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 denotea 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 thepitch circle.
Addendum Curve: The part of theprofile that's above thepitch 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 meshinggear is held in place. (This is normally measured inmodule units, alongthe pitch circle).
Bevel Gear: A conical gear, used to connect intersectingshafts.
Cage Gear: Another name for a lantern gear (which see).
Cam: A smooth solid imparting a specific motion to a so-calledfollowerin contact with it (often spring-loaded). Adisk cam, is a rotating cylinder whose lateral surfacedrives a flat follower, whereas the active surface of a cylinder camis actually helical... A cam in straight motion is called atranslation cam.
Center Distance: The distance between the two axles (parallel or not).
Centrode: The motion of arigid 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 calledtooth space.) Thecurvilinear distance between the centers of two adjacent teeth,as measured along thepitch circle. Inmodule units, thecicularpitch is always equal to .
Clearance: The amount by which thededendum of a gearexceeds theaddendum of another gear when both mesh.
Cog: Another name for thetooth 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 [thefillet of]a tooth, below thepitch circle.
Dedendum Curve: The part of theprofile that's below thepitch circle.
Elliptic gears: The family of compatible gearsdescribedbelow which are syntrepent to an ellipserotating about one of its foci (which can be viewed as the basic single-tooth pinionbelonging that family).
Elliptical gears: Meshing gears in the general shapeof syntrepent ellipses possibly endowed withsmall matching teeth on their respective circumferences to prevent slipping. Opposed to circular gears.
External gearsare regular gears,as opposed to internal orannular 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 relevantflank.)
Flank: The surface of the gear which comes in contact with meshing gears. We considerflank andface to be synonymous. However, some authors reserve the wordflank for the part of active surfacewhich is inside thepitch surface and callface thepart outside of thepitch surface.
Gear Ratio: The ratio of the (average) angular velocity of the input gear to the (average) angular velocityof the output gear. Unless the input and output axles are perpendicular,the notion of asigned 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 whichelliptic gears are an example) the tooth profiles in a family of compatible spur gear come in twodistinct genders. Only teeth of opposite genders can mesh externally.
Helical Gear: A gear whoseflank 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 gearsconnect two shafts that do not intersect. (The term is a contraction of "hyperboloid",which is thepitch surface for such a gearing.)
Internal Gear : Seeannular gear.
Isotrepent: The qualifier applying to a curve which issyntrepent with itselfwith respect to one of its points. Examples includeellipses 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 attachingrods 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 thepitch circledivided by the number of teeth. It's used to describe the toothprofile in general terms. In module units, thecircular pitch of a gear is always equal to .
Pinion: A small gear with few teeth (possibly, a single tooth). When discussing a pair of meshing gears, the smaller one is called thepinionwhereas the larger one is thewheel (or therack, for an infinite radius).
Pitch Circle: Loosely speaking, what the cross section of a straight spur gear wouldbecome without its teeth (seepitch 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 rotationcenters at a point P called thepitch point. (P = K if and only if there's no slipping).
Pitch Surface: The pitch surfaces of two meshing gears are the abstract surfacesattached to each of them which roll without slipping on each otherin 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 acrown gear, a cylinder for aspur gear,a cone for abevel gear, an hyperboloid for anhypoid gear.
Profile: The shape of a gear's tooth. The planar curve corresponding toits 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 linesparallel to the shaft or intersecting it. Opposed to helical gear.
Syntrepent Curves: Planar curves which roll on each other without slippingas 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) The ratio of the driver's rotation to the output rotation.
The gear ratio of any gear train is defined as theratio of the (average) angular velocity of the input gear to the (average) angular velocityof the output gear. Thus, if the driving gear rotates 5 times faster than the (final) driven gear,the gear ratio is 5.
That ratio is often taken only as a positive quantityinvolving 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 directionsof their rotations can be compared unambiguously. The gear ratio can then usefullybe given a sign (the same sign as the dot product of the relevantrotation vectors).
For a simple train of two spur gears, the algebraic gear ratio so definedis negative is the two gears are meshing externally and positive when they aremeshing 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) "Perfect" straight spur gears roll against each otherwithout slipping.
When two rigid planar curves roll against each other without slipping, the pointof contact has zero velocity with respect to either curve.
The planar cross-sections of two straightspur gears rotaterespectively around two points O and O'. If these curves roll against each other in the above sense, the velocity ofthe 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.
The polar coordinates of the point of contact (M) in the systems bound to either curveobey 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 anannular gear).
r d + r'd' = 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 samegender mesh internally with each other. Curves of opposite genders mesh externally.
If one curve meshes externally with itself (as shownnext 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) Syntrepent planar curves roll on each other without slipping as they rotatearound two fixed centers. A curve syntrepent to a copy of itself [with respect tomatching centers] is said to beisotrepent. An ellipse isisotrepent about its focus.
Both terms (French:courbes syntrépentes,courbe isotrépente) were coined by the French mathematician Auguste Miquel (1816-1851) in 1838.
Ellipses areisotrepent because congruent ellipses may roll on each other withoutslipping, as they rotate around their respective foci.In such a motion, the two ellipses are symmetrical about their tangent ofcontact, as illustrated above.
In this symmetrical configuration, the line joining two "opposite" foci goes through thepoint of contact. This may be proved using the fact that an ellipse reflectsany ray from a focus back to the other focus. ( Draw the four lines goingfrom the contact point to each focus, then deduce collinearity from angular relations.)
This gearing does not allow one pinion to drive the otherin 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 pointwhen the 4 foci are aligned. In the absence of a flywheel, the direction of rotation can indeed reverse itselffrom this dead position (both shafts may rotate in the same direction if thebar tying the moving foci remains parallel to the line joining the fixed foci).
(2005-12-10) (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'sconjugate action (which would make the driven gear rotate at a uniformrate if the driver does). These gears are thus more suited for unlubricated clockworkthan high-power lubricated machinery.
If a focus is used as origin, the polar coordinates () of an ellipse of eccentricity e and parameter p obey the equation:
= p / (1 +e cos )
Thus, the polar coordinates (r) 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:
(A) d = d So: d = d /( A/ 1 ) = p d /( A +e A cos p )
Introducing the variable t = tg(/2) we have d = 2 dt / (1+t) and cos = (1-t) / (1+t). Therefore:
d =
2p dt
A +e A (1-t2) p
Introducing n such that n2p2 = (A-p)2 - (Ae)2, this boils down to:
Polar equation of an elliptic gear of order n :
r =
n2p
[ n2(1-e2) +e2] ½ + e cos(n)
Closed contours are obtained when n is an integer (which is what we normally want for an actual gear, except in therare 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-e2) 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 y = r - R
This yields, neglecting relative errors proportional to 1/n2 or less,
n = n x/R = x ( 1-e2) ½/ p
The relation y = r-R then gives the cartesian equation of a sine wave :
y =
- pe
cos
( 1-e2) ½
x
1-e2
p
Let's express this in terms of the traditional notations a,b andc 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 describedcan roll without slipping on each other as one of the foci of theellipse 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 sinewaveof wavelength 2 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 Rn of an elliptic gear of order n as the half-sum of itssmallest and largest radius (i.e., the distances fromthe axle to the root of a tooth and to the tip of a tooth).
Rn = a [ n2(1-e2) +e2] ½ = [ n2b2 +c2] ½ = [a2+ (n2-1)b2] ½
Nominal radius of an elliptic gear of order n :
Rn = nb
1 +
e2
n2(1-e2)
In particular, for the basic ellipse (n = 1) we have: R1 = a = p / (1-e2 )
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) The pitch radius of a gear does depend on what it meshes with.
We'll useelliptic gears to quantify the distinction, which is often butchered.
The median radius or nominal radius Rn 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 onwhat it actually meshes with. If an n-tooth gear of median radius Rn meshes externally with an m-tooth gear of median radius Rm , their pitch radii are:
This is to say that the previous formulas remain true if we make the conventionthat an annular gear has a negative number of teeth and a negativeradius. With ourprevious expression ofthe 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 = Rn . That is also the case when m is infinite (an n-tooth gear meshing with a rack) as is readily seen by envisioningthe rack profile moving forward between two gears with the same number of teethmeshing externally...
As an m-tooth driver meshes with an n-tooth wheel, the quantity R'n - Rn (for a constant value of n) can be viewed as a function of the gear ratio x.
(2012-11-21) 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 contoursthat keep sharing a common tangent as they rotate about two fixed centersof rotation.
Such contours can .../...
Consider a plane where one of the two gears is fixed and the other orbitsaround .../...
The successive contours of the moving gear form a parametrizedfamily 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 positionsof the moving contour.
The contour of the moving gear will never intersect the contour of the fixedgear if and only if those trivial and nontrivial parts of the envelope never cross each other.
(2005-12-25) One-way gearing featuring rolling without slipping.
With the elliptic gears described above, one gear can drive the otheronly half of the time. By retaining only the active half-tooth, we obtain an asymmetrical designin which one gear pushes against the other all the time, in a predetermineddirection of rotation.
(2012-11-17) The acting surfaces of horological wheels are radial planes.
Fig. 77 A correct depth...
for a pinion with 8 teethmeshing 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 withmany 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 ogivalshapes 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) whereasBergeon-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 preservethe 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 contactsurfaces without any slipping or sliding (which would be impossibleto achieve with rotating gears obeying Euler's conjugate action law).
(2012-11-24) A two-cusped hypocycloid is a straight line.
The simplest result in the theory of rolling curves: If a circle rolls without slipping insidea fixed circle twice as big, then any point on it remains ona 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
In practical gears, at most half an arch of either gender of cycloidcan be used (whichever of the two gears is acting as the drivercan 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 cycloidinto genderless gears is attributed to theFrench mathematician Philippe de la Hire (c. 1694).
(2012-11-17) (Euler, c. 1754) Gears featuring a steady rotational speed ratio.
As shownabove, if two rotating curves are engagedin pure roll on each other (without any sliding) thentheir point of contact is on the straight line joining theirfixed centers of rotation. Also, the rate of rotation of eithercurve varies inversely as the distance from that point of contact to the centerof 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 distancefrom 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 atall times...
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 attachingrods radially to a solid disk.
(2005-12-11) 
