Lego

Gears — Meshing

GEAR MESHING CALCULATOR to appear here soon…

Kurt Baty: how can I contact you? (don't say FaceBook, I don't go there) you can try me at robert at cailliau dot org.

Meshing Concepts

Two gears mesh when their axles are close enough that their teeth overlap.  Consider the these two wheels:

axle distanceA tooth overlap D outerradius R rollingradius t innerradius r
two meshing gears [SVG]

A gear wheel has an outer radius R which is measured over the tops of the teeth, an inner radius r which is measured at the base of the teeth (bottom of the valley in between teeth), and a rolling radius t, which is somewhere in between.  A meshing pair has an axle distance A and a tooth overlap D.

Meshing can be too loose or too tight depending on the axle distance:

very loose meshing [SVG]
over-tight meshing [SVG]

Tooth Overlap

For this measure the rolling radii are not important.

Given a gear G1 with radii r1, R1 which meshes with a second gear G2 whose radii are r2, R2:  some meshing will occur as soon as the distance A between their axles is less than R1+R2.

D A r1 R1 R2 r2
meshing [SVG]

The tooth overlap is then:

D=R1+R2−A

Maximum Tooth Overlap

The height of the teeth of G1 is R1−r1 and those of G2 are R2−r2 high.  Normally these values are equal, or at least very close.  Lego gears do not all have the same tooth height; even variants of gears with the same number of teeth and rolling diameter may not have the same tooth height.  If the teeth of G1 are higher than those of G2 then they cannot penetrate more into G2's outer diameter than by the height of the teeth of G2:  the maximum overlap of teeth is given by the tooth height of the wheel with the less high teeth.

The maximum tooth overlap is then:

Dmax=min(R1−r1,R2−r2)

Some Lego gears (the double conical ones) have tooth shapes that prevent the teeth of other wheels to get to the bottom of the valley between them, despite the fact that they have the smaller tooth height.  These wheels have been given an effective inner radius that is greater than the actually observable one, so that the maximum overlap can still be calculated.

Good Meshing

For any meshing to occur the distance between the axles must be

A < R1+R2

The overlap has a maximum value which is the height of the lesser teeth.  We assume:

Good meshing occurs when tooth overlap is a significant fraction of the smaller tooth height.

The table gives overlaps as they occur in some simple combinations, all on axles in a single horizontal bearing brick, i.e. the axle distance is a multiple of 8mm:

G1G2overlaplesser height% of lesser height
normal 8normal 81.82.186
normal 8normal 241.82.186
normal 8normal 401.72.181
normal 24normal 241.82.282
normal 40normal 401.62.176

No overlap is less than 75% of the height of the smaller tooth.

Gears mesh well when their teeth overlap sufficiently.  The height of the teeth of a gear is R-r, and the overlap cannot be greater than the height of the teeth of the gear with the smallest height.

Given a gear with radii r1, t1, R1 which must mesh with a second gear whose radii are r2, t2, R2:  some meshing will occur as soon as the distance A between their axles is less than R1+R2.  The axles obviously cannot get closer than A=r1+R2 or A=R1+r2, whichever is greater (hence the need to define an effective r for the double conical wheels).

The overlap of teeth is R1+R2-A, which is 0 when A=R1+R2, and maximum for one of the situations A=r1+R2 or A=R1+r2, when it is either R1-r1 or R2-r2, i.e. it is at most the tooth height of the wheel with the smallest tooth height of the two.

For a given pair of gears the axles must be put at a distance that gives good meshing.  There are the known simple situations where both axles are in the same Technics bearing brick, but many other stacks of bricks may give acceptable distances.  The calculation of such stacks is the subject of the meshing page.

Caveat

Two gears may satisfactorily mesh when placed in axle holes that are in other positions than the ones found on a straight Technics bearing beam.  The quality of the meshing depends on the inner and outer diameters of the gears, i.e. the diameter as measured over the tops of the teeth and the diameter as measured at the bottom of the teeth.  These depend on the particular gear and are not directly related to the rolling diameter.  That is the reason for the gear meshing calculator.

Normal meshing

This is the case when they are used in the same Technics bearing brick, e.g. wheels of 8&8 teeth, one hole apart (4+4=1x8), or 8&40 teeth, three holes apart (4+20=3x8), and so on.

Practical meshing

Gears also mesh reasonably well when the distance between their axles is somewhat less or greater than the sum of their rolling radii.

Such positions of the axles can occur when the axles are not in the same bearing brick, viz. when the axle of the second gear runs in a bearing that is both vertically and horizontally offset from that of the first.  This may be achieved by inserting plates, using special bearing bricks, etc.  A certain mechanism may require the axles to be offset in such a fashion, and it may be the only way to mesh two gear wheels when the sum of their rolling radii is not a multiple of 8 (e.g. 8&16 teeth: 4+8=12)

If the distance between axles is not exactly the sum of the rolling radii, the abstract smooth wheels will not touch, but if the teeth of the real wheels overlap sufficiently, the mechanism will function nevertheless and the gear ratio will be the same as in the ideal case since the number of teeth has not changed.

Consider a gear wheel with radii R1, t1 and r1 and another with radii R2, t2 and r2. As soon as the distance A between the axles of these wheels becomes less than R1+R2, some meshing of their teeth will occur; operation will be as designed when the distance is exactly t1+t2, but meshing can no longer occur when that distance is less than the larger of the two values R1+r2 and r1+R2 because the teeth of one wheel will touch the bottom between the teeth of the other.

In practice good meshing only occurs for distances A between R1+R2-D and r1+r2+d where D and d are experimental values.

The value of d is less important than the value of D. I call D the tooth overlap. For the design case, the overlap is R1+R2-(t1+t2) but in general, for a given axle distance A the overlap is

D=R1+R2-A

A good overlap is not too far away from the design overlap.