Chainring size question for the physicists/engineers

John M said:

It seems that physics would predict that since a bicycle crank is a class two lever, that lower force would be required to turn the smaller chainring since the load (chain) would be closer to the fulcrum (BB spindle) and therefore the lever arm would be longer, thus increasing mechanical advantage.

You're overlooking the fact that what you gain by having a longer lever arm on the chainring end is lost by having a smaller lever arm on the cog end.

My guess is that larger chainrings and cassette cogs are going to have a slight advantage because the larger levers will result in less tension on the chain.

A couple of observations here, I wish you had included 42/14 in the list, thus showing the difference betwen a 39 and 42 is only one tooth at the rear cassette. 😄

As for larger chainring sizes, ask any TT or Tri rider why they prefer bigger chainrings..... its easier to pedal. This compact crankset fad is a fashion myth.

As for cranklength, generally, the shorter the crank, the higher the cadence, to a point where the law of diminishing returns applys. :eek:

gclark8 said:

As for larger chainring sizes, ask any TT or Tri rider why they prefer bigger chainrings..... its easier to pedal. This compact crankset fad is a fashion myth.

I'm not sure where you're getting this one. 79 gear inches is 79 gear inches, no matter how you get there. TT riders use big chain rings to go fast, they aren't using bigger cassettes as well. A 54x11 is not easy to pedal. The exception being 650c setups, where a higher ratio is required to get the same number of gear inches because the wheel radius is smaller. And how is a compact crankset a myth? It's pretty obvious that smaller chainrings will give you lower gear ratios given the same cassette options.

ps. The difference between a 39 and a 42 tooth ring is only one cog tooth at the 14-15 range. At the low end, where it matters, it's the difference between a 25 and a 27.

John M said:

OK, there has been a lot of debate around here recently about crank arm length and numerous separate discussions on compact cranksets. I have a a question for the mathematically inclined among you:

For a given crank arm length, does chainring size matter if the ratio of chainring to cassette is constant? I.e. does 36/12 = 39/13 = 54/18? For purposes of discussion, let us assume that chainline is the same.

It seems that physics would predict that since a bicycle crank is a class two lever, that lower force would be required to turn the smaller chainring since the load (chain) would be closer to the fulcrum (BB spindle) and therefore the lever arm would be longer, thus increasing mechanical advantage.

Also what are people's thoughts on the potential for increased friction with smaller chainring/cassette cogs?

No. It doesn't. Look at the equations below. In the equations, T is the torque at the location given by the subscript; F is the force at the subscripted location; and r is the radius at the subscripted location. The torque generated by the force applied to the crankarm is the same torque applied to the chainring; however at the chainring the force goes up because the radius of the chainring is smaller than that of the crankarm. Note that now the force applied to the chain is the same as that given by the force at the chainring. The torque at the cassette is the final torque. From the last equation you can see that the only things varying are the radius at the chainring and the radius at the cassette. The final torque is the same, and the radius of the crankarm is also unchanged. However, in your question you constrained the ratio of chainring to cassette radius to be the same (that ratio varies identically as the circumference, i.e. the number of teeth does), thus nothing changes.

That's the ideal situation. However, in the real world, friction losses increase as a chainring or cassette gets smaller, but it's those losses are pretty darned small.

Oh yeah , you have to click on the damned pic.

alienator said:

No. It doesn't. Look at the equations below. In the equations, T is the torque at the location given by the subscript; F is the force at the subscripted location; and r is the radius at the subscripted location. The torque generated by the force applied to the crankarm is the same torque applied to the chainring; however at the chainring the force goes up because the radius of the chainring is smaller than that of the crankarm. Note that now the force applied to the chain is the same as that given by the force at the chainring. The torque at the cassette is the final torque. From the last equation you can see that the only things varying are the radius at the chainring and the radius at the cassette. The final torque is the same, and the radius of the crankarm is also unchanged. However, in your question you constrained the ratio of chainring to cassette radius to be the same (that ratio varies identically as the circumference, i.e. the number of teeth does), thus nothing changes.

That's the ideal situation. However, in the real world, friction losses increase as a chainring or cassette gets smaller, but it's those losses are pretty darned small.

Oh yeah , you have to click on the damned pic.

Thanks. That is the clarity I was looking for. In my initial thinking, I neglected to consider that to keep the gear ratio the same, the cassette cog radius would vary in direct proportion to the change in chainring size, and therefore, as you state, the net final torque is the same at any given gear ratio. Thus, frictional losses, while small, would favor using larger chainrings and cassette cogs (at the expense of a few grams of increased weight).

I don't agree, that may be ok in a theoretical world.

I lean on previous experience. Look at the torque of a large wheel, say in an engineering shop driving a belt to a machine, the bigger the wheel the less the slip and the more useful energy reaches the load.

Also my experience in Auto Electrics tells me small diameter alternator pulleys are a problem, they slip! Many times I have fitted a larger pulley when changing from a 55 amp to an 80 amp alternator.

In my first reply, ask TT and Tri riders which uses less energy, big rings.

gclark8 said:

I don't agree, that may be ok in a theoretical world.

I lean on previous experience. Look at the torque of a large wheel, say in an engineering shop driving a belt to a machine, the bigger the wheel the less the slip and the more useful energy reaches the load.

Also my experience in Auto Electrics tells me small diameter alternator pulleys are a problem, they slip! Many times I have fitted a larger pulley when changing from a 55 amp to an 80 amp alternator.

In my first reply, ask TT and Tri riders which uses less energy, big rings.

Well, that's where science trumps agreement. Since bicycle is somewhere around 95-98% efficient (mechanical efficiency), the real world will follow theory pretty damned closely.

Alternator pulleys aren't toothed, are they now? That's right, they're not. They rely on the friction between the belt and the pulley to work. Same with the pulleys in an engineering shop. The reason they're getting less slip is because the belt is coming into contact with more pulley, so there is more friction between the two.

John M said:

at the expense of a few grams of increased weight

Careful now, of someone is going to come in here and argue that smaller cogs = shorter chain = less rotational mass and the whole thing "spins up" so much faster.😉

artmichalek said:

Careful now, of someone is going to come in here and argue that smaller cogs = shorter chain = less rotational mass and the whole thing "spins up" so much faster.😉

Better not let "Little jackie" read this topic, with her 24" wheel MTB and slicks. 😄