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, 25th 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.)
Nanjing Plusplus Software Co., Ltd.
Gears, textbooks & training: Salem Company,
Woodstown, NJ 08098
Videos:
Non-circular gears and planetary gear
|
Super Oval Flowmeter
How to make wooden gears by Matthias Wandel.
Wooden Gear Template Generator (video demo)
by Matthias Wandel.
Cutting Wood Gear & Clock Wheel Teeth
by Ronald Walters.
These Gears Really Work?
by Dr. Clayton Boyer.
Around the Corner:
How a differential gear works
(Jam Handy, 1937).
Making Gears
by TheMetalCutter.
Gears 101 (29:01)
by This Old Iony (2019年02月02日).
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:
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.
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). [画像: Polar coordinates for two planar gears ]
r dq + r' dq' = 0
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.
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).
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+t2 )
and
cos j =
(1-t2 ) / (1+t2 ). Therefore:
[画像: Come back later, we're still working on this one... ]
Introducing n such that n2 p2 = (A-p)2 - (Ae)2, this boils down to:
Polar equation of an elliptic gear of order n :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).
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 q
y = r - R
This yields, neglecting relative errors proportional to 1/n2 or less,
n q = n x/R = x ( 1-e2 ) ½ / p
The relation y = r-R then gives the cartesian equation of a sine wave :
Ellipse (definition of a and b) 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 = ¥)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.)
We may define the nominal radius Rn 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).
Rn = a [ n2 (1-e2 ) + e2 ] ½ = [ n2 b2 + c2 ] ½ = [ a2 + (n2-1) b2 ] ½
Nominal radius of an elliptic gear of order n :In particular, for the basic ellipse (n = 1) we have: R1 = a = p / (1-e2 )
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.
We'll use elliptic 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 on what 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:
R'n = ( Rn + Rm ) n / (n+m)
R'm = ( Rn + Rm ) m / (n+m)
The sum R'n + R'm = Rn + Rm 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 = ( Rn - Rm ) n / (n-m)
R'm = ( Rn - Rm ) 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 = Rn . 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 - Rn (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).
In the case of elliptic gears, this function is:
[画像: Come back later, we're still working on this one... ]
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 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.)
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.
for a pinion with 8 teeth meshing with a wheel of 32 teeth.
From the book:
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).
[画像: Come back later, we're still working on this one... ]
Designing
Cycloidal Gears (rack cutting) by Hugh Sparks (2012)
Cycloidal Gear Builder
by Dr. Rainer Hessmer (2012)
Gears for instruments and clockwork mechanisms.
Cycloidal type gears. Double circular arc type gears :
British Standard : BS 978-2:1952
Addendum 1:1959
Swiss standards : NHS 26702
|
NIHS 20.01, 20.02, 20.25 (formerly NHS 56702 and 56703)
P. P. Thornton
Watch & Clock Wheel and Pinion Cutters
|
Bergeon-Tecnoli Gear Cutters
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.
Proclus Diadochus (AD 411-485)
|
Nasir al-Din al-Tusi (1201-1274)
Nicolaus Copernicus (1473-1543)
|
Philippe de la Hire (1640-1718)
Tusi couple
|
Trammel of Archimedes (ellipsograph)
Secrets of the Nothing-Grinder (12:52)
by Burkard Polster (Mathologer, 2018年12月07日).
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).
[画像: Come back later, we're still working on this one... ]
Albrecht Dürer (1471-1528) | Philippe de la Hire (1640-1718) | Cycloidal Gear
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...
[画像: Come back later, we're still working on this one... ]
Conjugate tooth action by Douglas Wright | Leonhard Euler (1707-1783) by Walter Gautschi
The rack conjugate to an involute spur gear has straight flanks.
[画像: Come back later, we're still working on this one... ]
Involute (or "evolvent") | Involute gear | Gear tooth generation (rack cutting) by Douglas Wright
[画像: Come back later, we're still working on this one... ]
Harmonic drive | Clarence Walton Musser (1909-1998)
The Wildhaber-Novikov gears feature a large contact area between the convex and concave mating teeth.
[画像: Come back later, we're still working on this one... ]
US Patent 1601750 "Helical Gearing"
(1923年11月02日 / 1926年10月05日) issued to Ernest Wildhaber (1892-1979).
USSR Patent 109750 "Helical Gearing" (1956) issued to M.L. Novikov.
Wildhaver-Novikov gear geometry
by Stepan V. Lunin (2001)
| Gallery
[画像: Come back later, we're still working on this one... ]
Chinese standards (1981) for helical double circular arc (DCA) gear.
