Simple Physics Question re SS Gearing

[dddd]

Originally Posted by bikinchris

This. The chain itself will have to articulate less and needs less energy to operate. Also, the chain pulling the rear cog will be much straighter when leaving the rear cog. Larger cogs and chainrings WILL allow the chain drive to be more efficient.
The ten tooth cogs on the modern systems is completely wrong in my opinion.

True, there is a proportional (to chainring size increase) reduction in the articulation of the links.

But, there is also a proportional increase in the number of links articulating per second.

The increase in swing angle of the links moving around smaller cogs is accommodated entirely under the condition of dynamic friction, while any increase in the number of links flexing incurs more in the way of static-friction threshold events (static friction being much higher than dynamic friction).

As I said earlier, the theoretical considerations are complex.

And I do think that the OP's suggested increase from 17t to 21t would perhaps have much less effect on efficiency than the aforementioned increase from 14t to 19t.
So, well less than 1 watt difference even at the relatively high output of 250W. How might that equate to the added weight and air resistance of the bigger sprockets and longer chain I wonder?

Still, I would expect the larger sprockets to increase the power transfer stiffness/resistance noticeably, seemingly giving this rider more efficient use of my muscles at higher force levels and cadences.

[oldpotatoe]

Must be winter....

[marciero]

.dup

[.RJ]

Originally Posted by dddd

As I said earlier, the theoretical considerations are complex.

If you collect all of your marginal grains together in one basket, you can redeem them for valuable cash and prizes.

[marciero]

Originally Posted by ColonelJLloyd

Thanks. Aside from friction, I'm curious more about what the energy required to get them both rolling at the same speed. If I took this to the extreme and say a 63x25 or even larger gear.

Usually we are concerned with power and force. We all have unlimited energy right? We just dose it out at different rates!

For a given force at the pedals it will take longer to reach a given speed for a larger gear ratio. You might ask what that time is or how it changes with gear ratio.

Ignoring friction and assuming wind resistance is proportional to velocity, the time it would take to get to reach a given speed v is -(b/m) log(1 - vb/F), where m is mass, b is drag coeff and F is force at the rear wheel*. Of course F is proportional to the pedal force, and the formula is only valid between 0 and the max possible speed of F/b. If you go from gear ratio R0 to different gear ratio R1 with same force at the pedals the force at the rear wheel is F/(R1/R0) and so we now get 1 - vb(R1/R0)/F inside the log. You need values for the constants to get actual times and speeds but the ratios you are talking about are very close. *53/21 to 43/17 is only about 0.2% and would only increase the time to go from zero to half the max possible velocity (with the original force) by about 0.3%. You get more granularity with larger differences. If you increased your ratio by 25% the time would increase about 42%.

*If you are mathematically inclined you can solve mx'' +bx' = F, then fix v = x' and solve for t

[benb]

Doesn't the larger chainring/larger cog combination also wear out more slowly?

Over a long enough time that means less of your time wasted and a really small savings in money, and possibly a little better for the planet.

Now round these parts we probably sell the bike and buy a new one before that matters though right?

[Bob Ross]

I just want to note that back in 2006 I asked the exact same question as OP (although I wasn't specifically asking about single-speed bikes, just chainring+cog combinations that yielded the same gear-inches) and reading this thread today, >18 years later, is the first time I've seen a comprehensive answer!

Thanks Paceline!

[Mark McM]

Originally Posted by dddd

True, there is a proportional (to chainring size increase) reduction in the articulation of the links.

But, there is also a proportional increase in the number of links articulating per second.

It is true that the total articulation is the same (angle of link articulation x number of links articulating). But the with the larger chainrings/sprockets, the chain tension force is lower for a given torque. The force in the articulatin links means the friction force is smaller, thus resulting in less friction loss with the larger chainrings/sprockets.

Originally Posted by benb

Doesn't the larger chainring/larger cog combination also wear out more slowly?

Yes, due to both the lower chain force, and the distribution of wear across more teeth.