Modern Machine-Shop Practice, Volumes I and II
by Joshua Rose · English · Gutenberg #39225 · 4,774,982 chars · public domain (US)
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Title: Modern Machine-Shop Practice, Volumes I and II
Author: Joshua Rose
Release date: March 21, 2012 [eBook #39225]
Language: English
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| TRANSCRIBER'S NOTES |
| |
| Transcriptions used in this e-text: |
| italics text in the original work is presented here between |
| underscores, as in _text_; |
| bold-face text in the original work is presented here between |
| equal-signs, as in =text=; |
| small-capitals in the original work are presented here as ALL |
| CAPITALS; |
| fractions are transcribed as, for example, 2-1/2 for 2½; where |
| the author uses the form 1-64, this form has been retained, |
| except in tables; |
| superscript texts are transcribed as in ^{text}; |
| subscript texts are transcribed as in _{text}; |
| single Greek letters are transcribed as [alpha], [beta], etc.; |
| the (single) oe-ligature used in the book has been transcribed |
| as oe (Phoenixville); |
| multi-line in-line formulas and calculations from the original |
| work have been transcribed as single-line in-line formulas and |
| calculations, where necessary with the addition of brackets. |
| The following transcriptions are used for special characters and |
| symbols, where x can be any character: |
| [=x] x-macron; |
| [)x] x-breve; |
| [x.] x-dot-below; |
| [<--] left-pointing hand. |
| The author uses letters from a different font to describe shapes.|
| These are transcribed between square brackets: [V] or [V]-shaped,|
| [T] or [T]-shaped, etc. Where the original work uses regular |
| letters for the same purpose, this transcription has not been |
| used. Special cases are __|¯¯ for a stretched S-shape, [/\] for |
| an upside-down V, and [_|_] for an upside-down T. |
| |
| More extensive Transcriber's Notes will be found at the end of |
| this text. |
+------------------------------------------------------------------+
[Illustration: _VOL. I. MODERN MACHINE-SHOP PRACTICE FRONTISPIECE_
_Copyright, 1887 by Charles Scribner's Sons._
=MODERN AMERICAN FREIGHT LOCOMOTIVE.=]
MODERN
MACHINE-SHOP PRACTICE
BY
JOSHUA ROSE, M.E.
ILLUSTRATED WITH MORE THAN 3000 ENGRAVINGS
VOLUME I.
NEW YORK
CHARLES SCRIBNER'S SONS
1887
COPYRIGHT, 1887, BY
CHARLES SCRIBNER'S SONS
Press of J. J. Little & Co.
Astor Place, New York.
PREFACE.
MODERN 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.
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 acknowledgment 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 or
wood; III. The construction and use of hand tools for working metal or
wood; IV. The construction 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 to apprentices, and to owners
and managers of manufacturing establishments equally with their
employees, whether machinists, draughtsmen, wood-workers, engineers, or
operators of special machines.
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
lighten, while making more profitable, the labor of his fellow-workmen.
CONTENTS.
VOLUME I.
CHAPTER I.
=THE TEETH OF GEAR-WHEELS.=
PAGE
=Gear-Wheels.= Spur-wheels, bevel-wheels, mitre-wheels,
crown-wheels, annular or internal wheels 1
Trundle-wheels, rack and pinion-wheel and tangent screw, or
worm and worm-wheel 1
The diameter of the pitch circle of 1
=Gear-Wheel Teeth.= The face, the flank, the depth or height 1
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 epicycloidal 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 10
Graphical demonstration that the curves are correct independent
of either the respective sizes of the wheels, or of the curve
generating circles 11
=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
diameters 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;" definition 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 approach, 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 to be conveyed 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 diameter
of the wheels, the pitch of the teeth, and the diameter 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 16
=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
contact 17
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
approach 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 21
=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 spaces 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 interfere
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 result 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 28
The proper number of teeth in the worm-wheel 29
Locating the pitch line of the worm so as to insure durability 29
Rule for finding the best location for the pitch line of the
worm 29
Increasing the face of the worm to obtain a smoother action 29
=Worms=, to work with a square thread 29
=Worm-Wheels=, applications of 30
=Gear-Wheels= with involute teeth 31 to 34
=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 35
CHAPTER III.
=THE TEETH OF GEAR-WHEELS (Continued).=
=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 system 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 bevel gear-wheel engine or
cutting machine 45
The use of extra circles in marking the curves with compasses 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 values and setting numbers for Professor Robinson's
odontograph 51
Walker's patent wheel scale for marking the curves of cast
teeth 51
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 56
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 number 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 angle 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 69
Herring-bone angular teeth 69
For transmitting motion at a right angle by means of angular or
helical teeth 69
Cutting helical teeth in the lathe 69
For wheels whose shaft axes are neither parallel nor meeting 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 direction 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 different
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 adjustment
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 89
=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 91
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 93
United States standard for gas and steam pipes 93
Taper for standard pipe threads 95
Tables of the pitches and diameters at root of thread, of the
Whitworth thread 95
Table of Whitworth's screw threads for gas, water, and
hydraulic piping 96
Whitworth's standard gauges for watch and instrument makers 96
Screw-cutting hand tools 96
=Thread-Cutting Tools.= American and English forms of stocks
and dies 97
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 the same dies 99
Hobs for hobbing or threading dies 100
Various forms of stocks with dies adjustable to take up the
wear 101
Dies for gas and steam pipes 101
=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 tap for use in a screw machine 107
The alteration of pitch that occurs in hardening 108
Gauging the pitch after the hardening 108
Correcting the errors of pitch caused by the hardening 109
For lead 109
Elliptical in cross section 109
For very straight holes 109
Tap wrenches solid and adjustable 110
=Thread-Cutting.= Tapping 110
Appliances for tapping standard work 111
CHAPTER V.
=FASTENING DEVICES.=
=Bolts=, classification of, from the shapes of their heads 112
Classification of, from the shapes of their bodies 112
Countersunk 112
Holes for, classification of 112
For foundations, various forms of 113
Hook bolts 113
The United States standard for finished bolts and nuts 113
The United States standard for rough bolts and nuts, or black
bolts 114
The Whitworth standard for bolts and nuts 114
=Screws= 114
=Studs= 115
=Set Screws= 115
=Bolts= for quick removal 116
That do not pass through the work 117
That self-lock in grooves and are readily removable 117
Heads and their bedding 117
=Nuts=, the forms of, when they are to be steam tight 118
Various forms of 118
Jamb nuts and lock nuts 119
=Differential Threads= for locking purposes 119
For fine adjustments 119
=Nuts=, taking up the wear of 120
Securing devices 120
Securing by taper pins 121
Securing by cotters 121
Securing by notched plates 121
=Pins.= Securing for exact adjustments 121
And double eyes fitting 121
Fixed 122
Working 122
=Bolts=, removing corroded 122
=Nuts=, removing corroded 122
=Washers=, standard sizes of 122
=Wrench=, the proper angles of 123
Box 124
Monkey 125
Adjustable, various forms of 125
Sockets 125
Novel for carriage bolts 125
Pin 126
Improved form of 126
=Keys=, the various kinds of 126
The bearing surfaces of 126
=Set Screws=, application of, to hubs or bosses 127
=Keys=, with set-screws 127
The draught of 127
=Feathers=, and their applications 127
=Keys=, for parallel rods 128
=Taper Pins=, proper position of, for locking purposes 128
Improved method of fitting 128
CHAPTER VI.
=THE LATHE.=
=Lathe=, the importance and advantages of 129
Classification of lathes 129
Foot 130
Methods of designating the sizes of 130
Bench 130
Power 130
Hand 130
Slide Rest for 131
American form of, their advantages and disadvantages 132
English forms of 132
For spherical work 132
Methods of taking up lost motion of 133
=Engine Lathe=, general construction of 133
The construction of the shears of 134
Construction of the headstock 134
Construction of the bearings 134
Construction of the back gear 135
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 137
=Lathe Apron=, Construction of the feed traverse 138
Construction of the cross-feed motion 138
=Engine Lathe=, lead screw and change wheels of 139
Feed spindle and lead screw bearings 139
Swing frame for lead screw 139
Lead screw nuts 140
With compound slide rest 140
Construction of compound slide rest 141
Advantages of compound slide rest 141
For taper turning 142
Taper-turning attachments 142
With compound duplex slide rest 143
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 148
=Self-Acting Lathe=, English form of 148
=Pattern Maker's Lathe= 148
Brake for cone pulley 149
With wooden bed 149
Slide rest for 149
=Chucking Lathe=, English 149
Feed motions of 150
=Pulley Lathe= 150
=Gap or Break Lathe= 151
=Extension Lathe= 151
=Wheel Lathe= 151
=Chucking Lathe= for boring purposes 152
=Lathe= for turning crank axles 152
Construction of the headstock 153
Construction of the feed motions 154
For turning crank, Arrangements of the slide rests 154
Application of the slide rest to a crank 155
CHAPTER VII.
=DETAILS IN LATHE CONSTRUCTION.=
=Live Spindle= of a lathe, the fit of 157
With coned journals 157
Methods of taking up the end motion of 158
Arranging the swing frame for the change gears 158
Taking up the wear of the back bearing 158
The wear of the front bearing of 158
=The Taper= for the live centre 159
=Methods= of removing the lathe centres 159
=Tapers= for the live centres 159
=Methods= of removing the dead centre 159
=Driving Cone=, arranging the steps of 159
Requirements of proportioning the steps of 159
Rules for proportioning the diameters of the steps of, when the
two pulleys are exactly alike and are connected by an open
belt 159 to 161
When the two pulleys are unlike 161 to 164
=Back Gear=, methods of throwing in and out 165
=Conveying= motion to the lead screw 165
=Attaching= the swing frame 166
=Feed Gear.= Arrangement for cutting worm threads or tangent
screws 167
=Feed Motion= for reversing the direction of tool traverse
in screw cutting 168
For lathe aprons 168
=Slide Rest=, weighted elevated 168
Double tool holder for 169
Gibbed elevating 169
=Examples= of feed motions 170
=Feed Regulators= for screw cutting 171
The star feed 172
=Ratchet 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 182
Examples of various forms of 183
=Lathe Shears= with one V and one flat side 183
Methods of ribbing 184
The arrangement of the legs of 184
=Lathe Tailblock= 185
With rapid spindle motion 185
With rapid fastenings and releasing devices 185
The wear of the spindles of 185
Spindles, the various methods of locking 186
Testing, various methods of 187
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 Manufacturers' Lathe= 190
Special chucks for 190
Pump centre rest 190
=Lathe=, hand 191
Screw slotting 192
With variable speed for facing purposes 192
Cutting-off machine 193
Grinding Lathes 193
With elevating rest 194
Universal 195
Special chucks for 196
The Morton Poole calender roll grinding lathe 196
The construction of the bed and carriages 197
Principles of action of the carriages 197, 198
Construction of the emery-wheel arbors and the driving
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
Special, for pulley turning 211
=Boring and Turning= mill or lathe 211
Construction of the feed motions 213
Construction of the framing and means of grinding the lathe 214
Construction of the vertical feed motions 215
=The Morton Poole= roll turning lathe 215
Construction of the slide rest 216
The tools for 216
=Special Lathes= for brass work 216, 217
=Boring Lathe= with traversing spindle 218
For engine cylinders 219
Cylinder, with facing slide rests 219
With double heads and facing rests 220
=Lathe for turning Wheel= hubs 221
CHAPTER IX.
=DRIVING WORK IN THE LATHE.=
=Drivers=, carriers, dogs, or clamps, and their defects 222
Lathe clamps 222
Equalizing drivers 223
The Clements driver 223
Driver and face plate for screw cutting 223
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 225
=Centres= for hollow work 226
For taper work 226
=Lathe Mandrels=, or arbors 227
Drivers for 227
For tubular work 227
Expanding mandrels 227
With expanding cones 228
With expanding pieces 228
Expanding, for large work 228
For threaded work 228
For nuts, various forms of 229
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 233
Follower rests 234
=Chucks and Chucking= 234
Simple forms of chucks 234
Adjustable chucks for true work 235
Two-jawed chucks 236
Box body chucks 237
Reversible jawed chucks 237
Three and four-jawed chucks 237
Combination chucks 237
The wear of scroll chuck threads 237
Universal chucks 238
The wear of chucks 240
Special forms of chucks 241
Expanding chucks for ring-work 241
Cement chuck 241
Chucks for wood-working lathes 242
=Lathe Face Plates= 243
Face plates, errors in, and their effects 243
Work-holding straps 244
Face plate, clamping work on 245
Forms of clamps for 245
Examples of chucking work on 246, 247
For wood work 247
=Special Lathe Chuck= for cranks 248
=Face Plate Work=, examples of 249
Errors in chucking 250
Movable dogs for 250
The angle plate 251
Applications of 251
Angle plate chucking, examples of 251
Cross-head chucking 251-253
CHAPTER X.
=CUTTING TOOLS FOR LATHES.=
=Principles= governing the shapes of lathe tools 254
=Diamond-pointed=, or front tool 254
=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
keenness 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
diameters of work 257
=Round-nosed= tools 258
=Utmost Duty= of cutting tools 258
Judging the quantity of the tool from the shape of its cutting 259
=Square-nosed= tools 260
The height of lathe tools 260
Side 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= 268
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 273
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= 274
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
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 281
For very true work 281
=Chucking Reamer= 281
The number of teeth for reamers 282
Spacing the teeth of reamers 282
Spiral teeth for reamers 282
Grinding the teeth of reamers 282
Various positions of emery-wheel in grinding reamers 282
Chucking reamers for true work 283
Shell reamers 283
Arbor for shell reamers 283
Rose-bit or rose reamers 283
Shell rose reamers 284
Adjustable reamers 284
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 286
The spring of boring tools 286
Boring tools for small work 287
Boring tool holders 287
=Boring Devices for Lathes= 288
=Boring Heads= 288
=Boring Bars= 289
Boring bar cutters 289
Three _versus_ four cutters for boring bars 290
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 301
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
Recentring turned work 304
=Straightening Work.= Straightening machine for bar iron 304
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 removing 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
compounded 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
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 331
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
Screw cutting with hand tools 334
Outside and inside chasers 334
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
=Circle=, division of the 348
Troughton'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
=Classification= of the measuring tools used by workmen 354
=Micrometer Caliper= and its principle of construction 354, 355
=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, 362
Keyway calipers 363
The advantages of calipers 363
=Fitting.= The four kinds of fit in machine work 363
The influence of the diameter of the work in limiting the
application of standard gauges 363
The wear of tools and its influence 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
=Fitting.= Practicable methods of testing the fit of axle
brasses forced in by hydraulic pressure 366
Shrinkage or contraction fits 366
Allowances for 366
Gauge for 367
The shrinkage system at the Royal Gun Factory at Woolwich 367
Experiments by Thomas Wrightson upon the shrinkage of iron
under repeated heatings and coolings 368 to 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, 380
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, &c. 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 387
American sheet zinc 387
=Rifle Bore= gauge 387
=Strength of Wire=, Kirkaldy's experiments 387, 388
CHAPTER XVI.
=SHAPING AND PLANING MACHINES.=
=General description= of a shaping machine 389
=Construction= of swivel head 389
Slide 390
Vice chuck 390
Feed motion 390
=Hand= shaping machine 392
=Quick Return Motion=, Whitworth's 392
=Vice Chucks=, the principles of construction of plain, for
planing machine 392
The proper methods of chucking work in 393
Holding taper work in 394
Various forms of 394
Swiveling 395
Rapid motion 396
For vice work 396
=Centres= for shaping machines 397
=Traveling Head= in shaping machine 397
=Planer Shapers= or shaping machines, having a tappet motion
for reversing the direction of motion 398, 399
=Quick Return Motion= shaping machines, link 399
The Whitworth 400
Comparisons of the link motion and Whitworth 401
=Simple Crank=, investigating the motion of 401
=Planing Machines=, or planer 402
The 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 406
The belt shifting mechanism 406, 407
The automatic feed motions 408
=Sliding Head= 408
=Cross Bar= 409
=Slides of Planers=, the various forms of construction of 410
=Wear of the Slides= of planer heads, various methods of
taking up the 410
=Swivel Heads= 411
=Tool Aprons= 411
=Swivel Tool-holding devices= for planers 411
=Planer Heads=, graduations of 412
Safety devices for 413
Feed motions for 414
V-guideways for 414
Flat guideways for 415
Oiling devices for 415
=Planing Machine Tables= 415
Slots and holes in planing machine tables 416
Forms of bolts for planer tables 417
Supplementary tables for planer tables 417
Angle plates for planer tables 418
Chucking devices for planer tables 418
=Planer Centres= 418
=Planer Chucks= 419
For spiral grooved work 419
For curved work 420
Chucking machine beds on planer tables 420
For large planing machines 422
Chucking the halves of large pulleys on a planer 423
=Gauges= for planing V-guideways in machine beds 421
Planing guideways in machine beds 422
Gauge for planer tools 424
=Planer Tools=, the shapes of 424
For coarse finishing feeds 424
The clearance of 424
For slotted work 424
=Planer Tool Holder=, with tool post 425
Various applications of 425
Simple and advantageous form of 426
Examples of application of 426
CHAPTER XVIII.
=DRILLING MACHINES.=
=Drilling Machines.= General description of a power drilling
machine 428
Lever feed 428
With automatic and quick return feed motions 428
Improved, with simple belt and uniform motion, two series of
rates of automatic feed, and guide for boring bar 429, 430
Radial 430, 431
For boiler shells 436
Cotter or keyway 438
Drilling Machine, three-spindle 434
Four-spindle 434
=Drilling and Boring= machine 431
Feed motion of 432
=Combined Drilling Machine= and lathe 433
=Boring Machine=, horizontal 433
For car wheels 438
For pulleys 438
=Quartering Machine= 434
=Drilling and Turning Machine= for boiler makers 435
Feed motions of 436
CHAPTER XIX.
=DRILLS AND CUTTERS FOR DRILLING MACHINES.=
=Jigs or Fixtures= for drilling machines 439
Limits of error in 439
Examples of, for simple work, as for links, &c. 440
Considerations in designing 440
For drilling engine cylinders 440 to 441
For cutting out steam ports 441
=Drills and Cutters= for drilling machines 442
Table of sizes of twist drills, and their shanks 442
Flat drills for drilling machines 442
Errors in grinding flat drills 443
The tit-drill 443
The lip drill 443
Cotter or keyway drills 446
=Drilling holes= true to location with flat drills 444
Drilling hard metal 444
Table of sizes of tapping holes 445
=Drill Shanks= and sockets 445
Improved form of drill shank 446
Square shanked drills and their disadvantages 446
=Drill Chucks= 446
=Stocks and Cutters= for drilling machines 447
Tube plate cutters 448
=Stocks and Cutters.= Adjustable stock and cutter 448
Facing tool with reamer pin 449
Counterbores for drilling machines 449
Drill and counterbore for wood work 449
Facing and countersink cutters 449
Device for drilling square holes 450
Device for drilling taper holes in a drilling machine 451
CHAPTER XX.
=HAND-DRILLING AND BORING TOOLS, AND DEVICES.=
=The Brad-awl= 452
=Bits.= The gimlet bit 452
The German bit 452
The nail bit 452
The spoon bit 452
The nose bit 453
The auger bit 453
Cook's auger bit 453
Principles governing the shapes of the cutting edges of auger
bits 453
Auger bit for boring end grain wood 453
The centre bit 454
The expanding bit 454
=Drills.= Drill for stone 454
The fiddle drill 455
The fiddle drill with feeding device 455
Drill with cord and spring motion 455
Drill stock with spiral grooves 455
Drill brace 455
Drill brace with ratchet motion 456
Universal joint for drill brace 456
Drill brace with multiplying gear and ratchet motion 456
Breast drill with double gear 456
Drilling levers for blacksmiths 457
Drill cranks 457
Ratchet brace 457
Flexible shaft for driving drills 458
Drilling device for lock work 459
Hand drilling machine 459
=Slotting Machine= 459
Sectional view of 460
Tool holders 460, 461
Tools 461, 462
CHAPTER XXI.
=THREAD-CUTTING MACHINERY AND BROACHING PRESS.=
=Pipe Threading=, die stock for, by hand 463
Die stock for, by power 463
Pipe threading machines, general construction of 463
=Bolt Threading= hand machine 464
With revolving head 465
Power threading machine 465
With automatic stop motion 466
Construction of the head 466
Construction of the chasers 466
Bolt threading machine with back gear 467
Single rapid bolt threading machine 467
Double rapid bolt threading machine 467
Construction of the heads of the rapid machines 468
Bolt threading machinery, the Acme 468
Construction of the head of 468 to 470
Capacity of 470
=Cutting Edges= for taps, the number of 471
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 hobbing, 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 to 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.
_Facing_
_Frontispiece._ MODERN LOCOMOTIVE ENGINE. TITLE PAGE
PLATE I. TEMPLATE-CUTTING MACHINES FOR GEAR TEETH. 34
" II. FORMS OF SCREW THREADS. 85
" III. MEASURING AND GAUGING SCREW THREADS. 93
" IV. END-ADJUSTMENT AND LOCKING DEVICES. 120
" V. EXAMPLES IN LATHE CONSTRUCTION. 148
" VI. CHUCKING LATHES. 150
" VII. TOOL-HOLDING AND ADJUSTING APPLIANCES. 174
" VIII. WATCHMAKER'S LATHE. 188
" IX. DETAILS OF WATCHMAKER'S LATHE. 188
" X. EXAMPLES OF SCREW MACHINES. 200
" XI. ROLL-TURNING LATHE. 215
" XII. EXAMPLES IN ANGLE-PLATE CHUCKING. 252
" XIII. METHODS OF BALL-TURNING. 325
" XIV. STANDARD MEASURING MACHINES. 341
" XV. DIVIDING ENGINE AND MICROMETER. 354
" XVI. SHAPING MACHINES AND TABLE-SWIVELING DEVICES. 398
" XVII. EXAMPLES OF PLANING MACHINES. 404
" XVIII. EXAMPLES IN PLANING WORK. 422
" XIX. LIGHT DRILLING MACHINES. 428
" XX. HEAVY DRILLING MACHINES. 430
" XXI. EXAMPLES IN BORING MACHINERY. 434
" XXII. BOILER-DRILLING MACHINERY. 436
" XXIII. NUT-TAPPING MACHINERY. 475
MODERN
MACHINE SHOP PRACTICE.
CHAPTER I.--THE TEETH OF GEAR-WHEELS.
A wheel that is provided with teeth to mesh, engage, or gear with
similar teeth upon another wheel, so that the motion of one may be
imparted to the other, is called, in general terms, a gear-wheel.
[Illustration: Fig. 1.]
When the teeth are arranged to be parallel to the wheel-axis, as in Fig.
1, the wheel is termed a spur-wheel. In the figure, A 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.
[Illustration: Fig. 2.]
[Illustration: Fig. 3.]
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 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, 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; 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_,
_spindle_, or _round_, and by some a _leaf_.
[Illustration: Fig. 4.]
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.
[Illustration: Fig. 5.]
[Illustration: Fig. 6.]
When the teeth of a wheel are inserted in mortises or slots 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 sometimes 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 of 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.
[Illustration: Fig. 7.]
When the teeth of wheels engage to the proper distance, which is when
the pitch circles meet, they are said to 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 _pitch_, as it is called.
The designations of the various parts or surfaces of a tooth of a
gear-wheel are represented in Fig. 9, 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 distance from the pitch
line to the root of the tooth, and C the 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 P P. 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, P P is called the pitch line, but when the whole wheel is
referred to it becomes the pitch circle.
[Illustration: Fig. 8.]
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. 10, which is called the arc or circular pitch,
and across B, which is termed the chord pitch.
[Illustration: Fig. 9.]
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 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 from 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 explained 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.
[Illustration: Fig. 10.]
The first is objectionable because it involves a calculation to find the
pitch of the teeth; furthermore, if this calculation be 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; (sine of 15
is .25881). Then .25881 × 2 = .51762 × 4 (= radius of wheel) = 2.07048
inches = chord pitch.
TABLE OF NATURAL SINES.
+--------+--------++--------+--------++--------+--------+
|Degrees.| Sine. ||Degrees.| Sine. ||Degrees.| Sine. |
+--------+--------++--------+--------++--------+--------+
| 1 | .01745 || 16 | .27563 || 31 | .51503 |
| 2 | .03489 || 17 | .29237 || 32 | .52991 |
| 3 | .05233 || 18 | .30901 || 33 | .54463 |
| 4 | .06975 || 19 | .32556 || 34 | .55919 |
| 5 | .08715 || 20 | .34202 || 35 | .57357 |
| 6 | .10452 || 21 | .35836 || 36 | .58778 |
| 7 | .12186 || 22 | .37460 || 37 | .60181 |
| 8 | .13917 || 23 | .39073 || 38 | .61566 |
| 9 | .15643 || 24 | .40673 || 39 | .62932 |
| 10 | .17364 || 25 | .42261 || 40 | .64278 |
| 11 | .19080 || 26 | .43837 || 41 | .65605 |
| 12 | .20791 || 27 | .45399 || 42 | .66913 |
| 13 | .22495 || 28 | .46947 || 43 | .68199 |
| 14 | .24192 || 29 | .48480 || 44 | .69465 |
| 15 | .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 1 inch will accord with an arc pitch of 3.1416; 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 1/2, 1/3, 1/4, 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 1/2, 1/3, or
1/4, 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 3.1416 ÷
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 3.1416 inches being
represented by a diametral pitch of 1, a diametral pitch of 1/2 (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 1/5) we have 1/5 × 3.1416 = .628 = the arc
pitch. Second (discarding the numerator), we have 3.1416 ÷ 5 = .628 =
arc pitch. If we are given an arc pitch to find a corresponding
diametral pitch we again simply divide 3.1416 by the given arc pitch.
Example.--What is the diametral pitch of a wheel whose arc pitch is
1-1/2 inches? Here 3.1416 ÷ 1.5 = 2.09 = 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 3.1416 is to 1, so is
1.5 to the required diametral pitch; then 3.1416 × 1 ÷ 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 × 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 3-2/10
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 ÷ 8 = 5, and its full diameter
would be 5-1/4 inches, because the diametral pitch is 1/8, and this
multiplied by 2 gives 1/4, which added to the pitch circle diameter of 5
inches makes 5-1/4 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 4/10 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 | 1.57 | 1.75 | 1.79 |
| 2.25 | 1.39 | 1.5 | 2.09 |
| 2.5 | 1.25 | 1.4375 | 2.18 |
| 2.75 | 1.14 | 1.375 | 2.28 |
| 3 | 1.04 | 1.3125 | 2.39 |
| 3.5 | .890 | 1.25 | 2.51 |
| 4 | .785 | 1.1875 | 2.65 |
| 5 | .628 | 1.125 | 2.79 |
| 6 | .523 | 1.0625 | 2.96 |
| 7 | .448 | 1.0000 | 3.14 |
| 8 | .392 | 0.9375 | 3.35 |
| 9 | .350 | 0.875 | 3.59 |
| 10 | .314 | 0.8125 | 3.86 |
| 11 | .280 | 0.75 | 4.19 |
| 12 | .261 | 0.6875 | 4.57 |
| 14 | .224 | 0.625 | 5.03 |
| 16 | .196 | 0.5625 | 5.58 |
| 18 | .174 | 0.5 | 6.28 |
| 20 | .157 | 0.4375 | 7.18 |
| 22 | .143 | 0.375 | 8.38 |
| 24 | .130 | 0.3125 | 10.00 |
| 26 | .120 | 0.25 | 12.56 |
+----------------+----------+----------+----------------+
From this table we find that the nearest diametral pitch that will
correspond to an arc pitch of 4/10 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 to find
the circumference of the pitch circle, and divide this by the nearest
arc pitch that would divide the circumference 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, however, 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 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. 11, in which A, B, C, E are four wheels in gear, B and C being
compounded, 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 conversely that the velocity will be increased when the
driver is of greater radius than the follower of the compound pair.
[Illustration: Fig. 11.]
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 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 3.1416 feet
(that being the circumference of the pulley). Now suppose that the
addendum circle of either of the wheels were cut off 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 3.1416 feet as before.
It is obvious, however, that the addendum circle must be sufficiently
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 addendum of the
tooth or teeth equals 3/10 of the pitch, and its addendum circle
measures 19.943 inches; what is the diameter of the pitch circle? Here
the addendum on each side of the wheel equals (3/10 of 3 inches) = .9
inches, hence the .9 must be multiplied by 2 for the two sides of the
wheel, thus, .9 × 2 = 1.8. Then, diameter of addendum circle 19.943
inches less 1.8 inches = 18.143 inches, which is the diameter of the
pitch circle.
Proof.--Number of teeth = 19, arc pitch 3, hence 19 × 3 = 57 inches,
which, divided by 3.1416 (the proportion of the circumference to the
diameter) = 18.143 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 proportion 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 1. 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 the other 50 teeth, the
distance between their centres is 18 inches; what is the pitch radius of
each wheel?
Here 100 ÷ 50 = 2, and 2 + 1 = 3. Then 18 ÷ 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 32-1/2 inches; what are the radii
of the respective pitch circles? 90 ÷ 40 = 2.25 and 2.25 + 1 = 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.--Radius of small wheel = 10 × 4 = 40
---- ----
radius of large wheel = 22.5 × 4 = 90.0
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 + 6 = 18) between centres (or line of
centres, as it is termed) has become 18-3/8 inches. Then the acting
effective or operative radii of the pitch circles will bear the same
proportion to the 18-3/8 as the numbers of teeth in the respective
wheels, and will be 12.25 for the large, and 6.125 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
+ 1 = 3. Then 18.375 ÷ 3 = 6.125 = pitch radius of small wheel, and
18.375 - 6.125 = 12.25 = 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 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 1/4
or 1/2 inch at a step, and a pitch measuring about, say, 1-1/4 inch
chord pitch, would be known to be 1-1/4 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 1 inch would be 1-1/8,
or more often 1-1/4 inch, and the difference between the arc and chord
pitch of the smallest wheel would not amount to anything near 1/8 inch,
hence there would be no liability to mistake a pitch of 1-1/8 for 1 inch
or _vice versâ_. 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 diameter is reduced, the
difference between the arc and chord pitch increases, and further the
steps by which fine pitches advance are more minute (as 1/4, 9/32, 5/16,
&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
3.1416, and the diameter may be obtained by dividing the circumference
by 3.1416.
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 revolutions, 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 compounded 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 C = 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, making 100 lbs. at the centre of E.
Then 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 3-33/100 lbs., which will be the force exerted
by A on the perimeter of B. Now from the perimeter of B to its centre
(or in other words its radius) is 15 inches, hence the 3-33/100 lbs. at
its perimeter will become fifteen times as much at the centre G of B,
and 3-33/100 × 15 = 49-95/100 lbs. From the centre G to the perimeter of
C being 20 inches, the 49-95/100 lbs. at the centre will be only
one-twentieth of that amount at the perimeter of C, hence 49-95/100 ÷ 20
= 2-49/100 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 transmitted 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 3-33/100, which multiplied by 30 (the number of
teeth on B) becomes 99-90/100, which divided by 40 (the number of teeth
on C) becomes 2-49/100 lbs. as before.
[Illustration: Fig. 12.]
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 therefore 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 1 revolution per minute, how many will C
make, A having 60 teeth, B 30 teeth, and C 40 teeth? In this case we
have but one driving wheel A, and one driven wheel B, the driver having
60 teeth, the driven 30, hence 60 ÷ 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.
Suppose that the 4 wheels in Fig. 14 have the respective numbers of
teeth marked 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:--
Revolu- Teeth Teeth Teeth Teeth
tions. in first in first in second in second
driver. driven. driver. driven.
60 × 40 ÷ 60 × 20 ÷ 120 = 6-66/100
and a remainder of the reciprocating decimals. We may now prove this by
reversing the question, thus. Suppose the 120 wheel to make 6-66/100
revolutions per minute, how many will the 40 wheel make?
Revolu- Teeth Teeth Teeth Teeth
tions. in first in first in second in second
driver. driven. driver. driven.
6.66 × 120 ÷ 20 × 60 ÷ 40 = 59-99/100 =
revolutions of the 40 wheel, the discrepancy of 1/100 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:--
[Illustration: Fig. 13.]
Referring again to Fig. 10, it has been shown that with a 50 lb. weight
suspended from a 4 inch shaft E, there would be 30-33/100 lbs. at the
perimeter of A. Now suppose a rotation be made, then the 50 lb. weight
would fall a distance equal to the circumference of the shaft, which is
(3.1416 × 4 = 12-56/100) 12-56/100 inches. Now the circumference of the
wheel is (60 dia. × 3.1416 = 188-49/100 cir.) 188-49/100 inches, which
is the distance through which the 3-33/100 lbs. would move during one
rotation of A. Now 3.33 lbs. moving through 188.49 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 1 lb.), bearing in mind that
there will be a slight discrepancy due to the fact that the fractions
.33 in the one case, and .56 in the other are not quite correct. Thus:
188.49 inches × 3.33 lbs. = 627.67 inch lbs., and
12.56 " × 50 " = 628 " "
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. × 3.1416 = 125.666 cir., and
125.666 × 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 ×
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.
[Illustration: Fig. 14.]
Gear-wheels therefore form a convenient method of either simply
transmitting motion or power, as when the wheels are 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.
[Illustration: Fig. 15.]
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 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 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 B 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 proportion 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 proportion to that of the other
irrespective of the actual diameter of either of them.
[Illustration: Fig. 16.]
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 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.
[Illustration: Fig. 17.]
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 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 undue 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.
[Illustration: Fig. 18.]
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 epicycloid, 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 circumference of another 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
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 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.
[Illustration: Fig. 19.]
A hypocycloid is a curve traced or generated by a point on the
circumference of a circle rolling 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 its circumference
will trace the two curves, C C, both curves starting from the same
point, the upper having been traced by rolling the generating circle or
wheel A in one direction and the lower curve by rolling it in the
opposite direction.
[Illustration: Fig. 20.]
To demonstrate that by the epicycloidal and hypocycloidal curves,
forming the faces and flanks of what are known as epicycloidal 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 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 by reason of the contact of their circumferences, it is
evident that the length of the arc E G must be equal to that of the arc
G F, and that the motion of A (supposing it to be the driver) would be
communicated uniformly to B.
[Illustration: Fig. 21.]
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 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.
[Illustration: Fig. 22.]
A represents the pinion (or lantern), and B the wheel, and C, C, the
primitive teeth reduced in thickness to receive the pins on 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 operation maintained the same epicycloidal
curves, but brought them closer together and made them shorter. It is
obvious, however, that such a method of communicating rotary motion is
unsuited to the transmission of much power; because of the weakness of,
and small amount of wearing surface on, the points or rungs in A.
[Illustration: Fig. 23.]
[Illustration: Fig. 24.]
In place of points or rungs we may have radial lines, these 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 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.
[Illustration: Fig. 25.]
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 tooth brought closer together
to give room for the reception of the teeth upon A. We have here a pair
of gears that possess sufficient strength and are capable of working
correctly in either direction.
[Illustration: Fig. 26.]
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 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 corresponding 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 O´, N´, corresponding to
O, 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 generating, 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.
[Illustration: 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_ is 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 line 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 to wheel Q a pencil whose point is at _n_. If then rotation be
given to _a_ _a_ in the direction of the arrow _s_, all three wheels
will rotate in that direction as denoted by their respective arrows _s_.
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 outside 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 prolonged respectively to _p_ 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 O
_f_, and O _g_, curve O _f_ being the same as _m_ _x_ placed in a new
position, and O _g_ being the same as _m_ _y_, but placed in a new
position. Now since both these curves (O _f_ 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
_n_ 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 O
_f_ and O _g_ 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 at some point in
their lengths, it is evident that if instead of being mere lines, _m_
_y_ was the face of a tooth on _a_ _a_, and _m_ _x_ was the flank of a
tooth on _b_ _b_, the same uniform motion may be transmitted from _a_
_a_, to _b_ _b_, by pressing the tooth face _m_ _y_ against 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.
[Illustration: Fig. 28.]
In this illustration the diameter of the rolling or describing circle Q,
being less than the radius of the wheels _a_ _a_ or _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 describing, rolling, or curve-generating circle is
equal in diameter 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 _o_, it will have prolonged _m_ _y_ into _o_ _g_ and _x_ into
_o_ _f_, 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 _n_ _f_ or the arc _n_ _o_.
[Illustration: Fig. 29.]
[Illustration: Fig. 30.]
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 rolling or describing circle. When the describing 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_. 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_, _n_ _f_, and
_n_ _g_, are of equal length.
[Illustration: 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 D the generating circle. The wheel and
rack are shown with teeth _n_ on one side simply for 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.
[Illustration: Fig. 32.]
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 rolled around _a_ _a_ until it
had reached the position marked 1, then it will have marked the curve
from _e_ to _n_, 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 from _f_ 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 at _f_ 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 our
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
circumference 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.
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 B at B. 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 by the dotted teeth, nor would teeth
I and K if in the position denoted by their dotted lines.
[Illustration: Fig. 33.]
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 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 generating 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.
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 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
theoretically perfect motion so far as uniformity of velocity is
concerned, or, in other words, the motion of the driver will be
transmitted 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 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 C of the teeth on W, and the
motion will again be correct, providing that the same diameter (whatever
it may be) of rolling 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 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 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 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.
[Illustration: Fig. 34.]
[Illustration: Fig. 35.]
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 for the face of the teeth on
the largest wheel of the pair and for the 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 the large one, a generating circle whose
diameter equals the radius of the large wheel.
[Illustration: Fig. 36.]
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, N and O; P representing
the pitch circles, and Q representing the circles for the roots of the
teeth. Let X and Y represent the lines of centres, and A, H, I and K the
generating or rolling circle, whose centres are on the respective lines
of centres--the diameter of the generating 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 pitch 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 R and
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.
[Illustration: Fig. 37.]
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 circles of two wheels, then the generating circle
would be rolled within B, as at 1, 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 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 throughout. 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 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.
[Illustration: Fig. 38.]
In determining the diameter of a generating circle for a set or train
of wheels, we have the consideration that the smaller the diameter of
the generating circle in proportion to that of the 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 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, E.
This thrust 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.
[Illustration: Fig. 39.]
[Illustration: Fig. 40.]
What is known as the Willis system for interchangeable gearing, 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.
[Illustration: Fig. 41.]
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 revolutions 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 arc, or path, of contact is
longer as the generating circle is made larger.
[Illustration: Fig. 42.]
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 flanks
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 thickened 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 Ls, 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 therefore 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 respective 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 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 paths of the lowest points of flank contact,
points F and G, marking the initial lowest contact for the two
directions of revolution.
[Illustration: Fig. 43.]
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.
In Fig. 44 is represented portion of a pair of wheels corresponding 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 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, C
N, and that of sliding is the difference between the distance C N and
the length of the tooth face.
[Illustration: Fig. 44.]
If now we examine the distance C N in Fig. 42, we find that 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 correspondingly diminished the
sliding action of the tooth contact. But at 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.
[Illustration: Fig. 45.]
In an article by Professor Robinson, attention is called to the fact
that if the teeth of wheels are not formed to have correct 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 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 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 L; 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 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 L
M. 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 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.
[Illustration: Fig. 46.]
[Illustration: Fig. 47.]
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 considered most
suitable to the particular conditions; as, for example, 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.
[Illustration: Fig. 48.]
It may now be pointed out that the degrees of angle that the teeth move
through always exceeds the number of degrees of angle contained in the
paths of contact, or, in other words, exceeds the degrees contained in
the arcs of approach and recess combined.
[Illustration: Fig. 49.]
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 degrees of angle the pinion has been moved
through are therefore 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 contact is as many degrees as are
contained in one-half the thickness 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.
[Illustration: Fig. 50.]
With a given diameter of pitch circle and pitch diameter of wheel, the
length of the arc of contact will be influenced by the 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 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.
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. 33 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 proportion 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 importance.
[Illustration: Fig. 51.]
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.
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 time, it suffers the most from
wear, and hence 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 pitch finer, giving to a wheel, of a given
diameter, a greater number of teeth.
[Illustration: Fig. 52.]
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 Fig. 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 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.
[Illustration: Fig. 53.]
[1]This form is undesirable in that there is contact on 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 _a_ _b_. A parallel line _c_ _d_ will work at a
constant distance as at _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.
[1] From an article by Professor Robinson.
[Illustration: Fig. 54.]
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,
Fig. 53 may be regarded as representing that form of toothed gear which
will operate with less friction than any other known form.
[Illustration: Fig. 55.]
This statement is intended to cover fixed teeth only, and not that
complicated form of the trundle wheel in which the cylinder teeth are
friction rollers. No doubt such would run still easier, even with their
necessary one-sided contacts. Also, the statement 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 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 equal to that of B
their circumferences will be equal, and the angular and velocity ratios
will therefore be equal.
[Illustration: Fig. 56.]
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 intersection 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 radiating from E (which
is the common centre of motion for the two shafts) the circumference C
is less than that of D, rendering a rolling contact impossible.
[Illustration: Fig. 57.]
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 circumference 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.
[Illustration: Fig. 58.]
[Illustration: Fig. 59.]
The principles governing the forms of the cones on which the teeth are
to be located thus being explained, we may now consider 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
_d_), 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 epicycloid. 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
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 difficult 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:--
[Illustration: Fig. 60.]
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, 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
addendum and _c_ for the roots of the teeth, so that from _b_ to _c_
will represent the height of the tooth at that end.
[Illustration: Fig. 61.]
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 _p_, 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_, _l_, _m_, 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 of _d_ mark _f_ for the roots of the teeth at that end. The
distance between the dotted lines (as _p_) represents the full height of
the teeth, hence _h_ meets line _p_, being the root of tooth for the
large wheel, and to give clearance, the point of the pinion teeth is
marked below, thus arc _b_ does not meet _h_ or _p_. 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 _e_ _f_, 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_, _l_, _m_.
[Illustration: Fig. 62.]
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 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 few teeth the wheel may contain to work with a
given pinion. Now it is found that although the curves of the teeth in
internal 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 overlap teeth D, E, and F, and would therefore
prevent the 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 sufficient 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.
[Illustration: Fig. 63.]
By the introduction into the figure of a segment of a spur-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 therefore 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 _Engineering_ how to accomplish this in a simple
and yet exceedingly ingenious manner which may be described as
follows:--
[Illustration: Fig. 64.]
It is required to find a describing circle that will roll the 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.
[Illustration: Fig. 65.]
[Illustration: Fig. 66.]
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 may 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 whose 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, 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
intermediate 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 describing
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 rolled on the outside of the pinion and on the
outside of the annular wheel we may distinguish it as the exterior
describing circle.
[Illustration: Fig. 67.]
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
may be distinguished as the interior describing 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.
[Illustration: Fig. 68.]
[Illustration: Fig. 69.]
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 interior and
exterior describing circles derived from it give teeth that not only
engage properly and avoid the interference shown in Fig. 62, but that
will also have an additional arc of 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. Supposing 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
dotted portion of the exterior describing circle as in ordinary gearing.
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 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 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 interior and exterior
describing circles will vary in proportion, the interior becoming
smaller and the exterior larger, while from the very mode of
const
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