# The Mechanics of Bicycle Drivetrains

I bet you didn’t know that the basic bicycle was invented over 200 years ago. While many refinements have been added since then, the bicycle remains a marvel of mechanical simplicity.

Today we’ll talk about the drivetrain, that is, the gears, pedals and chain that make it go. The speed at which you travel is related to what gear you are in, and how fast you pedal. The front wheel is not attached to any part of the drivetrain, so it just follows along with what the rest of bicycle is doing. We won’t need to discuss the front wheel again today.

The assembly of front gears and crank arms — where the pedals are attached — is call the crankset. A crankset can have 2 or 3 different gear-like things, called chainrings, attached to it. Each gear on the rear gearing assembly is called a cog; the entire set of them is known as a cassette. The cassette is mounted to the axle of the rear wheel, so as the cogs rotate, the wheel rotates. A chain connects the crankset on the front to the cassette in the back. It allows the force you apply to the pedal to be transferred to the rear wheel, causing the rear tire to rotate and move you and the bicycle forward. Unlike the front wheel, the rear wheel in an integral part of the drivetrain.

Each thing that looks like a gear has teeth around the edge, where the chain catches on the gear and forces it to rotate. The teeth are always evenly spaced around the edge of the gear, because otherwise the chain wouldn’t smoothly travel along each gear. The bigger the gear, the more teeth it has.

Now consider a specific bicycle, one with 3 chainrings on the front and 8 cogs in the back. The number of teeth on the chainrings is 22, 32, and 42 respectively. On the cassette, the number of teeth on each cog is 30, 26, 23, 20, 17, 15, 13, and 11. We can now compare the ratio of the number of teeth on the chainring to the number of teeth on the cog:

Cog # Teeth

30 26 23 20 17 15 13 11

Chainring # Teeth

22 0.73 0.85 0.96 1.10 1.29 1.47 1.69 2.00

32 1.07 1.23 1.39 1.60 1.88 2.13 2.46 2.91

42 1.40 1.62 1.83 2.10 2.47 2.80 3.23 3.82

The entries in this table represent something called the “gear ratio”. It tells us for each complete rotation of the front chainring how many times the rear cog rotates. The number of times the rear cog rotates determines how often rear wheel rotates. Notice that some of the gear ratios overlap in value.

This bicycle has wheels with a radius of 35 cm. The circumference of the wheel is therefore 2*pi*r = 2*pi*35 = 70*pi, or about 219.8 cm. So one rotation of the cog corresponds to the wheel traveling one circumference, which is 219.8 cm of linear distance along the ground.

The gear of 1-1, the lowest gear, uses the smallest chainring on the front and the largest cog in the back. According to the table, every time the chainring rotates one complete revolution, the cog rotates 0.73 of a revolution (rev). That corresponds to the bicycle traveling 219.8 cm/rev * .73 = 160.5 cm/rev, or about 63 inches of linear distance.

The gear of 3-8, the highest gear, uses the largest chainring and the smallest cog. Every time the chainring rotates one complete revolution, the cog rotates 3.82 revolutions, corresponding to traveling a distance of 219.8 cm/rev * 3.82 = 839.6 cm/rev, or about 331 inches of linear distance.

In order to determine how fast I am going, I need to introduce the idea of cadence. This is the rate at which I’m pedaling. If I assume an average cadence of 60 rev/min, the range of speeds I can expect my bicycle to go are easy to calculate. For the gear 1-1, I will be traveling at 160.5 cm/rev * 60 rev/min = 9630 cm/min, or about 3.6 miles/hour. For the highest gear, I will be traveling at about 839.6 cm/rev * 60 rev/min = 50,378 cm/min, or about 18.8 miles/hour!

We take many modern devices for granted. The study of past inventions, like the bicycle, can teach our STEM kids to appreciate sophisticated engineering principles, with the help of a little mathematics. A little curiosity today about a relatively simple device may just inspire our budding science, technology, engineering and math students to invent more remarkable wonders in the future.