Which factor determines the number of teeth in a sprocket

Question

I have thought about this question, a couple of times.

1. The main shaft sprocket (Output shaft from the engine) (for two wheeler,i.e, the bikes) is having the less number of the teeth, while the rear wheel sprocket is having large number of teeth.

2. While, I have seen a bit different structure in the bicycle, where the rear wheel sprocket's teeth will be less in number (Even in gear cycles) than the main sprocket (which is attached to the pedal).

I am just curious and interested to learn, why the things are having different structure and the reason behind it.

Thanks.

Answer 1

It's all to do with gearing and matching the optimum rotational speed of the engine with the wheels (put simply).

Imagine you have an engine which develops it's peak power between 3500RPM and 5000RPM. Now imagine that vehicle isn't equipped with any gears. At 60mph, a roadwheel and tyre that is around 2ft in diameter is rotating at around 840RPM. Increase the speed to 80mph and that wheel is still only rotating at around 1120RPM.

You can see how peak power of the engine is not reached at anything like road legal speeds. Combine that with the fact that below say 600 RPM, most car engines will stall, the engine would never be able to get the car moving.

For this reason, ratios are picked that optimise peak power at speeds commonly required for road use with respect to accelerating, cruising, climbing hills and towing.

The actual decisions also involve factors such as engine peak economy, noise and comfort, use of the vehicle (i.e. "sports" cars typically have "close ratio" gearing), the number of gears and any number of other factors.

Answer 2

It all has to do with the distance traveled for 1 input revolution.

On a bike, there are 3 factors. The front gear, the back gear, and the tire diameter.

The ratio of teeth on the rear gear to the number of teeth on the front gear is the gear ratio. If the front gear is larger, the wheel will spin more than 1 revolution per input revolution. If the front gear is smaller than the rear gear, the wheel will spin less than once per revolution.

Torque is multiplied by the gear ratio. Gears are a balance between torque output and RPM required to move a given speed. This is why most vehicles have a transmission with multiple gears for better acceleration in lower gears and lower RPM (therefore better gas mileage and higher top speed) in higher gears.

Higher ratios:
smaller front, larger back
Higher torque
Higher RPM required for the same speed

Lower ratios:
larger front, smaller back
Loss torque
lower RPM required for the same speed
can achieve a higher speed at maximum RPM

The tire diameter plays a role also as the overall ratio is how far you move per input revolution. A larger diameter tire will be like having a lower gear ratio, while a smaller diameter tire will be similar to a higher gear ratio.

Lets throw around some numbers. nothing real word, just pulling numbers out of the air.

15 up front, 40 in the back, 22" tire:
22*pi*(15/40) = 25.9" per input revolution
22*pi*(15/40)/12/5280*1000*60 = 24.5 MPH / 1000 RPM

lets change it to 20 teeth up front:
22*pi*(20/40) = 34.5" per input revolution
22*pi*(20/40)/12/5280*1000*60 = 32.7 MPH / 1000 RPM

lets change it to 35 teeth out back:
22*pi*(15/35) = 29.6" per input revolution
22*pi*(15/35)/12/5280*1000*60 = 28.0 MPH / 1000 RPM

lets change the tire size to 24":
24*pi*(15/40) = 28.2" per input revolution
22*pi*(15/40)/12/5280*1000*60 = 26.822 MPH / 1000 RPM

Now that I think of it, this is completely ignoring the transmission. You have the engine through the transmission gear ratios, then the chain/belt gears/sprockets acting as a final drive ratio.

Answer 3

The number of teeth in a sprocket is constrained by the radius of the sprocket. Assuming a standard link distance (for bicycle chain) of 1/2 inch, the radius is some number based on n where n is an integer multiple of 1/2 inch lengths. Since circumference is 2*pi*r, the relationship 1/2*n = 2*pi*r must hold. Solving for r, r = 1/4 * n / pi.

Thus for the series n = 18,19,20...,43,

inches           n
1.432394488      18
1.511971959      19
1.591549431      20
1.671126902      21
1.750704374      22
1.830281846      23
1.909859317      24
1.989436789      25
2.06901426       26
2.148591732      27
2.228169203      28
2.307746675      29
2.387324146      30
2.466901618      31
2.546479089      32
2.626056561      33
2.705634033      34
2.785211504      35
2.86478897       36
2.944366447      37
3.023943919      38
3.10352139       39
3.183098862      40
3.262676333      41
3.342253805      42
3.421831276      43

Answer 4

I don't see any of the other answers being complete, i.e. describing the reason why bicycles differ from motorcycles.

The difference is the power source: humans versus modern combustion engines / electric motors.

Humans are like steam engines. Humans produce power at extremely low RPM. This results in poor power to weight ratio: humans produce from about fifth to third of a horsepower for a 70 kg human, power to weight ratio being 0.003 - 0.005 horsepower per kilogram.

In contrast, a 100 kg car engine produces 100 hp, power to weight ratio being 1 horsepower per kilogram. Electric motors are even better at this: Tesla's motor produces 362 horsepower for 32 kg or 11.3 horsepower per kilogram.

How do electric motors and internal combustion engines achieve such a huge power output per kilogram? The answer is RPM, rotational speed. Power is torque times RPM. Torque for a car engine is actually quite poor: a 100 kg car engine produces perhaps 170 Nm, or 1.7 Nm per kg. A 70 kg human using shoes that attach to pedals on 170 mm cranks produces 70 kg (weight) + 25 kg (pulling up from rear shoe) + 25 kg (pushing on the front shoe an equivalent amount) + 30 kg (pulling up from handlebar), or 150 kg or about 1500 Netwons times 0.17 meters, being about 250 Nm. So: 250 Nm per 70 kg, or 3.6 Nm per kg. That's more than a car engine!

Unfortunately, humans can pedal at only about 100 RPM. In contrast, car engines operate at up to about 7000 RPM, and electric motors at up to about 20 000 RPM.

The poor torque output of these man-made machines is compensated by operating at huge rotational speeds / RPMs. Humans can never achieve that.

To understand why the gearing is different on bicycles and motorcycles, see the accepted answer: it's all about the RPM. But to understand why the RPMs are different, I think my answer is needed.

By the way, the RPM mismatch is also the reason why electric bicycles took so long to materialize and why even today they are suboptimal. The human wants to pedal at leisurely 60 RPM, whereas a good electric motor would like to spin at 20 000 RPM. The mismatch is usually solved by building a poor electric motor that spins at only 200 RPM @ 25 km/h and putting it to the wheel hub. That's not a problem because these poor motors produce only 250 watts, a fraction of what they could produce if spinning at 20 000 RPM (25 000 watts = 33.5 horsepower). For mopeds (moped means motor-pedal vehicle), the mismatch was solved by eliminating pedals, so despite their name mopeds don't anymore have pedals at all.

Also, the reason why serious cyclists don't use internal gear hubs is the same: the peculiar steam engine like characteristics of humans. I think only the very expensive Rohloff one withstands 250 Nm at the bottom bracket... If the cyclist weighs more than 70 kg, you already exceed that. Additionally, the reason why bicycle parts fail at a great rate and car parts don't fail at such a great rate is the same. I presume steam engine parts also failed often.