radius of the wheels. Motion is supposed to have occurred in the direction of the arrows, and the tracing point to have moved from n to m. During this motion it will have marked a curve y m, a portion of the y end serving for the face of a tooth on one wheel, and also the line k x, a continuation of which serves for the flank of a tooth on the other wheel. In Fig. 29 the pitch circles only of the wheels are marked, a a being twice the diameter of b b, and the curve-generating circle being equal in diameter to the radius of wheel b b. Motion is assumed to have occurred until the pencil point, starting from n, had arrived at o, marking curves suitable for the face of the teeth on one wheel and for the flanks of the other as before, and the contact of tooth upon tooth still, at every point in the path of the teeth, occurring at some point of the arc n o. Thus when the point had proceeded as far as point m it will have marked the curve y and the radial line x, and when the point had arrived at 0, it will have prolonged m y into 0 g and x into of, while in either position the point is marking both lines. The velocities of the wheels remain the same notwithstanding their different diameters, for the arc n g

must obviously (if the wheels rotate without slip by friction of their surfaces while the curves are traced) be equal in length to the arc nf, or the arc n 0.

Fig. 29.

In Fig. 30 a a and b b are the pitch circles of two wheels as before, and c c the pitch circle of an annular or internal gear, and D is the roUing or describmg circle. When the describing

Fig. 30.

point arrived at m, it will have marked the curve y for the face of a tooth on a a, the curve x for the flank of a tooth on b b, and the curve e for the face of a tooth on the internal wheel c c

12

MODERN MACHINE SHOP PRACTICE.

Motion being continued m y will be prolonged to o g, while simultaneously x will be extended into o f and e into h v, the velocity of all the wheels being uniform and equal. Thus the arcs n v, nf, and ng, are of equal length.

Fig- 31-

In Fig. 31 is shown the case of a rack and pinion ; a a is the pitch line of the rack, b b that of the pinion, A B at a right angle to a a, the line of centres, and n the generating circle. The wheel and rack are shown with teeth n on one side simply for

rolled around a a until it had reached the position marked i, then it will have marked the curve from e to «, a part of this curve serving for the face of tooth c. Now let the rolling circle be placed within the pitch circle a a and its pencil point n be set to e, then, on being rolled to position 2, it will have marked the flank of tooth c. For the other wheel suppose the rolling wheel or circle to have started fromy"and rolled to the line of centres as in the cut, it will have traced the curve forming the face of the tooth d. For the flank of d the rolling circle or wheel is placed within b b, its tracing point set aty'on the pitch circle, and on being rolled to position 3 it will have marked the flank curve. The curves thus produced will be precisely the same as those produced by rotating all three wheels about their axes, as in oui previous demonstrations.

The curves both for the faces and for the flanks thus obtained will vary in their curvature with every variation in either the diameter of the generating circle or of the base or pitch circle of the wheel. Thus it will be observable to the eye that the face curve of tooth c is more curved than that of d, and also that the flank curve of d is more spread at the root than is that for c, which has in this case resulted from the difference between the diameter of the wheels a a and b b. But the curves obtained by a given diameter of rolling circle on a given diameter of pitch circle will be correct for any pitch of teeth that can be used upon wheels having that diameter of pitch circle. Thus, suppose we have a curve obtained by rolling a wheel of 20 inches circum- ference on a pitch circle of 40 inches circumference — now a wheel of 40 inches in circumference may contain 20 teeth of 2 inch arc pitch, or 10 teeth of 4 inch arc pitch, or 8 teeth of 5 inch arc pitch, and the curve may be used for either of those pitches.

Fig. 32-

clearness of illustration. The pencil point n will, on arriving at m, have traced the flank curve x and the curve y for the face of the rack teeth.

It has been supposed that the three circles rotated together by the frictional contact of their perimeters on the line of centres, but the circumstances will remain the same if the wheels remain at rest while the generating or describing circle is rolled around them. Thus in Fig. 32 are two segments of wheels as before, c representing the centre of a tooth on a a, and d representing the centre of a tooth on b b. Now suppose that a generating or rolling circle be placed with its pencil point at e. and that it then be

If we trace the path of contact of each tooth, from the moment it takes until it leaves contact with a tooth upon the other wheel, we shall find that contact begins at the point where the flank of the tooth on the wheel that drives or imparts motion to the other wheel, meets the face of the tooth on the driven wheel, which will always be where the point of the driven tooth cuts or meets the generating or rolling circle of the driving tooth. Thus in Fig. 33 are represented segments of two spur-wheels marked respectively the driver and the driven, their generating circles being marked at G and g', and x x representing the line of centres. Tooth A is shown in the position in which it commences its contact with tooth

THE TEETH OF GEAR-WHEELS.

•3

B at C. Secondly, we shall find that as these two teeth approach the line of centres x, the point of contact between them moves or takes place along the thickened arc or curve C x, or along the path of the generating circle G.

Thus we may suppose tooth D to be another position of tooth A, the contact being at F, and as motion was continued the contact would pass along the thickened curve until it arrived at the line of centres X. Now since the teeth have during this path of contact approached the line of centres, this part of the whole arc of action or of the path of contact is termed the arc of approach. After the two teeth have passed the line of centres x, the path of contact of the teeth will be along the dotted arc from x to L, and as the teeth are during this period of motion receding from x this part of the contact path is termed the arc of recess.

That contact of the teeth would not occur earlier than at c nor later than at L, is shown by the dotted teeth sides ; thus A and B would not touch when in the position denoted bythedotted teeth, nor would teeth I and K if in the position denoted by their dotted lines.

If we examine further into this path of contact we find that throughout its whole path the face of the tooth of one wheel has

It is laid down by Professor Willis that the motion of a pair of gear-wheels is smoother in cases where the path of contact begins at the line of centres, or, in other words, when there is no arc of approach ; and this action may be secured by giving to the driven wheel flanks only, as in Fig. 34, in which the driver has fully developed teeth, while the teeth on the driven have no faces.

In this case, supposing the wheels to revolve in the direction of arrow p, the contact will begin at the line of centres x, move or pass along the thickened arc and end at B, and there will be contact during the arc of recess only. Similarly, if the direction of motion be reversed as denoted by arrow Q, the driver will begin contact at X, and cease contact at H, having, as before, contact during the arc of recess only.

But if the wheel W were the driver and v the driven, then these conditions would be exactly reversed. Thus, suppose this to be the case and the direction of motion be as denoted by arrow P, the contact would occur during the arc of approach, from H to x, ceasing at x.

Or if W were the driver, and the direction of motion was as

Fig- 33-

contact with the flank only of the tooth of the other wheel, and also that the flank only of the driving-wheel tooth has contact before the tooth reaches the line of centres, while the face of only the driving tooth has contact after the tooth has passed the line of centres.

Thus the flanks of tooth A and of tooth D are in driving contact with the faces of teeth B and E, while the face of tooth H is in contact with the flank of tooth i.

These conditions will always exist, whatever be the diameters of the wheels, their number of teeth or the diameter of the generat- ing circle. That is to say, in fully developed epicycloidal teeth, no matter which of two wheels is the driver or which the driven wheel, contact on the teeth of the driver will always be on the tooth flank during the arc of approach and on the tooth face during the arc of recess ; while on the driven wheel contact during the arc of approach will be on the tooth face only, and during the arc of recess on the tooth flank only, it being borne in mind that the arcs of approach and recess are reversed in location if the direction of revolution be reversed. Thus if the direction of wheel motion was opposite to that denoted by the arrows in Fig. 33 then the arc of approach would be from M to x, and the arc of recess from x to N.

denoted by Q, then, again, the path of contact would be during the arc of approach only, beginning at B and ceasing at X, as denoted by the thickened arc B x.

The action of the teeth will in either case serve to give a theo- retically perfect motion so far as uniformity of velocity is con- cerned, or, in other words, the motion of the driver will be trans- mitted with perfect uniformity to the driven wheel. It will be observed, however, that by the removal of the faces of the teeth, there are a less number of teeth in contact at each instant of time ; thus, in Fig. 33 there is driving contact at three points, C, F, and J, while in Fig. 34 there is driving contact at two points only. From the fact that the faces of the teeth work with the flanks only, and that one side only of the teeth comes into action, it becomes apparent that each tooth may have curves formed by four different diameters of rolling or generating circles and yet work correctly, no matter which wheel be the driver, or which the driven wheel or follower, or in which direction motion occurs. Thus in Fig. 35, suppose wheel V to be the driver, having motion in the direction of arrow P, then faces A on the teeth of v will work with flanks B of the teeth on W, and so long as the curves for these faces and flanks are obtained with the same diameter of rolling circle, the action of the teeth will be correct, no matter

14

what the shapes of the other parts of the teeth. Now suppose that V still being the driver, motion occurs in the other direction as denoted by Q, then the faces C of the teeth on V will drive the flanks D of the teeth on w, and the motion will again be correct, providing that the same diameter (whatever it may be) of rolling

MODERN MACHINE SHOP PRACTICE.

different diameters of rolling circles may be used upon a pair of wheels, giving teeth-forms that will fill all the requirements so far as correctly transmitting motion is concerned. In the case of a pair of wheels having an equal number of teeth, so that each tooth on one wheel will always fall into gear with the same tooth on the

circle be used for these faces and flanks, irrespective, of course, of what diameter of rolling circle is used for any other of the teeth curves. Now suppose that w is the driver, motion occurring in the direction of P, then faces E will drive flanks F, and the motion

other wheel, every tooth may have its individual curves differing from all the others, providing that the corresponding teeth on the other wheel are formed to match them by using the same size of rolling circle for each flank and face that work together.

Fig. 35-

will be correct as before if the curves E and F are produced with the same diameter of rolling circle. Finally, let W be the driving wheel and motion occur in the direction of Q, and faces G will drive flanks H, and yet another diameter of rolling circle may be used for these faces and flanks. Here then it is shown that four

It is obvious, however, that such teeth would involve a great deal of labor in their formation and would possess no advantage, hence they are not employed. It is not unusual, however, in a pair of wheels that are to gear together and that are not intended to interchange with other wheels, to use such sizes as will give to

THE TEETH OF GEAR-WHEELS.

15

both wheels teeth having radial flanks ; which is done by using for the face of the teeth on the largest wheel of the pair and for tne flanks of the teeth of the smallest wheel, a generating circle equal in diameter to the radius of the smallest wheel, and for the faces of the teeth of the small wheel and the flanks of the teeth of

circles of two wheels, then the generating circle would be rolled within B, as at i , for the flank curves, and without it, as at 2, for the face curves of B. It would be rolled without the pitch line, as at 3, for the rack faces, and within it, as at 4, for the rack flanks, and without C, as at 5, for the faces, and within it, as at 6, for

Fig- 37.

Fig. 36.

the large one, a generating circle whose diameter equals the radius of the large wheel.

It will now be evident that if we have planned a pair or a train of wheels we may find how many teeth will be in contact for any given pitch, as follows. In Fig. 36 let A, B, and C, represent three blanks for gear-wheels whose addendum circles are M, Nando ; P representing the pitch circles, and Q representing the circles for the roots of the teeth. Let Xand Y represent the lines of centres, and G, H, I and K the generating or rolling circle, whose centres are on the respective lines of centres — the diameter of the gene- rating circle being equal to the radius of the pinion, as in the Willis system, then, the pinion M being the driver, and the wheels revolving in the direction denoted by the respective arrows, the arc or path of contact for the first pair will be from point D, where the generating circle G crosses circle N to E, where generating circle H crosses the circle M, this path being composed of two arcs of a circle. All that is necessary, therefore, is to set the compasses to the pilch the teeth are to have and step them along these arcs, and the number of steps will be the number of teeth that will be in contact. Similarly, for the second pair contact will begin at Rand end at S, and the compasses applied as before (from R to s) along the arc of generating circle I to the line of centres, and thence along the arc of generating circle K to S, will give in the number of steps, the number of teeth that will be in contact. If for any given purpose the number of teeth thus found to be in contact is insufficient, the pitch may be made finer.

When a wheel is intended to be formed to work correctly with any other wheel having the same pitch, or when there are more than two wheels in the train, it is necessary that the same size of generating circle be used for all the faces and all the flanks in the set, and if this be done the wheels will work correctly together, no matter what the number of the teeth in each wheel may be, nor in what way they are interchanged. Thus in Fig. 37, let A represent the pitch line of a rack, and B and C the pitch

flanks of the teeth on C, and all the teeth will work correctly together however they be placed ; thus C might receive motion from the rack, and B receive motion from C. Or if any number of different diameters of wheels are used they will all work correctly together and interchange perfectly, with the single condition that the same size of generating circle be used through- out. But the curves of the teeth so formed will not be alike. Thus in Fig. 38 are shown three teeth, all struck with the same size of

Fig. 38.

generating circle, D being for a wheel of 12 teeth, E for a wheel of 50 teeth, and F a tooth of a rack ; teeth E,F, being made wider so as to let the curves show clearly on each side, it being obvious that since the curves are due to the relative sizes of the pitch and generating circles they are equally applicable to any pitch or thickness of teeth on wheels having the same diameters of pitch circle. In determining the diameter of a generating circle for a set or

i6

train of wheels, we have the consideration that the smaller the diameter of the generating circle in proportion to that of the

MODERN MACHINE SHOP PRACTICE.

Fig. 39.

pitch circle the more the teeth are spread at the roots, and this creates a pressure tending to thrust the wheels apart, thus causing the axle journals to wear. In Fig. 39, for example, A A

would exist throughout the whole path of contact save at the point F, on the line of centres. This thrust is reduced in proportion as the diameter of the generating circle is increased ; thus in Fig. 40, is represented a pair of pinions of 12 teeth and 3 inch pitch, and c being the driver, there is contact at E, and at G, and E being a radial line, there is obviously a minimum of thrust.

What is known as the Willis system for interchangeable gear- ing, consists of using for every pitch of the teeth a generating circle whose diameter is equal to the radius of a pinion having 12 teeth, hence the pinion will in each pitch have radial flanks, and the roots of the teeth will be more spread as the number of teeth in the wheel is increased. Twelve teeth is the least number that it is considered practicable to use ; hence it is obvious that under this system all wheels of the same pitch will work correctly together.

Unless the faces of the teeth and the flanks with which they work are curves produced from the same size of generating circle, the velocity of the teeth will not be uniform. Obviously the revolu-

Fig. 40.

is the line of centres, and the contact of the curves at B C would cause a thrust in the direction of the arrows D, Ji. This thrust

Fig. 41.

tions of the wheels will be proportionate to their numbers of teeth ; hence in a pair of wheels having an equal number of teeth, the revolutions will per force be equal, but the driver will not impart uniform motion to the driven wheel, but each tooth will during the path of contact move irregularly.

The velocity of a pair of wheels will be uniform at each instant of time, if a line normal to the surfaces of the curves at their point of contact passes through the point of contact of the pitch circles on the line of centres of the wheels. Thus in Fig. 41, the line A A is tangent to the teeth curves where they touch, and D at a right angle to A A, and meets it at the point of the tooth curves, hence it is normal to the point of contact, and as it meets the pitch circles on the line of centres the velocity of the wheels will be uniform.

The amount of rolling motion of the teeth one upon the other while passing through the path of contact, will be a minimum when the tooth curves are correctly formed according to the rules given. But furthermore the sliding motion will be increased in proportion as the diameter of the generating circle is increased, and the number of teeth in contact will be increased because the

THE TEETH OF GEAR-WHEELS.

17

arc, or path, of contact is longer as the generating circle is made larger.

Thus in Fig. 42 is a pair of wheels whose tooth curves are from a generating circle equal to the radius of the wheels, hence the

of the teeth on the larger wheel B, have contact along a greater portion of their depths than do the flanks of those on the smaller, as is shown by the dotted arc I being farther from the pitch circle than the dotted arc j is, these two dotted arcs representing the

Fig. 42.

tlanks are radial. The teeth are made of unusual depth to keep the lines in the engraving clear. Suppose v to be the driver, W the driven wheel or follower, and the direction of motion as at P, contact upon tooth A will begin at C, and while A is passing to the line of centres the path of contact will pass along the thick- ened line to X. During this time the whole length of face from C to R will have had contact with the length of flank from C to N, and it follows that the length of face on A that rolled on c N can only equal the length of c N, and that the amount of sliding motion must be represented by the length of R N on A, and the amount of rolling motion by the length N C. Again, during the arc of recess (marked by dots) the length of flank that will have had contact is the depth from S to L, and over this depth the full length of tooth face on wheel V will have swept, and as L S equals C N, the amount of rolling and of sliding motion during the arc of recess is equal to that during the arc of approach, and the action is in both cases partly a rolling and partly a sliding one. The two wheels are here shown of the same diameter, and there- fore contain an equal number of teeth, hence the arcs of approach and of recess are equal in length, which will not be the case when one wheel contains more teeth than the other. Thus in Fig. 43, let A represent a segment of a pinion, and B a segment of a spur- wheel, both segments being blank with their pitch circles, the tooth height and depth being marked by arcs of circles. Let C and D represent the generating circles shown in the two respec- tive positions on the line of centres. Let pinion A be the driver moving in the direction of P, and the arc of approach will be from E to X along the thickened arc, while the arc of recess will be as denoted by the dotted arc from x to F. The distance E x being greater than distance x F, therefore the arc of approach is longer than that of recess.

But suppose B to be the driver and the reverse will be the case, the arc of approach will begin at G and end at X, while the arc of recess will begin at x and end at H, the latter being farther from the line of centres than G is. It will be found also that, one wheel being larger than the other, the amount of sliding and rolling contact is different for the two wheels, and that the flanks VOL. I. — 6.

paths of the lowest points of flank contact, points F and G, mark- ing the initial lowest contact for the two directions of revolution.

Thus it appears that there is more sliding action upon the teeth of the smaller than upon those of the larger wheel, and this is a condition that will always exist.

Fig- 43-

In Fig. 44 is represented portion of a pair of wheels corre- sponding to those shown in Fig. 42, except that in this case the diameter of the generating circle is reduced to one quarter that of the pitch diameter of the wheels, v is the driver in the direction

i8

MODERN MACHINE SHOP PRACTICE.

of V, and contact will begin at C ; hence the depth of flank on the teeth of V that will have contact is C N, which, the wheels, being of equal diameter, will remain the same whichever wheel be the driver, and in whatever direction motion occurs. The amount of rolling motion is, therefore, CN, and that of sliding is the difference between the distance C N and the length of the tooth face. If now we examine the distance c N in Fig. 42, we find that

train of gearing in which the generating circle equals the radius of the pinion, the pinion will wear out of shape the quickest, and the largest wheel the least; because not only does each tooth on the pinion more frequently come into action on account of its increased revolutions, but furthermore the length of flank that has contact is less, while the amount of sliding action is greater. In Fig. 45, for example, are a wheel and pinion, the latter having radial flanks and the pinion being the driver, the arc of approach

Fig. 44.

reducing the diameter of generating circle in Fig. 44 has increased the depth of flank that has contact, and therefore increased the rolling motion of the tooth face along the flank, and correspond- ingly diminished the sliding action of the tooth contact. But al the same time we have diminished the number of teeth in contact. Thus in Fig. 42 there are three teeth in driving contact, while in Fig. 44 there are but two, viz., D and E.

In an article by Professor Robinson, attention is called to the

fact that if the teeth of wheels are not formed to have conect curves when new, they cannot be improved by wear ; and this will be clearly perceived from the preceding remarks upon the amount of rolling and sliding contact. It will also readily appear that the nearer the diameter of the generating to that of the base circle the more the teeth wear out of correct shape ; hence, in a

is the thickened arc from C to the line of centres, while the arc of recess is denoted by the dotted arc. As contact on the pinion flank begins at point C and ends at the line of centres, the total depth of flank that suffers wear from the contact is that from C to N ; and as the whole length of the wheel tooth face sweeps over this depth C N, the pinion flanks must wear faster than the wheel faces, and the pinion flanks will wear underneath, as denoted by the dotted curve on the flanks of tooth W. In the case of the wheel, contact on its tooth flanks begins at the line of centres and ends at L, hence that flank can only wear between point L and the pitch line .s ; and as the whole length of pinion face sweeps on this short length L S, the pinion flank will wear most, the wear being in the direction of the dotted arc on the left-hand side V of the tooth. Now the pinion flank depth C N, being less than the wheel flank depth S L, and the same length of tooth face sweeping (during the path of contact) over both, obviously the pinion tooth will wear the most, while both will, as the wear proceeds, lose their proper flank curve. In Fig. 46 the generating arcs, G and G', and the wheel are the same, but the pinion is larger. As a result the acting length C N, of pinion flank is increased, as is also the acting length S L, of wheel flank ; hence, the flanks of both wheels would wear better, and also better preserve their correct and original shapes.

It has been shown, when referring to Figs. 42 and 44, when treating of the amount of sliding and of rolling motion, that the smaller the diameter of rolling circle in proportion to that of pitch circle, the longer the acting length of flank and the more the amount of rolling motion ; and it follows that the teeth would also preserve their original and true shape better. But the wear of the teeth, and the alteration of tooth form by reason of that wear, will, in any event, be greater upon the pinion than upon the

THE TEETH OF GEAR-WHEELS.

wheel, and can only be equal when the two wheels are of equal diameter, in which case the tooth curves will be alike on both wheels, and the acting depths of flank will be equal, as shown in Fig. 47, the flanks being radial, and the acting depths of flank being shown at j K. In Fig. 48 is shown a pair of wheels with a generating circle, G and G', of one quarter the diameter of the base circle or pitch diameter, and the acting length of flank is

shown at Lm. The wear of the teeth would, therefore, in this latter case, cause it in time to assume the form shown in Fig. 49. But it is to be noted that while the acting depth of flank has been increased the arcs of contact have been diminished, and that in Fig. 47 there are two teeth in contact, while in Fig. 48 there is but one, hence the pressure upon each tooth is less in proportion as the diameter of the generating circle is increased. If a train of wheels are to be constructed, or if the wheels are to be capable

Fig- 47-

of interchanging with other combinations of wheels of the same pitch, the diameter of the generating circle must be equal to the smallest wheel or pinion, which is, under the Willis system, a pinion of 12 teeth ; under the Pratt and Whitney, and Brown and Sharpe systems, a pinion of 15 teeth.

But if a pair or a particular train of gears are to be constructed, then a diameter of generating circle may be selected that is con- sidered most suitable to the particular conditions ; as, for example,

19

it may be equal to the radius of the smallest wheel giving it radial flanks, or less than that radius giving parallel or spread flanks. But in any event, in order to transmit continuous motion, the diameter of generating circle must be such as to give arcs of action that are equal to the pitch, so that each pair of teeth will come into action before the preceding pair have gone out of action.

It may now be pointed out that the degrees of angle that the teeth move through always exceeds the number of degrees of

Fig. 48.

angle contained in the paths of contact, or, in other words, exceeds the degrees contained in the arcs of approach and recess combined.

In Fig. 50, for example, are a wheel A and pinion B, the teeth on the wheel being extended to a point. Suppose that the wheel A is the driver, and contact will begin between the two teeth D and F on the dotted arc. Now suppose tooth D to have moved to position C, and F will have been moved to position H. The

Fig. 49.

degrees of angle the pinion has been moved through are there- fore denoted by I, whereas the degrees of angle the arcs of contact contain are therefore denoted by J.

The degrees of angle that the wheel A has moved through are obviously denoted by E, because the point of tooth D has during the arcs of contact moved from position D to position c. The degrees of angle contained in its path of contact are denoted by K, and are less than E, hence, in the case of teeth terminating in a point as tooth D, the excess of angle of action over path of con- tact is as many degrees as are contained in one-half the thickness

20

MODERN MACHINE SHOP PRACTICE.

of the tooth, while when the points of the teeth are cut off, the excess is the number of degrees contained in the distance between the corner and the side of the tooth as marked on a tooth at P.

With a given diameter of pitch circle and pitch diameter of wheel, the length of the arc of contact will be influenced by the

Fig- SO-

height of the addendum from the pitch circle, because, as has been shown, the arcs of approach and of recess, respectively, begin and end on the addendum circle.

If the height of the addendum on the follower be reduced, the arc of approach will be reduced, while the arc of recess will not be altered ; and if the follower have no addendum, contact

It is obvious, however, that the follower having no addendum would, if acting as a driver to a third wheel, as in a train of wheels, act on its follower, or the fourth wheel of the train, on the arc of approach only ; hence it follows that the addendum might be reduced to diminish, or dispensed with to eliminate action, on the arc of approach in the follower of a pair of wheels only, and not in the case of a train of wheels.

To make this clear to the reader it may be necessary to refer again to Fig. j,}, or 34, from which it will be seen that the action of the teeth of the driver on the follower during the arc of approach is produced by the flanks of the driver on the faces of the follower. But if there are no such faces there can be no such contact.

On the arc of recess, however, the faces of the driver act on the flanks of the follower, hence the absence of faces on the follower is of no import.

From these considerations it also appears that by giving to the driver an increase of addendum the arc of recess may be increased without affecting the arc of approach. But the height of addendum in machinists' practice is made a constant propor- tion of the pitch, so that the wheel may be used indiscriminately, as circumstances may require, as either a driver or a follower, the arcs of approach and of recess being equal. The height of addendum, however, is an element in determining the number of teeth in contact, and upon small pinions this is of import- ance.

In Fig. 51, for example, is shown a section of two pinions of equal diameters, and it will be observed that if the full line A determined the height of the addendum there would be contact either at C or B only (according to the direction in which the motion took place).

With the addendum extended to the dotted circle, contact would be just avoided, while with the addendum extended to D there would be contact either at E or at F, according to which direction the wheel had motion.

This, by dividing the strain over two teeth instead of placing it all upon one tooth, not only doubles the strength for driving capacity, but decreases the wear by giving more area of bearing surface at each instant of time, although not increasing that area in proportion to the number of teeth contained in the wheel.

Fig. 51.

between the teeth will occur on the arc of recess only, which gives a smoother motion, because the action of the driver is that of dragging rather than that of pushing the follower. In this case, however, the arc of recess must, to produce continuous motion, be at least equal to the pitch.

In wheels of larger diameter, short teeth are more permissible, because there are more teeth in contact, the number increasing with the diameters of the wheels. It is to be observed, however, that from having radial flanks, the smallest wheel is always the weakest, and that from making the most revolutions in a given

THE TEETH OF GEAR-WHEELS.

21

time, it suffers the most from wear, and lience requires the greatest attainable number of teeth in constant contact at each period of time, as well as the largest possible area of bearing or wearing surface on the teeth.

It is true that increasing the "depth of tooth to pitch line" increases the whole length of tooth, and, therefore, weakens it ; but this is far more than compensated for by distributing the strain over a greater number of teeth. This is in practice accomplished, when circumstances will permit, by making the

Fig- 52-

pitch finer, giving to a wheel, of a given diameter, a greater number of teeth.

When the wheels are required to transmit motion rather than power (as in the case of clock wheels), to move as frictionless as possible, and to place a minimum of thrust on the journals of the shafts of the wheels, the generating circle may be made nearly as large as the diameter of the pitch circle, producing teeth of the form shown in ^'ig. 52. But the minimum of friction is attained when the two flanks for the tooth are drawn into one common hypocycloid, as in Fig. 53. The difference between the form of tooth shown in Fig. 52 and that shown in Fig. 53, is merely due to an increase in the diameter of the generating circle for the latter. It will be observed that in these forms the

Fig. 53-

acting length of flank diminishes in proportion as the diameter of the generating circle is increased, the ultimate diameter of generating circle being as large as the pitch circles.

* This form is undesirable in that there is contact oh one side only (on the arc of approach) of the line of centres, but the flanks of the teeth may be so modified as to give contact on the arc of recess also, by forming the flanks as shown in Fig. 54, the flanks, or rather the parts within the pitch circles, being nearly half circles, and the parts without with peculiarly formed faces, as shown in the figure. The pitch circles must still be regarded as the rolling circles rolling upon each other. Suppose b a tracing point on B, then as B rolls on A it will describe the epicycloid ab. * From an ariicle by Professor Robinson.

A parallel line c d will work at a constant distance 2.s sX c d from a b, and this distance may be the radius of that part of D that is within the pitch line, the same process being applied to the teeth on both wheels. Each tooth is thus composed of a spur based upon a half cylinder.

Comparing Figs. 53 and 54, we see that the bases in 53 are flattest, and that the contact of faces upon them must range

nearer the pitch line than in 54. Hence, 53 presents a more favorable obliquity of the line of direction of the pressures of tooth upon tooth. In seeking a still more favorable direction by going outside for the point of contact, we see by simply recalling the method of generating the tooth curves, that tooth contacts outside the pitch lines have no possible existence ; and hence, J^'iT- 53 ^'^y be regarded as representing that form of toothed gear which will operate with less friction than any other known form.

This statement is intended to cover fixed teeth only, and not that complicated form of the trundle wheel in which the cylinder

Fig- 55-

teeth are friction rollers. No doubt such would run still easier, even with their necessary one-sided contacts. Also, the state- ment is supposed to be confined to such forms of teeth as have good practical contacts at and near the line of centres.

Bevel-gear wheels are employed to transmit motion from one shaft to another when the axis of one is at an angle to that of the other. Thus in Fig. 55 is shown a pair of bevel-wheels to transmit motion from shafts at a right angle. In bevel-wheels all the lines of the teeth, both at the tops or points of the teeth, at the bottoms of the spaces, and on the sides of the teeth, radiate from the centre E, where the axes of the two shafts would meet if produced. Hence the depth, thickness, and height of the tooth decreases as

22

the point E is approached from the diameter of the wheel, which is always measured on the pitch circle at the largest end of the cone, or in other words, at the largest pitch diameter.

The principles governing the practical construction of the curves for the teeth of the bevel-wheels may be explained as follows :—

In Fig. 56 let F and G represent two shafts, rotating about their respective axes ; and having cones whose greatest diameters are at A and B, and whose points are at E. The diameter A being

MODERN MACHINE SHOP PRACTICE.

equal to that of B their circumferences will be equal, and the angular and velocity ratios will therefore be equal.

Let C and D represent two circles about the respective cones, being equidistant from E, and therefore of equal diameters and circumferences, and it is obvious that at every point in the length of each cone the velocity will be equal to a point upon the other so long as both points are equidistant from the points of inter- section of the axes of the two shafts ; hence if one cone drive the other by frictional contact of surfaces, both shafts will be rotated at an equal speed of rotation, or if one cone be fixed and the other moved around it, the contact of the surfaces will be a rolling contact throughout. The line of contact between the two cones will be a straight line, radiating at all times from the point E. If such, however, is not the case, then the contact will no longer be a rolling one. Thus, in Fig. 57 the diameters or circumferences at A and B being equal, the surfaces would roll upon each other, but on account of the line of contact not radiat- ing from E (which is the common centre of motion for the two

Ek—

I

! '^

.4._D-— \ H

I

Fig- 57-

shafts) the circumference C is less than that of D, rendering a rolling contact impossible.

We have supposed that the diameters of the cones be equal, but the conditions will remain the same when their diameters are unequal ; thus, in Fig. 58 the circumference of A is twice that of B, hence the latter will make two rotations to one of the former, and the contact will still be a rolling one. Similarly the circum- ference of D is one half that of C, hence D will also make two rotations to one of c, and the contact will also be a rolling one ; a condition which will always exist independent of the diameters of the wheels so long as the angles of the faces, or wheels, or (what

is the same thing, the line of contact between the two,) radiates from the point E, which is located where the axes of the shafts would meet.

The principles governing the forms of the cones on which the teeth are to be located thus being explained, we may now con- sider the curves of the teeth. Suppose that in Fig. 59 the cone A is fixed, and that the cone whose axis is F be rotated upon it in the direction of the arrow. Then let a point be fixed in any part

of the circumference of b (say at <f), and it is evident that the path of this point will be as B rolls around the axis F, and at the same time around A from the centre of motion, E. The curve so generated or described by the point d will be a spherical epicy- cloid. In this case the exterior of one cone has rolled upon the coned surface of the other ; but suppose it rolls upon the interior, as around the walls of a conical recess in a solid body ; then a point in its circumference would describe a curve known as the

Fig. 60.

spherical hypocycloid ; both curves agreeing (except in their spherical property) to the epicycloid and hypocycloid of the spur- wheel. But this spherical property renders it very diificult indeed to practically delineate or mark the curves by rolling contact, and on account of this difficulty Tredgold devised a method of construction whereby the curves may be produced sufficiently accurate for all practical purposes, as follows : —

In Fig. 60 let A A represent the axis of one shaft, and B the axis of the other, the axes of the two meeting at w. Mark E,

THE TEETH OF GEAR-WHEELS.

23

representing the diameter of one wheel, and F that of the other (both lines representing the pitch circles of the respective wheels). Draw the line G G passing through the point w, and the point T, where the pitch circles E, F meet, and G G will be the line of contact between the cones. From W as a centre, draw on each side of G G dotted lines as p, representing the height of the teeth above and below the pitch line G G. At a right angle to G G mark the line J K, and from the junction of this line with axis B (as at Q) as a centre, mark the arc a, which will represent the pitch circle for the large diameter of pinion D ; mark also the arc b for the adden-

in the wheel is diminished, which is also the reverse of what occurs in spur-wheels ; as will readily be perceived when it is considered that if in an internal wheel the pinion have as many teeth as the wheel the contact would exist around the whole pitch circles of the wheel and pinion and the two would rotate together without any motion of tooth upon tooth. Obviously then we have, in the case of internal wheels, a consideration as to what is the greatest number (as well as what is the least number) of teeth a pinion may contain to work with a given wheel, whereas in spur- wheels the reverse is again the case, the consideration being how

Fig. 61

dura and c for the roots of the teeth, so that from b toe will repre- sent the height of the tooth at that end.

Similarly from P, as a centre, mark (for the large diameter of wheel C,) the pitch circle g, root circle h, and addendum i. On these arcs mark the curves in the same manner as for spur-wheels. To obtain these arcs for the small diameters of the wheels, draw M M parallel to j K. Set the compasses to the radius R L, and from P, as a centre, draw the pitch circle k. To obtain the depth for the tooth, draw the dotted line/, meeting the circle h, and the point W. A similar line from circle i to w will show the height of the addendum, or extreme diameter ; and mark the tooth curves on k, /, III, in the same manner as for a spur-wheel.

Similarly for the pitch circle of the small end of the pinion teeth, set the compasses to the radius S L, and from Q as a centre, mark the pitch circle d, outside of d mark e for the height of the addendum and inside aid marky for the roots of the teeth at that end. The distance between the dotted lines (as/) repre- sents the full height of the teeth, hence h meets line/, being the root of tooth for large wheel, and to give clearance, the point of the pinion teeth is marked below, thus arc b does not meet h or /. Having obtained these arcs the curves are rolled as for a spur-wheel.

A tooth thus marked out is shown at x, and from its curves between b c, a. template for the large diameter of the pinion tooth may be made, while from the tooth curves between the arcs ef, a template for the smallest tooth diameter of the pinion can be made.

Similarly for the wheel C the outer end curves are marked on the lines g, h, i, and those for the inner end on the lines k, I, m.

Internal or annular gear-wheels have their tooth curves formed by rolling the generating circle upon the pitch circle or base circle, upon the same general principle as external or spur-wheels. But the tooth of the annular wheel corresponds with the space in the spur-wheel, as is shown in Fig. 61, in which curve A forms the flank of a tooth on a spur-wheel P, and the face of a tooth on the annular wheel W. It is obvious then that the generating circle is rolled within the pitch circle for the face of the wheel and without for its flank, or the reverse of the process for spur- wheels. But in the case of internal or annular wheels the path of contact of tooth upon tooth with a pinion having a given number of teeth increases in proportion as the number of teeth

Fig. 62.

few teeth the wheel may contain to work with a given pinion Now it is found that although the curves of the teeth in interna wheels and pinions may be rolled according to the principles already laid down for spur-wheels, yet cases may arise in which internal gears will not work under conditions in which spur-wheels would work, because the internal wheels will not engage together. Thus, in Fig. 62, is a pinion of 12 teeth and a wheel of 22 teeth, a generating circle having a diameter equal to the radius of the pinion having been used for all the tooth curves of both wheel and pinion. It will be observed that teeth A, B, and c clearly

Fig. 63.

overlap teeth I), E, and F, and would therefore prevent rtie wheels from engaging to the requisite depth. This may of course be remedied by taking the faces off the pinion, as in Fig. 63, and thus confining the arc of contact to an arc of recess if the pinion drives, or an arc of approach if the wheel drives ; or the number of teeth in the pinion may be reduced, or that in the wheel increased ; either of which may be carried out to a degree suffi- cient to enable the teeth to engage and not interfere one with the other. In Fig. 64 the number of teeth in the pinion P is reduced from 12 to 6, the wheel W having 22 as before, and it will be observed that the teeth engage and properly clear each other. By the introduction into the figure of a segment of a spur-

24

MODERN MACHINE SHOP PRACTICE.

wheel also having 22 teeth and placed on the other side of the pinion, it is shown that the path of contact is greater, and there- fore the angle of action is greater, in internal than in spur gearing. Thus suppose the pinion to drive in the direction of the arrows and the thickened arcs A B will be the arcs of approach, A measuring longer than B. The dotted arcs C D represent the arcs of receding contact and C is found longer than D, the angles of action being 66" for the spur-wheels and 72° for the annular wheel.

On referring again to Fig. 62 it will be observed that it is the faces of the teeth on the two wheels that interfere and will prevent them from engaging, hence it will readily occur to the mind that it is possible to form the curves of the pinion faces correct to work with the faces of the wheel teeth as well as with the flanks ; or it is possible to form the wheel faces with curves that will work correctly with the faces, as well as with the flanks of the pinion teeth, which will therefore increase the angle of action, and professor McCord has shown in an article in the London Engi- neering how to accomplish this in a simple and yet exceedingly ingenious manner which may be described as follows : —

It is required to find a describing circle that will roll the

Fig. 64.

curves for the flanks of the pinion and the faces of the wheels, and also a describing circle for the flanks of the wheel and the faces of the pinion ; the curve for the wheel faces to work correctly with the faces as well as with the flanks of the pinion, and the curve for the pinion faces to work correctly with both the flanks and faces of the internal wheel.

In Fig. 65 let P represent the pitch circle of an annular or internal wheel whose centre is at A, and Q the pitch circle of a pinion whose centre is at B, and let R be a describing circle whose centre is at C, and which is to be used to roll all the curves for the teeth. For the flanks of the annular wheel we may roll R within P, while for the faces of the wheel we ijiay roll R outside of P, but in the case of the pinion we cannot roll R within Q, because R is larger than Q, hence we must find some other rolling circle of less diameter than R, and that can be used in its stead (the radius of R always being greater than the radius of the axis of the wheel and pinion for reasons that will appear presently). Suppose then that in Fig. 66 we have a ring wliose bore R corresponds in diameter to the intermediate describing circle R, Fig. 65 and that Q represents the pinion. Then we may roll R

around and in contact with the pinion Q, and a tracing point in R will trace the curve M N o, giving a curve a portion of which may be used for the faces of the pinion. But suppose that instead of rolling the intermediate describing circle R around P,

Fig. 65.

we roll the circle T around P, and it will trace precisely the same curve M N O ; hence for the faces of the pinion we have found a rolling circle T which is a perfect substitute for the iHtermediate

Fig. 66.

circle Q, and which it will always be, no matter what the diameters of the pinion and of the intermediate describing circle may be, providing that the diameter of T is equal to the difference between the diameters of the pinion and that of the intermediate describmg

THE TEETH OF GEAR-WHEELS.

circle as in the figure. If now we use this describing circle to roll the flanks of the annular wheel as well as the faces of the pinion, these faces and flanks will obviously work correctly together. Since this describing circle is roiled on the outside of

25

dotted portion of the exterior describing circle as in ordinary gear- ing. But in addition there will be an arc of recess along the dotted portion of the intermediate circle R, which arc is due to the faces of the pinion acting upon the faces as well as upon the flanks of

the pinion and on the outside of the annular wheel we may distinguish it as the exterior describing circle.

Now instead of rolling the intermediate describing circle R within the annular wheel P for the face curves of the teeth upon P, we may find some other circle that will give the same curve and be small enough to be rolled within the pinion Q for its teeth flanks. Thus in Fig. 67 P represents the pitch circle of the annular wheel and R the intermediate circle, and if R be rolled within P, a point on the circumference of R will trace the curve

V w. But if we take the circle S, having a diameter equal to the difference between the diameter of R and that of P, and roll it within P, a point in its circumference will trace the same curve

V W ; hence S is a perfect substitute for R, and a portion of the curve V W may be used for the faces of the teeth on the annular wheel. The circle .s being used for the pinion flanks, the wheel faces and pinion flanks will work correctly together, and as the circle s is rolled within the pinion for its flanks and within the wheel for its faces, it maybe distinguished as the interior describ- ing circle.

To prove the correctness of the construction it may be noted that with the particular diameter of intermediate describing circle used in Fig. 65, the interior and exterior describing circles are of equal diameters ; hence, as the same diameter of describing circle is used for all the faces and flanks of the pair of wheels they will obviously work correctly together, in accordance with the rules laid down for spur gearing. The radius of S in Fig. 69 is equal to the radius of the annular wheel, less the radius of the intermediate circle, or the radius from A to C. The radius of the exterior describing circle T is the radius of the intermediate circle less the radius of the pinion, or radius C B in the figure.

Now the diameter of the intermediate circle may be determined at will, but cannot exceed that of the annular wheel or be less than the pinion. But having been selected between these two limits the inferior and exterior describing circles derived from it give teeth that not only engage properly and avoid the inter- ference shown in Fig. 62, but that will also have an additional arc o. action during the recess, as is shown in Fig. 68, which represents the wheel and pinion shown in Fig. 62, but produced by means of the interior and exterior describing circles. Sup- posing the pinion to be the driver the arc of approach will be along the thickened arc of the interior describing circle, while during the arc of recess there will be an arc of contact along the VOL. I. — /,

the wheel teeth. It is obvious from this that as soon as a tooth passes the line of centres it will, during a certain period, have two points of contact, one on the arc of the exterior describing circle, and another along the arc of R, this period continuing

Fig. 69.

until the addendum circle of the pinion crosses the dotted arc of the exterior describing circle at z.

The diameters of the interior and exterior describing circles obviously depend upon the diameter of the intermediate circle, and as this may, as already stated, be selected, within certain limits, at will, it is evident that the relative diameters of the

26

MODERN MACHINE SHOP PRACTICE.

interior and exterior describing circles will vary in proportion, the interior becoming smaller and the exterior larger, while from the very mode of construction the radius of the two will equal that of the axes of the wheel and pinion. Thus in Fig. 69 the radii of S,T,equalAB, or the line of centres, and their diameters, there-

Fig. 70.

fore, equal the radius of the annular wheel, as is shown by dotting them in at the upper half of the figure. But after their diameters have been determined by this construction either of them may be decreased in diameter and the teeth of the wheels will clear (and not interfere as in Fig. 62), but the action will be the same

having 22 and the pinion 12 teeth), the diameter of the inter- mediate circle having been enlarged to decrease the diameter of s and increase that of T, and as these are left of the diameter derived from the construction there is receding action along R from the line of centres to T.

In Fig. 71 are represented a wheel and pinion, the pinion having but four teeth less than the wheel, and a tooth, J, being shown in position in which it has contact at two places. Thus at k it is in contact with the flank of a tooth on the annular wheel, while at L it is in contact with the face of the same tooth.

As the faces of the teeth on the wheel do not have contact higher than point /, it is obvious that instead oi having them -ft of the pitch as at the bottom of the figure, we may cut off the portion X without diminishing the arc of contact, leaving them formed as at the top of the figure. These faces being thus reduced in height we may correspond- ingly reduce the depth of flank on the pinion by filling in the portion G, leaving the teeth formed as at the top of the pinion. The teeth faces of the wheel being thus reduced we may, by using a sufficiently large intermediate circle, obtain interior and exterior describing circles that will form teeth that will permit o( the pinion having but one tooth less than the wheel, or that will form a wheel having but one tooth more than the pinion.

The limits to the diameter of the intermediate describing circle are as follows : in Fig. 72 it is made equal in diameter to the pitch diameter of the pinion, hence B will represent the centre of the intermediate circle as well as of the pinion, and the pitch circle of the pinion will also represent the intermediate circle R. To

as in ordinary gear, or in other words there will be no arc of action on the circle R. But S cannot be increased without correspondingly decreasing T, nor can T be increased without correspondingly decreasing s.

Fig. 70 shows the same pair of gears as in Fig. 68 (the wheel

obtain the radius for the interior describing circle we subtract the radius of the intermediate circle from the radius of the annular wheel, which gives A P, hence the pitch circle of the pinion also represents the interior circle R. But when we come to obtain the radius for the exterior describing circle (x), by sub-

THE TEETH OF GEAR-WHEELS.

27

tracting the radius of the pinion from that of the intermediate circle, we find that the two being equal give o for the radius of (t), hence there could be no flanks on the pinion.

Now suppose that the intermediate circle be made equal in diameter to the pitch circle of the annular wheel, and we may

obtain the radius for the exterior describing circle T ; by sub- tracting the radius of the pinion from that of the intermediate circle, we shall obtain the radius A B ; hence the radius of (1) will equal that of the pinion. But when we come to obtain the radius for the interior describing circle by subtracting the radius of the intermediate circle from that of the annular wheel, we find these

two to be equal, hence there would be no interior describing circle, and, therefore, no faces to the pinion.

The action of the teeth in internal wheels is less a sliding and more a rolling one than that in any other form of toothed gearing. This may be shown as follows : In Fig. 73 let A A represent the pitch circle of an external pinion, and B B that of an internal one, and P P the pitch circle of an external wheel for A A or an inter- nal one for B B, the point of contact at the line of centres being at c, and the direction of rotation P P being as denoted by the arrow ; the two pinions being driven, we suppose a point at C, on the pitch circle P P, to be coincident with a point on each of the two pinions at the line of centres. If P P be rotated so as to bring this point to the position denoted by D, the point on the external pinion having moved to E, while that on the internal

Fig. 73-

pinion has moved to F, both having moved through an arc equal to C D, then the distance from E to D being greater than from D to F, more sliding motion must have accompanied the contact of the teeth at the point E than at the point F ; and the difference in the length of the arc E D and that of F D, may be taken to re- present the excess of sliding action for the teeth on E ; for what- ever, under any given condition, the amount of sliding contact may be, it will be in the proportion of the length of E D to that of F D. Presuming, then, that the amount of power transmitted be equal for the two pinions, and the friction of all other things being equal — being in proportion to the space passed (or in this case slid) over — it is obvious that the internal pinion has the least friction.

Chapter II.— THE TEETH OF GEAR-WHEELS.— CAMS.

Wheel and Tangent Screw or Worm and Worm Gear.

IN Fig. 74 are shown a worm and worm gear partly in section on the line of centres. The worm or tangrent screw W is sim- ply one long tooth wound around a cylinder, and its form may be

determined by the rules laid down for a rack and pinion, the tan- gent screw or worm being considered as a rack, and the wheel as an ordinary spur-wheel.

Worm gearing is employed for transmitting motion at a right angle, while greatly reducing the motion. Thus one rotation of the screw will rotate the wheel to the amount of the pitch of its teeth only. Worm gearing possesses the qualification that, unless of very coarse pitch, the worm locks the wheel in any position in which the two may come to a state of rest, while at the same time the excess of movement of the worm over that of the wheel enables the movement of the latter through a very minute portion of a revolution. And it is evident that, when the plane of rotation of the worm is at a right angle to that of the wheel, the contact of the teeth is wholly a sliding one. The wear of the worm is greater than that of the wheel, because its teeth are in continuous contact, whereas the wheel teeth are in contact only when passing through the angle of action.

If the teeth of the wheel are straight and are set at an angle equal to the angle of the worm thread to its axis, as in Fig. 75, p p representing the pitch line of the worm, c rfthe line of centres, and d the worm axis, the contact of tooth upon tooth will be at the centre only of the sides of the wheel teeth. It is generally pre- ferred, however, to have the wheel teeth curved to envelop a part of the circumference of the worm, as in Fig. I (Plate I.), and increase the line of contact of tooth upon tooth, and thereby pro- vide more ample wearing surface.

In this case the form of the teeth upon the worm wheel varies at every point in its length as the line of centres is departed from. Thus in Fig. 76 (Plate I.) is shown an end view of a worm and a worm gear in section, c d being the line of centres, and it will be readily perceived that the shape of the teeth, if taken on the line ef, will differ from that on the line of centres ; hence the form of the wheel teeth must, if contact is to occur along the full length of the tooth, be conformed to fit to the worm, which may be done by taking a series of sections of the worm thread at varying distances from and parallel to the line of centres, and forming the wheel teeth to the shape so obtained. But if the teeth of the wheel are to be cut to shape, then obviously a worm may be provided with teeth, as shown in Fig. 3 (Plate I.), thus forming what is known as a hob, which is used as shown in Fig. 4 (Plate I.), in which A is the hob mounted on the shaft or arbor E. B is a blank worm wheel, which is shown mounted in a chuck in such a manner that it is

free to revolve. C and D are gear wheels, the former (C) being fast upon the shaft E, and the latter (D) geared with E, while at the same time so arranged in connection with the chuck or spiral head gearing that the wheel B revolves at or during one complete revo- lution of A, a portion of a revolution equal to the pitch of the worm of the hob. It is obvious that friction disks might be used instead of gear wheels, the only essential being that there be no slip be- tween them. Instead of using a continuous cutting worm or hob to produce the teeth on the wheel, a single cutter of the requisite tooth shape, but of the form of an ordinary milling machine cutter, may be used, providing that the mandril or arbor that drives it is set at the requisite angle to the plane of the face of the wheel to be cut.

This angle may be obtained by drawing a line equal in length to the worm circumference, and another line meeting it at one end and distant from it at the other end equal to the worm pitch. The pitch line of the wheel teeth, whether they be straight and are disposed at an angle, as in Fig. 75, or curved, as in Fig. 76, is at a right angle to the line of centres c d ; or, in other words, in the plane o{ g k in Fig. 76. This is evident because the pitch line must be parallel to the wheel axis, being at an equal radius from that axis, and therefore having an equal velocity of rotation at every point in the length of the pitch line of the wheel teeth.

If we multiply the number of teeth by their pitch to obtain the circumference of the pitch circle, we shall obtain the circumfer- ence due to the radius of ^ h from the wheel axis ; and so long as g h is parallel to the wheel axis, we shall by this means obtain the same diameter of pitch circle, so long as we measure it on a line

Fig- 7S-

parallel to the line of centres e d. The pitch of the worm is the same at whatever point in the tooth depth it may be measured, because the teeth curves are parallel one to the other ; thus in Fig. ^^ the pitch measures are equal at m, n, or 0.

But the action of the worm and wheel will nevertheless not be correct unless the pitch line from which the curves were rolled coincides with the pitch line of the wheel on the line of centres ; for although, if the pitch lines do not so coincide, the worm will at each revolution move the pitch line of the wheel through a distance equal to the pitch of the worm, yet the motion of the wheel will not

VOL. I.

WORM AND WORM WHEEL.

PLATE I.

I

Fig. 2.

Fig. 3

lig. I.

rm

t

■1^::::

^tf?"

I I

w

Fig. 4.

MODERN MACHINE SHOP PRACTICE.

THE TEETH OF GEAR-WHEELS.

29

be uniform because, supposing the two pitch lines not to meet, the faces of the pinion teeth will act against those of the wheel, as shown in Fig. 78, instead of against their flanks, and as the faces are not formed to work correctly together the motion will be irregular.

The diameter of the worm is usually made equal to four times

—V

Fig- 77-

the pitch of the teeth, and if the teeth are curved as in figure 76 they are made to envelop not more than 30° of the worm.

The number of teeth in the wheel should not be less than thirty, a double worm being employed when a quicker ratio of wheel to worm motion is required.

When the teeth of the wheel are curved to partly envelop the worm circumference it has been iiund, from experiments made

by Robert Briggs, that the worm and the wheel will be more durable, and will work with greatly diminished friction, if the pitch line of the worm be located to increase the length of face and diminish that of the flank, which will decrease the length of face and increase the length of flank on the wheel, as is shown

Fig- 79-

in Fig. 79; the location for the pitch line of the worm being determined as follows : —

The full radius of the worm is made equal to twice the pitch of its teeth, and the total depth of its teeth is made equal to -65 of its pitch. The pitch line is then