Contents

Gear Artwork
Here is some of my gear artwork.
Bucky Brain Gear / Sphere Gear Combo Kinetic Sculpture / Lamp  POVRay 3.6.1 version: 2/7/11, 3ds Max, MaxScript model: 10/5/11
242 Gear Sphere  Mathematica 4.2, POVRay 3.6.1, 9/26/11
Here is a set of 242 interlocking bevel gears arranged to rotate freely along the surface of a sphere. This sphere is composed of 12 blue gears with 25 teeth each, 30 yellow gears with 30 teeth each, 60 orange gears with 14 teeth each, and 140 small red gears with 12 teeth each. I also found 3 other gear tooth ratios that will work, but this one was my favorite because the small gears emphasize the shape of a truncated rhombic triacontahedron.

Involute Gears  new version: Mathematica 4.2, 5/23/12; old version: POVRay 3.6.1, 7/12/06
The ideal profile for a gear tooth is the involute of a circle. The exact shape of the involute curve can be defined by the number of teeth, pitch circle radius, and pressure angle. This animation shows a gear with only 6 teeth, which may not be enough teeth for most applications, but it demonstrates a good example of extreme undercut. Here is some Mathematica code to draw ideal gears:
(* runtime: 0.3 second *)
n = 9; r = 1.0; a = Pi/9; dtheta = Pi/n; dr = 2r/n; rbase = r Cos[a]; IR = r  dr; OR = r + dr;
Rz[x_] := {{Cos[x], Sin[x]}, {Sin[x], Cos[x]}};
Involute[t_] := rbase((Tan[a]  t){Cos[a  t], Sin[a  t]} + {Sin[a  t], Cos[a  t]});
Undercut[t_] := r {Sin[t]  t Cos[t], Cos[t] + t Sin[t]}  dr Rz[t].{Tan[a], 1};
t2i = Tan[a]; t1i = t2i  Sqrt[(OR/rbase)^2  1]; t2u = 2 t2i/n; t1u = t2u  Sqrt[rbase^2  IR^2]/r;
Clear[ti, tu]; {t2i, t1u} = {ti, tu} /. FindRoot[Involute[ti] == Undercut[tu], {ti, 0.9t2i, 1.1t2i}, {tu, 0.9t1u, 1.1t1u}];
plist = Join[Table[Involute[t], {t, t1i, t2i, (t2i  t1i)/8}], Table[Undercut[t], {t, t1u, t2u, (t2u  t1u)/8}]];
plist = Join[Reverse[plist], Map[(Rz[dtheta].{#[[1]], #[[2]]}) &, plist]];
plist = Join @@ Table[Map[(Rz[2dtheta i].#) &, plist], {i, 1, n}];
Table[Show[Graphics[Map[(sign = #; Line[Map[{sign r, 0} + Rz[sign theta].# &, plist]]) &, {1, 1}],AspectRatio > Automatic, PlotRange > OR{{2, 2}, {1, 1}}]], {theta, 0, 2dtheta(1  1/10), 2dtheta/10}];
Links
Gear Template Generator  Java applet, by Matthias Wandel
Lobe Compressor  SPH simulation by Simerics
Paradoxical Gear Set  gears rotating in same direction with 1, 2 and 3 teeth, by Jacques Maurel
Mechanical Involute Gears  Mathematica demonstration by Stephan Heiss, here is the source code

92 Gear Spheres  Mathematica 4.2, POVRay 3.6.1, 9/22/11 (old version: 2/7/11)
These images show different ways of arranging 92 bevel gears to rotate freely along the surface of a sphere. Each sphere contains 12 large blue gears, 20 large yellow gears, and 60 small red gears. The gears are slightly offcentered from the vertices of a geodesic sphere. When I first attempted this on 2/7/11, I used a 25:30:12 gear tooth ratio, but unfortunately, I couldn't get the teeth to mesh together. However, on 7/26/11, Taff Goch demonstrated that it is possible to get the teeth to mesh reasonably well using a 20:24:13 gear tooth ratio. As it turns out, there are other gear tooth ratios that can be made to mesh together reasonably well, as you can see here. I found these solutions by writing a program that analyzed many different ratios. For a given ratio, the location and radius of each gear was calculated so that there are no gaps between mating pitch circles. Then the gear phase angles were calculated and solutions with large phase errors were rejected. Finally, I did a visual inspection of the remaining solutions and presented some of my favorite ones here. Two more of my favorites are 15:15:8 and 30:30:16 because they only require two gear sizes. It is interesting to note that the number of blue gear teeth must always be divisible by 5 and the number of yellow gear teeth must always be divisible by 3, but the number of red gear teeth does not have to be divisible by anything (it can be a prime number).
Links
92 Gears, 92 Gears Caged  Taff Goch succeeded in getting the gear teeth to mesh, see also his 62 gears sphere

Bucky Brain Gear  POVRay 3.6.1, 1/27/11
32 Gear Sphere  Mathematica 4.2, POVRay 3.6.1, 2/7/11
Here is a set of 32 identical bevel gears arranged to rotate freely on the faces of a truncated icosahedron (Buckminsterfullerene). As a general rule, the number of blue gear teeth must always be divisible by 5 and the number of yellow gear teeth must always be divisible by 6. The configuration shown here is my favorite because all the gears are the same with 30 teeth. Click here to see some other ratios that will work.
Links
Mechaneu  3D printed 32 gear sphere, by Toru Hasegawa
Big Brain Gear Machine  repeating pattern of 10 gear spheres in 3D, by CAD Gill
Gears Heart  amazing heart composed of 12 paper gears, designed by Haruki Nakamura, see also his Gears Cube
Gears Heart  3D printed model based on Haruki's design, by David Bush, you can read more here, see also his movies and Gear Ball
Gear Sphere  template for making your own paper gear sphere, by Michael James, based on Haruki's design
OctaGear  3D printed 8 gear sphere, created by Jeffrey Pettyjohn using GearTrax, you can see more pictures on Shapeways, see also his 42 gear DecaGear and 26 gear PentaGear
Spherical Sets of Wood Gears and Metal Gears  creative designs by Kenneth Snelson
Gear Wheel IQ Cube  a Rubik's Cube with gears

182 Gear Spheres  Mathematica 4.2, POVRay 3.6.1, 9/28/11
Here is a set of 182 interlocking bevel gears arranged to rotate freely on the vertices of a geodesic sphere. Click here to see many other gear ratios that will work.

362 Gear Sphere  Mathematica 4.2, POVRay 3.6.1, 2/7/11
Here is a set of 362 interlocking bevel gears (122 large blue gears plus 240 small red gears) arranged to rotate freely on the vertices of a geodesic sphere. Note: The teeth do not mesh together on this design; this is a work in progress.

Conventional Differential  POVRay 3.6.1, 1/18/11
Automobiles use differentials to allow the left and right drive wheels to rotate at different speeds. This prevents the wheels from skidding when making sharp turns. The drive shaft is directly connected to the pinion gear (shown in red). The pinion gear turns the ring gear (shown in blue), imparting a net rotation to the wheel axles. The differential pinion carriers or "spider" gears (shown in yellow) allow for different rotations between the left and right differential side gears (shown in green). The differential side gears are directly connected to the wheel axles. This design fails when one wheel loses traction (see Torsen Traction differential). Notice that the pinion and ring gears are spiral bevel gears.

Torsen Traction Differential  POVRay 3.6.1, 1/21/11
The Torsen Traction differential was patented by Vernon Gleasman in 1958. It takes advantage of worm gears to prevent the tires from slipping. If designed properly, the worm gears (shown in green) can turn the worm wheels (shown in yellow), but they cannot be turned by the worms wheels due to friction. In 1982, Gleasman joined Gleason Works, producer of 90% of the world's automotive bevel gears. There are three types of Torsen differentials (T1, T2, T3). The one shown here is of type T1.
Links
The strange geometry of Gleason's Impossible Differential  interesting Popular Science article, dated February 1984
Lego Torsen Differential  by Rob Stehlik

Spiral Bevel Gears  POVRay 3.6.1, 1/15/11
Other Interesting Links
Curta Calculator  amazing mechanical calculator developed by Curt Herzstark while he was in a concentration camp, story narrated by Cliff Stoll
Gear Ring  the gears can actually rotate, by Kinekt Design
Gear Ring  3D printed by Susan Hinton
