Gears
A gear is a toothed wheel, rigidly attached to its axis of rotation.
It meshes with other gears to transmit rotary motions to other axes.
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 rotating ring.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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 produced by 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.
 - Lantern Gear :
A rudimentary wooden gear (at right)
consisting of two wheels connected by a few rods, which serve as gearing teeth.
- 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).
- Leaf :
The tooth of a gear, in watchmaking parlance.
- Miter Gear :
A conical gear transmitting rotation between two shafts intersecting at a right angle
(the most common type of bevel gearing).
- Module :
A unit of length equal to the diameter of the pitch circle
divided by the number of teeth.
It's commonly 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 in the case of 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 angular
average 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.
- Spur Gear :
A cylindrical wheel, with teeth cut across its circumference.
- Straight Gear :
A spur gear or a conical gear whose teeth are cut along straight lines
(either 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 :
The 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.
(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
Incidentally, two curves meshing with a third mesh internally with each other.
Two "genders" may thus be defined so that curves of the same
gender mesh internally with each other,
whereas curves of opposite genders mesh externally.
If one such curve meshes externally with itself (as shown
next in the case of an ellipse)
then all of them mesh internally and externally,
without gender distinction.
 (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 introduced
by the French mathematician Auguste Miquel 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)
Elliptical Spur Gears:
From Ellipse Pinion to Sine Rack
A simple family of curves that mesh perfectly with each other.
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 )
Thus, the polar coordinates (r,q)
of a planar curve which rolls without slipping around such an ellipse
as it rotates around a center at a distance a from its focus
obey the differential equation
(r-a)
dq
= r dj
which may be solved by introducing the variable
t = tg(j/2)
for which
dj
=
2 dt / (1+t2 ) :
| dq |
= |
- dj /
( a - p + ea cos j ) |
Introducing n such that
n2 p2 =
(a-p)2 - (ae)2, this boils down to:
| r = |
n2 p |
 |
| [ n2 (1-e2) + e2
] ½ +
e cos(nq) |
|
(2005-12-25)
Split Elliptical Gearing
One-way gearing featuring rolling without slipping.
With the elliptical 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.
(2005-12-26)
Cycloidal Gearing
Traditional Watchmaker Gears.
(2005-12-30)
Epiycloidal Gearing
Philippe de la Hire (1640-1718).
(2005-12-26)
Involute Tooth Profile
Involute tooth profiles provide constant rotational speed ratios.
(2005-12-25)
Harmonic Drive
(patented by C. Walton Musser in 1955)
A "wave generator" rolls against a flexspline
inside a circular spline.
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