Gears Tutorial

A complete tutorial on Lego gears, their advantages and disadvantagesas well as the basic laws of mechanics that apply to them.

When I describe my constructions or ideas, and when I explain theirfunctionality, I usually assume that readers have the basicunderstanding of mechanics and of the rules that apply to gears. Thisassumption, it seems, is sometimes wrong. Even though it may appearfrustrating at times, I see no real reason to ignore the people who havenot yet learnt how the gears work, nor to deny them the pleasure ofbuilding with Lego Technic. Having considered that, I prepared adocument in which I’ve attempted to cover my entire knowledge on gearsin an accessible manner. The tutorial you’re about to read shouldhopefully be useful both to beginners and to experienced builders. Forbetter clarity it was divided into sections.

1. Introduction to gears

What do we need gears for? A very usual answer is: to transfer thedrive from a motor to the final mechanism. It is true, but not entirelycorrect. The essential purpose of gears is to transform the propertiesof a motor to suit our purposes in the best way possible. Transferringthe drive is in fact a side-effect of this process.

Gears can be obviously used with all kinds of drive, be it anelectric motor, a manual crank, a wind turbine, a mill wheel, whatever.For the purposes of this document we assume that drive is provided by anelectric motor, because it’s a popular solution with Lego Technic, andone with constant properties that can be transformed with gears.

Every motor has its mechanical power, specific for a given type ofmotor. A number of types of Lego motor exists, some types offeringgreater power than the others. The important thing is that mechanicalpower of a motor consists of two factors: speed and torque. These arethe two properties we can transform using gears.

Speed is simply the number of rotations of a driveshaft that thegiven motor produces within a given time interval. The higher the speed,the more rotations we get. In mechanics, speed is usually measured withRPM, that is Revolutions Per Minute. One RPM means one revolution ofthe motor’s driveshaft per minute – which is really slow. Most of theLego motors offers more than 100 RPM.

Torque is the strength with which the driveshaft is rotated. Thehigher the torque, the more difficult it is to stop the driveshaft.Therefore motors which offer high torque are usually preferred to theother, because they can drive heavier vehicles or more complexmechanisms than the motors with low torque. The torque is measured inN.cm, and all we need to know is that the more N.cm, the stronger themotor.

The mechanical power is, in a certain simplification, the quotient oftorque and speed. If we increase torque and/or speed, the mechanicalpower will be increased accordingly. In fact, the torque of a motor isconstant – it can’t be changed without changing the motor’sconstruction. The speed, on the other hand, depends on the voltage atwhich the motor is powered. The higher the voltage, the higher thespeed, which allows to increase the motor’s mechanical power bymanipulating the voltage of its power supply. The official standard forLego motors is 9V voltage, which is equal to the voltage of six AAbatteries. The recently released Lego rechargeable battery provides7.4V. It means that the motors powered from the battery have lowermechanical power than the ones powered from the AA batteries, but thisis just a theory, because the voltage provided by the AA batteriesdecreases over time, and the voltage provided by the Lego batteryremains more or less constant. Some experiments are done with motorspowered at 12V, and though the motors produce higher mechanical powerunder these conditions, it should be noted that they were designed for9V, not 12V, and it may result in a fatal damage to the motors. In thisdocument we assume that all motors run at the same voltage, whether it’s9V or less. You can find an exhaustive description of the performanceof specific Lego motors here.

What do we need the speed and torque for? That is actually differentfor each mechanism. Consider a model of a sport car – we want it to belight and fast. It means that we certainly need large speed, but not thetorque, because a light vehicle requires little torque to move. Usinggears, we can transform torque into speed, or speed into torque. Thereare two very important, but very simple rules for that:

- if we drive a large gear with a small gear, we increasethe torque but decrease the speed (that is called gearing down)
- if we drive a small gear with a large gear, we increase the speed butdecrease the torque (that is called gearing up)

The best thing is that we can transform part of one property toincrease the other, we don’t need to transform all of it. In the case ofour sport car it means that we can pick a drive motor, and use thefirst of the aforementioned rules to gain extra speed at the cost ofsome needless torque. How much torque can we transform depends mainly onthe car’s weight, so it’s a different value for every model.Experienced builders can estimate the range of possible transformationknowing just the vehicle’s weight and the type of the motor used todrive it. The basic rule is: speed and torque are inverselyproportional. It means that if we lose 20% of speed, we gain 20% oftorque.

A different example would be a rail crossing barrier. We can raise orlower it with motor, but the nominal speed of any motor will beprobably too large. A barrier should take at least several seconds to befully raised or lowered, and most of the Lego motors run at more than100 RPM. We need to use gears to get rid of this needless speed, and inexchange for that we will get extra torque, which can be used to operatea longer and heavier barrier. In this case, we use the second of theaforementioned rules.

Now that we know what gears can do, let’s have some theory.

2. Basic rules

In the first section we have learned the two rules of transferringtorque into speed or speed into torque. We know what to use the gearsfor, and now we will learn how to use them. We will need a number ofnotions for that.

We can talk about using gears to transform motor’s properties whenthere are no less that two gears meshed, each set on a separate axle.The gear that is closest to the motor is called a driver gear. The gearthat receives the drive from it is called a follower gear. On thediagram below the driver and follower gear are marked green and redrespectively.

Almost every mechanism has its driver and follower gear. In everypair of meshed gears there is a driver gear and a follower gear. Itshould be sufficient to remember that the driver gear is the one thedrive is transferred from, and the follower gear is the one the drive istransferred to.

As you may have noticed, on the diagram above axles are marked withthe same colours as the gears. That is because we can talk about axlesin the same manner in which we have just described the gears. In fact,many mechanisms have covered or hidden gears but clearly visible axles,so this approach is often more convenient. In this case we call the axlewith the driver gear (green) an input axle, and the axle with thefollower gear (red) an output axle. That’s it: input and output, justlike the driver and follower. Most of the mechanisms have usually asingle input axle (because it’s difficult to drive many input axles witha single motor), but there are multiple output axles possible. Thepopular differential mechanism is a good example of one input / manyoutputs solution:

It doesn’t end just with the driver gear and follower gear: we havealso an idler gear. If there is a number of gears meshed one by one,then only the first one is the driver gear and only the last one is thefollower gear. All the gears in between are called idler gears, andthat’s because they could as well not exist. Their presence does notaffect how the torque and speed are transformed: only driver andfollower gear determine that.

On the diagram above the large gray gear is meshed with driver gearat one side and with follower gear at the other. This is typical foridler gears: being meshed with many gears at the same time. Idler gearsare usually meshed with two gears at the same time, while the driver andfollower gear are only meshed with one. This is an easy way to identifythe idler gears, but there are exceptions.

The diagram above shows two sets of gears. The left set contains adriver gear, a follower gear and two gears in between, each meshed with asingle gear only. These two gears are set on the same axle, which meansthat they can be idler gears (not possible if they had separate axles),and they are of the same size, which means that they surely are idlergears. That is because many gears of the same size set on the same axlealways act like a single gear – no matter whether there are 2 gears or200. The right set also contains a driver gear, a follower gear and twogears in between, except these two gears are of different size. If theyhave different sizes while sharing the same single axle, they can’t beidler gears. That is because the difference in their sizes affects howthe torque and speed are transformed between the driver gear and thefollower gear. More precisely, the size of a gear affects the torque ittransfers – we see that the gears share the same single axle, so theirspeeds must be equal, but their sizes are clearly different.

With this classification in mind, we can now have an exact look atthe types of Lego gears.

3. Types of gears

Lego has released plenty of various types of gears in the history ofTechnic line. Below is the list of the ones that are still in use:

As you can see there are 13 classic, round gears, and there is onespecial gear called a worm gear. Moreover, the round gears can bedivided into two groups: the regular ones with square teeth, and thebevel ones with rounded teeth. Practically any gear of the first groupcan be used with any gear of the second group. The unique property ofthe bevel gears is that they can be meshed in both parallel andperpendicular manner. They are also more convenient to use with liftarmsbecause of their size. However, they are not suitable for use with theLego chain.

Let’s have a short description of each gear on the list (bevel gearshave the word bevel in their names):

8 teeth gear – the smallest gear currently beingproduced, and a very fragile one. It’s not suited for high torque, butvery popular, especially for gearing down (being the smallest, it isobviously the most efficient at it). There are at least three differentvariants of this gear, and the most sought for one is reinforced byextra layer of plastic around the axle, between the teeth.

12 teeth gear (a single bevel one) – the smallest bevel gear currently being produced. It’s not reallyuseful for gearing down or up, but irreplaceable with differentialmechanisms and very popular when there is a need to transfer the drivein a perpendicular manner inside a limited space. Easily broken underhigh torque, which led to complete absence of differentials in e.g. sometrial trucks.

12 teeth gear (a double bevel one) – the smallestdouble bevel gear currently being produced. It’s much stronger than itssingle bevel counterpart, and is most usually used together with a 20teeth double bevel gear.

14 teeth gear – the predecessor of the 12 teethsingle bevel gear. It was the first gear designed specifically fordifferential mechanisms, but proved so very fragile that it was laterreplaced by the 12 teeth version. It is no longer used in the officialLego models and is unpopular with builders.

16 teeth gear (a regular one) – a reasonably strongand useful gear. This is the smallest gear that can be operated withLego chain, and a popular one thanks to its convenient size.

16 teeth gear (with clutch) –  available almostexclusively in dark gray, a gear designed specifically for gearboxes.It’s weaker than the regular version and doesn’t work well with Legochain (it has a tendency to slip on it because of shorter teeth).Instead, it has the unique ability to be engaged or disengaged by thetransmission driving ring. Without the ring, it remains loose on theaxle, but it can be meshed with an old-type halfbush (the one withteeth) and thus get fixed to the axle.

20 teeth gear (a single bevel one) – larger versionof the 12 teeth single bevel gear. Rare and not really popular becauseof its thin body which makes it snap under high torque.  Usually meshedwith a 12 teeth double bevel gear or 20 teeth double bevel gear.

20 teeth gear (a double bevel one) – very popular,strong and reliable gear. Most commonly used together with a 12 teethbevel gear, but useful in different setups too.

24 teeth gear (a regular one) – another popular,strong and reliable gear. There are at least three different variants ofthis gear, the newest ones being the strongest ones. One of the mostuseful gears ever.

24 teeth gear (with clutch) – a specific version ofthe 24 teeth gear, not related to the 16 teeth gear with clutch. It’salways white and dark gray in the middle, and it has the unique abilityto harmlessly slip around the axle if a sufficiently high torque isapplied. It makes it a very useful and sought for gear, although a rareone. Most usually it is used for end-to-end applications, that isapplications when motor can only run until it reaches a certain point.This includes for instance almost all steering mechanisms, where thewheels can be turned only by a limited angle, or the aforementionedrailroad barrier mechanism, where the barrier can be only raised orlowered to some degree. In this type of mechanisms this gear slips whenthat end point is reached, so that the motor can continue to run whilethe mechanism is stopped. Another example are winches in the officialLego sets with motorized winches (e.g. 8297), where this gear is used tomake sure that motor doesn’t get damaged when the end of winch’s stringis reached. Please note that this gear slips under a very specificamount of torque – and in most cases you will want it to slip only underextremely high torque (e.g. to make sure that the steering mechanismsstops turning when the end point is reached, not when a wheel meets anobstacle). This can be achieved by using this gear right after thedriver gear:

Some sources indicate that the internal construction of this gear hasbeen changed over the years, but this is not confirmed.

24 teeth gear (with crown) –  a really old design,the first gear among the regular gears which could be meshed in aperpendicular manner. Again, there are at least three variants of thisgear, the older and weaker ones gradually replaced with never andstronger versions. The arrival of bevel gears made it one of thecurrently most unpopular gears; it’s weak and inconvenient to use.Still, it can be sometimes useful due to it’s unusual shape.

Worm gear – a gear with a number of uniqueproperties. Firstly, it can be only used as the driver gear, never asthe follower gear. It comes in handy for mechanisms that need to e.g.lift something up and keep it lifted; in this case worm gear acts like alock that keeps the desired part of mechanism lifted without puttingits load on the motor. There is a lot of possible applications for thisworm gear’s property, for instance many types of cranes or forklifts,railroad barriers, drawbridges, winches, and basically every mechanismthat needs to keep something steady once the motor stops.

Secondly, the worm gear is extremely efficient for gearing down. Itis theoretically 8 times more efficient that the 8 teeth gear, becauseevery revolution of the worm gear rotates the follower gear by just asingle teeth. Therefore worm gears are used for gearing down wheneverthere is a very high torque or low speed needed and there is littlespace to use.

Finally, as the worm gear rotates, it has a tendency to push againstthe follower gear and slide along its own axle. Usually this tendencyhas to be stopped by a strong casing around the worm gear, but there arecertain mechanisms that use it to move worm gear from one place toanother, for instance my pneumaticautovalve or my automated trafficators system.

The worm gear can be used with all the listed gears. The most commonuse is to mesh it with a 24 teeth gear:

But it can be easily used with any other gear. You can see someexamples of worm gears enclosed with follower gears inside strongcasings here. With proper spacing, it can be used with bevelgears too:

On the diagram above, there are two 20 teeth double bevel gears used.But it can be just a single double bevel gear, or two single bevelgears, or even a single single bevel gear. It’s even possible to use theworm gear to drive racks, which may result in e.g. a very compact boomextending mechanism:

36 teeth gear (a double bevel one) – the largestbevel gear currently being produced, and the only one with no singlebevel counterpart. A convenient and surprisingly strong gear, but a rareone. Usually comes in black.

40 teeth gear (a regular one) – the largest regulargear currently being produced. Rarely used because of its immense size,but sometimes really useful.

That concludes the list of gears we can usually choose from (thereare some outdated gears, but they are so unique that I actually neverhad any in my hands). Now let’s see why the sizes of gears matter.

4. Gear ratios

According to Wikipedia, the gear ratio is the relationshipbetween the number of teeth on two gears that are meshed or twosprockets connected with a common roller chain, or the circumferences oftwo pulleys connected with a drive belt. We will not deal with pulleysin this document, and the ratios for sprockets connected with a commonchain are exactly the same as for the gears that are directly meshed.Hence a gear ratio is simply:

number of follower’s gear teeth : number of driver’s gearteeth

Since the spacings between each gear’s teeth are equal, counting thenumber of teeth is a simple mean of calculating the gear’scircumference. And gear ratio is basically the relationship betweengears’ circumferences.

What do we need the gear ratio for? Basically to easily calculate thefinal speed of the mechanism and the torque it provides. Consider an 8teeth driver gear and 24 teeth follower gear. We know from the section 1that this is gearing down: we gain some torque, but we loose somespeed. The gear ratio is 24:8, which is equal to 3:1. Please note thatit is a common practice to calculate ratios in such a manner that theyend with 1. Why? Because from looking at 3:1 ratio we can easily tellthat it means that the revolution’s speed is reduced three times, whichmeans that three revolutions of the driver gear / input axle result in asingle revolution of the follower gear / output axle. Since thedecrease of speed results in an inversely proportional increase oftorque, we know that torque is increased three times.

Consider an opposite example: we have a 20 teeth driver gear and 12teeth follower gear. The gear ratio is 12:20, which is equal to 0.6:1.It means that we need 0.6 revolution of the driver gear to get a singlerevolution of the follower gear. Hence we gain 40% of speed, but we lose40% of torque.

As you could have noticed, it is easy to tell gearing up from gearingdown looking at the gear ratio. If the first number of the gear ratiois greater than the second (like 3:1), this is gearing down – alsocalled a gear reduction. If the first number of the gear ratio issmaller than the second (like 0.6:1), this is gearing up – also called agear acceleration or an overdrive. If we have 1:1 gear ratio, speed andtorque remain the same, just as if we used idler gears.

We can already calculate gear ratios of two meshed gears, but what ifthere are more gears in the mechanism? In this case, we ignore all theidler gears and calculate ratios for all pairs of driver/follower gears.Then, in order to get the final gear ratio of the entire mechanism, wesimply multiply these gear ratios. Consider a mechanism from section 3,with two pairs of 8 teeth drivers and 24 teeth followers. The gear ratioof the first pair is 3:1, and so is the ratio of the second pair. If wemultiply these ratios, we get the final ratio equal to 9:1 – which istrue and accurate.

Now that we can calculate gear ratios, let’s go back to the exampleof idler and non-idler gear from section 2:

Consider the left set of gears. It consists of two pairs of gears: 8teeth driver gear with 16 teeth follower, and 16 teeth driver with a 20teeth follower (let’s assume we don’t know if there are idlers in thisset yet; we calculate ratio of each pair separately). The ratio of firstpair is 2:1, and the ratio of second pair is 1.25:1. If we multiplythese, we get the final ratio equal to 2.5:1. 2.5:1 is equal to 20:8 –that is the ratio of the first and the last gear only. As you see, theidler gears did not change the ratio at all, and this is why we canignore them.

Now consider the right set of gears. It consists of another two pairsof gears: 8 teeth driver gear with 16 teeth follower gear, and 24 teethdriver gear with 20 teeth follower gear. The ratio of first pair isagain 2:1, but the ratio of the second pair is  0.833:1. If we mutliplythese, we get the final ratio equal to 1.66:1 – which is not equal to2.5:1 (the ratio of the first and the last gear only). Here the middlegears were not idlers, so they affected the final gear ratio of thewhole set and they couldn’t be ingnored.

Finally, how do we calculate ratio if a worm gear is used? Well,that’s even simplier:

number of follower’s gear teeth : 1

And that’s  because as it was mentioned before, a single revolutionof a worm gear rotates the follower gear by a single teeth. Therefore ittakes 24 revolutions of the worm gear to rotate a 24 teeth gear once,and hence we get the ratio 24:1 which is true.

You can use thiscalculator to calculate the ratios of your Lego mechanisms.

5. Efficiency

We had some theory, now we need to get back to practice, which isunfortunately a bit sad. Every gear we use has some weight and generatessome friction that has to be overcome if we want the gear to rotate.Hence every gear in our mechanism uses part of the drive motor’s power,and efficiency of the gear tells us how much power is transferred andhow much is lost. Unfortunately, it’s extremely difficult to calculateindividual efficiency of each gear, and as far as I know there are noreliable specifications for the efficiency of Lego gears. But we knowhow the power is lost, so we can safely assume two basic rules formaximum efficiency:

- the less gears, the better
- the smaller gears, the better

Sadly, it means that e.g. gear ratio equal to 1:1 is onlytheoretical. If there are gears, there are losses, so the real ratio hasto be 1.something:1. The only mechanism in which the 1:1 ratio ispossible is a motor connected directly to the final gear – for examplein my modelof Leclerc T6 tank, the drive motors were connected directly to thewheels in order to achieve 1:1 efficiency.

What about gear acceleration? Yes, you can obviously use gears to gete.g. 1:6 gear ratio which will greatly increase your speed. However, the quotient of your final speed and torque will be smaller than thequotient of the motor’s original speed and torque – because of thelosses. Using gears always includes losses, therefore if you want totransform the speed and torque of a motor, you have to keep in mind thatsome of it will be lost.

There are two cases  of mechanisms in which the efficiency iscrucial. First is a gearbox with transmission driving rings. This typeof a gearbox uses a number of 16 teeth gears with clutch, and while allof these gears are driven, only some of them transfer the actual drive.It means some of these gears – majority of them, if the gearbox has morethan 4 speeds – use motor’s power for nothing. They are so-called deadgears, which is even worse than idler gears because idler gears areusually needed to transfer the drive from one place to another, whilethe dead gears are not needed at all. And they can’t be removed fromsuch a gearbox, because every gear selected uses a different set ofgears to transform the drive. It means that a certain gear can work as adead gear at 1st, 2nd and 3rd gear, but is needed to transform thedrive at the 4th gear. A gearbox with many dead gears always performsbetter at lower gears, when there is a large gear reduction – it makesthe drive motor use little of its power to actually do its primary task,so it has plenty of power to drive the dead gears. You can see from thevideo of my 10-speed manual gearbox that motor becomes moreand more strained as gears are shifted from 1st to 2nd, then to 3rd andso on. In fact, some time after this gearbox was published I have built a14-speed version, just out of curiosity. When I connected it to a PF XLmotor, it was stalled and could not drive the gearbox even at the 1stgear despite it’s excellent torque.

The second mechanism is… a worm gear. As mentioned before, a wormgear is popular because it offers an extremely high gear reduction. Butthis is actually the worst gear in terms of efficiency – some sourcesestimate that it loses almost one third of the motor’s power due to highfriction (friction is the very reason why worm gear can’t be a followergear) and its tendency to slide along its axle. The friction is highenough to make worm gears hot if they handle high torque for a prolongedperiod of time. Worm gears are irreplaceable for some applications, butin general they should be only used when necessary.

6. Backlash

Gear tooth backlash is generally a complex issue (more at Wikipedia). For the purpose of Lego mechanics wecan simply assume that backlash is the free space between the meshedteeth of two adjacent gears. In a perfect situation there should be nofree space at all, and the teeth should have full contact with eachother. This situation is unfortunately very difficult to achieve withstandard gears (it’s much easier with helical gears, but these areabsent in the Lego Technic world), and Lego gears always generate somebacklash. The general rules are:

- regular gears generate much greater backlash than thebevel ones
- the smaller the gear, the greater the backlash
- the backlashes of any two meshed gears sum up

You can easily guess that 8 teeth gear is a real dynamite when itcomes to generating backlash. Out of all regular gears, the 40 teeth onegenerates the smallest backlash. Among the bevel gears, differences aremuch smaller due to a different teeth design – any bevel gear generatesa backlash several times smaller than in case of the feared 8 teethgear. As pointed out above, the backlashes of meshed gears sum up,therefore it’s a good idea to use regular gears together with the bevelgears – the resulting backlash will be somewhat reduced.

How does it work for a worm gear? Again, this gear proves uniquegenerating practically no backlash. It doesn’t mean that mechanisms withthe worm gear have zero backlash – unfortunately, they still havebacklash of the follower gear. Therefore a mechanism with a worm gearand a 16 teeth follower gear will always have greater backlash than theone with a worm gear and a 24 teeth follower gear. And again, it isrecommended to use worm gear with bevel gears due to their relativelyinsignificant backlash.

Why is backlash bad? Consider a steering mechanism with big wheels,driven by a motor reduced 27 times – which means that three pairs of an 8teeth driver gear and 24 teeth follower gear have been used. Three 8teeth gears together generate a backlash so large that it will not onlydegrade the accuracy of steering – it will also make the steered wheelshave some margin of freedom, so that they can e.g. turn a bit when theymeet an obstacle.

Backlash is usually not a real problem for vehicles (except for thevery large ones), but it’s troublesome whenever accuracy is needed. Manysorts of e.g. cranes, drawbridges or turntables suffer from backlash.The best way to avoid it is to consider the use of pneumatics instead ofmechanics, or the use of linear actuators which currently have theleast backlash out of all the mechanical parts produced by Lego.

I hope you have found this tutorial useful, and that it helped you toenjoy the Lego Technic world a little more.