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Editor of “Practical Engineering * 0 


Second Edition 



First Published . 
Second Edition . 

. October 1940 
. April 1941 



Gear design and methods of cutting gears have progressed 
considerably during the last thirty years. Accurate gears 
are so essential to a Vide variety of machine tools and 
mechanisms that the time seemed opportune to produce 
this practical work dealing with gear design, types, functions 
and methods of cutting them, together with formulae 
necessary for the designer and the operator. 

Additional material deals with the making of end-mills, 
hobs, and gear-generating cutters. The old method of 
cutting spur gears by means of a rotary milling cutter, one 
tooth at a time, is still used in some shops and, therefore, 
is included in this book. Gear measurement is an important 
aspect of inspection, which is also dealt with. 

This book includes the articles on gear-cutting contri¬ 
buted to Practical Engineering by W. A. Tuplin, D.Sc. 

F. J. CAMM. 



I. Types, Cutting Methods and Terms . , 9 

II. Bevel Gears ..... 29 

HI. Worm Gears ..... 34 

IV. Gear Generation . . . .41 

V. Gear-Wheel Forms - . . . .66 

VI. Epicyclic Gear Trains . . .71 

VII. Methods of Mounting . . . .74 

VIII. Measuring Gears . . . .78 

IX. Hobs, End-Mills and Generating Cutters ioo 

X. Load Capacity of Gears . . .113 

XI. The Efficiency of Gears . . .117 

XII. Useful Formulae for Gears . . .119 

Index. 141 




Types, Cutting Methods and Terms 

The toothed gear has long been a familiar element in 
mechanical engineering and its importance seems likely 
to increase as development proceeds. Early examples of 
gears were, cumbrous, noisy and inefficient, owing to the 
lack of suitable materials, design technique and manu¬ 
facturing facilities. Improvements in these respects have 
more than kept pace with advancing demands and toothed 
gearing is-now confidently employed at speeds and power 
outputs that could not have been attempted a few decades 

Gears are also used to transmit power between rotating 
shafts, which are not in the same straight line and which 
have, in general, different speeds. Other means are possible 
in some circumstances, but where a compact drive is 

Fig. t.— The shafts and pinions of an automobile gearbox. 




[Fig. 2.—This type is known as the double-helical gear with 
continuous teeth. 

necessary, or where “ timing ” of the shafts is essential, 
the toothed gear usually surpasses all competitors in 
simplicity, reliability and efficiency. 

Applications. —Examples of the use of gearing occur 


in the conventional auto¬ 
mobile. For example, the 
camshaft which operates 
the valves has to run at 
half the crankshaft speed, 
and in many instances it 
is driven from the crank¬ 
shaft through gearing. 

In order to permit the 
engine to run within a 
reasonable economical 
speed-range whilst driving 
the vehicle at different 
speeds, the gearbox con¬ 
tains several pairs of gears, 
the drive being taken 
through any particular 
pair at will, to suit the 
conditions under which 
the vehicle is running. 

From the propeller shaft 
which runs parallel to the 
length of the vehicle, the 
drive is taken to the rear 
axle through another pair of gears, in which there is a 
reduction in speed besides a change in the direction of the 
drive. The rear axle gear assembly also contains the 
“ differential,” which is a gearing assembly designed to 
divide the driving effort equally between the two rear wheels. 

In contrast with the small gears used in automobiles, 
gearing is used in diameters up to 20 ft., for example, in? 
ship propulsion, where the economical speed of the steam 
turbine may be about 3000 r.p.m., and that of the propeller 
about 250 r.p.m. 

In industrial service, gearing is now very extensively 
used, largely because of the widespread application of 
electric-motor drives. The economical running speeds of 
electric motors of moderate output range from about 
750 r.p.m. to 1500 r.p.m., whereas the greater proportion 
of industrial machinery runs at considerably lower speeds, 
usually from about 50 to 200 r.p.m. In such applications, 
toothed gearing forms a compact and efficient transmission. 

Fig. 3.—An example of a single- 
helical gear. 


whose availability makes it possible to design driving and 
driven machines each to run at its own most economical 

Main Types of Gear. —Toothed gearing is first of all 
divided on the basis of relative shaft position. The 
sim plest examples are those used for parallel shafts as, 
for instance, in the conventional clock or in the automobile 
gearbox (see Fig. i). These are classed as spur gears if 
their teeth are parallel to the shaft centre lines, otherwise 
they are helical gears which may be either “ single-helical ** or 
double-helical” (Figs. 3 and 2). Triple-helical teeth (Fig. 4) 
are also occasionally used but are not to be recommended. 

If the shaft centre lines intersect, the bevel gear is 
employed. Here again there is a subdivision associated 
with tooth formation, “ straight bevel ” gears having 
teeth whose centre lines converge at the apex, i.e. the 

Fig. 4.—Triple-helical teeth are occasionally used, but are not to be 

Fig. 5.—An example of gears in which the centre lines of the 
teeth converge towards the apex (straight bevel). 

intersection of the shaft centre lines (see Fig. 5). When, 
as in Fig. 7, the tooth centre lines are set at an angle to 
the straight lines which converge at the apex, the gear is 
classed as a “ spiral bevel ” or as a “ single-helical bevel.” 
Bevel gears are usually applied to shafts whose centre lines 
are perpendicular to one another, but they can be manu¬ 
factured to suit almost any shaft angle. 

If the shaft centre lines are not parallel and do not 
intersect, the connection between the shafts may be 
effected by spiral gears (Fig. 6), which have, however, a 
comparatively small load capacity. Each gear in such a 
combination is actually a helical gear, but the fact that the 
shaft centre lines are not parallel is the distinction that 
leads to the description " spiral gears.” It also causes the 
contact between any pair of mating teeth to be confined 
to a theoretical point, this being the^ reason for the small 
load capacity. 

Of a pair of gears, the one with the smaller number of 
teeth is called the “ pinion ” and the other is the “ wheel.” 

By forming one member of a pair of spiral gears so that 
the teeth “ envelop ” the other gear, line contact between 
pairs of mating teeth is secured and the worm gear (Fig. 8) 



is produced. The gear of parallel formation is called the 
“ worm.” Its teeth are helical and are called " threads,” 

and it may have 
(and often has) only 
one thread. If the 
worm has more 
than one thread, the 
number of threads 
is often described as 
the “ number of 
starts.” This is to 
avoid confusion 
with screw-thread 
terminology. Occa¬ 
sionally the worm 
is made " hollow 
faced ” so as to 
envelop the wheel 
which also envelops 
the worm, but this 
" hour-glass ” worm 
gear has no advant¬ 
age that justifies the 
extra cost and com¬ 
plication involved 
in manufacturing it. 

Essential Re¬ 
quirements. —The 
first essential of any 
pair of gears is that 
of uniform angular 
velocity transmis¬ 
sion. This means 
that for each degree 
of angular move¬ 
ment of the driving 
gear the mating gear 
Fig. 6. —Spiral gears. These have a com- rotates through a 
paxatively small load capacity. certain multiple of 

one degree. 

The second is that despite the distortion of the 
teeth under load, the gears must be capable of 

Fig. 7.—A spiral bevel-gear unit. 

transmitting power without appreciable shock or 

The third requirement is that the gears must transmit 
the full rated torque without fear of tooth breakage and 
without showing signs of excessive wear on the working 
surfaces of the teeth. 

Temperature rise resulting from heat generation by 
friction must be kept within due limits and, particularly 
with high-speed gears, this may demand special cooling 

Uniform Angular Velocity Transmission. —Smooth 
cylinders mounted on shafts and pressed into contact with 
each other will rotate at relative speeds depending on their 
diameters, and will transmit power up to the relatively 
low limit which causes slip to occur. Toothed gears are 
required to operate in the same way as such “ rolling 
cylinders ” except that the intermeshing of the teeth 
prevents any possibility of slip. The rolling "cylinders 


corresponding to any pair of toothed gears are called the 
“ pitch cylinders.” Fig. g shows the pitch cylinders 
corresponding to a pair of helical gears, the flat strip 
representing the " imaginary common rack. The corre¬ 
sponding geometrical forms in the case of bevel^ gears are 
cones (see Fig. io) with a disc representing the imaginary 
common crown-wheel. A worm-wheel has a corresponding 
pitch cylinder which rolls, however, with a plane and this 
is the “ pitch plane ” associated with the worm. A worm 
has no pitch cylinder, properly so called, although it does 
contain an imaginary cylinder which is a useful basis on 

Fig. 8.—A typical example of a worm gear. 

which to work in calculating detail dimensions of the 
worm; it may be regarded as a " nominal pitch cylinder,” 
and is indicated in Fig. II. 

The mating pitch elements (cylinder, cone, plane, etc.) 
generally make contact along a line which is called the 
" pitch line ” ; in the case of spiral gears the contact is 
confined to a single " pitch point.” 

It can be shown that if a pair of gears is to transmit 
unif orm angular velocity the line which lies perpendicular 
to the tooth surfaces at any point of contact must intersect 
the pitch fine. Many geometrical forms will meet this 
requirement, and choice amongst them is influenced by 
ease of production with the necessary degree of accuracy. 


Fig. 10 {right ).—The cor¬ 
responding geometrical J Q 

forms in the case of ' 

bevel gears are cones. 

The tooth form used 
almost exclusively in 
modern practice is based 
on the involute. . 

Generation of Gear 
Teeth. —A small part 
of the pitch cylinder of 
a gear of very large 
size appears as a flat 
surface. If the gear 
were of infinite size, a 
small part of it would 
have a flat “ pitch 
surface ”; such a gear 
is known as a rack. 

A rack may mesh 
with a gear (usually 
called a pinion in this 
connection), and, rota- 

Nominal Pilch. GjUndei 
1 of Worn, 

Pitch. Plane, 
of Norm. ' 

Fig. 11 {above ).—The 
"nominalpitch cylinder’’ 
of a worm. 

Fig. 9 {left ).—A diagram¬ 
matic {sketch, showing 
the pitch cylinders corre- 
. sponding to a pair of 
helical gears. 





If we start with an actual rack, having straight-sided 
teeth as shown in Fig. 12, we may use this to " generate ” 
teeth in a cylindrical blank of plastic material. The rack 
may be set with its teeth lying in a direction parallel to 
the axis of the blank and situated in such a position that 
when the rack is moved parallel to its length its teeth will 
penetrate the blank. 

Now if the rack be given a uniform motion of this sort, 
and the blank be simultaneously rotated at a uniform 
rate, the relative movement of rack teeth and blank will 
cause the teeth to mould or generate teeth in the blank 
in the manner shown in Fig. 12 and Fig. 13. 


It can be proved that, within certain limitations, the 
teeth generated in this manner are of involute form. 

The Involute. —Fig. 14 represents the end view of a 
cylinder (radius r Q ) from which a string AB is being un¬ 
wrapped, the end B having been originally at C. The 
curve CB traced by the end of the string is an involute to 
the circle, which is called the “ base circle ” of the involute. 

When an involute gear is generated from a rack in the 
manner already described, the size of the base circle 
associated with the involute tooth profiles is determined 
by the relative motions of rack and blank and by the 
spacing of the rack teeth. 

The distance between corresponding flanks of adjacent 
rack teeth measured in a direction parallel to the plane 
containing the tips of the teeth is called the normal pitch. 



The distance between corresponding flanks of adjacent 
rack teeth measured in the direction perpendicular to the 
flanks is called the " base pitch/' 

If the gear blank, during generation, makes one revolution 
whilst the rack is moved through a distance equal to t normal 
pitches, the gear produced will have t teeth. The base circle 
of the involute profiles will be the one whose circumference 
is equal to t times the base pitch of the rack teeth. 

The " pitch circle of generation ” of the gear will be the 
circle whose circumference is equal to t times the linear 
pitch of the rack. 

The angle between a line drawn perpendicular to the 
flank of the rack tooth and a line joining the tips of the 
rack teeth is called the “ pressure angle/’ and it can be 
shown that diameter of the base circle of the gear is equal 
to the diameter of the pitch circle multiplied by the cosine 
of the pressure angle. 

It will be noticed that, so far, no relation has been speci¬ 
fied between the diameter of the blank and the diameter 
of the pitch circle. 

The root circle of the gears is defined by the distance 
by which the rack teeth are allowed to penetrate into the 

blank. If the 
root circle is 
larger than the 
base circle, the 
tooth profiles are 
purely involute. 
If, however, the 
root circle is 
much smaller 
than the base 
circle, part of the 
tooth profile is 
not involute, be¬ 
cause the in¬ 
volute can have 
no existence 
inside its base 

If the root 
circle is appreci- 


ably smaller than the base circle, the part of the tooth 
which lies inside the base circle is liable to be “ under¬ 
cut ” ; it is mechanically weak and liable to produce 
unsatisfactory meshing of the gear with its mating 

Tooth Proportions. On the British Standard System, 
the total depth of a tooth from tip to root is made 0*716 
times the normal pitch of the tooth. The depth to which 
the tip of a gear tooth penetrates into the space between 
adjacent teeth in the mating gear is 0-636 times the pitch. 
The difference between these fractions, i.e. 0-080 times the 
pitch, is called the “ clearance.” 

The distance from the circumference of the pitch circle 

of generation to the pitch circle is called the ,f addendum ” 
of tooth. The distance from the pitch circle to the root 
circle is called the “ dedendum.” 

Methods of Defining Pitch. —The method of describ¬ 
ing the spacing of the teeth by specifying the distance 
between them on the pitch circle is the most straightfor¬ 
ward one. This is sometimes called the “ circular pitch,” 
or more often simply the ** pitch.” When dealing with 
helical gears, 4 ‘ transverse pitch ” is a more suitable term. 

Other methods of specifying pitch are by. module and 
diametral pitch. 

Module is equal to circular pitch divided by 71. 


The advantage of defining pitch in this way is that the 
pitch diameter of a gear is simply the product of module 
and number of teeth. This system of measurement has 
been almost confined to continental practice, using the 
millimetre as the unit of length, but it is equally applicable 
on the basis of the inch. 

Diametral Pitch. —This is equal to n divided by the 
circular pitch. The pitch diameter of a gear is equal to 
the number of teeth divided by the diametral pitch. For 
most practical purposes it is possible to use a diametral 
pitch which is a whole number, whereas with the inch as the 
unit of length, pitch and module are very often fractional. 

Table I. gives standard pitches expressed in all three ways. 

Determination of Blank Diameter. —When the num¬ 
ber of teeth in the gear and the pitch of the generating 
rack are fixed, the size of the pitch circle is fixed and the 
diameter of the tip circle [i.e. the blank diameter) is 
found by adding twice the addendum to the diameter of 
the pitch circle. 

In the past it was the practice to make the addendum 
equal to the module, and although this practice had the 
merit of minimising calculation, it often led to weak and 
inefficient tooth shapes that could easily have been avoided. 

There is no need to restrict the addendum in this way, 
and more satisfactory tooth shapes are obtained if the 
addendum is varied according to the numbers of teeth in 
the mating gears. Too large an addendum produces 
pointed teeth; too large a dedendum produces “ under¬ 
cut” teeth with weak roots. For a pressure angle of 20 
degrees and a tooth depth equal to 0716 times the pitch, 
addenda defined by the following formulae lead to satis¬ 
factory tooth shapes. 

Addendum of pinion—(1-4—ovJi)-P 

T 7C 

Addendum of wheel—(o-6+0-4 i) - 

T 7C 

where t —number of teeth in pinion. 

T=number of teeth in wheel, 
p—pitch of cutter. 

In each case 



Fig. 15 represents the British Standard basic rack-form, 
and Figs. 16 and 17 show typical involute teeth generated 
from it. 

The greater the number of teeth in a gear, the greater 
is the size of the base circle for a given pitch, and as the 
depth of tooth is the same in all cases, it becomes only a 
small fraction of the involute when the number of teeth 
is very large. Consequently, gears of large numbers of 
teeth have profiles which are only slightly curved, and 
the tooth of the rack (which is really an infinitely large 

Fig. 16.—Typical involute teeth generated from the British Standard 

gear) has no curvature at all, i.e., it is straight-sided as 
already mentioned. 

Generating Rack. —The fundamental principle in¬ 
volved in moulding a gear in the manner previously 
described is used in nearly all modern gear-cutting 
processes. The generating member is made in the form 
of a cutter, and besides its moulding motion has another 
which causes its teeth to cut. 

In some cases the cutter itself is actually of rack-form. 
In other cases the cutter corresponds to some particular 
form of gear, and in the process of generating the teeth. 

types, cutting methods and terms 25 

the cutter and the gear are rotated just as the correspond¬ 
ing gears would actually do when in mesh. 

In all cases, however, it is convenient to examine the 
conditions of generation and meshing of gears as if a rack 
were present. Actual tooth forms are most easily deter¬ 
mined from that of the imaginary basic rack. 

Meshing of Double-Helical Gears.—If two helical 
gears are generated by the same basic rack, and have 
equal and opposite spiral angles, they may be meshed 

Fig. 17.—Another example of involute teeth generated from the 
British Standard rack-form. 

together with their shafts parallel. On any plane perpen¬ 
dicular to the axes of the gears, the tooth action is exactly 
the same as in a pair of spur gears. At any instant the 
teeth in one transverse plane are in a different phase of 
engagement from those in other transverse planes. If the 
width of the gears is great enough, every phase of engage¬ 
ment is occurring in some transverse plane or other. 
Consequently, helical gears operate much more smoothly 
than do spur gears of comparable dimensions. 

This can be appreciated also by considering how the 


load is applied to the tooth of a rotating gear. In the case 
of spur gears the full width of the tooth enters the zone of 
engagement at some particular instant, so that load is 
applied to the tooth over its whole width at once. Similarly, 
when the tooth passes out of the zone of engagement the 
load is removed instantaneously. 

In a pair of helical gears, however, one end of the tooth 
enters the zone of engagement first, and as it passes 
through, other sections of the tooth enter the zone, one 
after another. If the width of the gears is great enough, 
one part of the tooth may have passed completely through 
the zone of contact before Other parts have entered it at 
all. Thus the load is applied to each tooth first at one 
end and then over a line of gradually increasing length 
which passes uniformly along the length of the tooth 
before diminishing gradually to a point and disappearing 
at the other end of the tooth. This action is in contrast 
to the sudden application and sudden removal of load in 
the case of a spur-gear tooth. 

Noise in gears is produced by the rapid succession of 
slight impacts as load is imposed on the various teeth, 
and the smooth mode of application in the case of helical 
gears causes a very substantial reduction in the amount 
of noise produced. Consequently, helical gears may be 
run quietly at much higher speeds than would be per¬ 
missible in the case of spur gears. 

The essential advantage of the helical gear may be 
attributed to the fact that the longitudinal centre line of 
tooth is not perpendicular to its direction of motion, and 
this feature is advantageously copied in other types of 

Overlap of Helical-Gear Teeth.—In order to make sure 
that all phases of engagement do occur simultaneously, it 
is necessary that one end of any tooth should overlap, or 
should^ at least come in line with the opposite end of the 
adjacent tooth. The condition for this is that the face- 
width should be at least equal to the transverse pitch 
multiplied by the cotangent of the spiral angle. If for 
any reason the face width cannot be made so great as this, 
the smoothest possible action will not be obtained, but 
the gears will, nevertheless, be superior in action to spur 


End-Thrust.—The effective force exerted by one gear 
tooth on the mating tooth is perpendicular to the centre 
lines of the shafts. If the teeth are of helical form, however, 
equal and opposite forces are exerted on the two gears 
in the direction parallel to the shaft centre lines. These 
forces would cause the gears to slide axially out of mesh if 
nothing were done to prevent it, and consequently a thrust 
bearing is necessary on each shaft. 

As in some instances it may be difficult or impracticable 
to provide suitable thrust bearings, the double-helical 

form of tooth has been developed. A gear of this type has 
helical teeth of a particular spiral angle in one half of its 
face-width, whilst the other half has similar teeth of the 
opposite hand. Thus the end-thrust produced by tooth 
load in one half of the gear acts in the opposite direction 
to that produced in the other half, and if either of the 
mating gears is free to move axially, it will do so and will 
take up the position which causes the two end-thrusts to 
be equalised. 

Thus, not only is it unnecessary to provide a thrust 
bearing, but the tooth formation ensures that the total 


load is equally divided between the two halves of the face- 
width. Owing to bending and twisting of slender pinions 
under load, there is not the same certainty in this respect 
with spur gears or single-helical gears. 

Direction of Rotation.—In one direction of rotation, 
the teeth of a pair of helical gears move with the apex 
leading, whilst if the direction of rotation is reversed, the 
apex follows the remainder of the tooth. It is sometimes 
supposed that there is a particular advantage in one 
direction of rotation over the other, but this is not the 

There is a certain disadvantage if the apices " trail,” 
because there is then a tendency for the oil which lies on 
the tooth surfaces to be trapped at the centre of the face- 
width, and in some circumstances this can l$ad to appreci¬ 
able loss of power and to objectionable noise. 

Triple-Helical Gears.—The assumption that there is 
an advantage in having the apex leading has led to the 
use of the triple-helical gear for frequently-reversing 
drives, as, for example, to mine winders (see Fig. 4). The 
idea is that the advantage is gained in each direction of 
rotation when triple-helical gears are used., The truth is 
that the disadvantage of oil trapping may occur in both 
directions of rotation, and the triple-helical gear is in no 
way superior to the double-helical gear. 

On the contrary, it is impossible to manufacture the 
triple-helical with the degree of accuracy easily attainable 
in double-helical gears, and the only method of cutting 
that can be adopted is relatively slow and expensive. 
Triple-helical gears are now never adopted, except in a 
few special fields of application where tradition alone 
continues to demand them. < 


Bevel Gears 

Just as a pair of spur gears is the toothed equivalent of 
a pair of rolling cylinders, a pair of bevel gears is the 
counterpart of a pair of cones which roll together whilst 
rotating on fixed axes. It may be noted here that since 
the surface-speed of each cone is zero at its apex, the two 
cones cannot roll together unless the apices coincide. In 
other words, the apex of a bevel gear must be at the point 
of intersection of the shaft centre lines. 

The general principles involved in the generation of 
spur gears apply with slight modifications to the generation 
of bevel gears. Instead of a basic rack we have an imaginary 
basic crown-wheel whose teeth are straight-sided, and this 
corresponds to a flat disc capable of rolling simultaneously 
with two cones (see Fig. io). 

The pitch of the teeth of the crown-wheel must clearly 
diminish as the centre is approached and, so that there 
may be similar tooth action ■ at all points, the depth of 
tooth is made proportional to the pitch. Thus the pitch 
and depth of a crown-wheel tooth at any section are each 
proportional to the distance of that section from the apex. 
The pressure angle is the same everywhere. 

In generating spur gears a rack-shaped cutter reciprocates 


in the direction of its teeth and produces teeth which are 
of the same shape and size at all sections. It is not possible, 
in generating bevel gears, to reciprocate the whole crown¬ 
wheel in a similar manner, because different teeth run in 
different directions. It is not even possible to reciprocate 
a single complete tooth because depth and thickness have 
to diminish as the apex is approached. It is possible, 

however, to sweep out the surface of one flank of a straight- 
aded tooth of the crown-wheel by a straight-sided cutter 
blade reciprocated so that its tip follows the root-surface 
between two teeth of the crown-wheel. As the cutter 
blade moves towards the apex, its depth of immersion in 
the blank diminishes, but since the profile of the blade is 



straight, it produces a complete straight-sided profile 
everywhere. Here an important virtue of the straight¬ 
sided basic crown-wheel tooth is that while the profiles at 
large and small ends of the tooth are similar to each other, 
the smaller profile is simply a part of the larger one, so 
that formation of the entire profile by a single reciprocating 
cutter is possible. (See Fig. 19.) 

The fillet at the root of the tooth must, of course, have 

the same radius at each end, and in this respect the tooth 
is not geometrically similar throughout its length, but this 
is unimportant from the point of view of tooth action. 

Generation of Bevel-Gear Teeth.—The process of 
generating bevel-gear teeth from a crown-wheel is examined 
in a plane perpendicular to that of the crown-wheel. In 
the neighbourhood of the point of contact of crown-wheel 
plane and pitch cone of bevel gear, the section of the 
crown-wheel tooth on the plane mentioned may be regarded 
as moving in a straight line. The associated part of the 
bevel gear may be regarded as rotating about the point of 


intersection of the gear’s axis with the plane mentioned 
(see Fig. 20). The shape of tooth generated in the pinion 
is the same as that in a spur gear of pitch radius BA, and 
likewise the wheel-tooth shape is that associated with a 
spur gear of radius CA. These two radii are the “ virtual 
radii ” of the gears. Similar generating action on smaller 
scales takes place on parallel planes lying nearer to the apex. 

Tooth Form of Straight Bevel Gears.—The simplest 
type of bevel gear is that in which the centre lines of all 
the teeth converge at the apex, i.e. the point of inter¬ 
section of-the centre lines of the mating gears when they 
are correctly in mesh. To avoid excessive difference in 
size between the inner and outer ends of the teeth, it is 
usual to limit the face-width of bevel gears to about one- 
third of the “ cone distance.” 

The nominal dimensions (pitch, addendum, dedendum) 
of bevel-gear teeth apply to the outer end ; the correspond¬ 
ing dimensions at other points in the face are proportional 
to their distances from the apex, but the pressure angle is 
the same at all points in the length of the tooth. The 
tooth form of the imaginary crown-wheel is similar to 
that of the basic rack for spur and helical gears. 

Owing to the fact that the same cutting tool generates 
teeth of different pitch at different points in the face- 
width, it is not necessary to adhere to ' • standard ” pitches. 

Although bevel gears are most frequently used to 
transmit power between shafts whose centre lines are 
perpendicular, they can be equally well designed for a 
wide range of shaft angle. The limit to what can be done 
in this respect is the pitch angle of the pinion, for in many 
types of bevel-gear-cutting machine this cannot be less 
than about 5 degrees. Subject to certain limitations in 
reduction ratio, shaft angles varying from about 30 degrees 
to about 150 degrees are permissible. 

Like straight *spur gears, straight bevel gears are apt to 
be unduly noisy at high speeds and for the same reason, 
i.e. that the full length of each tooth enters the zone of 
engagement at the same instant. To overcome the noise 
difficulty recourse is had to single helical bevel gears or 
spiral bevel gears. 

Spiral Bevel Gears.—If the teeth of a bevel gear have, 
their centre lines not perpendicular to their direction of 



motion they are called " spiral bevels ” or “ single-helical 
bevels.” In the latter case the centre line of each tooth is 

straight but does not pass through the apex. In the spiral 
bevel gear (Fig. 7) each tooth centre line is a curve, which 
is usually a circular arc. 

In neither type of gear is the spiral angle exactly the 
same at all points in the face-width. This is not important, 
provided, of course, that the mating gears have the same 
spiral angles at corresponding points. The reason for 
the adoption of the circular arc form for the spiral bevel 

tooth is that it permits 
the use of a cutter hav- 
ing’a number of blades 
set *in a circular body 
and that this can oper¬ 
ate without reversal of 
the motion of the 
cutter. The teeth of 
straight bevel gears 
and single-helical bevel 
gears are generated by 
reciprocating cutters. 

The nominal spiral 

Fig. 21A.—Diagram showing the prin¬ 
ciple of the zerol bevel gear, the 
spiral angle of which is zero at mid¬ 
point of face. 

angle of spiral bevel 
gears is the inclination 
of the centre line*of 
the tooth at the mid¬ 

point of the face-width to the line joining that point to 
the apex. This is approximately the mean spiral angle 
and is used in calculating the end-thrusts and journal 
loads on the bearings. 

The “ Zerol ” Gear.—Comparatively recently there 
has been introduced the “ Zerol ” spiral bevel gear, so 
called because although the teeth are of the curved spiral 
type, the spiral angle at the middle of the face is zero. 
This confines end-thrust to the amount produced in the 
corresponding straight bevel gears, but nevertheless gives 
the smoother running of the spiral bevel gear. The zerol 
gear thus corresponds to the double-helical gear. 

Double-helical bevel gears have been used but the only 
practicable way of cutting the teeth (by means of end- 
mills) is comparatively slow, expensive and inaccurate. 



Worm Gears 

The tooth action in worm gears (Fig. 8) involves com 
siderably more sliding than occurs in the types of gear 
already considered, and in the past this caused worm 
gearing to be regarded with disfavour. Excessive friction 
, led to serious loss of power and to heavy wear of the tooth 
surfaces. Careful study of contact conditions and the use 
of appropriate materials have combined to permit the 
economical use of worm gears in a wide range of practical 

If the worm be imagined to be cut by a plane passing 

Fig. 22. —Section of worm and worm-wheel on central plane. 




through its axis and lying perpendicular to that of the 
worm-wheel the threads appear as a series of rack teeth. 
Rotation of the worm gives these teeth an apparent motion 
parallel to the axis of the worm, and the form of the worm- 
wheel teeth examined in the same plane must be such as 
to mesh accurately with such rack teeth (see Fig. 22). 

Fig. 23.—Section of worm and worm-wheel on offset plane. 



On planes parallel to this central one, the worm threads 
show different shapes and the worm-wheel teeth must also 
differ in form in such planes from that which they have 
in the central plane (see Fig. 23). 

The complex shape of worm-wheel teeth makes it 
impracticable to produce them by any method other than 
generation by a cutter of .the same form as the mating 
worm. In this respect the generation of worm-wheel 
teeth differs from the normal method of producing spur, 



helical or bevel gear teeth, for mating gears of these types 
are each generated by a cutter having a tooth form 
depending only on the normal base pitch required. One 
cutter can produce gears of any size provided only that 
the base pitch is specified. A hob for generating worm- 
wheel teeth can be used only if the mating worm is of the 
same essential dimensions as the hob. 

It is not difficult to choose a worm thread form which 
will lead to satisfactory worm-wheel tooth shapes in the 
central plane, but shapes in other planes may be unsuit¬ 
able unless special care is taken. The British Standard 
worm thread is of involute form in a plane perpendicular 
to the axis of the worm, or, in other words, the worm is 
actually a single-helical involute pinion. The kinematic 





L ^ 



Fig. 26. — Develop¬ 
ment of pitch 
cylinder of worm. 


properties of the involute are not utilised in the worm 
gear and this thread form is adopted .because it offers 
certain advantages in manufacture and checking for 
accuracy. The normal- pressure angle is 20° except in 
cases where certain geometrical considerations make a 
greater pressure angle necessary. 

As indicated in Fig. 24, an accurate thread profile of 
this type contains a straight line and consequently shape 
errors may be comparison with a straight¬ 
edge. No such facility exists in the case of the worm- 
wheel teeth, and the only practicable method of checking 
them is by mounting the worm-wheel in correct relation to 
an accurate worm and observing the nature of the contact 
between them by the marking on the worm-wheel teeth. 

Dimensions of Worm Gears.—The distance measured 
parallel to the worm axis between adjacent threads of a 
worm is called the " axial pitch.” The axial distance in 
which each thread makes a complete revolution is called 
the <f lead ” ; it is equal to the axial pitch multiplied by 
the number of threads. The angle between the thread 
at the surface of the nominal pitch cylinder and a plane 
perpendicular to the axis of the worm is called the “ lead 

It will be seen that if the lead angle is small, the sliding 
velocity of the worm threads on the wheel teeth is high 
compared with the apparent axial velocity of the threads. 
Thus frictional losses are comparatively large and the 
efficiency of the gears is low. 

A high lead angle is necessary for high efficiency. 

The trace of a complete revolution of a worm thread on 
the pitch cylinder is indicated in Fig. 26, from which it 
may be seen that 


where A=lead angle. 


d=diameter of nominal pitch cylinder of 

The velocity ratio of the gears is equal to the number 
of worm-wheel^ teeth divided by the number of worm 
threads, and this is equal to the circumference of the pitch 



circle of the worm-wheel divided by the lead of the worm. 
Hence the lead is fixed if the wheel diameter is fixed, and 
therefore the highest value of X is secured by using the 
smallest worm diameter. 

Contact Between Worm and Wheel.—At any instant 
each worm thread touches one or more worm-wheel teeth 
along a curved line. As the gears revolve, the various lines 
of contact change their positions in space in such a way as 
always to lie on a particular surface called the “ zone of 

contact.” The boundaries of the zone of contact are 
decided by the- thread form and by the diameters and 
widths of the worm and wheel. In some particular angular 
position of the gears the total length of‘the lines of contact 
is a minimum which is used in determining the load 
capacity of the gears. Consideration of the shape of the 
zone of contact makes it possible to decide the maximum 
useful face-widths of worm and wheel. 

Fig. 27 shows the shape of the zone of contact for a 
typical worm and worm-wheel. 


Each worm-wheel tooth enters the contact zone obliquely, 
with the result that load is applied to it in a gradual 
manner and smooth action is easily obtained. Given 
suitable materials and lubrication, worm gearing is the 
quietest type of toothed gear. The curvature of the 
surfaces at any line of contact is less than in spur, helical 
or bevel gears of the same general dimensions and the 
better " fitting ” of the surfaces leads to substantial load- 
capacity despite the fact that the worm-wheel is usually 
made from phosphor bronze, which is a comparatively soft 
material. , 

Owing to the fact that the worm-wheel teeth are pro¬ 
duced by a rotating cutter whose axis does not change its 
position, the axial position of the worm-wheel must be 
accurately adjusted in relation to the worm if correct 
contact is to be obtained. Distortion of worm and wheel 
and of the members which support them has a serious 
influence here, and special attention has to be directed to 
this point when worm gears are being mounted before 
setting to work. As the cross-sectional dimensions of the 
worm are the same throughout its length, accurate axial 
location is unimportant. 


Gear Generation 

Sunderland Process for Spur Gears.—The moulding 
generation process depicted in Figs. 12 and 13 appears in 
the Sunderland spur gear generator, with the single 
modification that the rack is made with sharp cutting 
edges and is reciprocated parallel to the axis of the blank so 
that the teeth cut their way into the metal instead of being 
pushed into the plastic blank in the manner already described. 

In the machine shown in Fig. 28 the cutter-box moves 
horizontally to and fro across the whole width of the 

Fig. 28.—Piano-generating machine for spur gears. 



blank, and the slide which guides it is given a slow vertical' 
movement in unison with rotation of the^ blank. Whilst 
the cutter moves downwards through a distance equal to 
the pitch of its teeth the blank turns through an angle 

equal to one revolution divided by the number of teeth 
required in the work, the “ index change gears ” and the 
“ pitch change gears ” being selected to secure this result. 

When the cutter has been fed so far downwards that its 
uppermost tooth is beginning to cut, the generating-action 



Fig. 30.—A Fellows-type machine cutting cluster gears. 

is automatically stopped. The cutter is arrested at one 
end of its cutting stroke, clear of the blank, and is moved 
upwards through a distance equal to a whole number of 
pitches without rotation of the blank. Its feed mechanism 
is then reconnected to that which rotates the blank, a 
short generating movement is given to cutter and blank 
to take up backlash, and the cutting strokes and “ rolling ” 
motions are resumed. This cycle is automatically repeated 
until the entire circumference of the blank has been treated. 

If the pitch of the teeth is small compared with the 
capacity of the machine, the full depth of tooth may be 
cut at once, and complete generation of all the teeth 
effected in one revolution of the blank. In general, how¬ 
ever, two or more revolutions are necessary, the cutter being 
reset after each revolution to sink more deeply into the blank. 



Fig. 31.—A Fellows-type machine cutting an internal spur gear. 

In some circumstances this type of machine may use 
a special cutter-box mounting, two cutters facing in 
opposite directions so that cutting takes place in both 
directions of motion of the cutter-box across the face of 
the gear blank. 

The cutter is sharpened by grinding its front face, the 
form of the cutting edges being unaffected. 

Fellows Process for Spur Gears.—The Fellows 
process resembles the Sunderland process in that it uses a 
cutter which reciprocates parallel to the axis of the gear, 
but differs from it in the form of the cutter. The effective 
form of the Fellows cutter is that of a spur gear, the teeth 
being tapered to give cutting relief on the profile and the tip. 

The necessary generation motion of cutter and blank 
is secured by continuous rotation of each of them at rates 
inversely proportional to their numbers of teeth, and t'here 


is nothing to correspond to the periodic resetting of the 
cutter that occurs in the Sunderland process. ’ 

To avoid rubbing of the edges of the cutter on its idle 
return strokes the work is drawn a short distance away 
from it at the end of each cutting stroke and replaced at 
the beginning of the next one. 

The Fellows process can be applied to a gear which is 
situated close to a shoulder and it can, therefore, handle 
cluster gears (Fig. 30). The Sunderland process suffers 
some restriction in this direction (unless a specially shaped 
cutter is used) because part of the cutter-box must be in 
front of the cutter. 

The Fellows process can be used to generate internal 
gears (Fig. 31), and except for the Sykes process, which 
differs from it in detail, is the only true generating process 
applicable to that type of gear. 

The Hobbing Process for Spur Gears.—The most 
accurate spur gear generating process is that based on the 
use of the cutting tool known as the "hob” (Fig. 32). 

Fig. 33.—Hobbing a spur gear. 

Fundamentally this is a spiral gear or worm provided with 
cutting edges. Very, often—as in the case of the one 
illustrated—the hob has only one thread (if it be regarded 
as equivalent to a worm) or one tooth (if it be looked upon 
as a spiral gear). In this connection it may be observed 
that the provision of “ gashes ” (or “ flutes ”) in the hob 
produces a number of cutting teeth, but this number has 
no essential relation to the number of teeth in the gear to 
which the hob corresponds. 

In the hobbing machine the hob is mounted in the 
same relation to the blank to be cut as would be the corre¬ 
sponding spiral gear to mesh.with the finished.spur gear. 
For starting the cut, the hob is set clear of the end face 
of the blank, and the two are then rotated, the hob at a 
rate which gives its edges a suitable cutting speed and the 
blank at the corresponding rate inversely proportional to 
its number of teeth. The hob is slowly fed parallel to the 



axis of the blank until it has traversed the whole face- 
width. If the pitch of the gear is small enough, the hob 
may be set to the full depth of tooth, and one traverse 
suffices to complete the gear; usually, however, two or 
more cuts are required. 

The fundamental accuracy of the hobbing process is 
associated with its continuous nature. For each cut the 
whole width and circumference of the blank are covered 
without change in speed in any part of the hobbing machine, 
and there is thus no possibility of introduction of error by 
variation of stress at any point. 

The hobbing process applied to the generation of spur 
gear teeth is shown in progress in Fig. 33. The cutting of 
the teeth has commenced at the upper edge of the blank 
and part of the face-width has been covered. The bottom 
surfaces of the grooves cut by the hob teeth slope gradually 
to the outer surface of the blank, this “ run-out ” resulting 
from the circular path of the tips of the hob teeth. When 
the hob has been moved vertically downwards until full- 
depth teeth have been cut throughout the width of the 



blank, the run-out of the hob teeth demands clearance 
which has to be taken into account when setting-up the 
blank on the hobbing machine. Also, in the case of a 
pinion made solid with its shaft, this consideration limits 
the diameter of collars situated near to the ends of the 
gear teeth. 

Grinding of Spur Gear Teeth.—To secure the greatest 
load capacity in spur gears, it is necessary to make them 
of case-hardened steel, and if a high degree of accuracy is 
demanded, the profiles must be finished by a grinding 
process in order to remove the inaccuracies resulting from 
distortion in hardening. Several types of such grinding 
machines operate on the generation principle, and a typical 
one is the Maag, illustrated in Fig. 34. 

The grinding of the teeth is done by the outer faces of 
the abrasive wheels, each of which is set to correspond to 
one flank of a tooth of the imaginary basic rack. The 
abrasive wheel is moulded to a saucer-shape, contact with 
the gear tooth being limited to a narrow band near the 
edge of the outer face. This band is trimmed by a 
mechanically controlled diamond and, when viewed in 
a direction parallel to the axis of the gear, appears as a 
straight line, just as would the flank of the basic rack 
tooth. The active band of the abrasive wheel thus lies in 
the plane flank of the basic rack tooth and by feeding the 
blank parallel to its axis the band has ultimately the 
same effect as if it were a continuous abrasive face but 
with the advantage that as contact between wheel and 
gear tooth is confined to two very small areas at any one 
instant, the load on the wheel is small. This is particularly 
important because the wheel is necessarily thin, to pass 
between adjacent gear teeth, and any considerable lateral 
load would deflect it appreciably, thus introducing 

Generating action, as already described in. connection 
with the Sunderland gear generating machine, would 
require the abrasive wheels to be moved perpendicularly 
to the axis of the gear in unison with its rotation. It is 
in general desirable, however, that a precision grinding- 
wheel spindle should remain stationary in order that the 
utmost degree of rigidity may be obtained. In the Maag 
machine, this ideal is achieved and the necessary generating 



action is secured by giving the blank both straight-line 
motion perpendicular to its axis and rotation about that 
axis at the correct relative rate. 

Obtaining the Double Motion.—This double motion 
is obtained, simply and accurately, by means of the 
segment and tapes shown in Fig. 34. The radius of the 
segment is that of the desired pitch circle of the gear teeth. 
The segment is mounted on the end of the work spindle 
into which the work arbor is fitted and four steel tapes 
are attached to it, two passing horizontally to a fixed 
bracket on one 
side of the mach¬ 
ine, the other two 
lying between 
them and extend¬ 
ing to a tension¬ 
ing lever on the 
other side of the 
machine. (The 
tapes are dupli¬ 
cated as a safe¬ 
guard against 
damage should 
one of them 
break.) Thus 
when the work 
head moves per¬ 
pendicularly to 
the axis of the Fig. 35.—Formed-wheel grinding of a spur gear, 
work spindle, the 

tapes cause the spindle to rotate as if the segment were rolling 
without slip on a flat surface. Hence the gear teeth have the 
same rolling action, and contact with the appropriately posi¬ 
tioned abrasive wheels thus produces ground involute profiles. 

Generating action is produced by lateral reciprocation 
of the work head, and to cover the whole face-width of the 
teeth concerned, the gear is slowly fed parallel to its axis. 
After it has moved clear of the abrasive wheels an auto¬ 
matic indexing motion occurs and on the next traverse 
two other tooth profiles are treated. When dealing with 
gears of large pitch the abrasive wheels may be situated 
in the same tooth space, but usually they are in adjacent 



Fig. 36.—Spur and helical Fellows-type cutters. 

spaces and when grinding gears of very small pitch they 
may have to be even more widely separated. 

The Maag machine incorporates special mechanism to 
offset inaccuracy that would otherwise result from wear 
of the abrasive wheels. A diamond-faced feeler periodically 
follows a path in which it should make contact with the 
active band of the abrasive wheel. If it fails to do so by 
reason of wear of the wheel, the position of the wheel is 
automatically adjusted until contact is established. By 
this means the effective face of the abrasive wheel is kept 
within one-quarter of a thousandth of an inch of its correct 
position, despite wear. 

The Formed-Wheel Process .—Certain types of spur- 
gear tooth grinding machines do not use the generating 
principle but depend upon direct reproduction of the shape 
of the abrasive wheel in the gear teeth. Machines of this 
nature are described as operating by the “ formed-wheel 
process.” The wheel is trimmed by diamonds under the 
control of mechanism whose movements depend on the 
guidance of plates cut to the shape of the required tooth 


profile enlarged about five times. It is usual to grind the 
two profiles and the root surface of each tooth space at 
once, and this makes the form-grinding process more rapid 
than generation grinding, where the abrasive wheels are 
less effectively used. (See Fig. 35.) The gear is moved 
parallel to its axis so that the abrasive wheel may grind the 
whole face-width ; after that, rotation of the gear through 
an appropriate angle causes the wheel to grind a different 
tooth space on the next traverse. 

An advantage of the formed-wheel process is that it 
facilitates slight modifications of tooth profile such as 
are sometimes found necessary in heavily loaded gears. 

Generation of Helical Gear Teeth .—The spur gear 

Fig. 37.—Hobbing a helical gear. 


generating processes already described are all adaptable, 
with some complication in mechanism, to the generation 
of helical gear teeth. For example, in an elaborated form 
of the machine shown in Fig. 28 the cutter-slide can be 
set at any desired angle to the axis of the gear and then 
helical teeth are generated with involute profiles on 
transverse sections. 

A similar adaptation is possible in the case of the 
Fellows process (Fig. 29). A helical guide has to be pro¬ 
vided so that vertical movement of the cutter gives it 
proportional angular movement, and the cutter itself 
must have helical teeth. Fig. 36 shows a group of Fellows- 
type cutters, some for producing straight tooth gears, the 
others for helical gears. The helix angle of the cutter (and 
therefore of the gear produced) depends on its diameter, 
because the lead of the cutter must be that of the helical 

Hobbed Helical Gears. —Adaptation of the hobbing 
process to the generation of helical gear teeth is com¬ 
paratively easy. The angular position of the hob spindle 
is adjusted so that the teeth, when cutting, are moving 
in the direction of the tooth spirals of the gear to be 
generated (see Fig. 37). Feed of the hob parallel to the 
axis of the blank introduces a complication because it 
must be accompanied by rotation of the blank through 
an angle inversely proportional to the lead of its teeth. 
Thus if 

T—number of teeth in gear 
L=lead of gear teeth (in.) 
t—number of threads in hob 
f=feed of hob (in.) per revolution of hob 
N=rotational speed of gear 
n=rotational speed of hob 
then if there were no feed of hob 

Since feed of the hob through a distance equal to the 
lead of the gear would require one additional revolution of 
the gear, the additional .speed of the gear corresponding to 

feed of the hob at nf inches per minute is -j- r.p.m. 



Hence the speed of the 
gear is 

N=f±£=n (*± *) 

and this relation deter- 
mines the ratio of the 
change-gears required in 
setting up the machine. 

The double sign is used 
because, according to the 

relation of direction of Fig. 38. — Staggered arrangement of 
feed of hob to hand of teeth in a hobbed double-helical 

gear teeth and direction gear ' 

of rotation it may be necessary in one case to use -f- and 
in another to use the — sign. 

By adding a differential gear to the mechanism of the 
gear-cutting machine it is possible to change the rate of 
feed or even to discontinue it altogether without stopping 
the rotation of hob and blank. This is often a great con¬ 
venience in practice, but the additional complication in 
the machine tends to have an adverse effect on the accuracy 
of the product. 

The hobbing process is widely used for the production 
of double-helical gears, and the Cf run-out ” of the hob 
demands a gap between the right- and left-hand helices. 
By adopting staggered teeth and permitting a chamfering 
of the ends of the teeth the loss of effective face-width at 

the centre is minimised (see Fig. 38). 

Plano-Generating Processes for Double-Helical 
Gear Teeth. —The Sunderland process, with rack-shaped 
cutters, may be used for double-helical gears, but it 
involves a special type of machine of which an example is 
shown in Fig. 39. Here the cutter-boxes are guided to 
move in directions oppositely inclined to the axis of the 
blank, but the cutters themselves are set perpendicular 
to it. One cutter moves towards the centre of the face- 

width as the other recedes from it; cutting occurs on the 
“ in ” strokes only. By setting the cutters so that they 
very slightly overrun the central plane, each cuts a 
clearance for the other, and it thus becomes possible to 



Fig. 39.—Piano-generating machine for double-helical gears. 

produce continuous double-helical teeth by a generating 
process. This type of tooth is somewhat stronger than 
the double-helical tooth with a central gap, but the ad¬ 
vantage is slight in gears of normal proportions. Further¬ 
more, the load capacity of few double-helical gears is 
limited by tooth strength. 

The Sykes System.-— The Sykes process for producing 
double helical gears is the counterpart of the Fellows 
process; it uses simultaneously two helical pinion-type 
cutters, one of each hand. The mechanical construction 
of Sykes machines is elaborate, inasmuch as there is a 
spiral guide for each cutter, a relieving motion for each 
cutter to avoid rubbing on the idle stroke, and a hollow- 
shaft mounting for one cutter encircling the shaft which 
drives the other cutter. Nevertheless, machines of this 
type can cut accurate gears at high rates of production, 
and this is a tribute to care in detail design and to excellence 
of manufacture. 


Profile-Grinding of Helical Gears— The Maag, and 
similar generation-grinding processes for spur gears, may¬ 
be adapted to profile-grinding of helical gears at the cost, 
of course, of some complication in machinery. The 
formed-wheel process may be similarly adapted, but a 
difficulty is that the normal section of the abrasive wheel 
is not exactly reproduced in the gear owing to interference 
between the wheel and the helical teeth. Whilst it is 
possible to determine by calculation the shape of an 
abrasive wheel necessary to grind a helical tooth which is 
of involute form on the transverse section, the operation 
is more involved than in the case of spur gears, and the 
formed-wheel process has had comparatively little appli¬ 
cation to the grinding of helical gear teeth. 

Generation of Straight Bevel Gear Teeth. —The 
fundamental principle underlying the operation of most 
bevel gear generating machines is that of “ rolling” the 

Fig. 40.—Straight bevel gear generating machine. 


blank with a crown-wheel which is imaginary save for a 
cutter blade or blades continually sweeping two of its 
togth flanks. 

In Fig. 40 two straight-sided cutter blades can be 
distinguished. These are reciprocated along lines con¬ 
verging at the apex of the machine and the cutter head 
which carries their guides is given partial rotation about 
the imaginary crown-wheel’s axis, which passes through 
the apex. Simultaneously the blank (whose axis also passes 
through the apex) is given proportional rotation so that 
blank and imaginary crown-wheel are correctly “ rolled ” 
together and the reciprocating cutter blades then form teeth 
which will mesh correctly with teeth in other gears generated 
in the same way. 

After the rolling has proceeded so far that the tooth 
flanks concerned are fully generated, the blank is with¬ 
drawn from the cutters by rotation of the work head about 
a vertical axis passing through the apex, and the blank is 
rotated through an angle corresponding to one tooth 
pitch. At the same time cutter head and blank are 
rotated back to the initial generating position. The 
blank is then returned to the cutting position, and genera¬ 
tion of another pair of tooth profiles begins. These opera¬ 
tions are entirely automatic and the cycle is repeated 
until all the gear teeth have been generated. 

The generation of the teeth of the mating gear proceeds 
in exactly the same way except, of course, that the machine 
is “set up ” to suit the different pitch angle and the 
different number of teeth. Thus the two gears are generated 
so as to mesh with a common imaginary crown-wheel, 
and as the teeth of such a wheel are symmetrical (neglect¬ 
ing clearance) about the pitch plane, the two gears will 
mesh correctly with each other. 

Generation of Single-Helical Bevel Gears. —The 
process just described may be applied with slight modi¬ 
fication to the production of bevel gears with helical 
teeth. The path of each cutter blade is still a straight line 
which, however, does not intersect the axis of the imaginary 
crown-wheel. The teeth of that wheel are thus straight 
but not radial, and the teeth of bevel gears generated from 
it are of tapered helical form. Such gears are described as 
“ single-helical bevel ” gears. 



Generation of Spiral Bevel Gears. —Consideration of 
tooth action in helical bevel gears does not demand that 
the teeth of the imaginary crown-wheel be straight. The # 
advantages of helical tooth formation are secured if the* 
crown-wheel teeth are of almost any form other than the 
straight radial one that characterises the straight bevel 
gear. A bevel gear whose corresponding crown-wheel 
teeth are not straight is called a ft spiral bevel ” gear. 

The Gleason spiral bevel gear is based on a crown-wheel 
whose teeth are of circular arc form (Fig. 41). The 
advantage of this is that the cutter blade may be carried 
by a revolving disc and may thus return to take further 
cuts by virtue of successive rotations of the disc. In 
practice the disc is fitted with a number of equally spaced 
cutter blades, so that cuts on any one tooth flank take 
place in quick succession and cutting proceeds more, 
smoothly than is the case in a machine having recipro¬ 
cating cutters. Each cutter blade acts on one tooth 


flank only, and successive blades deal with opposite 
flanks. This feature can be distinguished in Fig. 42, 
which shows a typical Gleason spiral bevel cutter having 
“ inside” and “outside” blades placed alternately along 
the edge of the circular body. Fig. 43 shows a Gleason 
spiral bevel gear generating machine operating on a wheel, 
and Fig. 44 shows a similar machine generating teeth in 
a small pinion. 

The rotation of the cutter about its axis causes its 
blades to sweep out one tooth space of the crown-wheel. 
Simultaneously the cutter is moved bodily in a manner 
corresponding to a partial rotation of the imaginary 
crown-wheel and the gear blank is given the appropriate 
rotation in unison with it. These movements continue 
far enough for complete generation of the flanks adjacent 
to one tooth space. Then the cutter is withdrawn from* 
the blank, the “ rolling ” motions are reversed and the 
blank “ indexed ” so that on the next cycle another tooth 
space is generated. The machine is fully automatic ; 
once set to work it generates all the teeth in the gear and 
then brings itself to rest. 

The Gleason process is the most widely employed method 
of producing spiral bevel gears, and it finds special favour 
in dealing with large quantities of the small gears of this 

Fig. 42.—A Gleason-type spiral bevel gear cutter. 

gear generation 


Fig. 43.—Gleason-type spiral bevel gear generating machine. 

type used in automobile rear axles. By certain modi¬ 
fications in the machine it can be arranged to produce 
“ offset spiral bevel ” gears to which the name “ hypoid ” 
has been applied. An example of this type of gear is 
shown in Fig. 45. 

Another modification of the Gleason spiral bevel gear 
is the “ formate ” gear. In this the teeth of the wheel are 
not generated at all, the blank and cutter head being 
held stationary whilst each tooth space is treated and the 
teeth, therefore, have the same shape as the cutter blade. 
The pinion teeth are generated substantially in the stan¬ 
dard Gleason manner. This process is a specialised 
one and can be applied econoifiically only when several 



Fig. 44.—Generating a Gleason-type spiral bevel pinion. 

hundreds of pairs of gears are to be made in one 

Worm-Wheels. —The only practicable way of produc¬ 
ing accurate worm-wheel teeth is by generation with a 
hob whose essential dimensions are the same as those of 
the worm which is to mate with the wheel. A typical 
worm-wheel generator is shown in Fig. 46. The hob is 
mounted horizontally and rotates at a rate which imparts 
a suitable cutting speed to its teeth. . 

The worm-wheel blank is arranged 
with its axis vertical, and its speed 
of rotation is that which the com- llg 

pleted worm-wheel would have when j||g ( |B 
meshed with a worm running at the l g|| \Jl 
hob’s speed. The head carrying the wSl fL 
hob and its driving mechanism is 
able to slide horizontally, and during 

the generating process it is caused to Fig. 45.—A hypoid gear. 


do this so that the hob penetrates more and more deeply 
into the blank until finally the distance between their 
axes is equal to the centre distance of the finished gears. 

A hob is an expensive tool to manufacture, and its cost 
may be prohibitive if it is required to produce only one or 
two worm-wheels. In such cases it is common to use a 
“ fly-hob,” which corresponds to one. tooth of the com¬ 
plete hob (see Fig. 47). In order that this tool may effect 
complete generation of the worm-wheel, the two must 
have the appropriate relative speeds as for worm and 
worm-wheel, and in addition the hob must be fed along 
its own axis so that it gradually penetrates more deeply 
into the blank, the centre distance between them remain¬ 
ing constant throughout the generating operation. This 



axial feed of the hob means that the worm-wheel blank 
must be given the corresponding rotation superimposed 
on the one which it already has in association with 
rotation of the hob. 

Generation by “ fly-hobbing ” is naturally much slower 
than is possible with a complete hob, because the single 
tooth has to do the amount of work that is spread over 
perhaps fifteen teeth in the complete hob. To obtain 
smoothly finished teeth the axial feed of the hob has to 
be slow. 

Axial feed is often adopted when using complete hobs 
with the object of securing a higher degree of tooth finish 
than is practicable with the direct “ in-feed.” For such 
applications the hob is usually made of tapered form over 
part of its length, the roughing-out of the worm-wheel teeth 
being effected by the teeth at the small end of the hob. 

Fig. 47.—The generation of a worm-wheel by means of a " fly-hob." 



Fig. 48.—Cutting a worm in a thread-milling machine. 

When quantities of worm-wheels are being generated it 
is usual to finish the cutting of the teeth in a separate 
operation from the initial rough-cutting. In such cases 
the finishing of the teeth is often carried out by means 
of a “ serrated hob ” which has a large number of finely- 
pitched teeth. 

Cutting of Worm Threads. —The worm threads are cut 
in a thread-milling machine (Fig. 48) or (in certain circum¬ 
stances) by hobbing. The thread-milling _ cutter rotates 
about an axis whose angular position is adjustable to suit 
the lead-angle of the worm to be produced. The table 
which carries the work spindle can slide parallel to the 
spindle and in actual operation this sliding occurs in unison 
with rotation of the blank, change-gears being provided in 
order to afford means of accommodating any required lead 
of worm. This combination of rotation and axial move¬ 
ment of the blank past the rotating cutter causes it to cut 


a helical groove in the blank. When the whole width of 
the blank has been covered, the blank is quickly returned 
to the starting-point, and after " indexing/' the process is 
repeated to cut another thread. Further repetition takes 
place until all the threads have been completed. 

Grinding Worms. —Worm threads are nearly always 

Fig. 49.—A worm-thread grinder, finishing the rough-cut worm. 

subjected to a grinding operation, and this is carried out 
on the type of machine shown in Fig. 49. The movements 
of the machine are similar to those of the thread-miller, 
but the milling cutter is replaced by an abrasive wheel 
and the operation is entirely automatic. Each flank of 
each thread has usually to be ground several times to 
bring the thread thickness down to the required figure. * 
In grinding worm threads of the involute helicoid type 

gear generation 65 

(the British Standard) the abrasive wheel may be flat-sided 
and this gives the advantage that the wheel may be 
trimmed by a diamond moving in a straight line. By 
setting the axis of the abrasive wheel at the appropriate 
angle it touches^ the worm thread along the straight line 
which characterises the involute helicoid worm. On this 
basis only one thread flank can be ground at once. 

Alternatively, two flanks may be ground at the same time, 
but in this case the abrasive wheel has to be shaped to 
two curved contours by diamonds moving under the 
control of specially shaped former plates. The shape of 
the plates is determined by calculation according to the 
particular thread form required. 

Finishing Worms. —In order to obtain the necessary 
mirror finish on worm threads the grinding operation 
must be followed by treatment with fine abrasive. To 
effect this the worm is rotated in mesh with a lap in the 
form of a wooden worm-wheel charged with abrasive 
paste. Power is applied to the worm which thus rotates 
the worm-wheel against the resistance of an adjustable 
brake and is consequently polished by the paste. To 
cover the whole face-width of the worm it is reciprocated 
in the direction parallel to its axis. 


Gear-Wheel Forms 

Small gears may be made in the form of solid discs, but 
as this is not economical of material the armed wheel is 
common in large sizes. An example is the cast-steel 
wheel shown in Fig. 2, and it will be noticed that the 
arms are of H-section to give adequate strength with a 
comparatively small amount of material. The particular 

gear shown has a 
“ split boss,” this 
form of construction 
being adopted in 
large gears in order 
to reduce the strains 
caused by contrac¬ 
tion of the metal as 
it cools in the mould 
when in the solid 
state immediately 
after casting. When 
the boss is made in 
one piece the arms in 
the finished wheel 
are in a state of 
tension; splitting of 
the boss avoids 
this. After the boss 
has been rough- 
machined, the splits 
are filled with molten 
brass, and after this 
has cooled the boss is consolidated with two steel rings, 
one shrunk on to each projecting end. 

An alternative way of diminishing contraction strains 
is to make the wheel in halves, and although this is more 
expensive by reason of extra material and machining, the 
reduction in dimensions of the largest piece to be handled 
often brings important advantages in transport and 

Fig. 50.—Cast-steel double-helical gear¬ 
wheel in halves. 


Fig. 51.—A double-helical gear-wheel built up by welding. 

erection. Fig. 50 shows a large gear-wheel made in halves 
and fastened together by means of bolts. 

Built-up Gears.—When loading conditions demand 
the use of a material of such hardness that it cannot be 
obtained as a casting, the wheel may consist of a forged 
steel rim shrunk on to a " centre ” made from cast-iron 
which is, of course, comparatively inexpensive. 

There is nowadays a tendency to build up steel gear¬ 
wheels by welding the rim to arms cut out of plates, and 
very often the design is such that the wheel is not easily 
distinguishable from a cast wheel of conventional design. 
Alternatively the arms may be made from standard 
rolled-steel sections (see Fig. 51). 

Because of the high cost of the phosphor bronze used 



for worm-wheel teeth, such wheels are usually made as 
rims secured to cast-iron centres by shrinking and pegging, 
by bolting, or by welding. 

Pinions. —The pinion of a high-ratio pair has often to 
be made solid with its shaft because a bore large enough 
to accommodate a separate shaft of adequate strength 
would leave insufficient metal below the roots of the teeth. 
In some cases ( e.g „ Fig. 52) the effective face-width 
necessary for load capacity is so great in relation to the 
diameter of the pinion that extra support must be provided 
by a third bearing separating the two halves of the 

Automobile Gears. —Weight considerations demand 



Fig. 52.—Turbine reduction gears, showing each pinion arranged 
for a third bearing between the right-hand and left-hand helices. 



Fig. 53.—Spiral bevel gears for automobile rear-axle drives, 

the minimum dimensions in automobile gears, and case- 
hardened steel is almost invariably used in such cases 
because of its high load capacity. Fig. 53 shows a typical 
group of rear-axle spiral bevel gears, the pinion being 
made solid with its shaft and the wheel bored to fit on 
to an intermediate shaft. In heavy automobiles the rear- 
axle drive is usually a pair of worm gears, such as is 
shown in Fig. 54. This type of gear is specially suited to 
the comparatively high reduction ratios (between 6 to 1 
and 10 to 1) required in such drives, and its quiet running 
qualities are specially valuable in trolley-buses because in 
that class of vehicle all other noises have been reduced 
to a negligible total. 

Automobile Change Gears. —These are now so accur¬ 
ately cut that the noisiness associated with them is a thing 
of the past. This is due not only to the use of helical 
gear teeth but also to ground teeth, the use of shafts of 
stouter diameter which do not whip, and to improved 
methods of mounting. 

The use of constant mesh gears instead of sliding dogs 
has also contributed to gearbox efficiency. 

In the synchromesh gearbox the constant mesh gears 


are fixed on the primary shaft and countershaft as also 
are. all the countershaft gears, but the second and third 
mainshaft gears are free to revolve on the mainshaft, 
and are engaged by means of dogs formed on the gears. 

It must be remembered that in replacing worn gears 
it is advisable to renew them in pairs, as replacement of 
one of the pair invariably tends to create noise and cause 
accelerated wear of the new gear. 

Mention must also be made of the overdrive. One 
type takes the form of two bevel pinions in the rear axle. 

Fig. 54.—Worn gears for automobile rear-axle drive. 

meshing with crown-wheels of different diameter. Either 
of the bevel pinions can be locked to the driving shaft by 
means of a dog clutch, so that either a high or a low back- 
axle ratio is available. It will be seen that with this 
arrangement, not only are two top gears available but 
also a low and high ratio for each of the other gears. 

The same effect is obtained in the case of certain large 
passenger and commercial vehicles by providing an 
auxiliary two-speed box behind the gearbox. A further 
alternative is to provide an additional ratio providing an 
overgeared top in the gearbox itself. 


Epicyclic Gear Trains 

A gear train in which any gear rotates about a centre line 
which is not fixed is called an “ epicyclic ” train. In the 
example in Fig. 55 the gear B runs on a pin carried by arm 
D which rotates about the axis XX. Gear A is carried by 
another shaft whose centre line is also XX. Gears A and B 
mesh together, and B also meshes with the internal gear C 
whose centre line is also XX. If C is prevented from 
rotating, rotation of A will cause rotation of D with a 
definite velocity ratio. Similarly, if any one of the members 
A, B, C, and D is prevented from rotating, the motions of 
the other three are governed by fixed relations. In the 
following table, each of the letters A, B, C, and D refers to 
the number of teeth in the corresponding gear: 

Revolutions of 

Gear A 

Gear B 

Gear C 

Arm D 

Case 1. 

D fixed 







Case 2. 

C fixed 

, A 
I+ C 

A A 

C B 




Case 3. 

B fixed 


A A 
B — C 



Case 4. 

A fixed 


A , 
B +I 

A , 
C +I 


Case 3 is included for the sake of completeness, although 
it is unlikely to occur in practice. The condition is that B 
is prevented from rotating although permitted to move in 
a circle in accordance with the rotation of D. 



To illustrate the use of the table, consider the assembly 
in which 

A =20 B=i8 C=56 
and C is fixed. Then 

A makes i-f ~ = 1*357 revolutions while 


B makes ^ = -0-754 revolutions and 

56 18 

D makes 



= 0*357 revolutions 

If A is the driving gear, and D the driven member, then 
* 1 * 3^7 

Velocity ratio = b 7 ^^ 3 * 8 ' 

r _j Gear A is called the 

" sun/' and B is a 
“ planet/' In order to 
obtain greater load- 
capacity and to im¬ 
prove the balance of D, 
more than one planet 
may be used. Such an 
arrangement cannot be 
assembled, however, 
unless (A+C) is exactly 
divisible by the number 
of planets. 

Compound epicyclic 

1 -1 gear trains are often 

used for obtaining very 
high velocity ratios. The revolutions of the gears in the 
particular compound train, shown in Fig. 56, are given, for 
several important cases, in the table overleaf. 

Case 2 is the one most frequently used in practice, with 
A the driving gear and E the driven gear. If B and C are 
nearly equal, the velocity ratio between A and E becomes 
very high. 

The method of constructing tables, such as those given 
above, is simple and is applicable to any epicyclic gear 
assembly. In the first instance, the member carrying the 

Fig. 55. — An 
example of 
an epicyclic 
gear train. 




B and C 




Case 1. 

F fixed 











Case 2. 

D fixed 

1+ D 

A A 


A A 





Case 3. 

A fixed 



A , 

B + 1 


D + I 

A C 
B E 

+ 1 


Case 4. 

E fixed 

, A C 
I+ B E 

A /C \ 
B (e z ) 

A C 

B E 




A C 
B E 

Table showing the revolutions of the gears in the compound train 
illustrated in Fig. 74. 

planetary spindle is assumed to be fixed. Any other 
member is imagined to have one revolution, and the corre¬ 
sponding revolutions 
of the remaining 
members are deter¬ 
mined exactly as 
for an ordinary (not 
epicyclic) train. 

This is Case 1. Now 
if any particular 
member is to be con¬ 
sidered at rest, the 
numbers of revolu¬ 
tions of all members 
are each reduced by 
the amount neces¬ 
sary to make that 
particular member’s 
revolutions equal to Fig. 56.—A compound epicyclic gear train, 
zero. Thus, in the 

first table, Case 2 is derived from Case 1 by subtracting 
A A\ 

— ^ {i.e. adding to each quantity. 

Although the compound epicyclic gear is a means of 
obtaining a high velocity ratio in small space, its efficiency 
is low if the ratio is high, and on that account it is suitable 
only for small powers. 


Methods of Mounting 

Gears have to be manufactured to a high degree of 
accuracy for two main reasons : 

(a) to permit of reasonably quiet running, and 

(b) to permit of adequate distribution of load 

across the face-width. 

The first of these requires that errors of tooth shape be 
kept within two or three ten thousandths of an inch, that 
pitch errors between adjacent teeth be less than about 
half a thousandth of an inch, and that the pitch error 
between any two teeth in the gear be less than one.or two 
thousandths of an inch. A reasonable approximation to 
uniformity of distribution of load requires that the tooth 

Fig. 57.—A self-contained single-reduction turbine gear unit (hobbed 
double-helical gears). 



spiral shall be correct within about one thousandth of an 
inch across the width of the gear. 

These degrees of accuracy can be achieved by appropriate 
care in manufacture, but they are quite unavailing unless 
the gears are mounted with equal accuracy. In some 
instances, particularly highly-stressed units such as auto¬ 
mobile rear-axle assemblies, distortion of the mounting 
under the bearing-loads produced by the gears is sufficient 
to affect the accuracy of meshing to a serious extent, and 
the initial “ setting-up ” has to be adjusted so as to 
counteract the full-load distortion. Such adjustment 

Fig. 58 .—A self-contained single-reduction double-helical gear unit. 

means that spread of contract over the whole face-width 
can occur only at full load or some selected fraction of 
full load and emphasises the desirability of the utmost 
rigidity in the mounting. 

Spur- and Helical-Gear Units.—This consideration in 
conjunction with the necessity for protection of the gears 
from dust and dirt and for their lubrication has led to the 
use of self-contained gear units. These are standardised 
in type and size so that manufacturing costs are minimised. 
Bearing details and lubricating arrangements are settled 
once and for all, and the only variations between indi¬ 
vidual units lie in the gear ratio. 


The parallel-shaft type of gear unit (Fig. 58) may contain 
spur, single-helical or double-helical gears and the bearings 
are usually of the ball or roller type. Unless the peripheral 
speed is higher than about 2500 feet per minute, lubrication 
of the gears is effected by allowing them to dip in a bath 
of oil contained in the lower part of the case. Oil is 
splashed on to the walls of the case, and troughs are pro¬ 
vided to guide into the bearings the small amount of 
lubricant that they need. 

Lubrication Problems. —Gear units of this type may 
be used in conjunction with steam turbines, but the high 
speeds concerned necessitate some modification in detail. 
White-metal-lined journal bearings are the rule in appli¬ 
cations of this sort and oil is fed to them under pressure 
to ensure the flow required to avoid excessive rise in 
temperature. Oil is sprayed on to the gears near to the 
point of mesh, oil bath lubrication being objectionable 
owing to the considerable power loss which occurs at very 
high peripheral speeds. In turbine reduction gears the 
peripheral speed may be as high as 15,000 feet per minute. 

Bevel-Gear Units. — For connecting shafts whose 
centre-lines are perpendicular, either bevel-gear or worm- 


gear units may be used, the former having some prefer¬ 
ence if the gear ratio required is lower than about 4 to 1. 
An example is shown in Fig. 7, the horizontal shaft 
being driven by an internal-combustion engine and the 
vertical shaft transmitting power to a centrifugal pump 
situated some distance below the engine-room* Installa¬ 
tions of a similar character are associated with deep bore¬ 
hole pumps, whose speeds are often three or four times as 
high as those of the engines which supply the power. 

Worm-Gear Units. —Largely because of the wide 
range of ratios that it can conveniently afford the worm 
drive is extensively used in industrial service. It also 
gives high torque capacity in small space and is con¬ 
sequently an inexpensive form of gear transmission. Its 
mechanical efficiency is lower than that of a helical-gear 
unit, but this disadvantage is insufficient to outweigh the 
worm gear’s compactness and low initial cost. A rough 
estimate of the overall efficiency of a worm-gear unit may be 
obtained by subtracting half the reduction ratio from 100 ; 
this rule apples for ratios between about 10 to 1 and 50 to 1. 

It is very often found that the load capacity of a worm- 
gear unit with natural cooling is limited by temperature- 
rise rather than by endurance of the teeth. In order to 
raise this limitation it is now usual to provide artificial 
cooling by mounting a fan on the worm shaft and arranging 
to direct its draught between fins cast on the gearcasing. 
Fig. 59 illustrates a typical worm-gear unit of this type. 


Measuring Gears 

A Gear dimension that is invariably checked either 
directly or otherwise is the tooth thickness. Given an 
accurate machine and cutter, the desired tooth thickness 
may be secured by sinking the cutter to the appropriate 
depth in the blank, and whilst in practice this is often 
done (with the aid of measuring devices built into the 
machine and indicating the movement of cutter-mounting 
relatively to the blank), the final check is usually made by 

Fig. 6o.— A vernier caliper for measuring the thickness of gear teeth. 

means of a gear-tooth caliper (Fig. 60). This has a vernier 
adjustment, setting the edge of a blade at the required 
height above two gauging points. The distance between 
these points is measured by a second vernier. The caliper 
thus measures the thickness of a tooth in a plane at any 
required distance from the tip of the tooth. 

The vernier graduations correspond to steps of *ooi in., 
but owing to the difficulty of reading such fine graduations 
it is not usually possible to guarantee a setting to less than 




*002 in. Even this order of accuracy more than meets 
nornal requirements, any error occurring on this account 
being easily absorbed in the backlash allowance which is 
rarely less than about 0*005 in. 

The gear-tooth caliper may be applied to any type of 
gear tooth and usually, though not essentially, it is set to 
measure the tooth thickness on the pitch cylinder of 

Micrometer Measurement of Tooth Thickness.— 
In certain circumstances the tooth thickness of involute 
spur gears may be determined from micrometer measure¬ 
ment over the teeth. In Fig. 61 are shown a number of 
tangents to the base circle of a spur gear. From the 
construction of the involute it follows that each of the 
tangents shown intersects the two " opposed ” involutes 
at right angles. Also the distance between the points of 
intersection of a tangent with the two involutes is equal 


to the distance PQ (measured round the base circle) 
between the origins of the two involutes. Hence the 
distances AB, CD, EF are all equal and consequently a 
micrometer measurement over any two " high points ” 
such as A and B on the opposed involutes X and Y gives 
the same dimension (see Fig. 62). This can be calculated 
from the tooth thickness and the number of teeth between 
X and Y. Obviously this method can be used only where 
the dimensions of the gear are such that a tangent to the 
base circle can be drawn to intersect two involute profiles 
in points to which a micrometer can be applied. Alterna¬ 
tively use may be made of a vernier caliper, but this is 
less accurate than a micrometer. 

Fig. 63.—Checking tooth thickness by micrometer measurement 
over rollers. 

Tooth thickness may also be determined from micro¬ 
meter measurement over rollers placed in two tooth spaces 
(see Fig. 63). 

Checking by Comparison.—When gears are being 
made in quantities, it is often more convenient to check 
tooth thickness by comparison with a master gear than 
by direct measurement. Fig. 64 shows this principle 
applied to the checking of thickness of worm threads. 
The worm is mounted between centres, and a ball held 
between two threads presses on the flat gauging anvil of 
a dial indicator as the worm is slowly rotated. The read¬ 
ing of the indicator is compared with that given in similar 
circumstances by a master worm. Repetition of the 
operation with the ball in each thread-space in turn affords 
a check on concentricity. 



For checking the thickness of worm-wheel teeth a ball 
micrometer (Fig. 64) is often used ; here again the reading 
obtained is compared with that derived from a master gear. 

Tooth Shape.—The simplest method of checking the 
tooth shape of spur, helical or straight bevel gears is to 
lay against a profile a correctly formed plate gauge. Such 
a gauge is made by filing until its optical projection (at 
25 to 50 magnifications) matches an accurate large-scale 

A comparable direct check on tooth form is afforded by 
using a gear-tooth caliper to measure tooth thickness on 
various planes be¬ 
tween tip and root 
and comparing the 
figures with those 
obtained by calcu¬ 

These methods 
are, however, in¬ 
sufficiently accur¬ 
ate for modern 
precision gears, and 
instruments have 
therefore been de¬ 
veloped by which 
the departure of 
the profile from 
the desired form is 
directly indicated. 

An example is the 
instrument shown 
in Fig. 66. Here 
the gear under in¬ 
spection is mounted 
on the same spindle 
as a disc of dia¬ 
meter equal to that 
of the base circle 
of the gear. The 
disc is pressed into 

contact with a p IG . g 4 .—Checking the thickness of worm 
straight-edge C by threads by means of a ball and indicator. 




means of screws acting through the rods A A and the 

bar B. Endwise movement of the bar causes the gear to 
roll along the straight-edge and (looking parallel to the 
axis of the gear) the point of intersection of a true 
involute profile with the straight-edge will not change its 

The checking of the profile is effected by a stylus placed 
so as to touch it on the line of the straight-edge and 
operating the plunger of a dial indicator reading in ten- 
thousandths of an inch. Departure of the tooth profile 
from the true involute form is shown directly by the 
reading of the dial indicator. 

A rapid check on accuracy of profile and concentricity 
of teeth and bore is afforded by the type of gear-tester 
shown in Fig. 67. Here the gear under test is slipped on 
to a spindle supported in a movable carriage spring-loaded 
in such a way as to press the gear into contact with a 
master gear mounted on a parallel spindle. The fit of 
the spindles in the gears is such that they can be 
rotated by hand, when any error in profile or concen¬ 
tricity causes variation in centre distance between the 

Fig. 65.—Checking the thickness of worm-wheel teeth by the use 
of a ball micrometer. 




Fig. 66 .—An involute-testing instrument. 

spindles. This variation is shown on a dial indicator, 
and a permament record may be taken by a pen which 
traces a line on a disc rotating in unison with the master 

A perfect combination of gears results in a circular 
track; any variation in centre distance as the gears 
revolve is reproduced with suitable magnification as a 
radial departure from the ideal circular form. 

The instrument shown in Fig. 67 is designed to accom- 


modate spur or helical gears. The same general principle 
applies in other machines arranged to take bevel gears and 
worm gears. 

Profile Checking of Worm Gears.—Some worm- 
thread profiles show straight sections in planes passing 
through the centre-line of the worm. In such cases, 
checking of shape may be effected by a gauge fixed to a 
V-block placed in contact with the tips of the worm 

Fig. 68 .—Checking a worm thread of the involute helicoid type 
with a straight-edge. 

threads. A more accurate method, however, is to mount 
the worm between centres and to apply to it a stylus 
pivoted on a block constrained by an adjustable guide to 
move parallel to the correct thread profile. Any error in 
profile is detected by the stylus and is made visible on a 
dial indicator. 

A worm thread of the involute helicoid type may be 
checked for shape in the manner shown in Fig. 68. Here 
a straight-edge is positioned so as to correspond to a 
generator of the thread, and the worm is rotated so as 

measuring gears 85 

S a thr , e ^ “ t0 contact with the straight-edge. 
Examination of the line of contact with a light placed on 
5®. ^ ar slde °f the straight-edge reveals proffle errors 
which, if large enough, may be estimated by use of feeler 
gauges. J 

The form of worm-wheel teeth is so complex that the 

Fig. 69.—The principle of an instrument for testing 
spur or helical gears. 

only practicable way of checking it is to mount the 
wheel in correct relation to a master worm whose 
threads have been lightly smeared with “blue marking” 
and after rotation of the gears to observe the distribution 
of marking on the wheel-teeth. 


Pitch-Testing.—The principle involved in the pitch¬ 
testing instrument shown in Fig. 69 is to bring into contact 
with two gear teeth a pair of styluses mounted in such a 
way as to show on dial indicators the amount by which 
the distance between the gauging points differs from the 
original setting. These errors are recorded for consecutive 
pairs of teeth until the whole circumference of the gear has 

been covered. The algebraic 
sum of the true errors (the 
accumulated error) is, of 
course, zero for the whole 
circumference, and the actual 
sum in any particular instance 
is a measure of the unavoid¬ 
able errors in setting the in¬ 
strument. The necessary 
corrections to the individual 
readings are conveniently 
made by recording them 
graphically as shown in Fig. 
70. The straight line AB 
joining the first and last points 
is the base line from which the 
accumulated pitch errors are 
measured. Thus the accumu¬ 
lated error between tooth 1 
and tooth 12 is represented 
by CD, i.e, 0-0003 in- 
In the instrument shown in 
Fig. 69 the gauging elements 
are carried on a pillar which 
is swung about its anchor-pin 
to bring them into and out 
of the measuring position. 
This is done by means of the 
handle shown, which is held 
against a fixed pin by a 
spring-loaded pawl to ensure 
uniformity of location when 
measuring. This move¬ 
ment of the gauging 
elements is necessary 

5 * * £ 


to permit of rotation of the gear for successive 

In order to deal with large gears which could not con¬ 
veniently be rotated about a fixed axis for measuring 
purposes, the pitch-testing instrument shown in Fig. 71 
may be employed. The framework is held in contact with 
the tips of the teeth of the gear to ensure its correct location 

in relation to the centre-line of the gear. Circumferential 
location relative to a tooth is by means of a ball attached 
rigidly to the frame, whilst a dial indicator measures the 
adjacent error between any two teeth that are not too 
widely separated. 

A check on accumulated error in gears which are small 
enough to be rotated on centres is shown in principle in 


Fig. 72. The instruments are set to measure the accumu¬ 
lated error between two teeth nominally separated by 
180 deg. By rotating the gear through an angle which 
causes the gauged teeth to interchange their positions, the 
instruments may be made to measure the errors in the 
two different half circles. The difference between these 
readings is equal to the difference between the true errors, 
whose sum must be zero. Hence the difference between 
the readings is equal to twice the numerical value of the 
accumulated pitch error in either half circle. 

Angular Division Tester.—Measurement of pitch 
errors in gear teeth may be effected by an optical method. 
A theodolite is mounted on the gear in such a position 
that its centre-line intersects that of the gear at right- 
angles. The mounting is such that relative angular move¬ 
ment of gear and theodolite can be accurately measured 
by means of a finely graduated dividing plate and a 
“ built-in ” microscope. 

The gear is first located angularly, by allowing a tooth 
to bear on a stylus, such as one of those shown in Fig. 69, 
accurate positioning being ensured by use of a sensitive 
dial indicator. The theodolite is sighted on to an illu¬ 
minated target situated at least ten feet away, and the 
reading on the dividing plate is recorded. The gear is 
then rotated until a different tooth is located by the 
stylus, the theodolite is redirected on to the target, and 
the difference between the new reading on the dividing plate 
and the previous one is the angle between the selected teeth. 

'An important point is that this procedure measures 
• accumulated error directly and not merely as the sum of 
measured adjacent errors. On this account the gain in 
accuracy of measurement is considerable. 

Spiral Angle of Helical Gears.—The instrument 
shown in Fig. 73 is used to check spiral angle and con¬ 
centricity of helical gears. It employs a gauging member 
in the form of a tooth of the basic rack corresponding to 
the gear teeth. This member is mounted at one end of a 
spindle which carries on its other end an arm having two 
projecting pins. These permit a micrometer or slip-gauges 
to be used to effect accurate angular setting of the gauging 
member, and when this has been done a dial indicator is 
set with its operating plunger in contact with one pin. 



The spindle assembly is then moved bodily towards the, 
gear until the gauging member has penetrated as far as 
possible into a tooth space. Any error in spiral angle acts 
on the gauging member to rotate the spindle, the amount 
of such movement being shown on the dial indicator. 

The distance between the spindle mounting and the 
axis of the gear when the gauging member is fully engaged 
is measured with a second dial indicator, and variation of 
this reading from one tooth space to another is a measure 

of eccentricity. 

Checking by Running Together.—The most trouble¬ 
some effect of errors in high-speed gears is that of noisy 
running. Consequently, if it is convenient to mount a 
pair of gears in bearings and to run them at full speed, 
this is the most satisfactory method of checking them. 
If they pass such a running-test it may be concluded that 
any errors they do possess are too small to be of any 

practical importance. 


Gear-Tooth Comparators. — Until comparatively 
recently the only means of measuring gear teeth was by 
means of the gear tooth-caliper, an instrument necessitating 
two settings for each pitch measured, and also a different 
setting for any variation in the number of teeth for a given 
pitch. That is to say, if a gear of 20 teeth, six d.p,, is being 
measured, a different setting is required for a gear of 30 
teeth of the same pitch. This complication arises out of the 
fact that the gear-tooth caliper measures the thickness at 
the pitch line, so that for gears of a given pitch, but of 
varying numbers of teeth, a calculation has to be made of 
the chordal thickness and also the chordal height. 

Fig. 74.—A diagram explaining the principle of the Sykes gear-tooth 
comparator or caliper. 

All workmen, unfortunately, are not able to make this 
calculation, and they are rather apt to set the instrument to 
measurements given in tables, which, of course, require 
correction according to the number of teeth; Another dis¬ 
advantage of this instrument is the fact that the measure¬ 
ment of the tooth thickness is taken on the comers of the 
two jaws, so that the wear is concentrated at these points, 
and leads to a rapid deterioration of the instrument. 

Two instruments which have largely overcome these 
difficulties are the Sykes' comparator and the Sykes' caliper 
for measuring gear teeth. The comparator and its method 
of application is shown in the accompanying illustrations. 

It will be observed that, in fine, the instrument consists 
of a substantial frame or beam carrying one fixed and one 


adjustable jaw, in addition to a specially designed dial test 
indicator. The movable jaw is provided with means for 
fine adjustment, and between the jaws the plunger of the 
dial indicator projects. Fig. 75 shows the instrument set to 
a master gauge block. Fig. 78 shows the instrument 
complete in case, with a set of master gauge blocks. 

The Working Principle.—The principle on which the 
instrument is based will be obvious to those possessing a 

knowledge of the property of the involute. Each gauge 
block represents an involute rack tooth of a particular 
pitch and pressure angle, and the inclined faces of the jaws 
are made to correspond. It is well known that any gear of 
a given pitch, irrespective of the number of teeth, will 
accurately gear with a rack of the same pitch and pressure 
angle. Therefore, when the comparator is set for a rack 
tooth (represented by the gauge block), the setting is 
correct for all teeth of the same thickness, pressure angle 
and addendum. It will readily be apparent that any tooth 


thicker or thinner than that setting will cause the dial 
indicator to show plus or minus, owing to the teeth entering 
to a lesser or greater distance between the jaws. 

Using the Comparator.—For convenience the dial of 
the indicator can be rotated so as to alter the position of the 
zero mark. When it is only desired to test teeth for 

uniformity, gauge blocks may be dispensed with, as a 
master gear may be used to set the comparator. The jaws 
of the instrument and also the gauge blocks are glass hard 
and are ground and lapped to a minute degree of error. 

The standard gauge blocks are supplied at 144 degrees 
pressure angle, but of course, blocks of any pressure angle 
can be supplied. 


One division on the dial represents *0005 in. for teeth of 
14J degree pressure angle. The instrument has a capacity 
for 12 d.p. to 1 d.p., or from J in. to 3 in. circ ular pitch. 

It must not be assumed that a gauge block is necessary 
for every pitch to be measured. Seven blocks will measure 
all pitches ; six blocks will measure all pitches between 

12 d.p. and ij d.p. and four blocks will measure every 
pitch between 12 d.p. and 3 d.p. 

Fig. 76 shows the method of testing for uniformity of 
tooth thi ckn ess; a series of three instruments may be 
used for testing tooth curves. Due to the principle of the 
ins tr um ent, no alteration or adjustment in any _way is 
necessary in cases where the addendum of the pinion is 
increased or decreased, provided that the addendum of the 


wheel into which the pinion gears is similarly decreased or 
increased. In either case the jaws automatically bear on 

the theoretical con¬ 
tact points of the 
tooth curves. 

Zeiss Optical 
Gear - Tooth 
Caliper.—This in¬ 
strument is similar 
in principle to the 
vernier gear-tooth 
caliper but differs 
from it in that 
the “ height ” and 
“ thickness ” scales 
are on glass and can 
be seen only through 
a magnifying glass 
built into the in¬ 
strument. They lie 
at right angles to 
each other and their 
Fig. 78.—The comparator complete in case, point of intersection 
with a set of master gauge blocks. is at the centre of 

the field of view. The longitudinal line of each scale acts 
as the fiducial line for the other. 

The Sykes Gear-Tooth Caliper.—The diagram. Fig. 74, 
also explains the principle of the caliper shown in Fig. 79, 
designed by the originator of the comparator. The lines 
aa , aa represent a rack-tooth space, and the dotted lines 
the jaws of the Sykes caliper, which it will be seen directly 
represent the rack. The line c is the pitch line of the rack, 
and d the pitch line of a particular pinion or wheel, bb and 
bb are the lines of pressure or lines of action, on which tooth 
contact must always occur, whatever the size or number 
of teeth of the gear. These lines are always the same for any 
given pressure angle, and it is well known that in the in¬ 
volute system the rack tooth has straight sides, and the first 
law of gear-tooth contact is the common normal to the tooth 
curves must pass through the pitch point. In the diagram 
the pitch point is fi } and the lines b pass through it and are 
normal to the rack-tooth profile. It will, therefore, be 



apparent that contact must always take place on the lines 
b t and that the position of the point of contact e depends 
only on the thickness of the tooth. In gears of any particular 

pitch the profile of the tooth is different for each number of 
teeth, but the point of contact, with the centre-line of the 
tooth and tooth space coincident, never varies. 

Base Circle.—In measuring helical or spur-gear teeth 
an important dimension is the base diameter of the gear. 
This is denoted by d 0 and its value is given by 

d 0 =—* cos <J/ t or Sfr -——^-—7 — r - r 

TZ 7TT v (cos 2 CT-ftan 2 ^ a ) 

where t —number of teeth in gear. 

p t =transverse pitch of teeth (measured in plane 
perpendicular to axis). 

p n =normal pitch of teeth (measured in plane per¬ 
pendicular to tooth spiral). 
t{; t =transverse pressure angle (measured in plane 
perpendicular to axis). 

^ n =normal pressure angle (measured in plane per¬ 
pendicular to tooth spiral). 
o- =spiral angle at pitch cylinder. 
o- 0 == spiral angle at base cylinder. 

L=Lead of spiral. 


The base spiral angle cr is determined from the given 
quantities by the relation 

sin c^-sin <r cos 


J tp t • tp n 

tan or sm cr~-jr~ 

a ^ so tan ^ t =tan sec cr. 

Another important mathematical quantity is the " in¬ 
volute function ” of an angle. 

Thus the involute function of 0 is denoted^ by inv. 0 and 
is equal to tan 0—0, where 0 is measured in radians, i.e. 
it is equal to the number of degrees in the angle multiplied 
by tc/i8o. Very often .this function is introduced in a 

form such as inv. sec -1 5 which means " Determine the 
d 0 

angle whose secant is g- (or whose cosine is -g°) and then 

find the involute function of that angle by subtracting its 
value in radians from the value of its tangent.” 

To determine the tooth thickness of involute spur or 
helical gears from micrometer measurement over the 
teeth (see Fig. 62): 

Let t=number of teeth in gear. 

T=number of teeth contained between micrometer 

M=micrometer measurement over teeth. 

S 1 =tooth thickness measured perpendicular to tooth 
spiral on cylinder of diameter d x . 
spiral angle at diameter d x . 


d # —base diameter. 

<r 0 —spiral angle at base cylinder. 

Then „ Ted. 


and o / ' M . d tc„ , 

Si= id^;- mv - sec d„- 5t<r—cos or 1 

This gives the normal tooth thickness on any desired 

For the determination of tooth thickness of involute 



spur gears from micrometer measurement over rollers (see 

Fig. 63): 

Let d—diameter of circle on which tooth thickness is 

d^diameter of base circle. 

Z=measurement over two equal rollers in opposite 
tooth spaces. 

D =diameter of each roller. 

W—circumferential thickness of tooth on circle of 
diameter d. 

t—number of teeth in gear (must be even). 


\xt a ( n 1 • -1 . d D\ 

W=d mv. sec 1 -mv. sec” 1 -^ J 

Assuming that the profile of the teeth is accurately 
involute, this formula applies with any roller diameter 
that causes the roller to make contact with the teeth at 
points on the involute part of the profile. 

If the roller is required to touch the teeth on the pitch 
circle where the pressure angle is i|» and where the width 
of tooth is equal to width of space, then the roller diameter 
may be determined from 

D=d 0 | tan (^-j- 2 L)—tan 4 » j- 

The expression for D is approximately equal to 

d, tan 22. and this value may be used in all ordinary cases. 
2 t 

The measurement of tooth thickness of involute spur or 
helical gears at the pitch cylinder by use of vernier calipers 
is accomplished in the following manner : 

Measurement of Tooth Thickness by Vernier Caliper 
(Fig. 60 ).—If tooth and space are to be of equal 
the pitch cylinder, the vernier caliper will fit the teeth with 
the settings specified below— 

Height Setting=Addendum—0-196 £2. 

Thickness Setting—^ sin ( 2 ?) 

where p n =normal pitch 

and t=number of teeth in gear. 



Alternative Formulae.—As an alternative to the 
measurement formulas given, the following formulae are 
offered as being simpler to apply. The fundamental 
difference is that they apply to measurement of tooth 
thickness at points which do not lie on the pitch cylinder. 
The formulae do not involve any mathematical approxi¬ 
mation but are quite- accurate. 

To determine the tooth thickness of involute spur 
or helical gears from micrometer measurement over 
teeth by the base tangent method (Fig. 62): 

Let p n =normal pitch. 

^—normal pressure angle. 

^—transverse pressure angle, 
c-—spiral angle, 
t—number of teeth. 


w—normal thickness of tooth at pitch cylinder. 
n=number of teeth between micrometer anvils. 

M—micrometer measurement over teeth. 


sin <r — t p n /L 
tan (p t =tan sec a- 

and M—cos |w-fp n [ (n—i)inv. <J> t Jj j- 

Here inv. tan <J; t —the angle t|/ t being measured 
in radians, i.e. (71/180) times the angle measured in degrees 
and decimals of a degree. 

To determine the tooth thickness of involute spur 
or helical gears by micrometer measurement over 
rollers (Fig. 63) ; 

The symbols are as above, except that w represents the 
required normal thickness of tooth at the pitch cylinder 
and then 

Roller diameter — (p n —w) cos 

and the micrometer measurement over the rollers, when the 
normal tooth thickness is actually w, is given by 

N=^- n sec cr—{— (p n —w) cos 

7 T 

This applies only to gears with even numbers of teeth. 


To determine the tooth thickness of involute gears 
of odd numbers of teeth by micrometer measure¬ 
ment over rollers: 

The symbols are as above, and micrometer measurement 
n =~F ? (i —3 sin 2 ^-)+(p„—w) cos <J>„. 

This applies to straight tooth gears (so that £*=circular 

Fig. 79a.—D iagram showing the principle of the involute gear¬ 
testing instrument. See also Fig. 66. 

pitch and \p n —transverse pressure angle) and to the con¬ 
dition in which one roller is in a tooth space as nearly as 
possible opposite to that occupied by the other roller. The 
number of teeth must be odd, and w represents the required 
tooth thickness at the pitch cylinder. 

Micrometer measurement of tooth thickness cannot be 
applied to helical gears with odd numbers of teeth. 


Hobs, End-Mills and Generating Cutters 

The hobbing process is the one that produces the most 
accurate cut gears, and it does demand a high standard of 
accuracy in the hob itself. To meet modern requirements 
the hob teeth must be finished by grinding on a machine 
in which the abrasive wheel head has a relieving motion 
whilst the hob is rotated and moved endwise. 

The material used for hobs of normal dimensions is high¬ 
speed tool steel, containing about 075 per cent, carbon, 
18 per cent, tungsten, 4 per cent, chromium, and 1-3 per 
cent, vanadium. This needs special care in forging to 
avoid the formation of cracks and, after turning and boring, 
careful inspection is required to detect any defect of this 

A boss about a quarter-inch wide is turned at each end 
of the hob in order to provide cylindrical surfaces for check¬ 
ing the true running of the hob in service. The keyway or 
driving slot is cut at this stage. 

Thread-Milling. —The thread (or threads) are then 
produced in a thread-milling machine. If quantities are 
involved, it may be worth while to make a milling cutter 
specially formed to produce the required thread section. 
Otherwise the most closely approximating available cutter 
is used to rough out the thread, which is afterwards modified 
to the required shape by use of a form-tool in the lathe. The 
thread shape is checked by comparison with a gauge or, 
if it is straight-sided, by the use of a sensitive tracer and dial 
indicator traversed in a straight line along a slide at the 
appropriate angle. 

Milling Flutes. —The flutes are then produced on a 
universal milling machine using a cutter of 30 deg. V- 
section, one side of which is perpendicular to the axis of the 
cutter. The vertical line through the plane containing 
these perpendicular edges is set to intersect the centre-line 
of the hob flank so that the cutting face of each flute 
contains radial straight lines. The radius at the tip of.the 




cutter is equal to about one-eighth of the normal pitch of 
the hob. 

The angular setting of the table of the milling machine 
is the spiral angle of the flute at the pitch cylinder of the 
hob. The change-gears between dividing head and horizontal 
feed screw are selected to produce the lead of the flutes. 

The hob is indexed from one flute to the next by use of 
the dividing head. The cutter is finally sunk into the blank 
so that the root of the flute is at a distance from the outer 
surface of the hob blank equal to 

Depth of hob tooth-frise of relieving cam-f- tip 
radius of cutter. 

Relieving. —The relieving of the teeth is carried out on 
a relieving lathe ; the tool is formed to match the thread 
form of the hob, a witness about 0*020 in. wide being left 
on each tooth. 

The hob is then hardened in accordance with the steel¬ 
maker's instructions for the particular brand of steel used, 
double-quenching and tempering frequently being specified. 

Grinding. —The next operation is to grind the bore, 
end faces and cylindrical registering surfaces. These 
surfaces are required to run truly in relation to each other 
to an accuracy of o*oooi in., and the diameter of the bore 
must not exceed the nominal size by more than 0*0002 in. 

The cutting faces of the flutes are next ground on a hob- 
sharpening machine set up for the lead of flute specified for 
the hob. Indexing of the flutes is required to be accurate 
within 0*0005 in. measured at the pitch cylinder. 

The relieved surfaces of the teeth are then ground on a 
hob-profile grinding machine. After the tip (which is 
usually straight) has been treated, the abrasive wheel is 
trimmed by successive trial to give the required hob tooth 
form. If this is of simple straight-sided type it may be 
checked by use of a fixture arranged to run a tracer-point 
along the required path. Otherwise an optical projector is 
used to compare a magnified shadow of the tooth with an 
accurate, drawing of the required normal section on the 
same enlarged scale. 

After one set of flanks has been satisfactorily^ shaped, the 
others are similarly treated. Further grinding is then done 
to bring the teeth to the required thickness. 

The error of axial position of any tooth relative to its 


neighbours should not exceed 0*0002 in. and the error of 
relative position between any two teeth should not exceed 
o*ooo2 in. plus o-oooi in. for every inch of axial distance 
between them. 

Sharpening. —The hob is sharpened when necessary by 
grinding the radial faces of the flutes on a hob-sharpening 
machine, accurately set up to give correct lead and indexing. 
This operation must be carried out with care if the hob is to 
reproduce the accuracy with which it was manufactured. 

Formulae used in Hob Design and Manufacture 

Let p n =normal pitch of tooth to be produced, 
t =number of threads in hob. 
d =pitch diameter of hob. 
j = outside diameter of hob. 

A. —lead angle of hob thread, 
or ^spiral angle of flutes. 

F =*=number of flutes. 

L —lead of threads. 

I —lead of flutes. 

For hob made with solid shank 

Minimum desirable value of d—4p a . 

For hob bored to mount on arbor 
Minimum desirable value of d—4p a +diameter of arbor. 
Outside diameter of hob j 

=d-|-2 X dedendum of gear tooth. 

Lead of hob thread L 

~W VCMla 2 ) 2 ] 

. , L 


Tip radius of flute-cutter—0-125 Pu 

Lead of flutes 

Spiral angle of flutes 0-=A.. 

Number of flutes F= ~^ ^ 

, , 1,3 P* 

(taken to nearest whole number). 

The number of flutes F and the number of threads t should 
preferably have no common factor. 



Rise of relieving cam^ ^^J j (approx.). 

Ratio of change-gears for relieving cam: 

Product of numbers of teeth in driving gears 
Product of numbers of teeth in driven gears 
T _ / Number of flutes \ Z-fL , Tr . . . 

= K x 'number of lobes in cam' T where K15 a constant 

for the machine. The number of lobes in the cam is usually 

The Manufacture of End-Mills for Gear-Cutting.— 

The end-mill process is usually employed for producing 

gear teeth only when more accurate processes are im¬ 
practicable, e.g. in the manufacture of triple-helical gears 
with continuous teeth. 

The end-mill cutter (see Fig. 80) of the shape necessary 
for cutting average tooth forms is a comparatively fragile 
and short-lived tool, and more than one may have to be 
used in completing a large gear. In the smaller sizes the 
cutter is made in one piece with the shank, which is tapered 
to fit into the spindle of the machine. In large sizes it is 
practicable to make cutter and shank as separate pieces. 


the one from high-speed tool steel and the other from plain 
carbon steel. 

Turning. —The tapered shank is turned in the lathe and 
a tapped hole made for reception of a draw-bolt when the 
cutter is in use. The body of the cutter is turned to a profile 
which fits a “ half-shape ” gauge, the diameter at a specified 
distance from the end being measured by a vernier gear- 
tooth caliper, and made to leave an allowance of about 
o-oio in. for grinding. The profiling is accomplished by 
feeding the cutting tool inwards by hand, whilst the saddle 
has a slow traverse away from the headstock ; the final 
shape is achieved by applying a hand-scraper, ground to a 
suitable radius, to the work whilst running in the lathe. 

Milling. —Cutting edges are next produced by a milling 
operation. As the edges must continue almost to the centre¬ 
line they are usually limited to four ; in large cutters four 
extra intermediate edges are sometimes formed on the 
curved part of the profile. A plain cylindrical milling cutter 
is employed and is guided by hand control of the vertical 
and horizontal movements of the milling-machine table. 
The shape of the surfaces produced by the cutter is not 
important provided that the radial cutting faces and the 
" lands " behind the cutting edges are each about J in. wide. 

The lands are then filed with a relief angle of about 10 
deg., leaving a witness about 0-020 in. wide at the cutting 
edges. This operation is not critical, as the edges are 
subsequently finished by grinding. 

Grinding. —After hardening, the tapered shank is 
ground to fit a standard taper gauge. 

The final operation of profile grinding the cutting edges 
is effected by mounting the cutter in the taper bored 
spindle of a fixture arranged to be moved by hand over a 
horizontal table. The spindle carries a dividing plate by 
which it may be indexed into four equidistant angular 

The table carries an abrasive wheel head arranged with 
the centre-line of the wheel slightly lower than that of the 
centre-line of the cutter. Vertically beneath the periphery 
of the abrasive wheel is a stop of the same shape as the 
plan view of the edge of the wheel. 

A former plate is attached to the cutter fixture in such a 
position as to correspond with the desired profile and to 


bear on the fixed stop. Thus when the fixture is guided by 
hand so that the former plate always makes contact with 
the stop, the abrasive wheel produces an edge of the 
correct shape on a cutting face which lies in a hori¬ 
zontal plane. The difference in height of centre-lines of 
cutter and abrasive wheel is such as to give a suitable 
relief angle. 

The grinding is continued equally on each cutting edge 
until the diameter at a certain distance from the end 
corresponds to the calculated figure. As a check on the 
shape of the profile, the diameter should be measured in two 
other planes as widely separated as possible and the figures 
compared with those obtained by calculation. 

Determination of the Shape. —The shape of the end- 
mill required to produce any helical gear teeth of involute 
form can be determined by calculation. Except in the case 
of spur gears the end-mill shape is not involute, although 
it approximates to it if the helix angle is small. The shape 
may be taken as that corresponding to a spur gear having 
pitch and pressure angle equal to the normal pitch and 
normal pressure angle of the actual gear and a number of 
teeth equal to the actual number multiplied by the cube 
of the secant of the spiral angle. If this latter number 
exceeds about 60 the approximation is adequate for good- 
class commercial gears. 

The cutter is re-sharpened when necessary by the method 
used for grinding its form in the first instance. 

Gear-Generating Cutter.— The four principal methods 
practised for cutting gear teeth are : (i) by means of a 
rotary disc cutter formed to the tooth profile ; (2) by means 
of an end-mill; (3) by means of a planer; and (4) by 

The rotary disc cutter has the disadvantage that practical 
considerations compel the use of the same cutter through 
a considerable range of sizes of gears, whereas to obtain a 
true tooth shape a cutter would be required for every size ; 
in one well-known system only eight cutters are supplied 
to cut all gears of a given pitch from 12 teeth to a rack, 
which latter may be considered as a gear of infinite radius. 
For the second method, it may be said that it is chiefly 
used for herring-bone teeth, and would not be used except 
in special circumstances for straight-cut teeth. 


The third practice is chiefly of a makeshift nature. The 
fourth, however, has now been accepted as the only correct 
method of cutting gears, and its introduction has made 
possible the cutting of gears of greater accuracy than by 
any other process. 

Before dealing with the actual manufacture of gear 
cutters it is necessary to understand certain principles 
and functions relating to generation. 

There are two principal methods of machining a con¬ 
toured or other surface, namely, forming or generating. 

To illustrate this 
point, a surface plate 
may be planed by 
using a tool having 
a perfectly straight 
cutting edge, the full 
width of the plate, 
feeding the tool 
gradually at every 
stroke; this is an 
example of forming a 

On the other hand, 
if a tool is fed 
across the work, and 
moved slightly in a 
parallel plane at each 
successive stroke, the 
,„ j surface would be 
Fig. 8i.—A typical gear cutter. generated. Similarly 

a cylinder may be 
formed on the milling machine, or generated in the lathe. 
It is evident that the generating method of obtaining a 
surface is the more accurate, whether applied to gears or 

Gear Generation.—The generation of gears is made 
possible by the use of a cutter which is in itself a gear 
(see Fig. 81), and the action of the tool is clearly shown in 
rig. 82. This illustrates a metal gear being rolled and in 
mesh with a material of plastic nature, such as wax. When 
rolled together as two gears in mesh, the generating gear 
will mould teeth in the plastic material of correct shape;» 


therefore, all gears so shaped will mesh with one another, 
or with their generator. 

Fig. 83 shows this principle as applied to the cutting of 
metal blanks. The cutter is reciprocated vertically by the 
ram of the gear shaper on which it is mounted. During this 
motion the tool is first gradually fed inwards until the 
correct depth has been reached, then the cutter and the 
’generated gear are revolved together exactly as two gears 
in mesh. The teeth of the revolving and reciprocating 
cutter thus form teeth of the correct form in the blank. 
Obviously, the teeth of the generating cutter must be 
extremely accur¬ 
ate, with proper 
clearance angles, 
in order to en¬ 
sure free cutting. 

The sequence of 
the tool move¬ 
ments will be 
gathered by re¬ 
ferring to Fig. 84. 

Fig. 82 (above ).—-A 
sketch showing the 
action of the tool 
illustrated in Fig. 

Fig. 83 (left ).—The 
cutter at work on a 
metal blank. 


Making the Cutter.— As is the practice with other gears, 
the cutter is firstly roughly machined, bored and faced on 
the turret lathe. The teeth are then profiled roughly either 
on the millin g machine or on the gear shaper by means of 
cutters, which leave a sufficient tolerance on the tooth 
thickness for final grinding to shape. 

After these preliminary roughing operations have been 
effected the cutter is hardened, and it will thus be seen that 
on account of the following finishing and final grinding 
operations there is no risk of the tool, being inaccurate. 
As the accuracy of subsequent operations is dependent 

on the bore, this is 
atank s' finished first, after 

\ which, the front and 
f^C\ \ back faces of the 

K\\ \ teeth and blank and 

/Hill 1 A \ an y succeeding 
\ \ operation that may 

, uTS "" I \ be necessary, are 

\ completed. 

\ The Involute 

/ ( / Grinder. — Figs. 85 

/ \ Cutter an( i 86 outline the 

Shape of \ machine and prin- 

chip ciples for grinding the 

' profile of the teeth 

^ , to a true involute 

FiG.84.—The sequence of the tool move- This opera _ 

ments, showing the shape of the chip ,. . ~ ^ , 

removed. tion is effected by the 

generating process, 

and it is this function which defines a generated cutter. 


Shape of 

Fig. 84.—The sequence of the tool move¬ 
ments, showing the shape of the chip 

Referring to the illustration, the cylinder A represents 
the rolling circle of the cutter, and it is, therefore, of the 
same diameter as the pitch circle of the generating gear. 
This cylinder oscillates on a plane surface B, and is ensured 
against slipping by the action of the steel tapes C, which 
are fixed to the cylinder and positively control the rolling 
motion. The cutter D (Fig. 85) is mounted on the end of the 
spindle, which is rigidly secured to the cylinder. As the 
cylinder oscillates up and down on the inclined plane the 
cutter is caused, by means of the steel tapes, to roll in 
exactly the same way as if it were a gear in mesh with 


the imaginary rack, F, as denoted by dotted lines in 
Fig. 88. 

The straight face of the grinding wheel, E (Fig. 85), 
realty represents one of the flat faces of the rack teeth, 
and since the rolling motion of the blank is controlled by 
the cylinder,, together with the angular plane surface and 
steel tapes, it will cause the revolving grinding wheel to 
profile a true involute curve on the face of the cutter tooth 
or work. In order that one face only of each of the teeth 
of the cutter may be ground in turn, a dividing head is 

fitted so that the blank can be indexed in relation to the 
spindle. When this operation is completed the cutter is 
taken off and placed on a machine which is “ left hand ” 
to the first, and the same procedure is repeated on the 
opposite faces or sides of the teeth. 

The face of the grinding wheel should be dressed to great 
accuracy by means of a diamond, which should be truly 
traversed across the face of the wheel. 

A string being unwound from a cylinder describes the 
involute curve, and in this grinding operation the steel 
bands impart a similar movement; thus, since the cutter 
motion is synchronous with that of the rack the correct 



shape must necessarily be transferred to the tooth being 

Pressure Angle Corrections.—The head which carries 
the grinding wheel has a swivelling movement, so that the 
cutters for generating helical gears may be ground. It must, 
of course, have a relative motion in a right-angled direction 
which permits the relief angles to be ground on each tooth 

With regard to relief angles, these are involved in the 
calculations for determining the pressure angle. For 
example, if the tooth has a 2 deg. relief angle, the pressure 

angle of the grinding wheel 
^£=====5^. is 14J- deg. However, the 

// relief angle will increase 

ff __ fK the pressure angle of the 

II \\ C cutter by 12' approxi- 

fHHHi Jr mately; thus, in order 

\\ If cu ^ er luay be 

\ produced with the required 

pressure angle, this discrep- 
ancy must be adjusted by 
inclining the grinding wheel 
head 14J 0 —I2'=I4° 18'. 

j In setting the grinding 

l _head to correspond to the 

Secf/b /7 X—X s P iral for a helical 

Fig. 86 .—A section on the line fatter, the pressure angle 

X_X in Fig. 85 . has again to be corrected ; 

it has to be transferred 
from the linear pressure angle into the normal pressure 

This is found by applying the following formula : tan of 
normal pressure angle=tan of linear pressure angle X cos 
spiral angle. 

In checking the chordal thickness of a helical cutter, it is 
easier and more convenient to measure it normally to the 
spiral angle. The normal thickness (NT)—circular thickness 
X cos spiral angle. 

It frequently happens that a cylinder, to which the steel 
tapes are fitted, required with a pitch diameter correspond¬ 
ing to that of the cutter desired to be produced, is not in 
stock. The cylinder having a diameter nearest to that 



figure required should be used and the pressure angle 
corrected to correspond. The following formula will give 
the corrected pressure angle : cos press, angle required x 
correct pitch radius divided by pitch radius of cylinder used. 

The normal pitch—cos spiral angle x circular pitch. 

The following pressure angle data will be of great assist¬ 
ance in the design of gear-generating cutters. 

Tan normal pressure angle (NPA)=tan linear pressure 
angle (LPA) x cos spiral angle (SA). 

tan NPA 
cos SA 

=tan LPA 

tan NPA 
tan LPA 

—cos SA 

tan LPA x cos SA 
tan NPA 
tan NPA 

cos SA x tan LPA' 

Addendum of the Cutter.— In measuring the thickness 
of these cutters it should be remembered that the addendum 


of the cutter is the dedendum of the gear tooth ; calipers 
should therefore be set to a corrected addendum and not 
to that obtained by the formula, 

diametral pitch (DP) 

which only gives the addendum of the teeth cut. The rule 
for the addendum of gear-generating cutters is ~jyp 

Chordal Thickness and Height. —Fig. 87 indicates 
clearly what is meant by the chordal thickness—the usual 
method of measuring the thickness on the pitch line, with 
the gear-tooth calipers set to the addendum of the gear. 
Standard formulas only give the length of the arc cut by 
the tooth on the pitch Ime, and it mil at once be evident 
that this is incorrect, as the true thickness must be the 
chordal length of the arc. Again, the curvature of the pitch 
line virtually increases the addendum at the point of 
measurement, and this increase is equal to the height of the 
arc. In making these corrections the following formulas 
should be applied: 

angle B “timber 0 f teeth 
Chordal thickness=sin BX2 rad. 

or—sin Bx pitch diam. 

Chordal height —pitch rad. x (i-cos B) 

Load Capacity of G ears 

When power is being transmitted by a pair of gears a force 
is exerted by each tooth on the one with which it makes 
contact, and the load capacity of the pair is determined bv 
the ability of the teeth to resist the effects of such forces. 

For example the force P applied to the gear tooth 
shown in Fig.. 89 causes local “surface stress” in the 
material pear its point of application and also tensile and 
compressive bending 
stresses” at A and B 
respectively. These 
stresses are applied and 
removed every time the 
tooth passes through the 
zone of contact, and the 
intermittency of the 
loading makes it more 
destructive than a con¬ 
stant loading of the same 

Repetition of excessive 
surface stress usually 
leads to destruction of 
the tooth surface by 
“pitting,” small pieces 
of metal breaking away 
to leave depressions of 
approximately spherical 
shape. If lubrication is 
inadequate, high surface pressure on the parts of the 
tooth profile which slide over the mating profile causes 
wear to occur in such parts. In extreme cases, the 
lubricant may be squeezed out, when sliding contact of 
metal on metal under heavy pressure may lead to partial 
welding together and “ dragging ” of the surfaces, leaving 
them too rough and inaccurate for smooth meshing. 

8 IJ 3 

Fig. 89.—The force P causes inter¬ 
mittent tensile and compressive 
bending stresses at A and B. 


In most cases the load capacity of a pair of gears is 
limited by surface pressure rather than by bending stress. 
In other words, the teeth are more likely to fail by 
destruction of the working, surfaces than by breakage of 
the teeth. If the material is case-hardened steel, however, 
the reverse may be true, because the surface is much 
harder than that of steel in its normal unhardened con¬ 
dition, and so the teeth can withstand relatively high 
surface stress whilst their bending strength is not 
proportionately increased. Also, teeth made from a 
brittle material such as cast-iron may snap off under a 
shock load which, if only occasionally repeated, might 
not have had any serious effect on the working surfaces. 

Shock Loading.—In designing gears it is necessary to 
take into account intermittency of loading, the possibility 
of shock loading such as occurs, for example, in rolling 
mill drives and the occurrence of continual load pulsation 
as, for example, in drives associated with engines or 
reciprocating pumps. 

The magnitude of the load that may safely be imposed 
on (say) one-inch width of a gear tooth depends on the 
following factors: 

(a) Frequency of application of load, i.e. speed of 

rotation of gear. 

(b) Nature of material of gear. 

(c) Shape and size of tooth. 

(d) Rigidity of mounting of gear. 

(e) Duration and frequency of loading periods. 

No further comment is needed on (a) and (b). So far 
as surface pressure is concerned the important feature of 
(c) is the “ relative radius of curvature ” of the teeth at 
the line of contact. This varies to some extent according 
to the position of the line of contact on the profile, but its 
average value is used as a means of determining the 
allowable surface pressure. Broadly it may be said that 
for gears of given normal pressure angle and spiral angle, 
the permissible load per inch width is proportional to the 
o*8th power of the pitch diameter of the pinion. The 
limiting figure in any particular instance may be calculated 
from, the dimensions of the gears with the aid of formulae 
published in the appropriate British Standard specification. 

Bending Stress.—So far as bending stress is con- 


cerned, the important features of (c) are the depth of the 
tooth and its thickness at the root. These are controlled 
largely by the pitch, but even for a given pitch the tooth 
thickness varies somewhat according to the number of 
teeth. Here again the British Standard specification 
indicates how to determine load capacity on the basis of 
" strength ” (as this is conveniently described in distinction 
from “ wear ”) of any pair of gears. 

It may be observed that if the materials, diameters and 
width of a pair of gears are fixed, the load capacity on the 
basis of " strength ” may be varied by varying the numbers 
of teeth, or, in other words, varying the pitch. Such a 
change, however, has no appreciable effect on the load 
capacity on the basis of “ wear,” and as this is usually 
the limiting factor it follows that nothing is to be gained 
by adopting excessively large pitches. 

Smoothness of running is improved by the use of a 
fine pitch, and as a general rule the best design practice 
is to use the finest pitch that will give adequate strength. 
Except when case-hardened steel is used this means that 
the most suitable number of teeth in the pinion is rarely 
less than about 25. 

Working Life of a Gear. —The rigidity of mounting 
of the gears affects the load capacity, because it has an 
influence on the distribution of loading across the face of 
the gear. It is not easy, or usual, to attempt any numerical 
assessment of this factor, and indeed its variation between 
different examples is not great if care is taken in design to 
provide a mounting of the utmost practicable rigidity. 

The standard basis of gear design gives a normal life- 
expectation of 25,000 hours, or, roughly, 2,000 working 
days of 12 hours each. If the operating time in any 
particular instance is known to be different from 12 hours 
per day, it is usual to modify the working stresses so as to 
keep the normal expectation of life at 2,000 working days. 
The probable life is approximately proportional to the 
reciprocal of the third power of the working stress; for 
example, if the required length of life were only one- 
eighth of 25,000 hours the working stresses might be 
double those applying to the standard 25,000-hour rating. 

In some instances, particularly in the case of worm 
gears, continuous operation may lead to such a rise of 


temperature (due to generation of heat by friction and oil 
turbulence) that the oil tends to lose its lubricating quali¬ 
ties, when rapid wear of the teeth commences. It can 
therefore happen that the limit of load that can be safely 
imposed on gears is set by temperature-rise rather than 
by stresses in the teeth. 

Fig. 89A. —Comparative Sizes of Involute Gear Teeth. 

Worm gears are especially prone to this limitation 
because of the relatively great amount of sliding associated 
with their tooth action, whilst high-speed gears of any 
type are liable to encounter it by reason of power loss in 

In order to raise this limitation, recourse may be had to 

The Efficiency of Gears 1x7 

artificial cooling devices in the form of fans or cooling 
coils. Alternatively, the oil may be sprayed on to the 
gears in the zone of contact (thus avoiding the considerable 
power loss otherwise caused by turbulence in the oil-bath), 
drained from the bottom of the gearcase, and passed 
through an oil-cooler before being returned to the gears. 

The accepted formulse relating to the strength and load 
capacity of gears are published in British Standard Speci¬ 
fications Nos. 436, 545 and 721. These have been prepared 
in collaboration with the leading gear manufacturers. They 
are obtainable from the British Standards Institution, 28 
Victoria Street, London, S.W.i. 

The Efficiency of Gears 

The amount of power lost by friction between the teeth 
of most gears is usually less than x per cent, of the 
transmitted power, but in estimating the efficiency of 
the whole gear drive it is necessary to take into account 
the power lost in bearing friction, oil-drag, and windage. 
Bearing friction may absorb only about the same amount 
of power as tooth friction, but oil-drag may be more 
serious, particularly if the peripheral speed is high. Windage 
is unimportant except at very high speeds. 

The overall efficiency of a single-stage gear assembly 
of any of the types mentioned above may be expected to 
lie between 97 and 98 per cent, at full load under favourable 
conditions. At fractional loads the efficiency is lower, 
largely because oil-drag (on the teeth and in the bearings) 
does not diminish as the load drops and therefore becomes 
a greater fraction of the transmitted power. 

The power lost by friction between the teeth of worm 
gears depends on the lead angle of the worm, and therefore 
indirectly on the reduction ratio. In this type of gear 
there is a greater amount of relative sliding than in other 


forms, and consequently tooth efficiency is lower. Oil-drag 
is also a cause of appreciable loss of power if the worm-shaft 
speed is high. To take all the factors into account is 
difficult, but as an approximate guide the overall efficiency 
of a worm-gear unit with a worm-shaft speed not higher than 
about iooo r.p.m, may be taken as equal to 100 minus 
half the reduction ratio, e.g. 30 to 1 would give 85 per cent. 
This assumes the use of a case-hardened and ground steei 
worm in conjunction with a phosphor-bronze worm-wheel. 
At higher worm-shaft speeds the efficiency tends to be 
slightly lower because of greater oil-drag. At worm-shaft 
speeds below about 100 r.p.m the efficiency again tends to 
be lower because of the rise in co-efficient of friction at 
low rubbing speeds. 

Lubrication of Gears.—Gears which run only very 
intermittently may be lubricated by grease, but in general 
oil lubrication is essential if the full load-capacity of the 
gear-materials is to be realised. Oil of high viscosity is 
necessary when the tooth-loading is heavy, but is unsuitable 
if the speed is high because, on the one hand, it may lead to 
senous power loss by churning, or may " channel ” to such 
an extent under the action of the gears that they are not 
satisfactorily lubricated. Fortunately, consideration of 
fatigue makes it impracticable to subject high-speed gears 
to very heavyloading, and it is therefore possible to lubricate 
them with the low-viscosity oil which is necessary to keep 
oil-drag loss within reasonable bounds. 

Up to peripheral speeds of about 2500 f.p.m., lubrication 
of gears is adequately effected by allowing one of a pair to 
dip into an oil-bath. At higher speeds oil-drag loss tends 
to become serious, and on this account it is usual to employ 
anoil-spray, keeping the gears clear of the oil in the s um p 
The value of a particular oil as a gear lubricant is not 
shown by viscosity alone, and it is necessary to make sure 
that the oil used for any important gears carries the 
approval of the gear manufacturer. This is especially 
important in the case of worm gears because they are 
liable to complete failure in a very short time if lubrication 
is defective, whereas the other conventional types of gear 
are rather less sensitive in this respect. 

Useful Formulae for Gears 

Spur Gearing.—The Standard gear terms and symbols, 
forming the basis of calculations concerning their tooth 
proportions, are shown in Figs. 90 and 91. It should be noted 
that circular pitch refers to the distance, measured round 
the pitch line, from the centre of one tooth to the centre 
of the next tooth; whereas diametral pitch refers to the 
number of teeth per inch of the pitch diameter. It is usual 
to refer to gears according to their diametral pitch. 

Module is found by dividing the pitch diameter by the 
number of teeth. Thus a 40-tooth gear with 20-in. pitch 
diameter would have 2 teeth per inch of its pitch diameter 
and would thus be : 



The module of the same gear would be : 

P/tcmC/pclE Dedemdl/m 

Fig. go.—Diagram explaining terms used in gearing. 

It is thus obvious that the module is the reciprocal of the 
diametral pitch, and vice-versa. The following formulae 
apply to spur gears, where the addendum is equal to 0-3183 
x circular pitch : 

N—number of teeth. 
s=addendum or module. 




2=thickness of tooth on pitch line. 

chordal thickness of tooth. 

/—clearance at bottom of tooth. 

D=outside diameter. 

D'—pitch diameter. 

D"=working depth of tooth. 

D"+/=whole depth of tooth. 

D'*—bottom diameter. 

P==diametral pitch. 

P'—circular pitch. 

H—height of arc. 

s /l '=distance from chord of pitch line to top of tooth, 
0—J angle subtended by circular pitch. 

The following formulae apply to the system in which the 
addendum is equal to 0*3183 x circular pitch. 

Fig. 91.—Diagram illustrating symbols used in spur gearing formulae. 



/= * 
^ 10 

p ' 

s +/=^r?»' or 0-3683 P\ 

B"= 2s; or|. 

D"-f-/==?^ 3 Z; or 0-6866 P'. 

D '=p , or 521 ; or 0-3183 P'N ; or . 

F 7 T N + 2 

D=D'+2s; or N+2 

0 = 22 .°. 


r=D' sin 0. 

D"’=D-2(D"+/); or 
N =D'P ; or DP-2. 

H=P-' (?~ C ?- S - 9 j. ; s'=s+H. 

B. & S. Involute Gear Cutters 
No. 1 will cut wheels from 135 teeth to a rack inclusive. 

2 „ 

55 „ 

„ 134 teeth 


3 » 

35 » 

>, 54 „ 


4 „ 

26 „ 

t* 34 >> 


5 „ 

21 „ 

„ 25 „ 

6 „ 

17 » 

„ 20 „ 


7 » 

14 » 

„ 16 „ 

8 „ 

12 „ 

13 .. 

Bevel Gears 
(Axes at Right Angles) 
N.=Number of teeth on gear. 
N* “Number of teeth on pinion. 
P=diametral pitch. 

P'—circular pitch. 



a,,—centre angle—angle of edge or pitch angle of gear, 
a*—centre angle=angle of edge or pitch angle of pinion, 
d—angle of top. 
p'=angle of bottom. 

angle of face of gear. 
g h —angle of face of pinion. 

^—cutting angle of gear. 

Zk=cutting angle of pinion. 

A—apex distance from pitch circle. 

A'=apex distance from large bottom of tooth. 
D=outside diameter, 

D'—pitch diameter. 

D"—working depth of tooth. 

5—addendum or module. 

2 —thickness of tooth at pitch line. 

/^clearance at bottom of tooth. 

D*+/=whole depth of tooth. 

2#—diameter increment. 

b= distance from top of tooth to plane of pitch circle 
F—width of face. 

Fig. 92 .—Diagram of symbols relating to right-angle bevel gears. 


tan a.—tan a a = 




2 sin a 

~~W~ P 





tan or£±- f 

N A 

g.= 90 °—(o.X/S); g s =90°—(a—/S). 

A=== ^n/N?W. 


A=—- * 
2 P sin a 

Ai=—^ or 


cos /?" 2P sin a cos /?’ 




sin (a+P) 


2 A sin a' J 


COS p. 

N-f-2 cos a tc 
° r-p-,or p . 

p# TC 

n ,_N nr NP ' nr DN 

JL) —-=i or f or ijj~r 1 

P iv N+2 cos a 

, or D= 

2 cos a 
P * 

Cutters for Mitre and Bevel Gears 

Diametral Pitch 

Diameter of Cutter 

Hole in Cutter 


3i in. 

i£ in. 


3 i i n - 

1} in. 


3} in. 



3 i i 11 * 

1} in. 


3 i in- 

ij in.. 


3 im. 

I in. 



I in. 





2J in. 



2J in. 



2| in. 


Table for Obtaining Set-Over for Cutting Bevel Gears 



00 1 M 

t 1- ■*f* 04 m to 04 vO O 

O CO CTi M w VO 4>.CO 

04 CM M CO CO ro CO CO 


!>>, M 

Ot CO O OO CO VO 0 0 
<0 OO O'. O M to t-*00 

04 04 04 COCOCOCOCO 

6 6 6 6 6 6 6 6 

\Q , W 

O O N >0 0 N OOO 
VOOOOO 0 O too vO 

04 04 04 CO CO CO CO CO 


Ov 0^3 N 4>*0O O w 
IOJ>00 0 O Th «0 VO 

04 04 04 CO CO CO CO CO 

< 1 ) 


tOi M 

OO t>» COCO 04 CO CO 04 

0 0.00 0.0 



00i00400 0 000 ^j* 
to {''•OO Ov O -tt* CO '«t* 

04 04 04 04 CO CO CO CO 
6666 6 6 66 



"$1 w 

NtCOvOOO C-. -rt-oo 
to 0-.00 O' Q\ CO CO CO 

04 04 04 04 04 CO CO CO 




4>* '^“OO CO VO Tf* O'. M 
tO O'- O- O'. O'. CO 04 CO 

04 04 04 04 04 CO CO CO 





^f-J M 

NC0<0 w toO Tf" Ot 

O O. tN O O <0 C4 M 



«H*. i_j 


VO 04 CO 0> COOO 0 Q\ 
to O' O'OO O'. 04 H 0 

04 04 04 04 04COCOC0 

to M M 10 O coco VO 
»o t"> i>.oo 0 n 0 a 




t}- 0 O OO 0 to OO OO VO 
tovo vo 00 00 m avoo 

04 04 04 04 04 CO 04 04 

CO» H 

Tt-VD VO to O H O' to 
too o x>-oo H OO o 

04 04 04 04 04 CO 04 04 

jo^no jo -ojvt 

M 04 CO Tf too 0"0O 


S 3 







•tJ * 


_ » 
53 8> 

£ 0 





s s 


. .a .2 

g HCV 



Relevant formulae are 
on next page. 




2(2=25 COS a. 

Bevel Gears 

(Axes at any Angle) 

The formulae for tooth parts on pages 121 to 124 apply in 
both cases. C=angle formed by axes of gears. 

tan a t = * m . C ; cot cot C. 

£-‘+cos C N « sm C 


tan <x» =g ^— - y cot — ^.^,-^ - 1 cot C. 

^•+cos C 

Where the indicating letters a and b are not used the 
formulae apply to gear and pinion. 

In the case of Fig. 93 the formulae given on pages 124 to 
126 apply. 

In the case shown by Fig. 95 : 

tan a , = - - sin - ( l8o - C ) ; tan *-**&£=£ L_ 

COS (180-C) Jj£—,cos (180-G) 

In the case shown by Fig. 94: 

a a = 90° ; a*=C—90° 

In the case shown by Fig. 96 : 

tan a a — 

sin E 


sin E 
—cos E 

Fig. 97 .—Symbols used in calculations relating to bevel gears 
with axes at any angle. 

Spiral or Screw Gearing 



=Number of teeth in gears. 
=centre distance. 

P' = circular pitch. 

P M = normal diametral pitch. 

P'„= normal circular pitch. 
y- wangle of axis. 

L= lead of tooth spiral. 

T = number of teeth marked on cutter. 



D'=pitch diameter. 

D=outside diameter. 

angle of teeth with axis. 

t —thickness of tooth, 
addendum or module. 
D''+/==whole depth of tooth. 
y=a fl +aa. 




P' n N 

: —--, Ui i r=--. 

7 u cos a N+2 cos a 

D=D , +2 s, or 

P'=^E.or P ' 

N ' cos a 
P *=P' cos a. 

P n =pr-(pitch of cutter). 

-U M 

P' n . i D—D' 

or pT' ——* 

TC V H 2 

, P'n 

D*+/- 25 +-L. 

J 10 

T N 
cos a 3 ' 




tan a* ° r P tan a 

In many small shops the Brown & Sharpe system of 
milling involute gear teeth is still used. In this system 
a series of eight cutters are required to cut all gears having 
12 or more teeth in any one pitch. The list of eight cutters 
supplied for each pitch appears on page 121. 



Worm and Worm Wheels 

L=lead of worm. 

N =Number of teeth in gear. 

M —turns per inch of worm. 

diameter of worm. 

&' =pitch diameter of worm. 
d"=diameter of hob. 

D—throat diameter. 

D'=pitch diameter of worm-wheel. 
B=blank diameter (to sharp corners). 
C=distance between centres. 
P=diametral pitch. 

Fig. 98 .—Worm proportions 
and symbols. 


P'—circular pitch for worm 
wheels or axial pitch 
for worms. 

r *j see Fig. 98. 

5= addendum or module. 

£=thickness of tooth at 
pitch line. 

4 =normal thickness of 

,/= clearance at bottom of 

D"—Working depth of tooth. 

Fig. 99 .—Hob proportions. 


D" -)-/=■whole depth of tooth. 

b =pitch circumference of worm, 
a—width of worm thread tool at end. 
w —width of worm thread of top and width of hob 
tool at end. 

S=angle of tooth of worm-wheel with its axis. 

F —min. length of worm. 

F 1 —length of hob. 

If the lead is for single, double or multiple start threads, 
L=P' 2 P', 3 P', etc. 

Fig. ioo . — Diagram 
illustrating hob and 
worm relations. 

a= 6o° to 90°. 



_ 7 zP 


6 = 7 u (d— 25), or 7 i i'. 

-tan S=t. 




- T~ ‘ 


Fig. ioi . — Diagram 
illustrating angle of 
tooth of worm-wheel 
with its axis. 



c=^- s , or£±l 
2 2 

B— D+2 (y’— r' cos 

d”=d-\- 2 f. 

^=0-3095 P' 

0-3354 P'. 

F=5s+2 v 's(D'- s ). 



cos S. 


Chordal Thickness of Teeth of Gears 

(Basis of 1 Diametral Pitch) 

N =Number of teeth in gear, 
t"—Chordal thickness of tooth. 

H 2 ==Height of arc. 

D'— Pitch diameter. 

R2—pitch radius. 

^'=90° divided by the number of teeth. 
t'^D 1 sin 8 . 

H 2 =R (1—cos /i 1 ). 

Chordal Thickness of Teeth of Gears 






No. 1—135 T—1 P 




No. 2— 55 T—1 P 




No. 3— 35 T—-1 P 




No. 4— 26 T—1 P 




No. 5—- 21 T—1 P 




No. 6— 17 T—1 P 




No. 7— 14 T—1 P 




No. 8— 12 T—r P 




11 T—1 P 




10 T—1 P 




9 T—1 P 




8 T—1 P 




Velocity Ratios of Gear Trains.—If all the gears in 
the train rotate about fixed centre 'lines, then 
Velocity ratio = 

Pr oduct of numbers of teeth in all driv en gears 
Product of numbers of teeth in all driving gears 

Revolutions of first driving gear= 

Velocity ratio x Revolutions of last driven gear. 


Sprocket Wheels.—Chain or sprocket wheels are gears 
made to mesh with chains, by means of which motion 
may be transmitted from one point to another. The teeth 

are of different formation 
from the ordinary involute 
gear, as they have to accom¬ 
modate the circular rollers of 
the chain. Ordinary gearing 
calculations for pitch line 
functions do not apply, for 
whereas with two gears in 
mesh with one another the 
circular pitch is the distance 
between two tooth centres 
measured round the pitch 
line, with a sprocket the 
circular pitch refers really to 
the chordal distance between 
Fig. 102. — Proportions of a two tooth centres. As in- 
sprocket wheel on block-centre formation relating to these 

calculations is extremely 
scanty, the following formulae for block-centre and roller- 
chain may be of interest. 

Sprocket Wheels for Block-Centre Chains.—Fig. 102 
shows a wheel as used for block-centre chains, and the 
following formulae give proportions and diameters relevant 

, l8o° _ 0 sin a 

to them, Ian p=g - 

5 + cos a 

Pitch diam. — . ^ —5- 
sm p 

Outside diam.—pitch diam.-ffc. Bottom diam.=pitch 
diam. —b. 

In calculating the diameter of sprocket wheels the bottom 
diameter is most important. In the above calculations 
N=number of teeth, b =diameter of round part of chain 
block, B=centre to centre of holes in chain block, A—centre 
to centre of holes in side-links. 


British Standard Roller Chain Wheels 

Measuring pin diameter 
Maximum roller diameter 

Bottom diameter=Pitch diameter minus roller diameter. 
Chordal distance—[| (pitch diameter of 'wheel having 
twice the number of teeth) minus roller diameter]. 

Measurement over Pins 

The bottom diameter of wheels with even numbers of 
teeth is checked by measuring over pins inserted in opposite 
tooth spaces. 

The bottom diameter of wheels with odd numbers of 
teeth is checked from chordal distance by measuring over 
pins inserted in the tooth spaces most nearly opposite. 

For even numbers of teeth =pitch diameter 
-t , plus roller diameter. 

Measurement p Qr num fo ers 0 f teeth=[ J (pitch diameter 

over pins. 0 f w h ee l having twice the number of 
teeth) plus roller diameter]. 

Tolerance on Cutting Sizes — 

Bottom Diameter and Measurement over Pins 
Plus o-ooo inch. 

Minus 0-004 inch per inch (mm. per mm.) of pitch dia- 


meter, with a maximum of 0-020 inch (0-51 mm.), subject 
to the provision that a tolerance up to 0-005 inch (0-13 mm.) 
shall be permitted on all chain wheels with, pitch diameter 
less than 5 inches (127-00 mm.). 

Formula for Centre Distances 

(Two point drives) 

N=number of teeth in wheel. 
n=number of teeth in sprocket. 
D== distance between centres. 
L=number of links in chain. 


2 xD N+n Px(N-n) 2 ' 

rT + ~ + —^oD~-. 


The result obtained from this expression will not 
ordinarily be a whole number of links; in that event the 
nearest even number should be taken and a recalculation 
made to find the corresponding centre distance from the 
following : 

\^(A+p'9B) x (A—o* 9B) J 
where A=2L—(N+n) and B—(N—n). 

This formula gives results correct to within 0-05 per cent, 
for wheels having a ratio of 2:1 at minimum centres. This 
is sufficiently ■ accurate for drives where adjustment is 


Block-Centre Chains j 



of block 

of cutter 

Centre to 
centre of 


Hole in j 
cutter J 




1 1/2 





3 1/2 

3 3/4 





1 1/4 

1 1/4 

Seven cutters are made for each pitch, for Numbers of 
teeth as follows : 8, 9, 10, 11, 12, and 13, 14 to 16, 17 to 
20, 21 and over. 

Roller Chains 



of rolls 

of cutter 

Hole in 






*306 or -308 

' at 







3 i 




3 l 



•5625 or *-625 

■ 3i 



•625 or *-750 


Xi . 


•75 or *-875 





• xi 





" Whitney Standard." 

measuring along the Pitch Circle. 
































Ps Ph 




7 -\ 00 

% P 

'rp h 

CO co 



00 ~ 

H 7 Q 



? % » 



II 1 





















d5 4-» 

*> Ph 


0 ' 

- I 

JJ § 
g co 
S 00 

cd H 


A . 0 ^ 

0 ' 

h-i *,d 

o O 



<o *5 0 

*Tj O Jh 

• P f»H 0 

> PnrO 

> 3 dJ 

fe co N 
2® p •£ j* 
§ T’S, 

• 2 O T 3 ' 

Q rP O rP 
t-w H-i {L H-> 

o 0 P0 
v £ ^£ 
»df H H 

*tn 5 ^ g ^ 

-m, d o 00 
o g 0 
Q . B 


3 a) * 




U I 


CO ,, , (/j 

' H ° P & 

00 +* 0 

. Q O r~ 

0 s g 

r ci r ^^ p 

g 2 

Cd q, 

o H H rQ 
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Pulley Calculations : General Rule.—A general rule 
which is not affected by the system of gears and/or pulleys 
employed—it applies whether the train is simple or 
compound—is that the product of the diameters (or the 
number of teeth in the case of gears) of the driving wheel 
and the number of revolutions per minute of the first 
driver is equal to the product of the diameters or the 
number of teeth of the driven wheels and the number of 
revolutions per minute of the last driven wheel. 

In the workshop, calculations regarding speeds of 
machine countershafts, etc., are frequent, and the following 
formulas apply : 

R.P.M. of Driven Pulley or Gear 

__ Diam. of Driver x R.P.M. of Driver. 

Diam. of Driven. 

Diam. of Driven Pulley or Gear 

__Diam. of Driver X R.P.M. of Driver. 

R.P.M. of Driven. 

R.P.M. of Driver Pulley or Gear 

__ Diam. of Driven x R.P.M. of Driven . 

Diam. of Driver. 

Diam. of Driver 

_ Diam. of Driven X R.P.M. of Driven. 



Addendum, 21. 

Angle, base spiral, 96. 

_involute, function of, 9*>- 

British Standard rack-form, 21. 
— Standard system, 21. 

Brown and Sharpe involute 
gear cutters, 121. 

Built-up gears, 67. 

— pressure, 20, 27. 

.— spiral, 27. > 

Angular division tester, 08. 

— velocity, 15- 
Automobile gearbox, 9. 

— gears, 68. 

Axial feed, 62. 

— pitch of worm, 38. 


Ball and indicator, checking 
worm teeth by, 81. 

— micrometer, checking worm 

teeth by, 82. 

Base circle, 20, 95. 

— pitch, 20. 

— spiral angle, 96. 

Bending stress, 113. 

Bevel-gear units, 76. 

— gears, 29, 121. 

-cutters for, 123. 

-double-helical, 33. 

-formulae for, 121. 

— — generation of, 31. 

-principles of, 29. 

-set over for, 124. 

-single-helical, generating, 


-spiral, 13, 15, 3 2 * 

-generating, 57. 

-straight, 12. 

-tooth form of, 32. 

Blank diameter, determining 
size of, 22. 

Block - centre chains, formula 
for, 132. 


Caliper, gear-tooth vernier, 78. 

— Zeiss, 95. 

Camshaft, gearing of, n. 

Chain sprocket, formula for, 132. 

— wheels, block-centre, formula 

for, 135. 

Chordal thickness, formula for, 
98, 131- 

Circular pitch, 21. 

-tables, 24, 136. 

Cluster gears, cutting, 43. 
Concentricity of teeth and bore, 

Contractions, strains, 66. 

Cutters for mitre gears, 123, 

— generating, making, 108 


Dedendum, 21. 

Diameter of blank, 20. 

— of pitch circle, 20. 
Diametral pitch, 22, 138. 
Double-helical bevels, 33. 

-gears, 12. 

-meshing of, 25. 

— piano-generating, 53. 


Efficiency of gears, 117. 
Electric motors, speed of, 11. 
End-mills, making, 103. 

-thrust, 27. 

Epicyclic gear trains, 71. 





Fellows gear generator, 42. 

— process, 44. 

Fly hob, 61. 

Formate gears, 59- 
Formed wheel, grinding, 50. 
Formulae for spur gearing, 120. 
Friction, 117. 


Gear cutters, B. and S., in¬ 
volute, 123. 

— generation, 17, 41. 

— generator. Fellows, 42. 

— grinder, Maag, 47. 

— single-reduction, 75. 

— teeth, checking by com¬ 

parison, 80. 

-Checking shape of, 81. 

-helical, overlap of, 26- 

-measuring, 78, ' 

-- comparative sizes of, 116. 

-tooth caliper, Sykes, 94. 

-Zeiss, 94. 

■-comparator, using, 90. 

— vernier, 78. 

— trains, epicyclic, 71. 

-velocity ratio of, 140. 

— types, 12- 

— units, bevel, 76. 

-worm, 77. 

— wheel, definition of, 13. 

-form, 66. 

-types of, 66. 

— units, spur, 75. 

-worm, 76, 77. 

— working life of, 118. 

— Zerol, 33. 

Gearbox, 9. 

Gears, applications of, 10. 

— automobile, 68. 

— bevel, 29. 

-cutters for, 123. 

-generating, 55 

-generation of, 31. 

— built-up, 67. 

— cause of noisy, 26. 

— checking by running together, 


— chordal thickness of, 135. 

— double-helical, meshing of, 25. 

Gears, efficiency of, 1x7. 

— formate, 59. 

— helical, generating, 51. 

-hobbing, 52. 

-profile grinding, 55. 

— hypoid, 59. 

*— load capacity of, 113. 

— loading of, 14. 

— lubrication of, 76, rxS. 

‘— measuring, 78. 

— mitre, cutters for, 123. 

— mounting, 74. 

— requirements of, 14. 

— root circle of, 20. 

— screw, formula for, 126. 

— shock loading of, 114. 

— spiral, 13. 

-formula for, 126. 

-Gleason, 57. 

— straight bevel, 12. 

-tooth form of, 32. 

— triple-helical, 12. 

— turbine reduction, 68. 

— types of, 9. 

— velocity ratio of, 38. 

— welded, 67. 

Generating cutters, making, 108. 
•— helical gears, 51. 

— rack, 23. 

— single-helical bevels, 56. 

— system, Sykes, 54. 

Generation of bevel gears, 31. 

— of gear, 17, 41. 

— pitch circle of, 20. 

Gleason spiral gear teeth, 57. 
Grinding, formed wheel, 50. 

— spur gears, 48. 

— worms, 64. 


Helical bevels, single, 33. 

— gear, single, 11. 

-teeth, overlap of, 26. 

-units, 75. 

— gears, direction of rotation of, 


-double, 12. 

-generating, 51. 

-hobbing, 52. 

-piano-generating, 53. 



Helical gears, profile grinding, 
55 * . 

-spiral angle of, measuring, 

88 . 

-Sykes generating system, 


-triple, 28. 

Helicoidal worm thread, 36. 
Hob, fly, 61. 

Hobbing helical gears, 52. 

— spur gears, 45. 

Hobs, making, 100. 

Hypoid gears, 60. 


Internal spur gear, 44. 

Involute curve, 19. 

— function of angle, 96. 

— gear cutters, B. and S., 121. 

— tester, 83, 99. 


Lead of worm, 38. 

Load capacity of gears, 113. 
Loading of gears, 14. 
Lubrication of gears, 76, 118. 


Maag spur gear grinder, 47. 
Meshing of double-helical gears, 

Micrometer measurement of 
tooth thickness, 79. 

Milling worm threads, 63. 

Mitre gears, cutters for, 123. 
Module pitch, formula for, 21. 
Mounting gears, 74. 


Parkson gear tester, 83. 
Pinion, definition of, 13. 
Pinions, making, 68. 
Pitch, 21. 

— base, 20. 

Pitch circle, diameter of, 20. 
-of generation, 20. 

— circular, 21, 136. 

— cylinder of worm, 37. 

— cylinders, 16. 

— diametral, 22. 

— line defined, 16. 

— methods of defining, 21. 

— module, formula for, 21. 

— of worm, axial, 38. 

— point defined, 16. 

— testing, 86. 

-- by optical method, 88. 

— transverse, 21. 

Pitches, tables of standard, 24. 
Piano-generating machine, 41. 

— processes, 53. 

Pressure angle, 20, 27. 

Profile grinding gears, 55. 

— of teeth, checking, 83. 

— of worm gears, checking, 84. 
Proportions of teeth, 21. 

Pulley calculations, 140. 


Rack - form, British Standard 
basic, 21. 

— generating, 23. 

— method of generation, 18. 
Roller Chain Wheels, 132. 
Rollers, measuring tooth thick¬ 
ness by, 80. 

Root circle of gear, 20. 

— of tooth, fillet at, 31. 
Rotation, direction of, 28. 


Screw gears, formula for, 126. 
Shafts and pinions of car gear¬ 
boxes, 9. 

Shock loading, 114. 

Single-helical bevels, 33. 

— gear, 11. 

-reduction gear units, 75. 

Spiral angle, 27. 

-measuring, 88. 

— bevel gears, 13, 15, 33. 

— -generating, 57. 

-- Zerol, 33. 

— gear, Gleason, 57. 



Spiral gears, 14. 

-formula for, 126. 

Sprocket wheels, formula for, 

Spur gear generator. Fellows, 

-■ grinder, Maag, 47. 

-internal, 44. 

-units, 75. 

— gears, definition of, 12. 

-generating, 41. 

-Fobbing, 45. 

Standard pitches, table of, 24. 
Stress, bending, 113. 

Sunderland process, 41. 

Sykes generating system, 54. 


Teeth, measuring by rollers, 

80, 97- 

Thread-milling, 63. 

Thrust, end-, 27. 

Tooth and bore, checking con¬ 
centricity of, 83. 

— depth of, 21. 

— fillet at root of, 31. 

— form of straight bevel gears, 


— proportions, 21. 

— shape, checking, 81. 

— thickness, finding from micro¬ 

meter measurements, 97. 

-measurement of, 79. 

Toothed gearing, 12. 

Transverse pitch, 21. 

..gears, 12, 28. 

:ion gears, 68, 



Velocity, angular, 15. 

— ratio of gears, 38, 131. 
Vernier, caliper, 78. 


Welding gears, 67. 

Worm and wheel, contact be¬ 
tween, 39. 

— dimensions of, 36. 

— gear, 14. 

—> -— units, 76, 77. 

— gears, 16, 34. 

-checking profile of, 84, 

-dimensions of, 38. 

-tooth action of, 34. 

— lead of, 38. 

— pitch cylinder of, 37. 

— teeth, checking by ball and 

indicator, 81. 

-micrometer, 82. 

— thread, helicoidal, 36. 

— threads, milling, 63. 

— wheel generator, 61. 

-pitch cylinder of, 16. 

— wheels, cutting, 60. 

-formula for, 129. 

-producing, 60. 

Worms, finishing, 65. 

— formula for, 129. 

— grinding, 64. 


Zeiss optical gear-tooth caliper, 

94 - 

Zerol, gear, 33. 

— spiral bevel, 33. 

Undercut, 21.