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Illustrated gearing tutorial
A detailed gearing tutorial illustrated with mock-ups and my own MOCs. The simple relationship between gear ratio and axle spacing in LEGO spur gear pairs is a very valuable tool.
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Abstract
Pitch measures the distance between successive teeth on the circumference of a gear. A gear with Z teeth and pitch P has an effective diameter or pitch diameter of D = P Z and hence a pitch radius of R = D / 2 = P Z / 2. For 2 gears to mesh at all, their pitches must be identical. For them to mesh well, their on-center axle separation X must equal the sum of their pitch radii.

One can learn a great deal about gearing from these 2 simple requirements alone, but it gets better: Since all LEGO® gears have the same 1 mm pitch, their pitch diameters are just their number of teeth Z in millimeters. This convenient fact makes easy work of predicting the ideal axle separation for any pair of LEGO® spur gears (including hybrids like the double bevel gears that also serve as spurs): Xideal = ˝(Zin + Zout) * 1 mm, where Xideal is the ideal axle separation in millimeters, and Zin and Zout are the tooth counts on the input and output gears, respectively. Axle separations under ideal by as little as 0.3 mm can lead to excessive friction and wear, but you can often get away with separations up to 1 mm over ideal if the added backlash and potential for skipping under high torque are acceptable.

Perhaps the most useful insight to come out of these simple relationships is what I call the Rule of Sixteens, which states that for Xideal to be a whole number of LEGO® units (LU, aka "studs"), the tooth sum Zin + Zout must be a multiple of 16 — specifically, 16, 32, 48, 64, 80, 96, or 112. However, the relationship Xideal = ˝(Zin + Zout) * 1 mm also simplifies the process of finding acceptable meshes for gear pairs mounted in pin holes off the whole-LU grid.




Introduction
Gears find many uses in technical LEGO®. Most commonly, they allow one to bring a motor to bear on a load at the most advantageous point on the motor's torque vs. speed curve. They also allow one to transfer torque from location to location within a build.

This tutorial focuses on true LEGO® spur gears and the many hybrid LEGO® gears that can be used as spurs. Most LEGO® gears fall into these 2 categories, which I'll lump together as "spurs" hereafter.

Simply put, spur gears transfer torque between parallel axles. They're by far the most commonly used gears in technical LEGO®, and they're also the easiest to understand. Bevel and crown gears, which transfer torque between non-parallel axles, are a bit more complicated.

Gear train design is usually an iterative process. The workflow goes something like this:
  • Think about the external loads the MOC is likely to face. Will it be lifting heavy loads or climbing over obstacles or up steep slopes? Will it be plowing through soft surfaces like sand or snow or fighting air or water resistance?

  • Get a feel for the MOC's internal losses. Will it have large, complex gear trains with many bearings? Will the wheel or track bearings be supporting a lot of weight?

  • Pick the actuators (motors, pneumatic cylinders, linear actuators) the MOC will need to overcome its internal and external loads.

  • Find gear ratios providing the best match between the chosen actuators and their loads. (This potentially time-consuming step may well require some knowledge of the actuators' speed vs. power and speed vs. torque or force characteristics.) Will the inevitable trade-offs be forgiving or critical?

  • Identify the combinations of available LEGO® gears best approximating the target gear ratios.

  • Keep track of all the rotation direction reversals and add idlers to the gear train if necessary.

  • Determine the necessary axle separations.

  • Anticipate the available pin holes after the most critical structural pins are in place. (Better to err on the side of too much axle support here.)

  • Build a test mock-up or take your chances on building straight into the MOC.

  • Test against the full range of conceivable loads.

  • Loop back to previous steps until you're comfortable with all the compromises you've made. (There will be compromises.)
For most of us, getting even a few of these guesses right the first time around would be nothing short of miraculous. Experience definitely helps, but in the end, success often depends more on the last step (and the necessary perseverence) than anything else.

There are some shortcuts, however. This tutorial addresses one of the most important — namely, the prediction of axle separations from gear ratios based on nothing more than the total number of teeth involved in a given meshing gear pair.

The ability to make such predictions quickly and accurately greatly simplifies a common and often frustrating task -- namely, the allocation of available pin holes between turning axles and structural pins without compromising either axle support or structural strength.




Part 1. Basic gearing concepts

It all comes down to pitch
Pitch — an important measure of tooth separation along the circumference of a gear — is the key to LEGO® gearing. To understand pitch and its many useful implications for the LEGO® system of gears, you need to know a little about gears and what it takes for 2 gears to mesh well — i.e., to run smoothly and quietly at high speed with minimum friction and no skipping under high torque.

Like everywhere else the LEGO® realm, trade-offs will almost always be with us here. The trick is to play them well.



Key gear attributes
Gears are basically wheels with teeth designed to prevent slippage when one gear wheel rolls on another. A gear train consists of one or more pairs of meshing gears collectively linking an input to an output shaft. Some use the term gearset as a synonym for gear train, while others restrict it a 2-gear train.

The most important gear attributes are
  • Gear type — i.e., spur, crown, bevel, or some combination thereof
  • Pitch (P), a measure of tooth separation
  • Effective (pitch) diameter (D)
  • Tooth count (Z)
  • Tooth height
  • Tooth profile
Pitch, pitch diameter, and tooth count are closely interrelated; the remaining gear attributes are more mix and match. Since gear type and number of teeth (Z) are by far the easiest attributes to determine, it would be nice to work in teeth and type whenever possible, and that's what this page is largely about.

The photo below shows a simple 40-tooth LEGO® spur gear (Z = 40). Though not apparent from the photo, I will show below that its pitch P = 1 mm. Its outside diameter is 42 mm, but its pitch diameter D = P * Z = 40 mm. Like all other LEGO® spurs, the teeth are 2 mm high with involute profiles.



The numerical equality between tooth count and pitch diameter in mm is found in all LEGO® gears.

This tutorial generally deals with external gears — i.e., those with outward-pointing teeth. I'll have to save the 2 LEGO® gears with inward-pointing teeth (aka annuli, internal gears, or ring gears) for another day, but I do show examples of their use here and here.



External LEGO® gear types: Spurs, crowns, bevels, and hybrids
Gear types are based on tooth crest orientation in 3D. The 3 basic types are spur, bevel, and crown, but many external LEGO® gears are hybrids of these.

Spur gears, the most common type within and beyond the LEGO® realm, are designed to mesh with other spur gears on parallel axles or with crown gears on perpendicular axles. Their teeth point radially outward, directly away from the axle.

The photo below shows all of the LEGO® simple spurs. (Note the internal 24-tooth gear inside the large Technic turntable at upper right.)



Crown gears have teeth pointing parallel to their axles (or perpendicular to their faces) so as to mesh with spurs on a perpendicular axle. Both LEGO® crowns have teeth that wrap around from face to edge to allow them to function as spurs as well.

The photo below shows both of the LEGO® crown-spur hybrids. The one on the right is on the original Technic differential.



Bevel gears fall between these extremes. Because they're meant for non-parallel axles that may or may not be perpendicular, their teeth point outward at an angle to one or both faces. The only pure LEGO® bevels the single bevels with teeth on only one face.

All of the LEGO® double bevels are actually double bevel-spur hybrids. The 28-tooth double bevel-spur hybrid is on the small Technic turntable.



Any gear with tooth crests falling in a plane containing the axle is said to be "straight-cut". All LEGO® gears are straight-cut (i.e., there are no helical gears), and most are either bona fide spur gears or can be used as such. (The latter group includes the newer double-bevels, which serve equally well as spurs and bevels).

All LEGO® gears usable as spurs — hereafter, just spurs or spur gears unless otherwise noted — share the same tooth height (2 mm) and involute tooth profile. This profile allows them to run reasonably well at axles separations up to 1 mm above ideal, though with some increase in backlash and potential for skipping under high torque.

Most importantly, all spurs also share the same tooth separation or pitch.

Single bevel gears have the same tooth separations and similar tooth heights and profiles.



Available LEGO® gears useable as spurs
By tooth count and type, the master list of available external LEGO® gears useable as spurs is

{8s, 12ds, 16s, 16sf, 20ds, 24s, 24cs, 24sc, 24sf, 28dst, 28csf, 36ds, 40s, 56st}

The curly braces denote a list. The letters appearing immediately after the tooth counts denote gear type: "s" for simple spur, "ds" for double bevel-spur hybrid, and "cs" for crown-spur hybrid. A terminal "c" indicates the presence of a built-in clutch (really a torque limiter), while a terminal "f" or "t" denotes a gear on a differential body or turntable, respectively. The list of gear types appearing at least once in the master list is therefore {s, sf, sc, st, ds, dst, cs, csf}.

The photo below shows all the gears usable as spurs in the order listed from bottom left to top right. The only one out of order is the 16sf gear on the closer differential body.



The list of tooth counts appearing at least once in the master list of LEGO® spur gears is {8, 12, 16, 20, 24, 28, 36, 40, 56}. This list is an important one, because it limits the gear ratios attainable with spur pairs. The only tooth counts appearing more than once are {16, 24, 28}.

Most LEGO® spurs are simple spurs with teeth pointing in only one direction — namely, radially outward. The sublist of simple spurs is {8s, 16s, 16sf, 24s, 24cs, 24sc, 24sf, 40s, 56st}. These gears can't function as crowns or bevels.

The 16sf and 24sf simple spurs are on opposite ends of the same Technic differential, whereas 56st spurs are found on the outer edges of all large Technic turntables. None of these differential- and turntable-based gears have their own axle holes, but central axle holes can be arranged, as shown below. The white 24sc has a built-in torque limiter designed to start slipping at 25 N mm (2.5 N cm). The rest of the simple spurs are basic one-piece gears with built-in axle holes.

For my money, the double bevel-spur hybrids {12ds, 20ds, 28dst, 36ds} are the strongest of the lot and generally also the most versatile WRT mounting. The 28dst is on the small Technic turntable.

The difference between a double bevel (36ds, above) and a simple spur (40s, below) is evident in the on-edge view below. The sandwich-like double bevel consists of a central spur flanked by single bevels facing in opposite directions. It's exactly 1 LU thick from face to face. The simple spur is thinner overall.



The sublists of differential- and turntable-based external gears are {16sf, 24sf, 28scf} and {28dst, 56st}, respectively. (The other half of the large Technic turntable has an internal 24-tooth spur.) The 28csf entry refers to the hybrid crown-spur gear on the original Technic differential body. The only other crown in the master list is the 24cs hybrid — a rather soft, weak gear that nonetheless has its uses in low-torque settings.

Adding a central axle to the black halves of the small and large Technic turntables allows them to function as regular gears. The 2 photos below show two ways to accomplish that. The LBG half of the small turntable could be removed at this point if desired.





Locking the differential-based gears onto an axle is as simple as loading the housing with the usual trio of 12-tooth bevels and running the axle all the way through, as shown in the next photo.





Pitch — a handy and critical measure of tooth separation
The most useful practicalities of LEGO® gearing involve effective (pitch) radius (R) or diameter (D), tooth count (Z), and the all-important notion of pitch (P). The concepts and equations involved are simplest for spur gears but also apply (with very minor complications) to crown and bevel gears, as we'll see below.

Every spur gear has an effective radius called the pitch radius, R, to be defined in a moment. The pitch radius in turn defines a pitch circle centered on its axle with a pitch diameter D = 2 R, and a pitch circumference C = 2 π R = π D.

The pitch circle always falls somewhere within the teeth. In LEGO® spur gears, the pitch circle happens to lie 1 mm inboard of the tooth crests and 1 mm outboard of the bottoms of the inter-tooth troughs — i.e., half-way between the two. Together, this geometry and the involute tooth profiles determine how far you can stray from the ideal axle separation for a given gear pair and still get a workable mesh.



Criteria for ideal meshing in spur gears
For 2 spur gears to mesh at all, it's essential that the arc length between corresponding points on adjacent teeth (e.g., tooth crests) be exactly the same on both. That arc length is given by the circular pitch Pc ≡ C / Z = π D / Z, where C is the pitch circumference, and Z is the tooth count.

A much more convenient measure for design purposes, however, is the diametral pitch (aka module and hereafter simply pitch) defined as P ≡ D / Z, where D is the pitch diameter.

NB: The pitch P is a length, here expressed in millimeters. To keep our units straight, Z must be just a number with no units. Unless otherwise noted, all lengths and separations will be in millimeters.

Two spur gears with the same pitch will mesh perfectly if and only if their pitch circles just touch. Hence, our criteria for an ideal mesh are
  • Pin = Pout
  • X = Rin + Rout, or equivalently, 2 X = Din + Dout
where X is the axle separation between the meshing gears, and the indices "in" and "out" point to the input and output gears, respectively.

From these 2 simple equations come many useful relationships that apply to LEGO® and real-world spur gears alike. Thanks to their involute tooth profiles, LEGO® gears also obey the Fundamental Law of Gearing, which requires that the angular velocities of both gears in a meshing pair remain constant throughout the mesh.

The photo below shows a number of gear pairs meshing at ideal separations. The black-red axle separation is 16 mm or 2 LU. Consistent with the Rule of Sixteens below, the sum of the teeth on each meshing pair is 32. The black-yellow and red-yellow axle separations are both 24 mm or 3 LU, and the sum of the teeth on each meshing pair is 48.






Part 2. Using gears — a LEGO® perspective


Gear roles: Drivers, followers, and idlers
In a meshing gear pair (2-gear train), the powered input gear, with tooth count Zin, is called the driver, and the output gear, with tooth count Zout, the follower or driven gear. By convention, the pair's tooth ratio is defined as Qz ≡ Zin / Zout.

If a 3rd gear on its own unpowered axle is inserted between the input and output gear (say, to relay power across a gap between their axles), the 3rd gear is called an idler. As we'll see below, one can insert any number of idlers between an input and output gear without changing the final ratio of the entire gear train, but each added idler will flip the output's direction of rotation relative to the input's.

Hence, idlers have at least 2 uses — (i) to bridge gaps between input and output axles, and (ii) to the control output rotation direction without affecting the final gear ratio.


Gear ratios
Since the pitch circles in a meshing pair roll on each other without slipping by design, it's easy to show that the speed ratio Q ≡ ωout / ωin = -Zin / Zout = -Qz, where the gear speeds ω are both measured in either radians/sec or RPM. The minus sign indicates that the meshing gears rotate in opposite directions. The speed ratio Q is also known as the velocity ratio or gear ratio.

In a gear train with multiple meshing pairs, the final gear ratio Q = ± (Π Zdrivers) / (Π Zfollowers), where ΠZdrivers is the product of the tooth counts on all of the drivers, and ΠZfollowers is the product of the tooth counts on all of the followers.

Consider a 3-gear train with a single idler. Since the idler is a follower WRT the input gear and a driver WRT the output gear, its tooth count Zidler appears in both numerator and denominator and therefore cancels out of the final speed ratio like so:

Q ≡ ωout / ωin = (-Zin / Zidler) * (-Zidler / Zout) = +Zin / Zout = +Qz

Hence, the idler's only effect on the final speed ratio in this example is a change in sign manifest as a change in the output gear's direction of rotation.

NB: Each idler inserted between an input and output gear flips the latter's rotation direction, but the absolute value of the final speed ratio remains unchanged, no matter how many idlers there might be.

A gear reduction or simply reduction occurs when Zout > Zin. The name reflects the fact that the speed of the output shaft is reduced by a factor of Zin / Zout < 1 relative to that of the input shaft. The output shaft is then said to have been "geared down".

A gear train whose final result in a decrease in shaft speed is a reducer or reduction gear. Reduction ratios are commonly expressed in |ωout / ωin|:1 format. Hence, when an 8s gear drives a 24s, the reduction ratio is written "3:1".

Conversely, when Zout < Zin, output shaft speed goes up by a factor of Zin / Zout > 1. The output shaft is then said to have been "geared up". A gear train whose final result in an increase in shaft speed is an overdrive. Overdrive ratios are usually expressed in 1:|ωin / ωout| format. Hence, when a 24s gear drives an 8s, the overdrive ratio is written "1:3". The 1:1 case is sometimes referred to as direct drive.

The next photo shows a 2-stage overdrive transmission in a no-frills propeller-driven cart. The final speed ratio Q ≡ ωout / ωin = (-36 * -36) / (12 * 12) = 9, with motor and prop both turning in the same direction.



Final ratios higher and lower than 1:9 resulted in lower top cart speeds.



Torque-speed trade-off
The gearing biz operates under an inescapable trade-off between torque and speed. The reason is simple: Rotary mechanical power is the product of torque and angular speed (measured in radians/sec), and that product is conserved from shaft to shaft in a lossless gear train. Mathematically,

Tout * ωout = Tin * ωin

where T is torque and ω is angular speed.

This ideal relationship holds quite well in ligthly loaded LEGO® gear trains with good axle support and few meshing gear pairs. In such cases, transmissions that reduce speed at the output shaft necessarily increase torque there by the same factor. The opposite is true of overdrive transmissions. The equation overestimates output torque when losses due to friction and deformation of gear train components become significant.

Playing the torque-speed trade-off well can make all the difference in an MOC's performance and durability. A Technic car geared up too high in the name of speed might well end up with too little drive wheel torque to overcome small bumps or inclines. Conversely, excessive reduction of the steering motor in an effort to slow the steering mechanism to a controllable speed might well generate enough torque to tear the mechanism apart at full lock. If satisfactory compromise ratios can't be found, the car may be in for some major design changes.

The photo below shows the right flipper arm of my Advanced PackBot. The arm is mounted on the black half of the large Technic turntable near center and driven by an NXT motor (not shown) that also rotates the left flipper arm.



The NXT motor produces more torque than any other LEGO® motor but not enough to lift and flip this heavy MOC properly without a substantial reduction. Driving the 56st spur on the turntable with an 8s pinion on the NXT motor shaft yielded a 7:1 reduction — enough for the task at hand but not enough to twist the motor shaft in two. The reduction also improved control by slowing the flipper arms down to a manageable speed.

Unfortunately, factors not directly related to performance can complicate the speed-torque trade-off, and scaling is often one of them. The photo below shows one of the geared wheel hubs used to propel my Mars Science Laboratory rover. The 1.67:1 reduction shown here leaves the rover much too fast for scale. A much larger speed reduction would have been preferable, but scaling restrictions demanding the smallest hub possible took precedence over the speed issue. (The extra torque wouldn't have hurt, but with 6 driven wheels, motive power wasn't the issue.) Given the limited selection of single-bevel gears and hub mounting options available, this was the best compromise I could muster.





Matching LEGO® motors to their loads: A black art
Let me say up front that finding the gear ratios that best match motors to their loads can be something of a black art, both in the LEGO® realm and beyond.

For fixed loads (e.g., a weight to be hoisted), the short answer is deceptively simple: You just fiddle around with the choice of motor and final gear ratio until the total load (including internal losses under the external load) slows the motor's shaft speed ω (measured RPMs or radians/sec) falls to ω0 /2, where ω0 is the motor's no-load shaft speed.

The photo below shows the turret rotation mechanism in my Little Black Tank. A largely hidden old red 9V micromotor at left drives black 20-tooth gear above, which in turn drives the 28-tooth gear ring on the small Technic turntable at right. This match-up was sheer luck. I had no other options for a turntable at this scale, but the resulting 1.4:1 reduction turned out to match the micromotor's low speed and torque to this role quite well.



The next photo shows the 8:1 worm drive hoist used to lift the counter-weight (CW) into cocked position in my floating-arm trebuchet. (The metal parts at bottom right are part of the CW assembly.) The M motor powering the hoist has a no-load shaft speed ω0 ≅ 300 RPM at the 7.2V delivered by a PF AAA battery box loaded with NiMH rechargeables. The external load on the hoist — the CW — is limited to ~0.5 kg by the strength of the trebuchet's throwing arm. To minimize the time required to cock a 0.5 kg CW, I'd have to adjust the hoist gearing to get a shaft speed ω ≅ ω0 / 2 ≅ 150 RPM. If the resulting cocking time wasn't fast enough for my taste, I might try an L motor instead, but I'd almost certainly have to re-optimize the gearing at that point.



My focus on the ω ≅ ω0 / 2 goal here reflects 2 facts: (i) All DC electric motors, and hence all LEGO® motors, deliver maximum mechanical power when ω ≅ ω0 / 2. (ii) All that really counts in an MOC is the mechanical power actually delivered to the external load after internal losses like friction, vibration, frame flexure, and other transient structural distortions. Hence, once you get acceptable performance at ω ≅ ω0 / 2 for the chosen motor, you're done with both motor selection and gearing.

Problem is, getting accurate shaft speed measurements with and without loads can be quite difficult. In a stationary MOC like the floating-arm trebuchet, an inexpensive hand-held laser tachometer like the one shown in the photo below can make quick work of the measurement part. (The bright strips on the 2x2 round bricks on the speedboat's motor shafts are pieces of timing tape.) However, the fiddling part can still be quite time-consuming — especially if you're juggling both the motor choice and the gearing.



The challenges escalate dramatically when the load itself becomes a variable, as in motorized boats with a number of different props to choose from. (Some props will load the motors more than others, even when the hull resistance is fixed.) There are many more options to juggle in such cases, and the problem is only compounded by the fact that the motor shaft has become a moving target for the tachometer.

Optimizing motor-gearing-prop combinations for the marine geology research vessel shown below and several motorized speedboats under development at the same time brought all these problems to the fore. (Some of the gory details can be found toward the bottom of the research vessel page. One of the speedboats appears two previous photos.) The speedboats were especially tough, because hull design options introduced additional variables. I can't recall a more frazzling technical LEGO® experience, but at least I'm in good company. Naval architects who match props to gear trains to engines to hulls for a living in real boats consider it a frustrating black art to this day, even with modern tools like computational fluid dynamics at their disposal.



Ready access to LEGO® motor characteristics — especially no-load shaft speed, stall torque, and peak mechancial power at 9V (or 7.2-7.4V if you're using rechargeable batteries) — is extremely helpful here, and there's no better place to go for everything you ever wanted to know about the subject than LEGO® 9V Technic Motors compared characteristics by Philippe "Philo" Hurbain, the undisputed guru WRT LEGO® motors and all other LEGO® electricals.




Part 3. Useful implications for LEGO® gearing

The universal pitch shared by all LEGO® gears
We know from experience that all LEGO® spur gears mesh well when properly separated, so all must have the same pitch P = Di / Zi = 2 Ri / Zi for every index i pointing into the list of LEGO® spurs gears. Hence, finding the value of D for one gear pair suffices to determine P for all LEGO® spur gears, and for that I chose a pair of 24s simple spurs.



It's easy to build a mock-up showing that these gears mesh perfectly at axle separation X = 3 LU * 8 mm/LU = 24 mm, where "LU" stands for LEGO® unit (aka "stud"). Since Rin = Rout here, we must have Rin = Rout = 24 mm / 2 = 12 mm, or Din = Dout = 24 mm. Hence, P = D / Z = 24 mm / 24 = 1 mm. (The "*" here denotes multiplication.)

Turns out that all LEGO® gears have this same pitch of P = 1 mm, regardless of type. In theory, then, all can be meshed if the axle separations and angles are right.

I have no doubt that LEGO® engineers chose this universal pitch quite intentionally, as it greatly expands the utility of LEGO® gears and simplifies their use as well.



The Rule of Sixteens and other useful shortcuts
The fact that P = 1 mm universally has several valuable practical implications for spur gear pairs.

NB: From here on out, the "∑" symbol will indicate summation, so that ∑R = Rin + Rout, ∑D = Din + Dout, and ∑Z = Zin + Zout.

The most important of these implications are as follows:
  • Pitch diameter rule: The pitch diameter D of any LEGO® gear is just its tooth count in millimeters, because D = Z * P = Z * 1 mm.

  • The outside diameter (OD — i.e., the diameter between tooth crests) of any LEGO® gear is 2 mm greater than its pitch diameter.

  • Average rule: The ideal X in millimeters is the average number of teeth in millimeters, because X [mm] = ∑R = ∑D / 2 = P * ∑Z / 2 = average(Zin, Zout) * 1 mm.

  • The ideal X in LU (studs) is just ∑Z in millimeters divided by 16, because X [LU] = X [mm] * 8 mm/LU = P * ∑Z / 2 = P * ∑Z / 16.

  • Rule of Sixteens: Whole-LU ideal axle separations occur only when ∑Z is a multiple of 16 — specifically, {16, 32, 48, 64, 80, 96}.
The Rule of Sixteens and the Average Rule are arguably the most useful tools this tutorial has to offer.

The photo below illustrates the 2 mm difference between the 42 mm OD of the 40s simple spur and the 40 mm pitch diameter given by D = P Z. Likewise, all 3 24-tooth gears have ODs of 24 * 1 mm + 2 mm = 26 mm, and that's why none of them fit in 3 LU-wide spaces — e.g., within the 5x7 and 5x11 Technic frames in the short direction. One of the more endearing qualities of the 20ds double-bevel (22 mm OD) is that it does fit in such spaces.





Gearing on the whole-LU grid
When laying out gear trains with multiple parallel axles in studless setting, it's often useful to imagine a discrete rectangular (orthogonal) grid or coordinate system with points — i.e., pin holes — centered at 1 LU intervals in 3 perpendicular directions. We'll call such a 3D array a whole-LU grid. If all of your axle separations match up with holes paralleling an edge (axis) of the 3D grid, then you're gearing on the grid. As we'll see below, gearing off the grid has many important uses, but most MOCs are based largely if not entirely on this straightforward on-grid approach, mine included.



The photo above shows three 2D pin hole arrays. The center array is embedded in a raft of 7 alternating DBG and LBG 1x7 liftarms arranged in parallel without gaps. Think of the pin holes as defining one face of a 3D whole-LU grid. The rows and columns of holes on this face run parallel and perpendicular to the liftarms, respectively. On-grid axle separations on this face span two holes in the same row or column. The axles themselves run perpendicular to the face — i.e., in the 3D grid's 3rd dimension.

NB: It's essential to distinguish between (i) the outside dimensions of pin hole-bearing LEGO® parts and (ii) the range of on-center hole-to-hole distances they provide. For example, the maximum axle separation on a 1x7 LU liftarm is 6 LU, so what we really have here is a 7x7 raft bearing a 6x6 whole-LU grid of potential axle separations. Bottom line:

When counting out pin hole separations on the grid, you must start counting at zero!

Studless MOC frameworks are usually built on 3D whole-LU grids. In static MOCs and many moving MOCs at rest on level surfaces, one grid face is typically horizontal and the other two are vertical and perpendicular to each other.

My Advanced PackBot (APB) is a case in point. When at rest with its flipper tips off the ground, the top of its embedded NXT is horizontal. The frame is built on a 3D grid with faces paralleling the 3 sides of the NXT. As shown in the 2 photos below, this grid can assume almost any orientation in space in operation. The important thing is that the grid moves with the APB.





In studless constructions, every pin hole in the 3D whole-LU grid is potentially available for axle mounting, but that's rarely if ever true for studded structures. In the simple stack of 6 studded Technic beams on the right in the photo at the top of this section, the only holes still on the grid face defined by the studless liftarm raft in the center are those in the top and bottom beams. All the rest are off-grid. I inserted pairs of plates between the beams on the left to bring the beam holes into alignment with every other row of holes in the liftarm raft. All these holes are on the whole-LU grid face defined by the raft, but half of the on-grid holes are still missing.

In the photo below, the same cluster of 16s and 20ds gears found ideal or near-ideal axles separations in the liftarm raft and left beam-plate stack but couldn't be mounted on the right beam-only stack. Both of the {16s, 16s} separations in each cluster are ideal and on-grid. The diagonal {16s, 20ds} separations are neither, but they're very close to ideal, and, practically speaking, their meshes are just as good. Note that the gear cluster works on the left beam-plate stack only because none of the required axle separations span an odd number of grid face rows.

I could have mounted a {16s, 16s} pair on any row of holes in the right beam-only stack, but that's it. The right beam stack has its uses nonetheless. For example, it alone provided a workable separation for the {36ds, 40s} gear pair shown. The mesh is actually quite good.

Studded beam-plate stacks open up many axle separation possibilities not easily accommodated in purely studless constructions. Mixed studless and studded structures open up even more, but the grid concept remains a useful mental guide in all these cases. The reason is simple: A full third of all possible LEGO® spur gear tooth count pairs (Z-pairs) have whole-LU ideal axle separations, as we'll see in the next section.





The Rule of Sixteens on the whole-LU grid
Recall the list of tooth counts occuring at least once among LEGO® spurs: {8, 12, 16, 20, 24, 28, 36, 40, 56}. I'll now step through all the relevant multiples of 16 — namely, {16, 32, 48, 64, 80, 96, 112} — to see which available Z-pairs have whole-LU ideal axle separations Xideal according to the Rule of Sixteens.

The Z-pair concept is quite useful, as it focuses on the tooth counts (Zin and Zout) involved without regard to gear type. Since all LEGO® gears have the same pitch P = 1 mm, ideal separation and speed ratio depend only on Zin and Zout for any spur gear pair. Gear type has nothing to do with it. A single Z-pair often stands for several actual LEGO® spur gear pairs. For example, the {8s, 24s}, {8s, 24cs}, {8s, 24sc}, and {8s, 24sf} gear pairs all fall under the {8, 24} Z-pair.

For consistency, I've written each Z-pair below with the smaller tooth count first and calculated its ratio as a reduction ratio given by (Zlarger / Zsmaller):1.
  • ∑Z = 16: The only available Z-pair with Xideal = 1 LU is {8, 8}. Its ratio is of course 1:1. (Beware of this one: It's weak and introduces lots of backlash.)

  • ∑Z = 32: The available Z-pairs with Xideal = 2 LU are {8, 24}, {12, 20}, and {16, 16}. Their reduction ratios are 3:1, 1.67:1, and 1:1, respectively.

  • ∑Z = 48: The available Z-pairs with Xideal = 3 LU are {8, 40}, {12, 36}, {20, 28}, and {24, 24}. Their reduction ratios are 5:1, 3:1, 1.4:1, and 1:1, respectively.

  • ∑Z = 64: The available Z-pairs with Xideal = 4 LU are {8, 56}, {24, 40}, and {28,36}. Their reduction ratios are 7:1, 1.67:1, and 1.29:1, respectively.

  • ∑Z = 80: The available Z-pairs with Xideal = 5 LU are {24, 56} and {40, 40}. Their reduction ratios are 2.33:1 and 1:1, respectively.

  • ∑Z = 96: The only available Z-pair with Xideal = 6 LU is {40, 56}. Its ratio is 1.4:1.

  • ∑Z = 112: The only available Z-pair with Xideal = 7 LU is {56, 56}. Its ratio is 1:1.
In descending order, the list of reduction ratios occurring at least once above is {7, 5, 3, 2.33, 1.67, 1.4, 1.29, 1}:1.

A total of 45 unique Z-pairs can be drawn from the list {8, 12, 16, 20, 24, 28, 36, 40, 56} of tooth counts occuring at least once among available LEGO® spur gears. Of these, 15 were just found to have ideal separations of 1-7 LU on the whole-LU grid: {{8, 8}, {8, 24}, {8, 40}, {8, 56}, {12, 20}, {12, 36}, {16, 16}, {20, 28}, {24, 24}, {24, 40}, {24, 56}, {28,36}, {40, 40}, {40, 56}, {56, 56}}.

In other words, a full third of all possible LEGO® spur gear Z-pairs have whole-LU ideal axle separations.

Several practical points are worth noting here.
  • You just can't get even-numbered reductions like 2:1, 4:1, or 6:1 with spur gear pairs at whole-LU axle separations.

  • The 16s gear meshes ideally only with itself at whole-LU separations.

  • The 7:1 reduction offered by the {8, 56} Z-pair is the largest you can get from a pair of LEGO® gears. As noted above, the high-torque flipper arms on my Advanced PackBot took advantage of that fact.

  • Practically speaking, 3 LU separations are often harder to arrange than 2 LU and 4 LU separations for a variety of reasons.

  • The total mounted width W of a Z-pair is W = ∑Z + 2 in millimeters, or just over 2 X in LU. Hence, Z-pairs with X > 3 LU are going to take up a lot of precious space in your MOC.



Over- and under-separated axles: Backlash and skipping and friction! Oh, my!
The photo below shows two 24s simple spurs meshing at ideal separation. As you can see, there's a lot more wiggle room for lengthening the separation than there is for shortening it. Actually, there would have been even less room for shortening if these axles had been properly supported. Axle separations under ideal by as little as 0.3 mm can lead to binding, excessive frictional losses, and accelerated wear. In short (pun intended), under-separated axles are likely affect your MOC's performance in intolerable ways.



Though often less problematic, over-separation of axles also comes at a price. The involute tooth profiles on most gears both within and beyond the LEGO® realm are designed to continue to run smoothly just above ideal separation. LEGO® spurs, for example, run smoothly at separations up to 1 mm above ideal. (At 2 mm above ideal, of course, their teeth don't overlap at all.)

Smooth operation, however, isn't the only thing at stake here. Over-separation has 2 potentially significant downsides: (i) Increased backlash, and (ii) reduced torque capacity.

In this context, backlash is the angle a gear can rotate without moving another gear meshed with it. Backlash can be a big problem in some MOCs — especially when reversible precision motions are required. The smaller the gears in your gear train, and the larger the number of gears, the more backlash the gear train as a whole will have — even with ideal axle separations throughout. Over-separated axles only exacerbate the problem.

Torque capacity, on the other hand, is the maximum amount of torque that can be transmitted from input to output axle without causing the gear teeth in between to skip, bend, or break. If you stick with technical LEGO® long enough, you'll eventually discover that it's all too easy to build large, motorized MOCs plagued by gear skipping — even if all the axle separations are ideal. ABS plastic, after all, is only so strong. Over-separated axles make that worse, too, but beefing up axle support can counteract that effect to some extent.

In short, under-separating axles carries a risk of excessive friction and wear, while over-separating them adds to the gear train's total backlash and susceptibility to skipping under high loads.



Gearing on whole-LU grid diagonals
There's nothing wrong with gearing off the grid of whole-LU axle separations in perpendicular directions. In fact, it opens up an array of gear ratios that would otherwise be very difficult to obtain. It just takes more effort to put together a workable set of gears and pin holes and avoid excessive under- and over-separations at the same time.

One useful way to gear off the grid is to stick with pin holes on the whole-LU grid but use axle separations diagonal to it. The geometry is that of a right triangle with an axle at each end of its hypoteneuse. The photo below shows {8, 28}, {12, 24}, and {16, 20} Z-pairs mounted along hypoteneuse of an 8 x 16 x 17.89 mm (1 x 2 x 2.24 LU) right triangle. From the Average Rule, all 3 Z-pairs have the same ideal separation of X = average(Zin, Zout) * 1 mm = 18 mm. In each case, the actual 17.89 mm separation is close enough to ideal to provide an excellent mesh. The {8, 28}, {12, 24}, and {16, 20} Z-pairs yield reduction ratios of {3.5, 2, 1.25}:1, respectively.

Clearly, if you find one Z-pair that works for a particular diagonal axle separation, any other Z-pair with the same tooth average or sum will mesh equally well.



The only truly ideal diagonal axle separation using pin holes on the whole-LU grid is the 80 mm (5 LU) hypoteneuse of a 24 x 32 x 40 mm (3 x 4 x 5 LU) triangle. The corresponding Z-pairs are {24, 56} and {40, 40} with 2.333:1 and 1:1 reductions, respectively.

The hypoteneuse of a 24 x 24 x 33.94 mm (3 x 3 x 4.24 LU) triangle is a near-ideal separation for the {28, 40} Z-pair with a reduction of 1.43:1. The {8, 36} pair finds a somewhat loose but potentially useful separation on a 16 x 16 x 22.63 mm (2 x 2 x 2.83 LU) triangle with a large reduction ratio of 4.5:1. Ditto for the {36, 56} pair on a 24 x 40 x 46.65 mm (3 x 5 x 5.83 LU) triangle with a reduction ratio of 1.56:1.

The next 2 photos show a planetary geared wheel hub with a carrier based on the 8 x 16 x 17.89 mm (1 x 2 x 2.24 LU) right triangle discussed above. The black compound wheel — a pair of hard plastic wheels with small cleats and flanges (64712) joined nose to nose — can take a 94.8 x 44 R balloon tire (54120).





Each half of the wheel has a central pin hole and a 48si simple internal spur gear or annulus inside its cleated rim. The M motor drives the central 24s sun gear, which in turn drives the two 12ds planet gears. The planets are mounted on the 2-arm DBG carrier, which in turn mounts on the vehicle's suspension or frame.

Of the 3 Z-pairs compatible with the carrier's 17.89 mm sun-planet axle separation, however, {12, 24} is the only one also meeting the planetary meshing criterion Zsun + 2 Zplanet = Zannulus. In this "fixed-carrier" configuration, the planet gears serve only as idlers relaying drive from the sun gear to the wheel via the latter's 48si annulus. The wheel turns freely on 8L sun axle seen in the upper photo. This planetary gear train yields a 2:1 reduction at the hub.



Gearing on the half-LU grid
Another useful off-grid technic involves placing one of the axles of a gear pair on the whole-LU grid, and the other, an odd-numbered multiple of 4 mm (O.5 LU) away in a direction parallel to the grid rows or columns. For lack of a better term, I refer to this tactic as gearing on the half-LU grid. Ideal axle separations then occur when ∑Z is an odd-numbered multiple of 8 — specifically, {24, 40, 56, 72}.

Many available Z-pairs fill the bill here:
  • ∑Z = 24: The available Z-pairs with X = 1.5 LU are {8, 16} and {12, 12}. The reduction ratios are 2:1 and 1:1, respectively.

  • ∑Z = 40: The available Z-pairs with X = 2.5 LU are {12, 28}, {16, 24}, and {20, 20}. Their reduction ratios are 2.33:1, 1.5:1, and 1:1, respectively.

  • ∑Z = 56: The available Z-pairs with X = 3.5 LU are {16, 40}, {20, 36}, and {28, 28}. Their reduction ratios are 2.5:1, 1.8:1, and 1:1, respectively.

  • ∑Z = 72: The available Z-pairs with X = 4.5 LU are {16, 56}, and {36, 36}. Their reduction ratios are 3.5:1 and 1:1, respectively.
In descending order, the list of reduction ratios occurring at least once above is {3.5, 2.5, 2.33, 2, 1.8, 1.5, 1}. Note how little overlap there is with the corresponding list for whole-LU axle separations: {7, 5, 3, 2.33, 1.67, 1.4, 1.29, 1}. Notably absent from both lists are the 6:1 and 4:1 ratios.

As noted above, a total of 45 unique Z-pairs can be drawn from the list {8, 12, 16, 20, 24, 28, 36, 40, 56} of tooth counts occuring at least once among available LEGO® spur gears. Of these, we just found that 10 are on the list of Z-pairs with ideal separations only on a half-LU grid. These are {{8, 16}, {12, 12}, {12, 28}, {16, 24}, {16, 40}, {16, 56}, {20, 20}, {20, 36}, {28, 28}, {36, 36}}.

Another 15 Z-pairs are on the corresponding Rule of Sixteens list for the whole-LU grid: {{8, 8}, {8, 24}, {8, 40}, {8, 56}, {12, 20}, {12, 36}, {16, 16}, {20, 28}, {24, 24}, {24, 40}, {24, 56}, {28,36}, {40, 40}, {40, 56}, {56, 56}}. This list doesn't overlap with the one above.

The combined list for Z-pairs with ideal separations on a whole- or half-LU grid accounts for 25 of the 45 possible Z-pairs for spurs: {{8, 8}, {8, 16}, {8, 24}, {8, 40}, {8, 56}, {12, 20}, {12, 28}, {12, 36}, {16, 16}, {16, 24}, {16, 40}, {16, 56}, {20, 20}, {20, 28}, {20, 36}, {24, 24}, {24, 40}, {24, 56}, {28, 28}, {28,36}, {36, 36}, {40, 40}, {40, 56}, {56,56}}.

The 20 remaining Z-pairs with no ideal separations on the half- and whole-LU grids are {{8, 12}, {8, 20}, {8, 28}, {8, 36}, {12, 16}, {12, 24}, {12, 40}, {12, 56}, {16, 20}, {16, 28}, {16, 36}, {20, 24}, {20, 40}, {20, 56}, {24, 28}, {24, 36}, {28, 40}, {28, 56}, {36, 40}, {36, 56}}.

Ideal axle separations for all of them could be found on a quarter-LU grid if you were so inclined. As we saw above, 7 of them — namely, {8, 28}, {8, 36}, {12, 24}, {16, 20}, {28, 40}, {36, 56}, and {56, 56} — find near-ideal or somewhat loose but potentially useful separations on whole-LU grid diagonals. Many if not most of the rest — e.g., {8, 20}, {12, 16}, {16, 24}, {24, 40}, {24, 56}, {28, 36}, {28, 56} — can be used if diagonal and half-LU approaches are combined.

The next 3 photos show different ways to achieve half-LU ideal axle separations with studless and studded constructions.







As shown in the photo below, combining the diagonal and half-LU approaches on an 8 x 12 x 14.42 mm (1 x 1.5 x 1.80 LU) triangle provides slightly loose but useful separations for 2 of the remaining 13 Z-pairs incompatible with both whole-LU and half-LU grids — namely, {8, 20} and {12, 16}.



The {8, 20} pair can be used in a planetary gear train with the 48si annulus shown in a previous example. Here, the 8s serves as sun, and the 20ds as the planets. The {12, 16} pair doesn't meet the Zsun + 2 Zplanet = Zannulus planetary meshing criterion.

It may well be possible to put some of the 20 Z-pairs with neither half- nor whole-LU ideal separations to use in studded settings as well, but I haven't looked into that.



Bevels and crowns: Think folded grid
Up till now, we've dealt with spur gears running on parallel axles. Many MOCs and real-life machines need nothing else. But the need to transfer rotary power between non-parallel axles comes up quite often in a variety of settings, and that's where bevel and crown gears come into their own.

Case in point: The photo below shows the rear undercarriage of my Rescue Truck make-over. The 24s spurs at far right belong to the truck's transmission, which is located just aft of the motor (not seen). The longitudinal black propeller shaft (right center) transfers power from the closest of the transmission's 24s spurs at right to the 24cs crown-spur hybrid near the center.

The 24cs is functioning as a crown here, driving the 16s spur on an axle perpendicular to its own. The 16s is just an idler relaying drive to the 16sf spur on the body of the differential, which in turn relays power to the drive wheel axles via the trio of small 12-tooth single bevels inside. (The 12ds double-bevel seen to the left of the differential isn't part of the drivetrain.)



Crowns and bevels are again seen working together in the undercarriage shot of my BFT-9000 Haul Truck below. The gear trains at center power linear actuators used to raise and lower the dump body at right, here seen in raised position. The M motor at center drives the DBG 8s spur to its right via a longitudinal shaft. The 8s in turn drives the flanking DBG 24s spurs, which in their turn drive the LBG 24cs crown-spur hybrids via hidden 8s spurs.

The crowns share transverse axles with the LBG 12-tooth single bevels near the centers of the 5x7 Technic frames. These bevels then mesh with hard-to-see black 12ds double bevels on the linear actuator input shafts. In all, power is transmitted through 2 right angles here. The final reduction from motor to linear actuator shaft is 9:1.



The next photo shows 12-tooth and 20-tooth single- and double-bevels meshing on perpendicular axles at ideal separations. Note that ideal axle separation now depends on both the Z-pair and the gear thicknesses involved. The single- and double-bevels are respectively 8 mm = 1 LU and 4 mm = 0.5 LU thick.

Going around a 90° corner with bevels generally adds 12-16 mm (1.5-2 LU) to the ideal separation given by the Average Rule, which implicitly assumes a flat grid and is based on Zin and Zout alone.



The photo below shows the {12ds, 36ds} bevel-spur hybrids functioning as bevels rather than spurs at top. At bottom is a 20ds double-bevel engaging the 28csf crown-spur hybrid on the original Technic differential body. The differential is locked, and the 28scf is functioning as a crown with a reduction ratio of 1.4:1. Both gear pairs are on perpendicular axles at ideal separations. I'll have more to say about bevel and crown axle separations below.



The next photo shows the 28-tooth single bevel on the body of the latest Technic differential. As shown, this 3L differential fits inside a 5x7 Technic frame, whereas the 4L original Technic differential does not.

The central block in the new differential has its good and bad points. On the plus side, it holds the internal single-bevels in place when its axles are removed. On the minus side, it doesn't allow the new differential to be locked by running an axle all the way through it.



The photo below shows 3 different double- and single-bevel combinations mounted inside specialized Technic pieces designed for that purpose. The reduction ratios are 1:1, 1.67:1, and 1:1 from left to right. []



The next photo shows two 24cs crown-spur hybrids functioning as crowns in ideal meshes 16s and 8s spurs. The reduction ratios are 1.5:1 and 3:1, respectively.



[bevel and crown axle separations]




Concluding remarks
This tutorial was originally intended as an introduction to the Rule of Sixteens and the Average Rule, but it soon took on a life of its own. The good news: It covers the use of spur gears in primarily studless settings fairly exhaustively. The bad news: It's definitely exhausting to read.

It could use more illustrations here and there, but it's time to move on. That said, please drop me a comment if you catch an error or have thoughts on ways to improve the presentation -- other than cutting the page in half.

At some point, I hope to post an introduction to the LEGO® epicyclic gear trains made possible by the 24-tooth annulus inside the large Technic turntable and the 48-tooth annulus inside the rim of the hard plastic wheel with small cleats and flanges (64712). Here's a sneak preview....



The planetary gear train test bed below has no external load. An original 9V Technic motor (2838, lower right) drives the LBG 3-arm planet carrier at center. The surrounding black 48si annulus is fixed to the frame. The freely spinning 16s DBG planets on the carrier in turn drive the 16s LBG sun at center. The latter is keyed to the output axle at upper left.

This fixed-annulus planetary configuration yields a final overdrive of 1:4. The smooth black wheels on the input and output shafts bear strips of reflective timing tape for laser tachometer measurements. At 9V, the motor reaches a top shaft speed of ~1,400 RPM, or ~32% of its no-load speed of ~4,400 RPM at 9V. The output shaft spins at ~5,600 RPM.

The 2838 motor packs more peak mechanical power (up to 0.9 watts at 9V) than most 9V motors but has no internal gearing and hence little torque. The planetary gear train runs very smoothly, but frictional losses keep motor shaft speed well below the target 50% of no-load speed marking the peak of the motor's mechanical power vs. speed curve.




Useful references
  • The Unofficial LEGO® Technic Builder's Guide by Pawel "Sariel" Kmiec, No Starch Press, 2013. Sariel's one of the most talented and prolific Technic builders around, period. I can't recommend this book highly enough. IMO, there should be a well-worn copy in every technical LEGO® junkie's workshop, regardless of experience or skill level. You can also learn a great deal from Sariel's well-illustrated web site and countless YouTube videos.

  • LEGO® 9V Technic Motors compared characteristics by Philippe "Philo" Hurbain, the undisputed guru of everything electrical in the LEGO® realm — especially the motors.

  • Wikpedia is a superb resource for LEGO® engineers, with good explanations of everything discussed here. You're unlikely to find a science, engineering, or math topic that isn't covered well.




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