Gear Ratios

[pinkbikes]

Sheldon's gain ratio takes the crank length into account. I think it's better than gear inch. Besides, it is dimensionless.

Sheldon's formula takes crank arm length into account and works on gain ratio, which gives the added benefit of taking into account the greater/lesser lever arm of longer/shorter cranks. Strictly speaking it sort of introduces a "feel meter" for two sets of identical gearing with different crank lengths.

I find this is only a useful add-on if you are using it compare perceived difficulty of the same gearing ratios on bikes with different lengths of crank arm. Which is of course completely valid and very useful, especially if you are thinking of changing your crank arm length and want to know how it may make your existing gearing "feel" compared to how it used to.

But if you have your set of gears already and you are already set up with a suitable crank arm length for your anatomy (or even have multiple bikes with the same length of crank arms) then all it adds is mathematical complexity and it obscures the straight comparison of the actual gear ratios on different bikes.

Don't get me wrong - Sheldon was an absolute genius and his formula has valid uses - but really most of the world talks gear inches when comparing apples with apples (even in Europe and Oz where metric is next to Godliness! Gear centimetres???).

I'm not sure why anybody would prefer a "dimensionless" answer because frankly a dimension is much more meaningful in outright terms. A gain ratio of 6.6 means very little unless you are comparing it to a gain ratio of 8.4 or 3.1. Whereas a 100inch gear is a gear where you will travel pi x 100inches for each time you turn your pedals through one rotation. And hell - that is a pretty long way whatever length your crank arm is!!!

So..... for the life of me I have always pondered why they didn't just slap the damned pi into the formula there and make the dimension of gear inches actually mean something more representative - like how many inches you will travel for the one full crank rotation (maths geek - quite the waste of time really to ponder these things)?

I guess it is just that the numbers would have been so much larger and the pi is only a constant anyway, so why bother? Hmm - still a waste of time to ponder.....

[Over50Newbie]

Woo-hoo! Thanks PinkBike - I'm so glad that I did it right.

And, once again, thank you to everyone that responded to my question. I love that we all help each other out on this site.

Lynette

[wildeny]

I'm not sure why anybody would prefer a "dimensionless" answer because frankly a dimension is much more meaningful in outright terms.

A dimensionless quantity is more universal. No matter where you live, you get the same gain ratio from the same configuration.

In US/UK, you use gear inch: gear size = Wheel diameter (in inch) x Chainring teeth/Cassette teeth
but people in other places may use gear meter, which is the "roll-out" distance (or gear centimeter )

Gain ratio is about leverage. More precisely, the torque produce by your feet vs the torque by the rolling friction. The minimum torque you need to apply is equal to the torque by the rolling friction.

Torque = force * lever arm, which leads us to the minimum force you apply:

F_c = gain ratio * F_r [c = cyclist, r = rolling friction]

For the same rolling friction, the higher the gain ratio, the more forceful your pedaling. (or just consider F_r = 1)

This combines with the cadence is related to your power (that's what the power meter measures).

Gear inch/Gear meter tells you how far you can go when pedal one turn, while gain ratio tells you how much effort(work) you make(do).

However, no matter which method you use for gear size, all of them don't consider the effect from the tire (surface roughness, pressure).

[pinkbikes]

A dimensionless quantity is more universal. No matter where you live, you get the same gain ratio from the same configuration.

In US/UK, you use gear inch: gear size = Wheel diameter (in inch) x Chainring teeth/Cassette teeth
but people in other places may use gear meter, which is the "roll-out" distance (or gear centimeter )

Gain ratio is about leverage. More precisely, the torque produce by your feet vs the torque by the rolling friction. Two should be equal to each other under the ideal condition.

Torque = force * lever arm, which leads us to

F_c = gain ratio * F_r [c = cyclist, r = rolling friction]

For the same rolling friction, the higher the gain ratio, the more forceful your pedaling. (or just consider F_r = 1)

This combines with the cadence is related to your power (that's what the power meter measures).

Gear inch/Gear meter tells you how far you can go when pedal one turn, while gain ratio tells you how much effort(work) you make(do).

However, no matter which method you use for gear size, all of them don't consider the effect from the tire (surface roughness, pressure).

As an engineer I am pretty comfortable with the physics of torque. But I think you may be a little confused in your explanation here. In ideal conditions I think you would certainly NOT want the torque created by your feet and the torque created by the rolling friction to be equal, or you would not overcome the rolling friction and you wouldn't go anywhere!

You then give a formula where you insert the gain ratio so that the torque produced by the cyclist is equal to the gain ratio multiplied by the torque from rolling friction. So what you say there is that in ideal cases they are NOT equal? Which is it?

Also I think that you may be a little confused - gear inch/metre does not tell you how far one revolution takes you because the formula does not involve pi, which is required to develop the circumference of the wheel from the diameter.

Wildeny, I don't disagree with you that Sheldon's formula is great and I am quite comfortable with the physics involved in its many applications. But the OP just wanted to know which gear was higher, not compare different rolilng frictions, surfaces etc, so gear inches is just a simpler method of doing what she wanted without all the rocket science!

[wildeny]

pinkbikes, I do not disagree with your reason for using gear inch. As you already pointed out.

My previous reply was to explain the dimensionless quantity and the principle behind the gain ratio.

In ideal conditions I think you would certainly NOT want the torque created by your feet and the torque created by the rolling friction to be equal, or you would not overcome the rolling friction and you wouldn't go anywhere!

You then give a formula where you insert the gain ratio so that the torque produced by the cyclist is equal to the gain ratio multiplied by the torque from rolling friction. So what you say there is that in ideal cases they are NOT equal? Which is it?

Thanks for pointing it out. I should say that that is the minimum force you need to apply so that the wheels can start to turn. The minimum force is just to let the object remain at its status (if it's at rest, it's still at rest; if it's moving, it still moves with the same speed).

Also I think that you may be a little confused - gear inch/metre does not tell you how far one revolution takes you because the formula does not involve pi, which is required to develop the circumference of the wheel from the diameter.

I checked the definitions in "The Long Distance Cyclists’ Handbook" by Simon Doughty before writing the previous post. There it said the gear size in meters is "development", i.e. the "roll-out" distance. Also in Sheldon's page. Maybe I shouldn't use the term of gear meter here.

Even though gear inch does not include "pi" in the formula, it is based on the same idea without multiplying "pi". (imo, neglecting the multiplication of pi is jut to make the calculation easier)

What I'm trying to say is that gain ratio is not more complicate than gear inch. Their formulas only differ by the crank length, although the thinking is slightly different:

Gear inch[/Gear meter] tells you how far you can go when pedal one turn, while gain ratio tells you how much effort(work) you make(do).

That difference is important when comparing the gear inches from different bikes. Of course, for someone who always uses the same crank length, he/she doesn't need to consider the effect from the crank length, as you already mentioned. But, why not try to understand the effect of the crank length in gear size and know how to quantify it? [so that one can understand why his/her taller fellow can pedal with ease by the same gear inch]
(Note: the crank length is related to bike fitting too)

My intention in my first post was only to point out Sheldon's gain ratio. I'm sorry if my following explanations confuse people or make people feel more uncomfortable with gain ratio.

[pinkbikes]

The sun did not shine.
It was too wet to play (ride).
So I sat in the house
All that cold, cold, wet day.

Hmm - the lengths rain-induced boredom will drive me to! I think we basically agree. I just get a bit OCD when it comes to accuracy in mathematics and physics. I guess it is my job to care about such things!

I think we agree that there are different tools and that each has benefits used in certain applications and I have no discomfort in understanding or applying any of them in the circumstances that fit. I have certainly not meant to imply otherwise.

A wise(a$$?) engineer who sits near me at work and comes out with sage sayings is fond of saying "An engineer is somebody who is trained to do with $2 what anybody could do with $100." I have been conditioned therefore to select the method of least complexity (needing fewer terms) and greatest efficiency to get the job done (answer the OP's question).

Your choice to stretch the mind further with the choice of a different tool is I'm sure the result of a different set of conditioning in your makeup! I'm sure between us at least we've improved the world's statistics on mathematical illiteracy just a little?

[Over50Newbie]

As I said before, that's why I love this site.

Thank you, ladies, for such an intelligent and respectful discussion.

Way over my head, but I'm sure very valuable to those who understood it!

And, as a bonus, I now know the answer to my original question!

Lynette

[OakLeaf]

A wise(a$$?) engineer who sits near me at work and comes out with sage sayings is fond of saying "An engineer is somebody who is trained to do with $2 what anybody could do with $100."

As a college math major and daughter of a high school math teacher, that reminds me of the old joke about the engineer, the physicist and the mathematician.

They're all staying in a hotel for a convention. In the middle of the night, the engineer wakes up to find that her trash can is on fire. She grabs the ice bucket off the dresser, runs to the bathroom, fills it with water, dumps it on the fire, and goes back to sleep.

Later the same night, the physicist wakes to find that her trash can is on fire. She glances at the amount of waste paper in the can, estimates the distance between her bed and the bathroom sink, hurriedly boots up her computer, furiously works calculus equations. Then she gets up, grabs the ice bucket off the dresser, runs to the bathroom, draws the exact amount of water necessary to put out the fire, dumps it on the trash can, and goes back to sleep.

Last, the mathematician wakes to find her trash can on fire. Without delay she boots up her computer and furiously works equations. After long minutes pass, she exclaims: "The problem has a solution!" And goes back to sleep.


/hijack

[pinkbikes]

Rotfl!

WAY to many good engineer jokes... and unfortunately far too many grains of truth in all of them!!