Gear Ratios – Pt 2

Find Gear Rations Pt 1 Here.

I’ve finally updated my gear rations spread sheet (almost 8 years later) to include 1x drivetrains. I also built it out for 2x drivetrains, but as 1x has become the defacto standard in the mountain bike industry, I’m going to skip the 2x’s for now. If anyone really wants to see the 2x spreadsheet, I’ll be happy to share it. Also, for the sake of brevity, I am focusing on 1×12 drivetrains only. If you want to see 1×11 or 1×10, I can share that as well.

Obviously the 1x spreadsheet is far simpler to view than the 3x. That said, lets jump in.

Columns A & B are inputs and their descriptors. A describes the data required, and B is the input field. Inputs are; tooth count of all 12 cassette cogs, chainring tooth count, crank arm length in mm, and wheel diameter (with tire, under load).

Columns D – I are the same as the 3x drivetrain sheet. Quickly, they are;

  • Column D: Simple Gear Ratio
    • Front chainring size (in teeth), divided by rear cassette cog size (in teeth).
  • Column E: Gear Inches
    • How a particular ratio equates to an old style high wheeler (penny-farthing) which were measured by the height of their front wheel.
  • Column F: Inches of Development
    • How far (in inches) you will travel along the ground in a given gear.
  • Column G: Feet of Development
    • How far (in feet) you will travel along the ground in a given gear.
  • Column H: Sheldon Brown Method
    • Sheldon Brown’s “radius ratio”
  • Column I: Sequence
    • Sequential order of gears. This is a hold over from my 3x sheet.

As you can see by comparing the two sheets, there is no way to get the same range from even a 1×12 as a 3×10, or even a 3×9. Your only adjustment is the change your chainring size and by doing so you will either sacrifice low end (climbing) or top end speed. That said, the tallest ratio on a 3×10 is really pretty theoretical. There are very few instances where you’re able to push a 4:1 ratio, unless you’re a pro athlete or on a long steep decent. And with the weight advantages of a 1x system (I shaved over a pound by converting one of my older bikes) I think the positives far outweigh the negatives.

Positives:

  • Weight
    • As I’ve already stated, there is a pretty significant weight savings by swapping over to a 1x system.
    • By moving from a 3×9 to a 1×10, I saved 1.1lbs on my old Bullit. This includes removing 2 chainrings and bolts, front derailleur, shifter pod and cable and housing. This also includes the increase of 1 rear cog and, obviously, a thinner chain. This weight savings is NOT inclusive of changing crank arms to a 1x specific set and the removal of the bolt bosses.
  • Better Chainline
    • All gears are usable, you’re not carrying anything you won’t or can’t use. Therefore the chainline is better set to work across the entire rear cog set.
  • Ridability
    • Without having to deal with the front mech, you are less likely to have a missed shift, dropped chain, chain suck and can more than likely, get to the gear you want quicker and more accurately.
  • No more dropped chains
    • I mentioned this above, but it bears further mention. By losing additional chainrings, we are now able to run Narrow/Wide chainrings which substantially reduce dropped chains.

Negatives:

The only real negative that I can think of (after being on 1×10 and 1×11 systems for over 3 years now), is really more of a perception, which is the reduced range of gears available. With the correct chainring for your riding style and terrain, you end up with 11 or 12 very usable gears. With a 1×11 on my Nomad, I can climb nearly anything here in the Santa Monica Mountains, and I can still hit the high 30s and (very) low 40s if I really try. And thats running out of leg, not gear.

Stolen Santa Cruz Nomad

This week was the Sea Otter Classic. My first time there. My friend Jim and I went up to experience the bike festival, and I was going to do my first official race; the Sea Otter Enduro. With the bikes safely locked to the Thule T2, we headed out.

 

Next stop, Sea Otter Classic. #seaotterclassic #seaotter2017

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Unfortunately, on our drive up, we stopped for lunch before heading into Monterey and the campgrounds. We pulled into the In-N-Out in Salinas, CA and since the drive-through line was long, we opted to go inside, eat and use the rest rooms. We were inside for no more than 15 minutes. When we returned, the Nomad had been ripped from the rack. Thankfully Jim’s Enduro was left behind.

What we’ve learned;

  • Bikes are stolen off of this block all the time. It makes sense, it’s a high traffic area, right off the freeway with lots of high dollar bikes on racks coming through town.
  • There are multiple hotels and drive-throughs so, high target area.
  • The guy that took my bike lives in the neighborhood (we think). Based on eye-witnesses (who we found and the Salinas PD never bothered to call [more on that later]) we know he knows the neighborhood well.
  • He uses drop bikes. He cruises the neighborhood on lower dollar bikes (also stolen) and when he finds a target he ditches that bike in the bushes or hedges, steals the bike and bugs out.
  • Eye-witnesses don’t slow him down. 2 women were parked next to my truck and watched saw him. When they tried to say something to him, he shot them a look, said something nasty and then split on my bike.
  • The Salinas PD is completely inept. To be fair, Salinas is crime ridden and they are severely understaffed. We never actually met a sworn officer, but we met with no less than a half dozen “community volunteers” who took my report, came out and picked up the drop bike, spoke to Jim, etc. Unfortunately, when I called back several days later, they had lost my report! It took a day or 2 from there to get my report in the system. Then as of several weeks later, they still had not contacted the witnesses. Based on conversations I had with several business managers in the area (in-n-out, 3 hotels, mc d’s), there are upwards of 20 bike thefts on that block every week; over 1000 bikes a year! Even if the average bike is worth 1/4 what mine is worth, that’s over $2million dollars a year in bikes being stolen, on that ONE block! Salinas PD really needs to so something about this.

Here is the link to Bike Index report

I got this bike directly from the Santa Cruz factory, through a special deal where my bike was built up by Doug Hatfield (Santa Cruz Syndicate team mechanic), I got to ride with the team and attend the factory holiday party. In addition, my top tube was signed by the Syndicate riders; Steve Peat, Josh Bryceland and Greg Minnaar.

Doug Hatfield building up my bike!

Steve Peat, Josh Bryceland & Greg Minnaar
Finished beast

My custom built wheels; Chris King limited edition Sour Apple hubs on Derby DH carbon hoops, built up with DT Swiss spokes & nipples by none other than Southern Wheelworks.

 

Limited edition sour apple @chriskingbuzz x #Derby #carbonrims = #baller #wheelbuilding #handbuiltwheels #southernwheelworks

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@chriskingbuzz #LimitedEdition #SourApple #wheelbuilding #southernwheelworks

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@chriskingbuzz #LimitedEdition Sour Apple is in the house! #wheelbuilding #southernwheelworks

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Gear Ratios

In as much riding as I have been doing lately, I’ve been thinking a lot about gear ratios. Specifically, how gear ratios on a bicycle overlap and how exactly the sequencing ends up laying out. In addition, I’ve noticed that I don’t necessarily use all of my available gears, but rather the lowest gear on the small chainring, the two or three tallest gears on the large chainring, and everything else I manage to do on the middle chainring. So, are there effective gears that I’m not using?

A typical modern mountain bike (at the time of this writing) has three chain rings in the front, and nine cogs on the rear cassette. By bike store math that makes it a 27 speed bike (3×9). But this is a bit of a misnomer as eight of those combinations are unusable due to the extreme angles and stresses that the chain would be under in order to accommodate certain ratios. For example, you should never put your chain on your large chainring and your largest rear cog at the same time; this is known as cross chain. By doing so you are subjecting the chain to undue lateral stress and shortening the life of your drivetrain. Of course, the opposite is also true (smallest chainring and smallest cog). So in effect, by not using the 4 largest rear cogs for the largest chainring, and the 4 smallest cogs for the smallest chainring, you’re left with a 19 speed bike and increasing the life of your drivetrain components.

My assumption has always been that there must be a lot of overlap within those ratios. That I could find, mathematically, two equivalent ratios, one on the small chainring and the other on the middle chainring. And again between the middle chainring and the largest chainring. Thus further eliminating additional ratios.

First, lets talk about the ways in which we measure these ratios, specific to a bicycle. Each more complicated and obscure than the last.

Simple Ratio
The first is a simple ratio which only takes into account front chainring size (in number of teeth) divided by rear cog size (also in number of teeth). E.g. 44/11 = 4.0:1. For every time you turn the cranks once, the rear wheel goes around 4 times.

Gear Inches
The second method of measuring these ratios is in “gear inches”. This is a holdover from those olden days bicycles that had one giant wheel in the front and one little wheel in the back. You see, those bikes dint have any gears, they were a single fixed speed with the pedal cranks were tied directly to this giant wheel, in effect, giving it a very “tall” gear. The taller the wheel, the taller the gear.

To compare a modern bicycle to one of these bikes, the above ratio (4.0:1) on a bicycle with a 27″ total  wheel height ( this equates to a 26″ mountain bike rim with a 26″x2.35″ tire) would have a gear inch measurement of 108″ or, would be equivalent to riding a Penny-Farthing or High-Wheeler with a 108″ front wheel diameter. This, of course would be impossible, unless you have an inseam of about 60 inches. The maximum height Penny-Farthing was about 60″ (for an extremely tall rider), but averaged closer to 50″. Obviously, geared bicycles already have a pretty big mechanical advantage over those old bikes.

The equation for finding gear inches is simple: Diameter of drive wheel (under load) in inches x number of teeth in front chainring / number of teeth in rear cog. Or; 27 x 44 / 11 = 108.

Development
One can take this method a step further and calculate distance traveled per crank revolution in a given gear, also called “Development”. This is also a pretty simple calculation, simply take your gear inches and multiply by π. Or, as in the above example, 108 x 3.14 = 339.12 inches of development. This means that when in this gear (44/11) you move forward 28.26 feet for every full revolution of the cranks. Traditionally this method is used by European cyclists and is referred to as Meters of Development. To find MoD, simply multiply your final inches of development by 0.0798. In this case we are making 27.061776 MoD.

Sheldon Brown Method
Mr Brown has long been considered an expert in all things cycling. He developed a unique way of determining development which references the distance a pedal moves, rather than a single revolution to calculate distance traveled.  He uses crank arm length in conjunction with wheel radius to to create a single measurement he calls a radius ratio. In Sheldon’s words;

“This ratio would be calculated as follows: divide the wheel radius by the crank length; this will yield a single radius ratio applicable to all of the gears of a given bike. The individual gear ratios are calculated as with gear inches, using this radius ratio instead of the wheel size.”

For example, to find the radius ratio on my particular bicycle, divide the wheel radius (13.5″) by the crank arm length in inches (6.8898″).

13.5 / 6.8898 = 1.9594 radius ratio

NOTE: It should be noted that crank arms are almost universally referenced in mm. To convert your crank arm length to inches simply multiply crank arm length in mm (in this case 175) by 0.0393700787. Which gives us 6.8898″ for our crank arms.

Next, in order to find development, we use the same calculation as in gear inches, replacing wheel size with this new radius ratio.  For example;

1.9594 x 44 / 11 = 7.8376

I know this seems complicated, but here is the beauty of Sheldon’s system:

“This number (in this case 7.8376) is a pure ratio; the units cancel out. I call this a “gain ratio”. What it means is that for every inch, or kilometer, or furlong the pedal travels in its orbit around the bottom bracket, the bicycle will travel 7.8376 inches, or kilometers, or furlongs.”

Pretty cool. But is it more accurate than simple Development (inches, feet, meters)? Or Gear Inches?

To help put this into context a little bit, I’ve built a ratio calculator. The screen shot below takes into account all of the above methods for my bike’s gear sets.

In rows 2, 13 and 24 you have my chainring sizes. In column J you’ll see my cassette cogs, repeated for each chainring. In columns D – H you have; Simple Ratio, Gear Inches, Inches of Development, Feet of Development and Sheldon Brown’s ratio.

You will also notice some coloring of certain areas. Red cells are gear pairs that are not used due to high lateral drivetrain loads (described above). Green rows are overlapping ratios shared by the small and middle chainrings (22 and 32 respectively). Orange rows are overlapping ratios shared by the middle and large chainrings (32 and 44 respectively).

In the “I” column, I have sequenced the gears; taking into account the gears which are eliminated (red) and overlap (green and orange), what this represents is the sequential order one would shift through all available gears going from lowest to highest.

While interesting to see a visual representation of our gears sets, it should also be noted that there are likely 2-4 more gears which could be eliminated from the “usable gears” within the overlap areas. I mention this because of the extremely tight nature of the gears in those areas. For example, using IoD, lets look at the differences between each gear all the way through the range;

  1. First Gear has an IoD of 58.2863″ of forward movement per revolution of the cranks.
  2. The shift from 1st to 2nd, increases that by 8.3266″ to 66.6129″.
  3. 2nd to 3rd: 11.1021″ to 77.7150″
  4. 3rd to 4th: 7.0650″ to 84.78.00″
  5. 4th to 5th: 4.0371″ to 88.8171″
  6. 5th to 6th: 8.0743″ to 96.8914″
  7. 6th to 7th: 6.7286″ to 103.6200″
  8. 7th to 8th: 9.4200″ to 113.0400″
  9. 8th to 9th: 16.1486″ to 129.1886″
  10. 9th to 10th: 21.5314″ to 150.7300″
  11. 10th to 11th: 18.8400″ to 169.5600″
  12. 11th to 12th: 24.2229″ to 193.7829″
  13. 12th to 13th: 13.4571″ to 207.2400″
  14. 13th to 14th: 18.8400″ to 226.0800″
  15. 14th to 15th: 7.0650″ to 233.1450″
  16. 15th to 16th: 13.4877″ to 246.6327″
  17. 16th to 17th: 19.8187″ to 266.4514″
  18. 17th to 18th: 44.4086″ to 310.8600″
  19. 18th to 19th: 28.2600″ to 339.1200″

Now, if we graph out each step, it looks like this. First we’ll look at the IoD of each gear;

It looks pretty smooth, right? Now, what if we look at the delta between each gear?

Now, what you’re looking at is the jump of each gear. My point being, if we eliminated just a couple of these steps, specifically if we didn’t use the gears I have called 5 and 7 as well as 13 and 15, we have a much smoother curve for each graph. We also have a much simpler shifting pattern by staying in the middle ring as much as possible, as well as the benefit of even greater reduction in drivetrain wear by reducing angles on a couple more of those gears.

But, what about those overlapped gears? Since there are a few sets that are, or are nearly identical, is there any advantage to one over the other? The simple answer is, no. A mechanical ratio of 2:1 is a mechanical ratio of 2:1. But what about that lever we are pushing on; What effect does that have? Given a constant lever length (crank arm), is there a benefit to 22/11 (2:1) gear set over a 32/16 gear set (also 2:1)? The only way this could be possible (and Im not sure if it is) would be because of the relative position of the 22 tooth chainring vs the 32 tooth chain ring, as they relate to the length of the lever (crank). Obviously, the 32 tooth is a fair distance closer to the end of the lever than the 22 is. So, what do you think?

  • Are these very similar ratios actually fundamentally same?
  • Does one have a mechanical advantage over the other? Can one “feel” the difference?
  • For that matter could mathematically identical ratios have a mechanical difference based on the relationship between the chainring diameter and the crank arm length?
    • For example, a 22/14 gear combination has a ratio of 1.5714:1 and 42.4286 gear inches given a 27″ diameter wheel (26×2.35 mtb tire). But so does a gear combo 44/28. For that matter, 22/11 and 32/16 are the same simple ratio of 2.0:1, with 169.56″ of development. But are these the same in terms of effort output?
      • Do they take the same amount of energy to make go around, or is one “easier” (even in perception) than the other?
      • Does effective chain length play any role?
      • How does crank arm length actually effect the effort to rotate it?

Ultimately, I think I’d like to build a rig to measure this. This could be done through simple  via foot pounds, or inch pounds of torque the effort each ratio requires to move a given amount of weight. But I wonder if this measurement would be sensitive enough.