Chapter 3: Gears

The robotics club at a school in Nagpur built a small four-wheeled robot for their annual science day. The frame was neat, the wiring was tidy, and the motors were connected straight to the wheels. On the smooth floor of the lab, the robot shot forward the moment the switch was pressed. It looked perfect.

Then the students carried it out to the corridor, where a low wooden ramp led up to the stage. The robot rolled up to the ramp, the motors whined loudly, the wheels spun — and the robot stayed exactly where it was.

Nothing was broken. The battery was full. The motors were turning as fast as ever. The problem was that fast turning is not the same as strong turning. A motor that spins quickly usually pushes weakly, and climbing a ramp needs a strong push, not a quick one.

The students needed a way to trade some of that speed for strength. A pair of toothed wheels did the job, as shown in Fig. 3.0-1.

Diagram

Fig. 3.0-1 – a simple robot chassis seen from the side; a small motor with a small toothed wheel on its shaft meshing with a much larger toothed wheel fixed to the rear wheel axle; label the small wheel “small gear on motor”, the large wheel “large gear on axle”, and add arrows showing the two wheels turning in opposite directions

Curiosity Corner

Q1. Why can a motor that spins very fast still fail to move a heavy robot? Q2. How can two toothed wheels change the turning force a robot can apply? Q3. If someone says a gear pair has a gear ratio of 3:1, what does that tell you? Q4. Why do two meshing gears always turn in opposite directions? Q5. Why do machines use so many different shapes of gears instead of just one shape?

This chapter answers each of these questions as it goes along.

3.1 Why a Robot Needs Gears

Think of opening a tight bottle cap. If you twist it quickly with two fingers, nothing happens. If you grip the whole cap and twist slowly but hard, it opens. The cap does not care how fast you turn. It cares how hard you turn.

Machines behave the same way. A turning shaft can be judged in two separate ways: how fast it spins, and how hard it twists. These two are not the same thing, and a robot builder has to think about both.

The turning effort itself has a name. Torque is the turning force that a shaft applies to whatever is attached to it. A screwdriver that barely moves a stiff screw is giving low torque; a spanner with a long handle gives high torque, which is why a longer handle makes a stiff nut easier to loosen.

Definition

Torque is the turning effect produced by a force acting on a rotating part. The greater the torque, the harder the shaft can twist its load.

Fast but weak

Most small electric motors used in school robots are built to spin quickly. Left to itself, such a motor turns its shaft many times a second, but the twisting effort at that shaft is small. That is fine for a toy fan, where the load is light. It is not fine for a robot that must climb a ramp, lift an arm, or push a load across a floor.

This is exactly what happened to the club robot in the story at the start of the chapter. The wheels spun, so the motor was clearly working. But the twisting effort reaching the wheels was too small to lift the weight of the robot up the slope, so the wheels slipped and the robot stayed put.

Curiosity Corner

Q1. Why can a motor that spins very fast still fail to move a heavy robot?

A. Speed and torque are two different things. A small motor is built for high speed, and at that speed the torque at its shaft is low. Moving a heavy robot, especially up a slope, needs high torque. So the wheels can keep spinning fast while still failing to push the robot forward.

Trading speed for strength

You cannot get more speed and more torque out of the same motor at the same time — the motor only supplies so much power. What you can do is give up some of the speed you do not need and receive extra torque in return.

A ceiling fan regulator does something similar in spirit: you accept less air movement in exchange for a quieter, cooler-running fan. In a robot, you accept a slower wheel in exchange for a wheel that can actually climb. See Fig. 3.1-1, which compares the two ends of this trade.

Diagram

Fig. 3.1-1 – two side-by-side panels of the same robot on a ramp. Left panel labelled “motor connected straight to wheel”: a long curved arrow marked “fast turning” and a short arrow marked “small torque”, with the robot stuck at the bottom of the ramp. Right panel labelled “motor connected through toothed wheels”: a short curved arrow marked “slower turning” and a long arrow marked “large torque”, with the robot part way up the ramp

Passing the turning on

There is a second, quieter problem. The motor cannot always sit exactly where the movement is needed. In a robot arm, the joint that must turn may be far from a comfortable place to bolt a heavy motor. In a four-wheeled robot, one motor may need to drive an axle that sits behind it.

So a robot needs a way to carry turning motion from one shaft to another shaft some distance away. This is called transmission of motion and force. Increasing the turning effort along the way is called amplification of force.

Definition

Transmission means passing turning motion and force from one rotating part to another. Amplification means increasing the turning force during that transfer, which always costs some speed.

A pair of toothed wheels does both jobs at once. It carries the turning across from the motor shaft to the wheel axle, and while doing so it can trade speed for torque in whatever proportion the builder chooses. The next section looks at what these toothed wheels actually are and how a pair of them behaves.

3.2 What a Gear Is

Imagine two plain wheels pressed against each other, rim to rim. Turn the first one and the second one turns too, dragged along by friction at the point where they touch. It works, but only until the load gets heavy. Then the rims slip against each other, the first wheel keeps turning, and the second one falls behind.

Cutting teeth into both rims removes that problem. The teeth of one wheel drop into the gaps of the other, so the two are locked together. Nothing can slip. Every turn of the first wheel is passed on exactly.

A wheel with teeth cut into its rim, made to work with another such wheel, is called a gear. When the teeth of two gears sit inside each other’s gaps and move together, we say the gears mesh. See Fig. 3.2-1.

Diagram

Fig. 3.2-1 – a single spur gear drawn face-on: a circular disc with evenly spaced teeth around the rim and a hole at the centre. Label the teeth, the central hole as “shaft hole”, and the flat body of the gear as “gear blank”. Beside it, a second smaller drawing of two such gears meshing, with the meshing teeth circled and labelled “teeth in mesh”

Definition

A gear is a rotating machine part with cut teeth that mesh with the teeth of another part, so that motion and force are passed on without slipping.

The gear is fixed to a shaft through the hole at its centre, so the gear and the shaft always turn together. Turn the shaft and the gear turns; push the gear round and the shaft turns with it.

How It Works

a pair of meshed gears What it is: Two toothed wheels sitting side by side, with the teeth of one filling the gaps of the other. How it works: When one gear turns, each of its teeth pushes against a tooth of the second gear, forcing it to turn as well. Because the teeth cannot slide past one another, the two gears stay in step for as long as they remain meshed. Used for: Carrying the turning of a motor shaft across to a robot’s wheel axle. Watch out: Two gears only mesh properly if their teeth are the same size and shape. Gears with mismatched teeth will jam, ride over each other, or wear out quickly.

Because gears cannot slip, the relationship between the two shafts is fixed and predictable. If you know how many teeth each gear has, you can say exactly how the second shaft will behave. That fixed relationship is what makes gears so useful, and Section 3.3 puts a number on it.

The next two sub-sections look more closely at a meshed pair — first at how the two gears move together, then at the parts of a single gear and the words used to describe them.

3.2.1 How Two Gears Work Together

In any meshed pair, one gear is being turned by a motor or by a hand crank, and the other is simply carried along. The gear that supplies the turning is the driver gear. The gear it turns is the driven gear.

Watch what happens at the point where they touch. As the driver gear rotates, its teeth move in one direction at the mesh, and they push the teeth of the driven gear the other way round. So the second gear turns the opposite way to the first. See Fig. 3.2-2.

Diagram

Fig. 3.2-2 – two meshed spur gears of different sizes on separate shafts. The smaller gear is labelled “driver gear (on motor shaft)” with a clockwise arrow; the larger gear is labelled “driven gear (on wheel axle)” with an anticlockwise arrow. At the meshing point, show two short straight arrows pointing in opposite directions to indicate the pushing action of the teeth

Curiosity Corner

Q4. Why do two meshing gears always turn in opposite directions?

A. The two gears touch on their facing sides. At that meeting point the teeth of the driver gear are moving one way and they push the teeth of the driven gear that same way. Since the driven gear is pushed on its facing side, it must swing round the opposite way, just as pushing the near edge of a turntable to your right sends the far edge to your left.

Counting teeth instead of turns

Here is the useful part. Because the teeth cannot slip, every tooth that leaves the mesh on one gear has been matched by exactly one tooth on the other. In one full turn of the driver gear, the number of teeth passing through the mesh is simply the number of teeth on that gear.

So think about a small gear with few teeth driving a large gear with many teeth. The small gear must go round several times before it has fed enough teeth through the mesh to take the large gear round once. The large gear therefore turns more slowly than the small one.

Turn the arrangement around and the opposite happens. A large driver gear takes a small driven gear round several times for each of its own turns, and the small gear spins faster.

Common Error

assuming both gears in a pair turn at the same speed What goes wrong: Students sometimes say that because the gears are locked together, they must rotate at the same rate. The fix: What is shared is the movement of the teeth at the mesh, not the number of turns. A gear with fewer teeth completes a turn sooner, because it has fewer teeth to feed through. Only two gears with the same number of teeth turn at the same rate.

Speed lost, torque gained

Slowing a shaft down through gears is not a waste. Recall from Section 3.1 that a motor gives out a fixed amount of power, and that speed and torque can be traded. When a small driver gear turns a large driven gear, the driven shaft turns more slowly, and the torque at that shaft is larger than the torque the motor gave. That extra torque is what the club robot needed to climb its ramp.

The reverse is also true, and it costs you. Using a large gear to drive a small one gives a faster shaft with less torque than the motor supplied. Neither arrangement creates anything from nothing — some torque is also lost as heat and friction at the teeth, so the gain is always a little less than the tooth counts alone suggest.

Safety Note

The meshing point of a gear pair pulls anything near it inwards, and it will not stop for a finger, a loose sleeve, or long hair. Keep hands clear of a running gear set, and always disconnect the battery or power supply before moving, adjusting, or rewiring any gear or motor on a robot.

3.2.2 The Parts of a Gear

If you buy a screw, you do not just ask for “a screw” — you ask for a length and a thickness, so that it fits the hole. Gears are the same. Two gears will only work together if they are described precisely, and engineers use a small set of names to do that. Fig. 3.2-3 shows most of them on one gear.

Diagram

Fig. 3.2-3 – a spur gear drawn face-on with three concentric dashed circles marked on it. The outermost circle passing through the tooth tips is labelled “tip circle”; the middle circle passing through the middle of the teeth is labelled “pitch circle”; the innermost circle passing through the bottoms of the tooth gaps is labelled “root circle”. A double-headed arrow across the pitch circle through the centre is labelled “pitch diameter”. One tooth is labelled “tooth”

The three circles

The teeth are the protrusions around the rim. They are what actually push against the next gear, so their shape, size and number decide how strong the gear is and how smoothly it runs.

The pitch circle is an imaginary circle drawn through the point where the teeth of two meshing gears effectively touch. It sits roughly halfway up the teeth. It is the circle that really matters, because the two pitch circles of a meshing pair must just touch each other for the gears to run correctly.

Definition

The pitch circle is the imaginary circle through the effective contact point of the teeth, used to describe a gear’s working size.

The root circle passes through the bottoms of the tooth gaps, marking the inner edge of the teeth. Teeth break at their roots, so this circle matters when an engineer works out how much load a gear can take.

The tip circle passes through the tips of the teeth and marks the outer edge of the gear. It tells you the overall space the gear needs, and how much clearance to leave around it.

Pitch diameter and module

The pitch diameter is simply the diameter of the pitch circle. Because the two pitch circles must touch, the distance between the centres of two meshing shafts is half of one pitch diameter plus half of the other.

The module describes how big the teeth themselves are. It is the pitch diameter divided by the number of teeth:

\[\text{module} = \frac{\text{pitch diameter}}{\text{number of teeth}}\]

Here the pitch diameter is measured in millimetres and the number of teeth is a plain count, so the module comes out in millimetres. A large module means big, chunky teeth; a small module means fine teeth.

Definition

The module of a gear is its pitch diameter divided by its number of teeth. It is a measure of tooth size, and two gears can only mesh if they have the same module.

That last condition is the whole point of the module. It gives manufacturers a standard set of tooth sizes, so a gear bought from one supplier will mesh with a gear bought from another as long as both are the same module. Table 3.1 shows what that means in practice for a common module.

Table

Table 3.1 – pitch diameter of module 2 mm gears with different tooth counts

Number of teeth	Module (mm)	Pitch diameter (mm)
15	2	30
20	2	40
25	2	50
30	2	60
40	2	80

Every gear in Table 3.1 has teeth of the same size, so any two of them will mesh. Adding teeth does not make the teeth bigger — it makes the gear wider across.

Example 3.1

finding the module and the distance between two shafts

Given: A gear with 20 teeth and a pitch diameter of 40 mm. A second gear with 30 teeth is to mesh with it.

Find: The module of the gears, and the distance between the centres of the two shafts.

Working:

Module of the first gear = 40 mm ÷ 20 = 2 mm.

The second gear must have the same module to mesh, so its pitch diameter = 2 mm × 30 = 60 mm.

The pitch circles touch, so the centre distance = half of 40 mm + half of 60 mm = 20 mm + 30 mm = 50 mm.

The module is 2 mm and the shaft centres must be 50 mm apart.

The angle of the push

One more measurement describes the shape of the tooth face. When one tooth pushes another, the push acts along a particular line, called the line of action. The pressure angle is the angle between that line of action and the tangent to the pitch circle.

You do not need to calculate it, but you do need to know it exists, for the same reason you need the module: a pair of gears must share the same pressure angle, or they will not run smoothly. Most general-purpose gears made today use a pressure angle of 20 degrees, and older machines often use 14.5 degrees. The angle affects how fat the tooth is at its root, how the load spreads across the tooth, and how quietly the pair runs. See Fig. 3.2-4.

Diagram

Fig. 3.2-4 – close-up of two meshing teeth. Show the pitch circles of both gears as dashed arcs touching at one point, a straight line through that point labelled “tangent to pitch circle”, another straight line through the same point along which the tooth force acts labelled “line of action”, and the angle between the two lines marked “pressure angle”

Common Error

buying two gears with the same number of teeth and expecting them to mesh What goes wrong: Tooth count alone says nothing about tooth size. Two 20-tooth gears of different modules have teeth of different sizes and will not mesh. The fix: Match the module and the pressure angle first, and only then choose tooth counts to get the behaviour you want.

3.3 Gear Ratio

Suppose two students describe the same gear pair. One says, “the small gear has ten teeth and the big one has thirty.” The other says, “the big one turns once for every three turns of the small one.” Both are describing the same fact, and the second way is more useful, because it tells you straight away what the pair does.

That comparison has a name. The gear ratio compares the number of teeth on the two meshing gears, and it tells you how the turning of one shaft relates to the turning of the other.

Definition

The gear ratio of a meshing pair is the number of teeth on the driven gear compared with the number of teeth on the driver gear. It decides how much the speed and the torque are changed between the two shafts.

A gear ratio is written with a colon, as 3:1, and read aloud as “three to one”. Order matters. The first number always refers to the driven gear — the output side — and the second to the driver gear, which is the input side. Fig. 3.3-1 shows the pair described above.

Diagram

Fig. 3.3-1 – two meshed spur gears. The smaller one, labelled “driver gear, 10 teeth”, is drawn with a motor symbol on its shaft. The larger one, labelled “driven gear, 30 teeth”, is on a second shaft. Below the pair, print the label “gear ratio 3:1”

Reading a ratio

A ratio of 3:1 means the driven gear has three times as many teeth as the driver gear. The driver must go round three times to take the driven gear round once. This is called a reduction, because the output shaft is slower than the input shaft.

A ratio of 1:3 is the opposite arrangement. The driven gear has one third of the teeth, so it turns three times for every single turn of the driver. This is a step-up, giving a faster output shaft.

A ratio of 1:1 means both gears have the same number of teeth. The output shaft turns at the same rate as the input shaft, though still in the opposite direction. Such a pair changes no speed at all — it is used only to carry motion across from one shaft to another.

The two sub-sections that follow show how to work a gear ratio out from tooth counts, and what it actually does to the speed and the torque of the output shaft.

3.3.1 Working Out the Gear Ratio

Finding the gear ratio needs nothing more than counting. You count the teeth on the driver gear, count the teeth on the driven gear, and compare the two. Manufacturers usually print the tooth count on the face of the gear, so often you only have to read it.

The comparison is a division:

\[\text{gear ratio} = \frac{T_{\text{driven}}}{T_{\text{driver}}}\]

Here \(T_{\text{driven}}\) is the number of teeth on the driven gear (the output side) and \(T_{\text{driver}}\) is the number of teeth on the driver gear (the input side). Both are plain counts, so the ratio itself has no unit.

A ratio worked out this way comes to a single number, and that number is then written in the “something to one” form. A result of 3 is written 3:1. A result of 0.5 is written 1:2, because half of one is the same comparison as one against two.

From ratio to shaft speed

Speed of rotation is usually measured in revolutions per minute, written rpm — the number of complete turns a shaft makes in one minute. A motor labelled 300 rpm makes three hundred turns every minute.

Once you know the gear ratio, the output speed follows at once:

\[\text{output speed} = \frac{\text{input speed}}{\text{gear ratio}}\]

The input speed is the speed of the motor shaft and the output speed is the speed of the driven shaft, both in rpm. Divide by a reduction ratio and the output is slower; divide by a ratio smaller than one and the output comes out faster.

Algorithm

finding the output speed of a gear pair Input: The number of teeth on the driver gear, the number of teeth on the driven gear, and the speed of the driver shaft in rpm. Output: The gear ratio and the speed of the driven shaft in rpm. Steps:

1. Count the teeth on the driver gear.
2. Count the teeth on the driven gear.
3. Divide the driven gear's teeth by the driver gear's teeth. This is the gear ratio.
4. Write the ratio in the form "number : 1".
5. Divide the driver shaft's speed by the gear ratio.
6. The result is the driven shaft's speed, in rpm.

Why it works: The teeth cannot slip, so one tooth of the driver always moves exactly one tooth of the driven gear. The tooth counts therefore fix how many turns of one shaft go with one turn of the other.

Example 3.2

speed of a robot’s wheel axle

Given: A motor turning at 300 rpm carries a driver gear with 12 teeth. It meshes with a driven gear of 36 teeth fixed to the wheel axle.

Find: The gear ratio and the speed of the wheel axle.

Working:

Gear ratio = 36 ÷ 12 = 3, written as 3:1.

Output speed = 300 rpm ÷ 3 = 100 rpm.

The gear ratio is 3:1 and the wheel axle turns at 100 rpm.

Now take the same pair and swap the gears over, so that the 36-tooth gear is driven by the motor and the 12-tooth gear sits on the output shaft.

Example 3.3

the same two gears used the other way round

Given: A motor turning at 300 rpm carries a driver gear with 36 teeth, meshing with a driven gear of 12 teeth.

Find: The gear ratio and the speed of the output shaft.

Working:

Gear ratio = 12 ÷ 36 = 0.333…, written as 1:3.

Output speed = 300 rpm ÷ 0.333… = 900 rpm.

The gear ratio is 1:3 and the output shaft turns at 900 rpm.

The same two pieces of metal give a slow shaft or a fast one, depending only on which one the motor drives. Table 3.2 sets out a few pairs, all driven by the same 300 rpm motor.

Table

Table 3.2 – gear ratio and output speed for different tooth counts, with a 300 rpm motor

Driver teeth	Driven teeth	Gear ratio	Output speed (rpm)
10	30	3:1	100
12	24	2:1	150
20	20	1:1	300
30	15	1:2	600
40	10	1:4	1200

Notice the middle row. Equal tooth counts give a 1:1 ratio, and the output shaft simply matches the motor.

Common Error

dividing the tooth counts the wrong way round What goes wrong: Dividing the driver’s teeth by the driven gear’s teeth turns a 3:1 reduction into a 1:3 step-up. The arithmetic looks fine, but the robot ends up designed for the opposite behaviour. The fix: Always ask which gear the motor is turning. That one is the driver, and its tooth count goes underneath in the division.

Try Yourself

Mesh two cardboard gears of different sizes on a board, and mark one tooth on each with a pen. Turn the small gear slowly and count how many turns it takes for both marks to line up again. Check the count against the gear ratio you work out from the teeth.

3.3.2 Trading Speed for Force

Think of a bicycle on a steep road. In a low gear the pedals go round many times for each turn of the wheel, and you move slowly — but the climb becomes possible. Shift to a high gear on flat ground and the wheel turns much more per pedal stroke, so you go fast, but you could never start that way on a slope.

A gear pair on a robot does exactly this. The gear ratio you calculated in Section 3.3.1 does not only change the speed of the output shaft. It changes the torque at that shaft by the same factor, in the opposite direction.

Torque is measured in newton metres, written N·m. It is the size of the force multiplied by the distance from the centre of the shaft at which that force acts, so a small force acting far from the centre gives the same torque as a large force acting close in.

\[\text{output torque} = \text{input torque} \times \text{gear ratio}\]

The input torque is the torque the motor gives at its own shaft, and the output torque is the torque available at the driven shaft, both in N·m. Fig. 3.3-2 shows the trade for a 3:1 reduction.

Diagram

Fig. 3.3-2 – two horizontal bar pairs, one above the other. Top pair labelled “motor shaft”: a long bar labelled “speed” and a short bar labelled “torque”. Bottom pair labelled “output shaft, 3:1 reduction”: a bar one third as long labelled “speed” and a bar three times as long labelled “torque”. A downward arrow between the two pairs is labelled “gear pair”

Example 3.4

torque at a robot’s wheel axle

Given: A motor supplies 0.20 N·m of torque at its shaft. It drives a gear pair with a ratio of 3:1.

Find: The torque at the output shaft, ignoring losses.

Working:

Output torque = 0.20 N·m × 3 = 0.60 N·m.

The output shaft receives 0.60 N·m, three times the motor’s torque.

Curiosity Corner

Q2. How can two toothed wheels change the turning force a robot can apply?

A. A small driver gear must turn several times to take a larger driven gear round once. The output shaft is therefore slower, and in exchange the torque at that shaft is larger — multiplied by the gear ratio. The gears do not create energy; they hand back as extra turning force what they take away in speed.

Curiosity Corner

Q3. If someone says a gear pair has a gear ratio of 3:1, what does that tell you?

A. It tells you the driven gear has three times as many teeth as the driver gear. The output shaft therefore turns at one third of the motor’s speed and carries about three times the motor’s torque. It also tells you the two shafts turn in opposite directions.

Nothing is free

Gears cannot make energy. A pair only redistributes what the motor already supplies, and it always keeps a little for itself. Some of the input is lost as heat and sound where the teeth rub, and a well-made spur gear pair typically passes on about 95 per cent of what it receives. So the real output torque is a little less than the formula predicts.

Example 3.5

allowing for losses in the same gear pair

Given: The output torque calculated above is 0.60 N·m. The gear pair passes on 95 per cent of what it receives, a typical figure for well-made spur gears.

Find: The torque actually available at the output shaft.

Working:

Useful torque = 0.60 N·m × 95 ÷ 100 = 0.57 N·m.

About 0.57 N·m reaches the output shaft.

The loss is small for one pair, but it adds up. A gearbox with three pairs one after another loses a share at every step, which is one reason engineers use no more reduction than the job needs.

Compare Table

Table 3.3 – reduction versus step-up

Feature	Reduction (for example 3:1)	Step-up (for example 1:3)
Driven gear	More teeth than the driver	Fewer teeth than the driver
Output speed	Slower than the motor	Faster than the motor
Output torque	Larger than the motor’s	Smaller than the motor’s
Suited to	Climbing, lifting, pushing a load	Light, fast movement
Typical use in a robot	Wheel drive, arm joints, grippers	A light spinning tool or fan

Safety Note

A high reduction gives a slow shaft that is much stronger than it looks, and a geared arm or gripper can crush a finger without the motor sounding any different. Never test a geared joint by holding it, and switch the power off before putting your hands near it.

3.4 What Gears Do in a Robot

Open up almost any working robot and you will find gears close to every motor. They are doing four different jobs, and it helps to name them separately, because a designer usually chooses a gear pair with one of the four in mind.

Most of the time these gears are not loose on the frame. Several pairs are built one after another inside a small sealed housing bolted to the motor, and the whole unit is called a gearbox. Ready-made motors with a gearbox already fitted are sold as geared motors, and these are what school robots normally use. See Fig. 3.4-1.

Diagram

Fig. 3.4-1 – photo of a small geared DC motor of the type used in school robots, with the plastic gearbox housing at the front and the output shaft projecting from it; the motor body and the gearbox should be clearly distinguishable

How It Works

a geared motor What it is: An electric motor with a small gearbox fixed to its front, sold as one unit. How it works: The motor shaft drives the first gear inside the housing. That gear drives the next, and so on through several pairs, each one reducing the speed a little further. The final gear is fixed to the output shaft that leaves the housing. Used for: Driving the wheels of a mobile robot, where a bare motor would spin far too fast to be useful. Watch out: The reduction is built in and cannot be changed. If a geared motor is too slow or too weak for the job, the whole unit has to be replaced with one of a different ratio.

Speed reduction

The first job is the one from the story at the start of the chapter. A motor spins too fast for the task, so a gear pair brings the output shaft down to a speed the robot can use. A robot that crosses a room in a controlled straight line needs wheels turning at a sensible rate, not wheels spinning as fast as the motor can manage.

Torque modulation

The second job is adjusting the turning effort to suit the work. Modulation here simply means changing something to the level you want. A gripper closing on a plastic bottle needs enough torque to hold it and not so much that it crushes it. A robot arm lifting a load needs a great deal. A light scanning head that only has to swivel needs very little, and gearing it down would waste both speed and space.

Direction control

The third job is changing which way things turn. You already know that two meshed gears rotate in opposite directions, so a single pair reverses the direction of a shaft without any change to the wiring.

Gears can also change the axis a shaft turns about. A motor may be mounted lengthways along a robot’s body while the wheels have to turn about a side-to-side axis. A suitable gear pair takes the turning round the corner, letting a designer place the motor where it fits rather than where the movement is needed. Section 3.5 describes the gears built for this.

Positioning and precision

The fourth job is the quiet one, and it follows from the fact that gear teeth cannot slip. Every position of the motor shaft corresponds to one fixed position of the output shaft, always, for as long as the gears stay meshed.

That is what allows a robot arm to stop at a chosen angle. The controller counts the turns of the motor, and because the gear ratio is known, it knows exactly where the joint is. A reduction improves this further: with a 20:1 gearbox, a whole turn of the motor moves the joint by only a small angle, so the joint can be set far more finely than the motor alone would allow.

Common Error

expecting a gearbox to hold a position perfectly What goes wrong: Real gear teeth are cut with a small gap between them so they do not jam. That gap, called backlash, lets the output shaft move slightly before the teeth take up the load, so a joint can sit a little off its commanded position. The fix: Choose gears with less backlash for precise work, and where possible measure the joint’s actual position with a sensor rather than trusting the motor count alone.

Taken together, these four jobs explain why gears appear wherever a motor meets a robot. What varies is the shape of the gear, and that is the subject of the next section.

3.5 Types of Gears and Their Uses

Everything so far has used a picture of two flat toothed wheels side by side. That picture is the simplest one, and it works only when the two shafts lie parallel to each other, like the two rails of a ladder.

Real machines are not always so obliging. Sometimes the two shafts meet at a right angle, as in the drive of a hand drill. Sometimes they cross each other without meeting at all, one passing above the other. Sometimes there is no room to place two gears side by side. And sometimes the pair works well but makes far too much noise. Fig. 3.5-1 shows the three common shaft arrangements.

Diagram

Fig. 3.5-1 – three simple line sketches in a row, each showing two shafts as plain rods without teeth. Sketch 1 labelled “parallel shafts”: two rods side by side pointing the same way. Sketch 2 labelled “intersecting shafts”: two rods meeting at a right angle in the same plane. Sketch 3 labelled “shafts that cross without meeting”: one rod passing above another at a right angle, with a gap between them

Each of these situations needs teeth cut in a different way. That is the whole reason there is more than one kind of gear: the tooth shape is chosen to suit how the shafts sit and what the machine has to do.

Curiosity Corner

Q5. Why do machines use so many different shapes of gears instead of just one shape?

A. Because gears have to do more than one kind of job. The shape of the teeth decides which shaft arrangements a gear can work with — parallel, meeting at an angle, or crossing without meeting — and it also decides how quiet the pair is, how much load it can carry, and how much space it takes up. No single shape is best at all of these at once, so each type is a different trade-off.

The six sub-sections that follow describe the types you are most likely to meet, what each one is good at, and where it is used. As you read them, keep two questions in mind for each type: how are the two shafts arranged, and what does this shape buy that a plain flat gear cannot give?

3.5.1 Spur Gears

The spur gear is the plain toothed wheel you have been picturing all through this chapter. Its teeth are straight and run parallel to the shaft, so if you look at the rim edge-on, each tooth is a simple straight ridge. See Fig. 3.5-2.

Diagram

Fig. 3.5-2 – photo of a pair of metal spur gears of different sizes, viewed at a slight angle so that the straight teeth running parallel to the shaft hole are clearly visible

Because the shape is so simple, spur gears are the easiest and cheapest gears to make, and they pass on power between parallel shafts very efficiently. They are the type you will find inside a school robot’s gearbox, in kitchen mixers, in clocks and in countless machines around a workshop.

The simplicity has a cost, and it shows up at speed. On a spur gear the whole width of a tooth comes into contact all at once, and then leaves all at once. Each engagement is a small knock. At low speed the knocks are gentle; at high speed they arrive quickly and add up to noise and vibration.

How It Works

spur gear What it is: A flat gear with straight teeth cut parallel to its shaft. How it works: The straight teeth of one gear drop into the straight gaps of the other, so the two parallel shafts turn together in opposite directions. Used for: The reduction gearbox of a small robot drive motor. Watch out: The whole tooth engages at once, so a spur pair grows noisy and starts to vibrate when it is run fast.

3.5.2 Helical Gears

The knocking of a spur gear comes from the whole tooth landing at once. So cut the teeth at a slant instead. In a helical gear each tooth is cut at an angle to the shaft, winding around the rim in a shallow spiral called a helix. See Fig. 3.5-3.

Diagram

Fig. 3.5-3 – photo of two meshed helical gears, positioned so that the slanting teeth winding around the rim are clearly visible against the straight teeth of an ordinary gear

Now the contact begins at one end of a tooth and rolls smoothly across to the other end as the gear turns. There is no sudden landing. One tooth is still taking up the load while the next has already begun, so the push passes on gradually.

Two things follow from that. The pair runs far more quietly and with much less vibration than a spur pair at the same speed. And because more than one tooth is in contact at any moment, the load is shared, so a helical pair carries heavier loads than a spur pair of the same size. This is why helical gears are used in heavy industrial machinery and in car gearboxes.

The slant has a price. Because the teeth are angled, part of the pushing force acts sideways, along the shaft, instead of round it. This side push is called thrust, and it has to be taken up by the bearings that hold the shaft. A helical drive therefore needs sturdier bearings than a spur drive, which makes it costlier to build.

How It Works

helical gear What it is: A gear whose teeth are cut at an angle to the shaft, forming a helix around the rim. How it works: Contact rolls along each slanted tooth instead of landing all at once, so the load passes on smoothly and several teeth share it. Used for: Gearboxes in heavy industry and in vehicles, where quiet running and high load capacity both matter. Watch out: The angled teeth push sideways along the shaft, so the design needs bearings able to take that thrust.

Compare Table

Table 3.4 – spur gear versus helical gear

Feature	Spur gear	Helical gear
Tooth direction	Straight, parallel to the shaft	Slanted, winding round the rim
Contact	Whole tooth at once	Rolls along the tooth
Noise at speed	Higher	Lower
Load carried	Lower for the same size	Higher for the same size
Side push along the shaft	None	Present, needs stronger bearings
Cost to make	Lower	Higher

3.5.3 Bevel Gears

Spur and helical gears both need parallel shafts. But a motor lying along the length of a robot’s body has to drive an axle that runs across it, and no flat gear can turn the movement through that corner.

A bevel gear solves it. Its teeth are cut not on the rim of a flat disc but on the sloping surface of a cone, so the gear looks like a wheel with its face angled inwards. Two such cones placed tip to tip mesh with their shafts meeting at an angle — most often a right angle, though a bevel pair can be made for whatever angle the machine needs. See Fig. 3.5-4.

Diagram

Fig. 3.5-4 – two bevel gears meshing at a right angle. Draw each as a cone-shaped wheel with teeth on the sloping face, their tips pointing towards a common meeting point. Label the smaller gear “pinion” and its shaft “input shaft”, the larger gear “gear” and its shaft “output shaft”, and mark the right angle between the two shafts

In a bevel pair the smaller gear has a name of its own. It is called the pinion, and the term is used for the smaller member of many gear pairs, not only bevel ones.

Bevel gears carry high torque well, and they do the one thing no parallel-shaft gear can do: they change the axis of rotation. The best known use is the differential of a car. When a car turns a corner, the outer wheel travels further than the inner one, so the two wheels on the same axle must be free to turn at different speeds. A set of bevel gears in the differential allows exactly that, while still driving both wheels from one engine shaft.

How It Works

bevel gear What it is: A gear with teeth cut on a cone-shaped surface, used in pairs on shafts that meet at an angle. How it works: The two cones meet along their sloping faces, so the turning of one shaft is passed on to a shaft pointing in a different direction, usually at a right angle. Used for: The differential of a car, and any robot where the motor lies along a different axis from the part that must turn. Watch out: The two shafts must be lined up accurately and held firmly. If either shaft shifts, the teeth meet unevenly and wear out fast.

3.5.4 Worm Gears

Getting a large reduction from spur gears means using several pairs one after another, which takes up room and loses a little at every step. A worm gear does the whole job in one meshing.

A worm gear system has two parts. The worm is a cylinder with a helical thread cut along it, so it looks very much like a long screw. The worm wheel is a toothed wheel whose teeth are curved to wrap around that screw. The worm is driven by the motor, and its thread pushes the teeth of the wheel round. See Fig. 3.5-5.

Diagram

Fig. 3.5-5 – photo of a worm and worm wheel in mesh, with the screw-like worm lying across the face of the toothed wheel so that the curved teeth wrapping around it are visible

The two shafts here cross each other without meeting — the worm’s shaft passes to one side of the wheel’s shaft, at a right angle to it. This is the third arrangement shown in Fig. 3.5-1.

One turn, one tooth

On the commonest kind of worm, the thread winds round only once. Each full turn of the worm pushes the wheel forward by exactly one tooth. So a wheel with forty teeth needs forty turns of the worm to go round once, giving a reduction of 40:1 from a single pair.

Example 3.6

output speed of a worm drive

Given: A motor turns a worm at 600 rpm. The worm has a single thread, and the worm wheel has 30 teeth.

Find: The gear ratio and the speed of the worm wheel.

Working:

One turn of the worm advances the wheel by one tooth, so 30 turns of the worm give one turn of the wheel.

Gear ratio = 30:1.

Output speed = 600 rpm ÷ 30 = 20 rpm.

The gear ratio is 30:1 and the worm wheel turns at 20 rpm.

Such a large reduction in one step brings a large torque increase with it, which is why worm drives appear in lifting gear, conveyor drives and heavy rotating tables.

The drive only works one way

A worm drive has an unusual property. The worm turns the wheel easily, but the wheel can rarely turn the worm — the thread is too shallow for the wheel’s push to make it rotate. The drive is said to be self-locking.

That is useful. A load lifted by a worm drive stays where it is when the motor stops, with no brake needed. It is also a limitation, because the joint cannot be moved by hand while the motor is off.

How It Works

worm gear What it is: A screw-like worm meshing with a toothed worm wheel, on shafts that cross at a right angle without meeting. How it works: Each turn of the worm advances the wheel by one tooth, so a single pair gives a large reduction in speed and a matching increase in torque. Used for: Lifting mechanisms and conveyor drives, where a load must be held in place when the motor stops. Watch out: The thread slides across the teeth instead of rolling, so a worm drive rubs, heats up and wastes noticeably more of the motor’s power than a spur or helical pair.

3.5.5 External and Internal Gears

Every gear drawn so far has had its teeth on the outside. Teeth can also be cut on the inside of a ring, and that small change makes the pair behave differently.

An external gear has teeth cut on the outer surface of a cylindrical or conical blank — the blank being the plain metal disc the teeth are cut into. All the spur, helical and bevel gears you have met are external gears. An internal gear has its teeth cut on the inner surface of such a blank, so it is a ring with teeth pointing inwards. A smaller external gear sits inside the ring and meshes with it. See Fig. 3.5-6.

Diagram

Fig. 3.5-6 – two labelled sketches side by side. Left, labelled “external pair”: two ordinary toothed wheels meshing outside each other, with curved arrows showing opposite directions of rotation. Right, labelled “internal pair”: a large ring with teeth on its inner surface, a smaller toothed wheel meshing inside it, and curved arrows on both showing the same direction of rotation

Look at the arrows in the two sketches. In an external pair the two gears turn in opposite directions, as you know. In an internal pair the pinion is pushed from inside the ring, so both turn the same way. If a design needs the output shaft to keep the motor’s direction, an internal pair does it without adding a third gear.

The arrangement also saves space. The pinion sits inside the ring rather than beside it, so the two shafts are much closer together and the whole unit is more compact for the same gear ratio. This is why internal gears turn up inside gearboxes, in vehicle transmissions, and in the actuators and joints of robots, where room is always short.

How It Works

internal gear What it is: A ring with teeth cut on its inner surface, meshing with a smaller external gear placed inside it. How it works: The ring’s inward-facing teeth push the pinion from the inside, so both parts turn in the same direction and the two shafts can sit close together. Used for: Compact gearboxes and robot joints where space is tight and the direction of rotation must be kept. Watch out: Cutting teeth on an inside surface is harder than cutting them on a rim, so internal gears cost more, and the parts must be assembled from the end of the ring rather than dropped in from the side.

This ring with inward teeth is worth remembering, because it is one of the three parts of the gear system described next.

3.5.6 Planetary Gears

In every arrangement so far, two gears have shared the load between them. That single meshing point is where the load can fail, and it also means the input shaft and the output shaft never line up — they always sit side by side.

The planetary gear system solves both problems at once by using several gears together. It has three kinds of part. A sun gear sits at the centre. Around it, usually three or four planet gears mesh with the sun and are held at fixed spacing by a frame called the carrier. Around all of them, an internal ring gear encloses the set, with its inward-facing teeth meshing with the outer sides of the planets. See Fig. 3.5-7.

Diagram

Fig. 3.5-7 – a planetary gear set drawn face-on. A small toothed gear at the centre labelled “sun gear”; three equally spaced toothed gears around it labelled “planet gears”, each meshing with the sun; a large ring with inward-facing teeth around the outside labelled “ring gear”, meshing with all three planets; a light triangular frame joining the centres of the three planets labelled “carrier”

The name comes from the picture: the planets circle the sun as they spin, much as planets move around the Sun in the sky. Such a set is also called an epicyclic gear system.

Two things follow from this layout. First, the load is shared between all the planets instead of resting on one pair of teeth, so a planetary set of a given size handles far more torque than an ordinary pair of the same size. Second, the sun, the carrier and the ring all turn about the same centre line, so the input shaft and the output shaft are in line with each other rather than offset.

That combination — high torque in a small, in-line package — is exactly what a robot joint needs. Planetary gearboxes are used in robotic joints and actuators, in the drive trains of autonomous vehicles, and in industrial robots that move heavy material and must stop precisely.

How It Works

planetary gear system What it is: A central sun gear, a set of planet gears held by a carrier, and an outer ring gear with inward teeth, all sharing one centre line. How it works: The sun drives the planets, which roll around inside the ring. Holding one of the three parts still and driving another makes the third turn, giving a large reduction in a small space. Used for: The joints and actuators of robot arms, where high torque is needed and the gearbox must sit in line with the joint. Watch out: There are many more parts than in a simple pair, all of which must be made and fitted accurately, so a planetary gearbox costs considerably more.

Side Note

The same idea is older than robots. Epicyclic gearing has been used in machinery for well over two centuries, and it appears in bicycle hub gears, where the whole set is hidden inside the rear wheel’s hub.

3.6 Choosing a Gear for a Robot

Knowing six types of gear is only useful if you can pick the right one. In practice a designer does not start from the gear at all. They start from the job, and let the job rule most of the types out.

Table 3.5 gathers the six types in one place, so that the choice can be made by comparison rather than from memory.

Compare Table

Table 3.5 – the six gear types at a glance

Gear type	Shaft arrangement	Best at	Main drawback
Spur	Parallel	Simple, cheap, efficient drive	Noisy and rough at high speed
Helical	Parallel	Quiet running and heavy loads	Pushes sideways along the shaft
Bevel	Meeting at an angle	Turning the drive through a corner	Needs accurate shaft alignment
Worm	Crossing without meeting	Large reduction in one step, holds its load	Rubs, heats up, wastes more power
Internal	Parallel, one gear inside the other	Compact, keeps the direction of rotation	Costlier to cut and to assemble
Planetary	All in line about one centre	High torque in a small in-line package	Many parts, high cost

Working through the choice

The order of the questions matters. Some answers rule out whole rows of the table at once, and there is no point comparing costs between gears that cannot even fit the machine.

Algorithm

choosing a gear type for a drive Input: How the two shafts are arranged, the reduction needed, the space available, and how the drive must behave. Output: A gear type suited to the drive. Steps:

1. Look at how the shafts sit. If they meet at an angle, choose bevel.
   If they cross without meeting, choose worm.
2. If the shafts are parallel or in line, work out the reduction needed.
3. If a very large reduction is needed in one step, choose worm.
4. If the load must stay put when the motor stops, choose worm.
5. If space is tight or the shafts must be in line, choose planetary.
6. If the direction of rotation must be kept, choose an internal pair.
7. If the drive must run fast and quietly under heavy load, choose helical.
8. If none of these apply, choose spur.

Why it works: The shaft arrangement and the space are fixed by the machine and cannot be argued with, so they are settled first. Everything after that is a preference, and the spur gear is the sensible default when no preference applies.

That last step is worth taking seriously. The spur gear is not a poor relation. It is cheap, efficient and easy to replace, and most school robots need nothing else.

One example

Consider a robot arm that lifts a tray onto a shelf and must hold it there while a second robot takes it. Run through the questions. The joint is heavy-duty, so the reduction must be large. The arm is thin, so the gearbox must be small and sit in line with the joint. And the tray must not sag when the motor switches off.

Two types answer parts of that. A planetary set gives the torque in a small in-line package, which suits the arm’s shape. A worm drive gives the large reduction and holds its position by itself. Which one wins depends on which need is stronger — a designer who cannot spare a brake will take the worm and accept the lost power; one who needs efficiency and speed will take the planetary set and add a brake.

Common Error

choosing the gear with the biggest ratio available What goes wrong: A very large reduction makes a robot strong but painfully slow, and every extra pair of gears adds friction, weight and cost. The fix: Work out the torque the task actually needs, then choose the smallest ratio that supplies it with a little to spare.

Try Yourself

Look at a hand drill, a manual eggbeater or a bicycle with hub gears. Find where the turning changes direction or changes speed, and decide from Table 3.5 which type of gear is doing it. Do this only with the tool held still and not connected to any power supply.

3.7 Summary

Speed and torque are two different things. A small motor spins fast but twists weakly, which is why a fast motor can still fail to move a heavy robot.
Gears transmit turning motion from one shaft to another, and can amplify the turning force along the way by giving up some speed.
A gear is a wheel with cut teeth. Because the teeth of two meshed gears cannot slip, the turning is passed on exactly.
In a meshed pair, the driver gear is turned by the motor and the driven gear is carried along. The two always turn in opposite directions.
A gear with fewer teeth completes a turn sooner, so a small gear driving a large one gives a slower output shaft, and the reverse gives a faster one.
A gear is described by its pitch circle, root circle, tip circle, pitch diameter, module and pressure angle. Two gears mesh only if their module and pressure angle match.
The module is the pitch diameter divided by the number of teeth, and it measures how big the teeth are.
The gear ratio is the driven gear’s teeth divided by the driver gear’s teeth, written in the form 3:1. Output speed is the input speed divided by the gear ratio.
Output torque is the input torque multiplied by the gear ratio, less a small amount lost to friction. Gears redistribute what the motor supplies; they never create it.
In a robot, gears are used for speed reduction, torque modulation, direction control, and positioning and precision.
A gearbox holds several gear pairs in one housing. A motor sold with one already fitted is called a geared motor.
Spur gears have straight teeth and parallel shafts. They are cheap and efficient, but noisy at speed.
Helical gears have slanted teeth. They run quietly and carry heavier loads, but push sideways along the shaft.
Bevel gears have teeth on a cone and join shafts that meet at an angle. They are used in car differentials.
A worm gear system pairs a screw-like worm with a worm wheel on shafts that cross without meeting. It gives a large reduction in one step and holds its load, but it rubs and heats up.
External gears have teeth on the outside; internal gears have teeth on the inside of a ring. An internal pair is compact and both parts turn the same way.
A planetary gear system has a sun gear, planet gears on a carrier, and a ring gear. It shares the load between several planets and keeps the input and output shafts in line.
The gear type is chosen from the job: the shaft arrangement and the space settle it first, and the spur gear is the sensible default when nothing else applies.

3.8 Key Terms

Amplification – Increasing the turning force while passing motion from one shaft to another, always at the cost of some speed.
Bevel gear – A gear with teeth cut on a cone-shaped surface, used in pairs on shafts that meet at an angle.
Driven gear – The gear that is turned by the driver gear, on the output side of a pair.
Driver gear – The gear that is turned by the motor or crank, on the input side of a pair.
External gear – A gear with its teeth cut on the outer surface of the blank.
Gear – A rotating machine part with cut teeth that mesh with another such part, passing on motion and force without slipping.
Gearbox – A housing holding several gear pairs, usually bolted to a motor.
Gear ratio – The number of teeth on the driven gear compared with the number on the driver gear, written in the form 3:1.
Helical gear – A gear whose teeth are cut at an angle to the shaft, winding round the rim as a helix.
Internal gear – A ring with teeth cut on its inner surface, meshing with a smaller gear placed inside it.
Mesh – The state in which the teeth of two gears sit in each other’s gaps and move together.
Module – The pitch diameter of a gear divided by its number of teeth; a measure of tooth size.
Pinion – The smaller gear of a meshing pair.
Pitch circle – The imaginary circle through the point where the teeth of two meshing gears effectively touch.
Pitch diameter – The diameter of the pitch circle.
Planet gears – The gears that mesh with the sun gear and roll around inside the ring gear of a planetary system, held in place by a carrier.
Planetary gear – A gear system made of a sun gear, planet gears on a carrier, and a ring gear, all sharing one centre line.
Pressure angle – The angle between the line of action of the tooth force and the tangent to the pitch circle.
Ring gear – The internal gear that encloses the planets in a planetary gear system.
Root circle – The circle passing through the bottoms of the tooth gaps, marking the inner edge of the teeth.
rpm (revolutions per minute) – The number of complete turns a shaft makes in one minute.
Spur gear – A flat gear with straight teeth cut parallel to its shaft, used on parallel shafts.
Sun gear – The central gear of a planetary gear system.
Tip circle – The circle passing through the tips of the teeth, marking the outer edge of the gear.
Torque – The turning effect produced by a force acting on a rotating part, measured in newton metres.
Transmission – Passing turning motion and force from one rotating part to another.
Worm – A cylinder with a helical thread cut along it, resembling a screw, which drives a worm wheel.
Worm gear – A gear system in which a worm meshes with a worm wheel on shafts that cross without meeting.
Worm wheel – The toothed wheel whose curved teeth wrap around and mesh with the worm.

3.9 Objective Questions

Which type of gear is used to carry power between two shafts that meet at a right angle?
1. Spur gear
2. Helical gear
3. Bevel gear
4. Internal gear
Fill in the blank: The ____________ of a gear is found by dividing its pitch diameter by its number of teeth.
State whether true or false: Two meshing external gears always turn in the same direction.
A drive must hold a lifted load in place when the motor is switched off, without a brake. Which gear system is best suited to this?
1. Spur gear pair
2. Worm gear system
3. Helical gear pair
4. Internal gear pair
Match each gear type in Column A with the feature in Column B that describes it best.

Column A	Column B
a. Spur gear	i. Both gears of the pair turn in the same direction
b. Helical gear	ii. Straight teeth, cheap to make, noisy at speed
c. Planetary gear	iii. Slanted teeth, quiet, pushes sideways along the shaft
d. Internal gear	iv. Sun, planets and a ring sharing one centre line

A motor turning at 400 rpm drives a gear pair with a ratio of 2:1. Write down the speed of the output shaft.
Which of these is not one of the four jobs gears do in a robot?
1. Speed reduction
2. Direction control
3. Increasing the motor’s total power
4. Positioning and precision

3.10 Concept Questions

Define torque and state the unit in which it is measured.
Two gears have the same number of teeth but were bought as different modules. Explain why they will not mesh.
Give two differences between an external gear pair and an internal gear pair.
A motor supplies 0.15 N·m of torque at 240 rpm and drives a gear pair with a ratio of 4:1. Ignoring losses, work out the speed and the torque at the output shaft, showing your steps and units.
Explain how a gear reduction helps a robot arm stop at a chosen angle more finely than the motor could on its own.
A student fits a 40:1 gearbox to a small line-following robot that only has to move across a flat floor. Give a reason why this is a poor choice, and say what you would do instead.

3.11 Case-Based Questions

A school canteen uses a short conveyor belt to carry trays from the counter to the washing area. The motor that drives it turns at 900 rpm, but the roller that moves the belt must turn at 45 rpm so that the trays do not slide off. The gear fitted to the motor shaft has 12 teeth. The space beside the roller is narrow.
1. Work out the gear ratio needed between the motor and the roller.
2. If a single spur gear pair were used, how many teeth would the driven gear need? Explain why this answer makes the spur pair unsuitable here.
3. Suggest a gear system that could give this reduction in one step, and give one drawback of your choice.
A robotics club builds an arm that lifts a book onto a shelf and must keep holding it there after the motor stops. The values they measured and chose are given below.

Quantity	Value
Torque at the motor shaft	0.25 N·m
Gear ratio of the gearbox	8:1
Torque needed to lift the book	1.5 N·m
Share of torque passed on by the gears	95 per cent

Work out the torque available at the output shaft, first ignoring losses and then allowing for them.
State whether the arm can lift the book, giving the figures you compared.
The club finds the arm sags as soon as the power is switched off. Name a gear system that would hold the load by itself, and state one cost of using it.