Everything around the topic of gearboxes

Gears were essential to the development of human civilization: They made it possible to build large buildings, draw water or lift and transport heavy objects. Later, they were used to generate energy (treadwheels, water wheels, windmills, steam engines), pump water (Machine de Marly) or drive vehicles. Gears also played an important role in measuring time: The creation of pendulum clocks and fine mechanical gears allowed people to build clocks that could tell time more precisely than using a sundial.

Definition

What exactly is a gear? A gear is a technical component (also called a “machine element”) that can be used to change kinetic quantities. What does this mean?

The movement of an object can be described by its direction, its path (or “location”), its speed, and its type (rotational movement, back and forth movement). A gear changes one or more of these properties, which we also call kinetic quantities. Each gear has an “input”, or drive, to which a crank, a motor or another machine element, for instance, transmits force, and (at least) one “output”, where a movement or force is transmitted to another machine element. 

We can visualise this using what is probably the simplest gear there is, a lever.

A lever consists of a rigid body (such as a bar) which is mounted in a flexible manner at one point. Imagine a seesaw on a playground – it is a lever. The two parts of the seesaw, which stick out to the left and right of the attachment point, are called the lever arms. One of the lever arms is the output, the other the input. If you sit on one side of the seesaw, your lever arm moves down and the other moves up – a seesaw (a lever) changes the direction of movement. It also changes the path of the movement, since your movement of one lever arm is transmitted to the end of the other lever arm. And the lever can also change the speed of the movement: If your lever arm is longer than the output lever arm, then you move farther while see sawing than the end of the other lever arm – in the same time. The movement of the output slows down.
Something fascinating happens: The force your weight exercises on the input lever arm of the seesaw is transmitted to the output lever arm. If it is shorter than the input lever arm, the force is increased. You have certainly seen this before: If a larger child is sitting on one side of the seesaw than on the other, then you can balance out the seesaw if the larger child shortens their lever arm by pushing it forward (toward the pivot joint). A gear can do one other thing: It can amplify force.

Observations of the movements of the planets inspired the development of mechanical clocks. In 1900, Greek divers discovered a highly complex mechanism (the “Antikythera mechanism”) in the remains of a sunken Greek ship from the first century B.C. consisting of over 40 toothed gears that was reconstructed as an astronomical clock. However, the knowledge required to build it was lost once again. Only around 1,500 years later, in the second half of the 14th century, were the first clock towers created with complex gears for astronomical displays (such as the phases of the moon). 

Gears flourished during the Renaissance, during which the antique writings of the Greeks and Romans were re-discovered. The drawings of Leonardo da Vinci (1452-1519), in particular, show many different kinds of gears for construction machinery, war machines and the first aeroplanes
We have the Dutchman Christiaan Huygens (1629-1695) to thank for the first pendulum clock, which was invented in 1657. The precision gear achieved an exactness of just a few seconds per day, which was unbelievable for the time. After the invention of the balance wheel, pocket watches came into fashion. John Harrison (1693-1776) used one such precision pocket watch, which lost only four seconds over the course of a multi-month sea journey, to solve the problem of longitude in 1759 – how to precisely determine longitude on the high seas.

Gears became even more important in the 19th and 20th centuries with the development of motorised “automobiles”. They were designed to transmit the drive force of the motor to the wheels with the highest possible effectiveness. To do so, they required manual and differential gears.

Spur gears

Spur gears use toothed gears to transmit motion. The toothed gears interlock together as the teeth engage. fischertechnik toothed gears are named based on the number of teeth they have: A Z20, for instance, is a toothed gear with 20 teeth.

Spur gears make it easy to calculate the gear ratio, which is the change in the speed of movement. The ratio between the rotational speeds of two axles of a gear is the inverse of the ratio of the teeth of the interlocking toothed gears on these axles. Example: If there is a Z10 on the input axle and a Z30 on the output axle, then the input axle will turn three times as quickly (30:10 = 3:1) as the output axle – you would have to turn the crank on the input axle three times for the output axle to revolve once.

In spur gears, the edges of the teeth rub against one another. This friction results in a loss of force of around 10%. The friction can be reduced by increasing the “play” (the distance) between the teeth. However, this causes the transmission of force to become less precise: You can move one of the axles slightly without the other axle moving.

Specially shaped teeth are used in practice to attempt to reduce this disadvantage. Spur gears are primarily used in geared motors.

Chain gears

Instead of having the teeth of the toothed gears interlock directly, they can be connected via a chain – this is the kind of gearing used on a bicycle. This does not change the direction of motion; the input and output axle (the crank axle and rear axle) turn in the same direction.

As with spur gears, the gearing ratio here can be determined from the ratio of the teeth on the toothed gears. Because the teeth interlock with the chain links, the loss of force due to friction is significantly less than in a spur gear.

Belt drives

In a belt drive, force is transmitted from the input to the output axle via a belt. In fischertechnik sets, this is a specialised rubber band. A household rubber band can also be used as an alternative. A belt drive has a built in “overload protection”: If the output is not strong enough to drive a connected machine element, then the belt slips once the force acting against static friction is greater.

In contrast to chain gears, you can change the direction of the output movement by crossing the belt (in the shape of an “8”). If the belt is flexible, the distance between the input and output axles can even be reduced or increased while the gear is in “operation”.

The loss of power in a belt drive is significantly less than that of a spur gear, since there is no significant friction in this case. It is even less than that of a chain. Because of this, belts used to be used in cars, and there are bicycles and motorcycles with belt drives. However, belts wear more quickly than a (well cared-for) chain, and must be exchanged more frequently.

Worm gears

Worm gears transmit the rotational movement of an axle to the teeth of a toothed gear via a worm thread. As the worm thread turns, the toothed gear turns one tooth further. Since the output axle must run vertically to the input axle, a worm gear also changes the location of the movement.

Calculating transmission for a worm gear is very simple: The rotational speed of the input axle (the worm gear) is n times as great as the output axle, if n is the number of teeth on the toothed gear on the output axle. Therefore: If the worm gear drives a Z30, then you would have to turn the input axle 30 times for the output axle to turn once. Worm gears therefore are very useful for gearing down, and allow for compact gears with high gear ratios.

A worm gear is “self-locking”, and therefore only works in one direction: From the worm gear to the toothed gear. A worm gear is always used for gearing down. 

The disadvantage of a worm drive is the significant loss of power of up to 30%, since the worm gear continuously rubs against the teeth of the toothed gear.

Gear rack

When a rotational movement is transmitted via a toothed gear to a gear rack, the location of the movement changes (the axle with the toothed gear is vertical to the direction of movement of the gear rack) and the type of movement changes (a rotational movement is changed into a translatory motion).

Gear racks also experience frictional loss, generally more than that of a spur gear (over 10%). In contrast to the other gearing types we have learned about, the effectiveness of this gear is limited by the length of the gear rack: once it reaches the end, the rotational movement of the drive can no longer be transmitted. Therefore, rack and pinion gears are used, for instance, in sliding doors or forklifts, and cog railways also use the same principle.

Crankshafts

A crankshaft gear also converts a rotational movement into a horizontal movement. It does so continuously, however, in a back and forth motion. It is not limited. However, the gear ratio is not even, unlike the other gears we have already learned about: while the crank rotates, the speed of the output axle changes. The gear is not self-locking like the worm gear, but if the input is a back and forth movement, the gear does have a “dead point” at each end of the back and forth motion. If it stops exactly there, the movement is arrested and cannot be continued.

Crankshaft drives with back and forth movement as the input play a key role in steam engines, and are still used in motors today: They convert the up and down motion of the piston into a rotational movement. Multiple drive pistons working at a slight offset to one another are used to overcome the dead point.

Fundamentals (Secondary level I+II)

Torque and force amplification

From the example of the seesaw, we learned that gears can also amplify force. This is true in particular of all gears that result in a gearing down – such as chain drives, spur gears, belt drives or worm gears. 

This property follows directly from the principle of the lever: Force times the length of the arm to which force is applied = Load times the length of the load arm. The product of the effective force F and length of the arm to which force is applied r is the torque (M).

Remember the seesaw: If there is double the force (= double the weight) on one side, then the arm to which force is applied only needs to be half as long to lift the same load on the other side. When they are balanced, both torques are the same and cancel one another out. 

In a seesaw, the effective force on both sides is the same – the force of gravity. However, the lever principle applies to all force. This can also explain force transmission in a spur gear: When the black toothed gear in the image is driven, a downward force (or an upward force, with the opposite direction of rotation) acts on the teeth of the red toothed gear. The larger the red toothed gear is, the longer the lever arm (in relation to the axle of the red toothed gear) – and the less force is required to drive the toothed gear. The black toothed gear must be turned more frequently – in relation to the ratio of the lengths of the levers, the radii r of the toothed gears to one another. This ratio, in turn, is identical to the ratio of the circumferences of the toothed gears to one another (for circumference U: U = 2 r π). 

As we can easily calculate, the ratio of the rotational speeds of the two gear shafts (input to output) is inversely proportional to the effective force. In our example, the axle of the black toothed gear (10 teeth) turns three times as quickly as that of the red toothed gear (30 teeth) – if we do not consider frictional loss, three times the force acts on the axle of the red toothed gear. We can use this kind of gear drive to increase the force acting on an axle in a targeted manner.

The same is true of belt and spur gears: The inverse of the ratio of the radii of the input and output wheel describes the slowing of the rotation of the axle, and at the same time the force amplification. 

Wheel and axle

A simple gear that directly utilises the lever principle has been used since ancient times: the wheel and axle. In this type of gear, a winch is fitted with multiple long levers. If the levers have a length of R and the winch has a radius of r, then the wheel and axle amplifies the force of the winch operator by the ratio between the radii, or the factor of Rlr.

Pulley

A pulley is another type of gear already widely used in ancient times, and it also amplifies force. It can be used to control the force required to complete linear work – lifting a certain weight to a defined height – via the length of the stroke path travelled. 

The linear work is defined as the product of force and path: If the stroke path is longer, therefore, less force is required to complete the same linear work. For example, it does not matter whether you walk up a steep path to a high peak or choose a less steep route – you will ultimately do the same linear work (you move the weight of your body up to the same elevation). You need less (linear) force if you take the flatter path – but the path will be longer. 

A pulley works in roughly the same way. It artificially lengthens the stroke path, or more specifically: the length of the pull rope which is wrapped around the axle to perform the linear work. Because of this, less force is needed to perform the linear work. The price you pay for this “force amplification” is having to pull (or crank) for a longer time. Generally, when people say “pulley” they mean a block and tackle, which extends the rope length through rope slings and rollers (see Fig.) 

Even a simple pulley with just one sling doubles the length of the tow rope pulled in to complete the stroke, cutting the force required to do so in half. A person who can lift a maximum of 50 kg can use such a pulley to lift up to a 100 kg load with the same tensile force. The force amplification can be increased by adding additional rope slings: The tensile force required to complete the linear work FZ drops by n rope paths (= pulleys) to an n-th of the weight force FL of the load.

F_Z=F_L/n
 
Pulleys also have a positive side effect: They stabilise the tow rope by making it more difficult for the rope to twist up: They make it possible to pull an object up very straight. The more rope slings there are, the more resistant the pulley is to torsion.

Ultimately, this reduces the load on the tow rope, since only a fraction of the weight force of the lifted object acts on each individual strand of the rope. A pulley can therefore be used to lift even very heavy objects with a relatively thin rope.

Differential gears

Differential gears play a very important role in vehicle technology. In every steerable vehicle, the “inner” wheels in the steering direction travel a shorter path than the “outer” wheels. In an independent suspension (such as a horse carriage), this is not a problem – but it can be a problem if one pair of wheels is driven by a motor on a rigid axle. In this case, depending on the traction, the inner wheels may spin or the outer wheels may slip.

We can avoid this by driving just one single wheel. Practically, however, this is not a satisfactory solution. A differential gear works much better, distributing the drive force when going around curves to the inner and outer wheel, so that both are driven at the appropriate speed. This is exactly what a differential gear does.

The best way to understand how a differential gear work is to build one. The drive force is transmitted to a crown wheel (Z32) via a Z15. The four cone gear wheels distribute the drive force to the two separate output axles (the wheels). 

If the resistance is higher on one wheel (for example when driving around a curve), part of the force acting on this axle is automatically shifted to the other axle until the load is balanced out. 

Differential gears also have one other interesting property: If we stop the input and turn one of the two axles (wheels), the other will turn at the same speed, but in the opposite direction.