# Bicycle Gearing Question

1. Apr 6, 2014

### ldalcomune

Hey guys, I read through the other two bicycle gearing threads and to be quite frank I'm not sure if my question was answered.

So to put it in it's most basic element, let's assume there's only one change, the gearing. Same bike, same crank length, pedal input, wheelsize, chain length. And let's not include inertia, chain deflection, etc...

So the bicycle has a 38 tooth front sprocket and a 11-36 cassette. We swap it out to a 34 tooth sprocket and keep the same cassette. So obviously torque will be increased and speed decreased because of the gearing change, but the question is; Is there any increase in mechanical advantage, or torque at the rear wheel by going with a smaller front sprocket alone? (Assuming cassette was changed to keep the ratio the same)

Hope that makes sense. Thanks

Alternatively and probably more simply. Assume a single front and single rear a sprocket.

40 front, 20 rear.

And

34 front, 17 rear

Both are a 2:1, so as far as gearing goes torque at rear wheel should be the same? Or is there a mechanical advantage with the 34

Last edited: Apr 6, 2014
2. Apr 6, 2014

### Staff: Mentor

Changing the front sprocket to one with fewer teeth and changing the cassette (which in olden days was called the freewheel) to one with smaller gears would make essentially no difference. I say "essentially" because reducing the sprocket by four teeth, the ratios with the smaller gears in the cassette would probably not be the same. After all, you can't run a cog in the cassette with a fractional number of teeth - the number of teeth has to be a whole number.
Yes, exactly the same. With a 2:1 ratio, the crank goes around twice for each revolution of the rear wheel. If you had a 34 front and 34 cog in the cassette, the ratio would be 1:1, meaning that the crank would go around once for each revolution of the rear wheel.
No, not with the corresponding change in the cassette.

3. Apr 6, 2014

### ldalcomune

Thanks a lot. That's what I figured but it still lingered.

4. Apr 6, 2014

### sophiecentaur

A terminology thing here: We may as well try to get it right, for when the problem gets more advanced.
The ratios that you are using give the Velocity Ratio and not the Mechanical Advantage.
Efficiency = MA/VR
And it's never Unity - the value will depend upon the actual ratio in each case, for a start.

5. Apr 6, 2014

### ldalcomune

Yeah my question was about mechanical leverage.

For that given velocity ratio vs the other, does either provide a change in mechanical leverage, advantage, etc

6. Apr 6, 2014

### sophiecentaur

That is not an 'official' term, I think. MA and VR are generally accepted terms, used for all types of Machine, from inclined plane to gearbox and are formally defined. Counting sprocket teeth can only give you VR. The efficiency with different ratios will depend upon specific details of the setup.
In the case of a bicycle, the dimensions of the rider's legs and his / her preferred loading and pedalling speed will affect the suitability of any particular design. I really don't know whether it's possible to arrive at an optimum for a given rider without some road testing.

7. Apr 6, 2014

### ldalcomune

Sounds good. Thanks for the info

8. Apr 6, 2014

### jim hardy

you say "...smaller front sprocket alone..." and then say "..Assuming cassette was changed to keep the ratio the same""

which is it?

9. Apr 6, 2014

### ldalcomune

Both.

Let's assume I have a 38t now. I swap to a 34 and have a theoretical cassette that leaves the same ratios.

Velocity ratios being the same, the only tangible change being the smaller front sprocket.

Does the smaller sprocket alone lead to an increase in torque?

10. Apr 6, 2014

### ldalcomune

Taking the cassette out of the equation it's basically the single speed problem.

40 front, 20 rear
34 front, 14 rear

Same velocity ratio.

But does the smaller front ring apply a different amount of torque to the rear

11. Apr 6, 2014

### jim hardy

The first thing the smaller front sprocket does is pull harder on the chain.
Next the chain pulls on the rear sprocket.

(the following assumes constant pedal effort and your sprocket radii are in proportion to number of teeth )

making the front sprocket smaller applies more tension to the chain for a given pedal effort,
torque = chain tension X sprocket radius = pedal effort X crank length

but when the rear sprocket is also made smaller to keep gear ratio the same, its smaller diameter turns that increased chain tension back into same torque as you had initially.

Front sprocket went from 38 to 34, so to keep same ratio rear must've gone from 19 to 17

at front sprocket, chain tension = torque /sprocket radius and that ratio went up by 38/34
and at rear sprocket, torque = tension X sprocket radius and that ratio went down by 17/19

so the smaller front sprocket will only give more torque at wheel if you keep the big rear sprocket.

Surely somewhere among the twenty-one or so speeds you can find two that have same overall ratio and run an experiment?

12. Apr 6, 2014

### jim hardy

well, i guess i finally read the question:

operating at higher chain tension i would think only accelerates wear and increases friction .

To be scientific about it -
One could assert that the chain only engages the leading or trailing tooth at each sprocket, and friction loss there is in proportion to load(chain tension).... then sit back and await rebuttals to that assertion..

13. Apr 7, 2014

### sophiecentaur

The friction force is probably affected as you (so reasonably ) say but the actual work done against the friction relates to the distance moved (circumference of the sprockets etc.) so it can't be as simple as you suggest. (Sneaky - eh?)
But all of that's only dealing with the mechanical part of the system, though. Imo, the extra factor of the rider himself (body dimensions etc.) is going to make a lot of difference to the performance of bike+rider. It's more a matter of matching than anyone seems to be acknowledging here. After all, 'good' cyclists all have a favourite bike configuration and that must be more than just the colour of the frame making them feel better. Length of pedal crank will contribute to the leg / lever system and 'cm per turn of the pedal' will relate to the metabolism in the individual's muscles.

What I am suggesting is that, whilst there will be some very broad limits for bike dimensions that encompass a 'good setup' there is no way to determine a 'best' setup until the rider has also been characterised. i.e. this is not just an Engineering problem.

If we wanted to know the ideal design for a motor car transmission system, the very first thing we would want to know would be the characteristics of the engine that's going to be used.

14. Apr 7, 2014

### jim hardy

hmmm indeed, chain friction force X circumference of sprocket = friction energy per revolution....
sneaky maybe, brilliant certainly.

Your observation about characterizing the operator is spot on. I would prefer less chain tension for no real reason other than longevity and perhaps smoother shifting when going uphill.

In a motorcar transmission i like really wide spacing of the ratios so i can creep up a steep driveway. A street racer might prefer narrow spacing so he can zoom through traffic whilst keeping engine rpm in the range where exhaust resonates loudly. (Ever heard a Royal Enfield 700 with Dunstall megaphones? Ahhh nostalgia...)

Last edited: Apr 7, 2014
15. Apr 7, 2014

### sophiecentaur

Brilliant, maybe, for once - but I haven't quoted Lavoisier and Bacon for some while. (Unlike some of our learned PF members )

16. Apr 8, 2014

### Baluncore

If the tooth count ratio of the front sprocket to the selected rear sprocket remains the same, then there will be no overall difference to the mechanical advantage.

Given that varied people may ride the same bicycle, there are two constraints on sprocket selection.

Firstly, the number of teeth on the smallest rear sprocket that can be used will be fixed by the roller chain geometry. I expect it is about 12 teeth.

Secondly, (for a fixed pedal crank length), the minimum front sprocket diameter will be set by the strength of the chain. Weaker chains need to travel faster on bigger sprockets, or they will break.

The angle between adjacent chain links changes several times per circuit of the chain, but only two of those rotations occur while under full tension. With bigger sprockets, each chain pin rotates through a lesser angle in the roller. The chain tension is also less with bigger sprockets, so bigger sprockets reduce chain wear. Significant chain wear takes place within the rollers, a chain therefore gets longer as it wears. It then runs higher on the tooth faces. If not replaced when worn, then when under load, the chain will climb progressively out of the sprocket and skip one link.

17. Apr 8, 2014

### ldalcomune

Sounds good guys. Thanks a lot

18. Apr 12, 2014

### sophiecentaur

So, what is your definition of Mechanical Advantage? Mechanical Advantage is supposed to be a term that refers to the actual effort needed to do a job. Where does the efficiency (dead weight and friction etc) come into your statement? The rest of your post is concerned with practical matters so why ignore them in your (misplaced) use of the term MA?
(I am only 'nit picking' in the same vein as you find everywhere else in PF with examples like Mass and Weight, Force and Momentum, Heat and Temperature etc.. Do we want to get these things right or not?)

19. Apr 12, 2014

### Baluncore

I disagree, you are nit picking because your analysis of roller chain friction was quite unrealistic. You are now attempting to deflect attention by putting others down through criticism of a term used in the OP.
“Effort” is an interesting term. I wonder how many different meanings that might have in a scientific context.

Wikipedia has a better description of Mechanical advantage in it's first paragraph.