Analytical Mechanics of Gears by
Earle Buckingham (1887-1978)
McGraw-Hill,
1949. Dover Publications, 1963 & 1988 (ISBN 0-486-65712-4).

Wheel and Pinion Cutting in Horology by
J. Malcolm Wild,
FBHI (b. 19??)
The Crowood Press Ltd,
2001. Hardcover reprint 2012, 253 pp. (ISBN 0-978-1-86126-245-5).
Appendix includes extracts from relevant Swiss standards (NIHS)
20-01, 20-02, 20-10, 20-25 and British standard BS 978-2 (1952).

Machinery's Handbook, 25^{th} edition (1996)
by
Erik Oberg 1881-1951,
Franklin D. Jones 1879-1967,
Holbrook L. Horton 1907-2001,
Henri H. Ryffel 1920-2012.
Editors: Robert E. Green and Christopher J. McCauley.
Industrial Press Inc., New-York 1996 (ISBN 0-8311-2575-6).
(First published in 1914.
The 29th edition was released on January 2, 2012.)

A gear is a toothed wheel, rigidly attached to an axis of rotation (axle).
It meshes with other gears to transmit rotary motions to other axles.

The following geometrical study is mostly concerned with the exact shape of ideal gears.
In this context, we may use the word pinion to denote
a single-tooth gear which may lack the axial symmetry of gears with several teeth.
This usage is more restrictive than the ordinary meaning of the word,
which is part of the following mechanical jargon:

Addendum^{ }:
For a straight spur gear, this is the maximum height of a tooth above the
pitch circle.

Addendum Curve^{ }:
The part of the profile that's above the pitch circle.

Annular Gear^{ }:
A gear whose teeth are cut on the inside of a ring.

Axle^{ }: The axis around which a gear revolves.
Also called shaft.

Backlash^{ }:
The amplitude of the back-and-forth motion allowed in one gear when a meshing
gear is held in place. (This is normally measured in module units, along
the pitch circle).

Bevel Gear^{ }:
A conical gear, used to connect intersecting shafts.

Cage Gear^{ }: Another name for a
lantern gear (which see).

Cam^{ }:
A smooth solid imparting a specific motion to a so-called follower
in contact with it (often spring-loaded).
A disk cam, is a rotating cylinder whose lateral surface
drives a flat follower, whereas the active surface of a cylinder cam
is actually helical...
A cam in straight motion is called a translation cam.

Center Distance^{ }:
The distance between the two axles (parallel or not).

Centrode^{ }:
The motion of a rigid plane in a fixed plane,
is either a pure translation or a rotation about a so-called
instantaneous center of rotation
(which has zero speed in either plane)
whose path is dubbed centrode.
The centrode in one plane rolls without slipping on the centrode in the other.

Circular Pitch^{ }:
(Also called tooth space.)
The curvilinear distance between the centers of two adjacent teeth,
as measured along the pitch circle.
In module units, the cicular
pitch is always equal to p.

Clearance^{ }:
The amount by which the dedendum of a gear
exceeds the addendum of another gear when both mesh.

Cog^{ }:
Another name for the tooth of a gear (especially for wooden gears).

Crown Gear^{ }:
A wheel with [straight or helical] teeth on its flat side.

Dedendum^{ }:
For a straight spur gear, this is the maximum depth of [the fillet of]
a tooth, below the pitch circle.

Dedendum Curve^{ }:
The part of the profile that's below the pitch circle.

Elliptic gears^{ }: The family of compatible gears
described below which are syntrepent to an ellipse
rotating about one of its foci (which can be viewed as the basic single-tooth pinion
belonging that family).

Elliptical gears^{ }: Meshing gears in the general shape
of syntrepent ellipses possibly endowed with
small matching teeth on their respective circumferences to prevent slipping.
Opposed to circular gears.

External gears ^{ }are regular gears,
as opposed to internal or annular ones.

Face of a Gear^{ }:
See flank.

Fillet^{ }:
The deep part of the teeth (near the dedendum)
which is never in contact with meshing gear.
(As opposed to the kinematically relevant flank.)

Flank^{ }:
The surface of the gear which comes in contact with meshing gears.
We consider flank and face to be synonymous.
However, some authors reserve the word flank for the part of active surface
which is inside the pitch surface and call face the
part outside of the pitch surface.

Gear Ratio^{ }:
The ratio of the (average) angular velocity of the input gear to the (average) angular velocity
of the output gear. Unless the input and output axles are perpendicular,
the notion of a signed gear ratio (or
algebraic gear ratio can usefully be introduced.

Gear Train^{ }:
Any system of two or more meshing gears.

Gender^{ }:
Except in the special case of genderless families
(of which elliptic gears are an example)
the tooth profiles in a family of compatible spur gear come in two
distinct genders. Only teeth of opposite genders can mesh externally.

Helical Gear^{ }:
A gear whose flank spirals around the shaft.

Herringbone Gear :
Two helical gears of opposite handedness, side by side on the same shaft
(to cancel the axial thrust of a single helical gear).

Hypoid gears^{ }
connect two shafts that do not intersect.
(The term is a contraction of "hyperboloid",
which is the pitch surface for such a gearing.)

Internal Gear : See annular gear.

Isotrepent^{ }:
The qualifier applying to a curve which is syntrepent with itself
with respect to one of its points. Examples include
ellipses and logaritihmic spirals
(French: courbe isotrépente).

Lantern Gear :
Also called: cage gear.
A wooden gear (at right)
consisting of two disks connected by rods that serve as gearing teeth.
Omitting one of the supporting disks turns this into a pinwheel gear,
which can also be used as a crown gear
(usually in a mitter gear arrangement).
The term pinwheel may also denote a rudimentary gear obtained by attaching
rods radially to a solid disk.

Leaf^{ }:
The tooth of a gear, in clockmaking parlance.

Miter Gear^{ }:
A conical gear transmitting rotation between two shafts intersecting at a right angle
(the most common type of bevel gear).

Module^{ }:
A unit of length equal to the diameter of the pitch circle
divided by the number of teeth.
It's used to describe the tooth profile in general terms.
In module units, the circular pitch of a gear is always equal to
p.

Pinion^{ }:
A small gear with few teeth (possibly, a single tooth).
When discussing a pair of meshing gears, the smaller one is called the pinion
whereas the larger one is the wheel
(or the rack, for an infinite radius).

Pitch Circle^{ }:
Loosely speaking, what the cross section of a straight spur gear would
become without its teeth (see pitch surface, below,
for more precision).

Pitch Point^{ }:
Let K be the point of contact of two planar gears as they mesh.
Their common normal through K intersects the line of the two rotation
centers at a point P called the pitch point.
(P = K if and only if there's no slipping).

Pitch Surface^{ }:
The pitch surfaces of two meshing gears are the abstract surfaces
attached to each of them which roll without slipping on each other
in a uniform rotation equal to the angularaverage of the actual motion.
A pitch surface is always a ruled surface of revolution, namely:
a plane for a crown gear, a cylinder for a spur gear,
a cone for a bevel gear, an hyperboloid for an hypoid gear.

Profile^{ }:
The shape of a gear's tooth. The planar curve corresponding to
its cross-section in the case of a straight spur gear.

Rack^{ }:
A toothed bar, which may be viewed as a gear of infinite radius.

Shaft^{ }: The axis around which a gear revolves.
Also called axle.

Spur Gear^{ }:
A cylindrical wheel, with teeth cut across its circumference.

Straight Gear :
Spur gear or conical gear whose teeth are cut along straight lines
parallel to the shaft or intersecting it. Opposed to helical gear.

Syntrepent Curves^{ }:
Planar curves which roll on each other without slipping
as they rotate about two centers.
(Miquel 1838.
French: courbes syntrépentes).

Tooth Profile^{ }:
Shape of a tooth (the same shape is repeated for all teeth).

Tooth Space^{ }:
See circular pitch.

Worm Gear^{ }:
An endless screw driving an helical gear perpendicular to it.

(2012-11-21) Gear Ratio _{ } ( signed
or unsigned )
The ratio of the driver's rotation to the output rotation.

The gear ratio of any gear train is defined as the
ratio of the (average) angular velocity of the input gear to the (average) angular velocity
of the output gear.
Thus, if the driving gear rotates 5 times faster than the (final) driven gear,
the gear ratio is 5.

In the automotive realm,
"overdrive" denotes a gear ratio less than 1
(French: surmultipliée ).
A gear ratio of 1 is called "direct drive".

That ratio is often taken only as a positive quantity
involving the magnitudes of the rotation rates, irrespective of their directions.

However, if the input and output axles are not perpendicular
(in particular, when they are parallel) the directions
of their rotations can be compared unambiguously. The gear ratio can then usefully
be given a sign (the same sign as the dot product of the relevant
rotation vectors).

For a simple train of two spur gears, the algebraic gear ratio so defined
is negative is the two gears are meshing externally and positive when they are
meshing internally (one of the gears is annular in the latter case).

The gear ratio is zero if the driver is a rack
in rectilinear motion, unless the output gear is itself also a rack
(in which case the gear ratio is undefined).
If the output is a rack and the driver isn't, the gear ratio is infinite.

(2005-12-11) Gears which Roll without Slipping_{ } "Perfect" straight spur gears roll against each other without slipping.

When two rigid planar curves roll against each other without slipping, the point
of contact has zero velocity with respect to either curve.

The planar cross-sections of two straight spur gears rotate
respectively around two points O and O'.
If these curves roll against each other in the above sense, the velocity of
the point of contact M is perpendicular to both OM and O' M.
This implies that M is on the line OO' joining the two centers of rotation.

Some slipping is thus necessarily involved in gear pairs
(see involute gears)
which hold the rotational velocity ratio strictly constant.
(Otherwise, the point of contact would maintain constant distances from both centers
of rotation, because such distances would have a fixed sum and a fixed ratio...)

The polar coordinates of the point of contact (M) in the systems bound to either curve
obey the following differential equation.
The distance a
between the centers or rotation is r+r' for external gearing,
and | r-r' | for internal gearing
(where one of the gears is an annular gear).

r dq + r'
dq'
= 0

Genders :

If two curves mesh with a third,
they'll mesh internally with each other.
Two genders are thus defined so that profiles of the same
gender mesh internally with each other.
Curves of opposite genders mesh externally.

If one curve meshes externally with itself (as shown
next in the case of an ellipse)
then all curves that mesh with it do so both internally and externally,
thus forming a genderless family of compatible gears.

(2005-12-11) Ellipses are Isotrepent Curves_{ } Syntrepent planar curves roll on each other without slipping as they rotate
around two fixed centers. A curve syntrepent to a copy of itself [with respect to
matching centers] is said to be_{ }isotrepent.
An ellipse is isotrepent about its focus.

Both terms (French: courbes syntrépentes,
courbe isotrépente) were coined by the French mathematician
Auguste Miquel (1816-1851) in 1838.

Ellipses are isotrepent because congruent ellipses may roll on each other without
slipping, as they rotate around their respective foci.
In such a motion, the two ellipses are symmetrical about their tangent of
contact, as illustrated above.

In this symmetrical configuration, the line joining two "opposite" foci goes through the
point of contact.
This may be proved using the fact that an ellipse reflects
any ray from a focus back to the other focus.
(HINT: Draw the four lines going
from the contact point to each focus,
then deduce collinearity from angular relations.)

This gearing does not allow one pinion to drive the other
in practice, since it pushes against the other for only half of each cycle.
Instead, the same motion can be reproduced in a gear-free mechanism,
by tying the two moving foci with a rigid rod...
This tranfers rotary motion from one shaft to the other in a 1:1 ratio.

Unfortunately, that simple mechanism retains a dead point
when the 4 foci are aligned.
In the absence of a flywheel, the direction of rotation can indeed reverse itself
from this dead position (both shafts may rotate in the same direction if the
bar tying the moving foci remains parallel to the line joining the fixed foci).

(2005-12-10) _{ } From Ellipse Pinion to Sinewave Rack
(Michon, 1975)
Elliptic spur gears roll on each other as they rotate (no sliding).

This family of gears involves only pure roll (no sliding or slipping)
at the expense of Euler's conjugate action
(which would make the driven gear rotate at a uniform
rate if the driver does).
These gears are thus more suited for unlubricated clockwork
than high-power lubricated machinery.

I devised this as a teenage student (in 1974 or 1975) mostly to test my
calculus proficiency.
I was convinced that the same idea must have occurred to many people
and left it at that.
It seems that nobody ever bothered to publish it, though.
The simplicitity of the final result could
be expressed and/or justified purely in geometrical terms,
but I'll derive it here using the same differential approach
as my younger self:

If a focus is used as origin, the polar coordinates (r,j)
of an ellipse of eccentricity e and parameter p obey the equation:

r
= p / (1 + e cos j )

This polar equation applies to any conic section.
The scaling parameter (p) specifies the size of the curve
and the eccentricity (e)
speciies its shape: e=0 for a circle, e<1 for an ellipse,
e=1 for a parabola, e>1 for an hyperbola.
Here, we assume 0<e<1.

Thus, the polar coordinates (r,q)
of a planar curve which rolls without slipping on that ellipse,
while rotating around a center orbiting at distance A from the origin,
obey the following differential equation:

(r-A) dq
= r dj
So:
dq
=
- dj /
( A/r - 1 )
=
- p dj /
( A + e A cos j - p )

Introducing the variable
t = tg(j/2) we have
dj = 2 dt / (1+t^{2 })
and
cos j =
(1-t^{2 }) / (1+t^{2 }). Therefore:

dq =

- 2p dt

A + e A (1-t^{2 }) - p

Introducing n such that
n^{2 }p^{2} =
(A-p)^{2} - (Ae)^{2}, this boils down to:

Polar equation of an elliptic gear of order n :

r = ^{ }

n^{2}_{ }p

[ n^{2 }(1-e^{2}) + e^{2
}]^{ ½} +
e cos(nq)

Closed contours are obtained when n is an integer
(which is what we normally want for an actual gear, except in the
rare situations when the gear will never execute a complete turn while meshing with another gear).

For given values of the parameter (p)
and eccentricity (e)
elliptic gears form a genderless family:
Every curve meshes with any other, either externally or internally
(for different values of n in the latter case).

Sinusoidal rack meshing with elliptic gears :

For very large values of n, the gear's median radius is nearly equal to:

R = n p / ( 1-e^{2 })^{ ½}

The limit of such a gear is best described as a straight rack
whose cartesian equation is obtained, as n tends to infinity, via the substitutions:

x = R q
y = r - R

This yields, neglecting relative errors proportional to 1/n^{2} or less,

n q = n x/R =
x ( 1-e^{2 })^{ ½}/ p

The relation y = r-R then gives the cartesian equation of a sine wave :

y = ^{ }

- p e

cos [

( 1-e^{2 })^{ ½}

x ]

1-e^{2}

p

Let's express this in terms of the traditional notations
a, b and c for, respectively, the
major radius, minor radius and focal radius
of the matching ellipse:

Sinusoidal Rack (n = ¥)

y = - a e cos ( x / b )
= - c cos ( x / b )

The unessential appearances of a negative sign and a
cosine (rather than a sine )
come from choosing the origin of x
at a point where y is smallest.

To summarize, the ellipse and the sinewave so described
can roll without slipping on each other as one of the foci of the
ellipse remains at a fixed distance from the axis of the sinewave.

This fact implies that the
perimeter of the ellipse
is equal to the length of one full arch of a sinewave
of wavelength 2p b
and amplitude c = a e.
(The distance between the two foci is 2c.)

Family of compatible elliptic gears :

We may define the nominal radius R_{n}
of an elliptic gear of order n as the half-sum of its
smallest and largest radius (i.e., the distances from
the axle to the root of a tooth and to the tip of a tooth).

In particular, for the basic ellipse (n = 1)
we have: R_{1} = a = p / (1-e^{2 })

The axle of an elliptic gear of order 1 is
at a focus of its elliptical contour. This isn't the center of symmetry
of that single-tooth gear.

If the axles of two compatible elliptic gears (i.e., same p and same e )
are separated by a distance A equal to the sum (resp. the difference)
of their nominal radii, those gears can roll externally (resp. internally)
on each other [without any sliding] as they rotate about their respective shafts.

(2012-11-21) Pitch Radius vs.
Nominal (Median) Radius_{ } The pitch radius of a gear does depend on what it meshes with.

We'll use elliptic gears to quantify the distinction, which is often butchered.

The median radius or nominal radius R_{n}
is an intrinsic mesurement of an n-tooth gear.
You can measure it on a given gear without knowing anything about the rest of the mechanism.

On the other hand, the pitch radius of a gear depends on
what it actually meshes with. If an
n-tooth gear of median radius R_{n} meshes
externally with an
m-tooth gear of median radius R_{m} ,
their pitch radii are:

R'_{n} = ( R_{n} + R_{m }) n / (n+m)
R'_{m} = ( R_{n} + R_{m }) m / (n+m)

The sum R'_{n} + R'_{m} =
R_{n} + R_{m} is the distance between the two axles.

For gears meshing internally, that distance is the difference between
the radius of the larger gear (the annular one) and that of
the smaller one:

R'_{n} = ( R_{n} - R_{m }) n / (n-m)
R'_{m} = ( R_{n} - R_{m }) m / (n-m)

This is to say that the previous formulas remain true if we make the convention
that an annular gear has a negative number of teeth and a negative
radius. With our previous expression of
the nominal radius of an elliptic gear as an odd function of its order,
we may simply view annular gears as gears of negative order.

In any genderless family of gears, if
m = n , then R'_{n} = R_{n} .
That is also the case when m is infinite
(an n-tooth gear meshing with a rack) as is readily seen by envisioning
the rack profile moving forward between two gears with the same number of teeth
meshing externally...

As an m-tooth driver meshes with an n-tooth
wheel, the quantity R'_{n} - R_{n}
(for a constant value of n)
can be viewed as a function of the gear ratio x.

We have |x| = n/m.
Recall that the gear ratio x is negative
for external meshing, positive for internal meshing and zero if the driver is a rack).
x goes from -n to n
(x = 1 being the dubious case of a frozen
gear meshing internally with itself).

(2012-11-21) _{ } Envelope of an orbiting gear
How to detect flawed meshing, the way Euler could have done it...

So far, we've been considering gears only as pairs of smooth mathematical contours
that keep sharing a common tangent as they rotate about two fixed centers
of rotation.

Such contours can .../...

Consider a plane where one of the two gears is fixed and the other orbits
around .../...

The successive contours of the moving gear form a parametrized
family of curves whose envelope
consists of two parts:

The trivial part is simply the contour of the fixed gear.

The nontrivial part consist of other locally extreme positions
of the moving contour.

The contour of the moving gear will never intersect the contour of the fixed
gear if and only if those trivial and nontrivial parts of the envelope never
cross each other.

They are allowed to be tangent to each other at certain points.
(This happens with zero-tolerance designs, either when there's no
backlash or when the tips of a gear touch the roots of the other.)

(2005-12-25) Split Elliptic Gearing_{ } One-way gearing featuring rolling without slipping.

With the elliptic gears described above, one gear can drive the other
only half of the time.
By retaining only the active half-tooth, we obtain an asymmetrical design
in which one gear pushes against the other all the time, in a predetermined
direction of rotation.

(2012-11-17) Traditional Watchmaker Gears_{ } The acting surfaces of horological wheels are radial planes.

Fig. 77 A correct depth...

for a pinion with 8 teeth
meshing with a wheel of 32 teeth.

In traditional clockwork, the protruding gear teeth are called
leaves (French: ailes, meaning wings).
The wheels (i.e., the large driven gears with
many leaves) have flat contact surfaces.
The pinions have
ogival profiles (so-called) matching such planar contact surfaces.

No specialized tools are required for machining the wheels but the ogival
shapes of pinion leaves require horological pinion cutters.
As far as I know, only two manufacturers are still supplying those nowadays
(see footnotes). Those tools aren't cheap in either case,
but you can easily obtain a single size from
P. P. Thornton (UK) whereas
Bergeon-Tecnoli (Switzerland) sells only expensive complete sets.

In horology, the gears are not at all expected to rotate at a constant
instantaneous rate.
Therefore, there's absolutely no reason to invoke Euler's
conjugate tooth action to preserve
the constancy of rotational speed from one gear to the next.
As conjugate action is not required, neither is the
involute gearing based on it
(which is virtually mandatory for lubricated high-speed machinery).

Horological mechanisms must work without any lubrication.
Their gear teeth could thus be designed to roll on each other's contact
surfaces without any slipping or sliding (which would be impossible
to achieve with rotating gears obeying Euler's conjugate action law).

(2012-11-24) _{ } La Hire's theorem (1694)
A two-cusped hypocycloid is a straight line.

The simplest result in the theory of rolling curves:
If a circle rolls without slipping inside
a fixed circle twice as big, then any point on it remains on
a straight line (others point attached to the moving circle describe ellipses).

Using modern nomenclature: An hypocycloid of ratio 2 is a straight line.
An hypotrochoid of ratio 2 is an ellipse.

This is to say that any acting part of the tooth profile outside
the pitch circle is an arc of an epicycloid,
whereas any acting part of the tooth profile inside
the pitch circle is an arc of an hypocycloid, whereas

Both genders of cycloids are mathematically generated by two congruent
circles that roll with slipping on the pitch circle.
Switching genders at the pitch circle (where the tangents of both
cycloids are purely radial) is just a practical
necessity. Otherwise tooth profiles would feature
points with infinite curvature, pointing either outward
(hypocycloid, yang) or inward (epicycloid, yin).

In practical gears, at most half an arch of either gender of cycloid
can be used (whichever of the two gears is acting as the driver
can only "push" the other; it cannot "pull" it).

The cycloidal shapes were first described by Albrecht Dürer
around 1525.
The idea to combine the two genders of cycloid
into genderless gears is attributed to the
French mathematician Philippe de la Hire (c. 1694).

(2012-11-17) Law of conjugate action _{ }
(Euler, c. 1754)
Gears featuring a steady rotational speed ratio.

As shown above, if two rotating curves are engaged
in pure roll on each other (without any sliding) then
their point of contact is on the straight line joining their
fixed centers of rotation. Also, the rate of rotation of either
curve varies inversely as the distance from that point of contact to the center
of rotation.

Therefore, the ratio of the rates of rotation of two such gears
cannot be constant (except when both are circles,
in which case the point of contact does remain at a fixed distance
from either center of rotation). However,
if the curves are allowed to slide tangentially to each other,
some profiles can maintain a constant rate of rotation of both gears at
all times...