Treebeard's Stumper Answer

Gearing Up For a Ride
Winter is turning into spring. It's a perfect time to tune up your bike and go for a ride! The most confusing part of biking for kids is surely the gears. My old mountain bike has 3 chain wheel gears in front that turn with the pedals and have 28, 38, and 48 teeth. I also have 7 freewheel gears in back with 13, 15, 17, 20, 23, 26, and 30 teeth. That makes3 x 7 = 21 combinations. How does this work? What is the shifting pattern from low (for climbing) to high (for cruising)? Which gear combination is in the middle? Would even more gears make biking easier?
This is a live image map. Click on a bike part name to get some good practical info from UtahMountainBiking.com. Bicycles are efficient and elegant machines, but they are only "simple" for an engineer. Kids find the gears complicated, but I'm more confused by the hidden bearings in the headset and the bottom bracket. By the way, this bike is way too clean to be mine! The real stumper is to understand your own bike. I put a paper clip on one gear tooth of each cog and counted my way around one by one. Then find your calculator and figure the gear ratios.
My bike's low granny gear uses the small cog in front and the large rear cog for a gear ratio of
28 / 30 = 0.93 , so every full pedal is less than one full turn of my wheels. That's power for climbing. My highest gear has a ratio of48 / 13 = 3.7 , so I go four times farther with every pedal, good for cruising. But in between, all successive shifts involve using both front and rear shifters in a complex pattern. It's easy enough to find a gear that works, but for competitive biking, those complicated shifts might make a splitsecond difference. Keep reading for the details.Notes:
It's easy to figure the gear ratios on your bicycle. Count the number of teeth on the front chainrings and the rear freewheel cogs. If your front chainring has 48 teeth and the rear freewheel cog has 24 teeth, then every time you pedal around once, the chain moves around 48 teeth. That means your 24tooth rear cog  and your rear wheel  goes around twice. That would provide a ratio of 1to2 or 1:2.00. If your rear cog has 12 teeth, the ratio would be 1to4 or 1:4.00, so you would go farther with every pedal. A higher ratio means your effort is applied over a greater distance, which is good for cruising but not for climbing. The gear ratios aren't as neat on my bike, but it works the same way.
I started to figure my gear ratios with a calculator, but then I found Barry Masterson's handy Gear Inch & Shifting Pattern Calculator web site that does it all at once. The hard part is counting the teeth on each gear cog, but you only have to do it once.
First I submitted the data for my bike:
The result is a confusing table of data, but it's simpler than it looks. The important numbers are the gear combination (CRxFW) and the gear ratio (RT). All 21 gear combinations are shown in order from low (for climbing) to high (for cruising):
 3 front chain wheel cogs with 28, 38, and 48 teeth
 7 rear freewheel cogs with 13, 15, 17, 20, 23, 26, and 30 teeth
 26 inch diameter wheels
Wheel Diameter: 26.00 inches Gears: 28/38/48 13151720232630  SP CRxFW GI GIdf DI diff DF PRPM RT MPH  1> 28x30 24.27 15.38% 76.24 11.73 6' 4.24" 831.10 1:0.93 4.33 2> 28x26 28.00 13.04% 87.96 11.47 7' 3.96" 720.29 1:1.08 5.00 3> 28x23 31.65 4.05% 99.44 4.02 8' 3.44" 637.18 1:1.22 5.65 4> 38x30 32.93 10.53% 103.46 10.89 8' 7.46" 612.39 1:1.27 5.88 5> 28x20 36.40 4.40% 114.35 5.03 9' 6.35" 554.07 1:1.40 6.50 6> 38x26 38.00 9.47% 119.38 11.31 9'11.38" 530.74 1:1.46 6.78 7> 48x30 41.60 2.94% 130.69 3.84 10'10.69" 484.81 1:1.60 7.43 8> 28x17 42.82 0.31% 134.53 0.42 11' 2.53" 470.96 1:1.65 7.64 9> 38x23 42.96 11.74% 134.95 15.84 11' 2.95" 469.50 1:1.65 7.67 10> 48x26 48.00 1.11% 150.80 1.68 12' 6.80" 420.17 1:1.85 8.57 11> 28x15 48.53 1.79% 152.47 2.72 12' 8.47" 415.55 1:1.87 8.66 12> 38x20 49.40 9.84% 155.19 15.27 12'11.19" 408.26 1:1.90 8.82 13> 48x23 54.26 3.21% 170.47 5.46 14' 2.47" 371.69 1:2.09 9.69 14> 28x13 56.00 3.78% 175.93 6.65 14' 7.93" 360.15 1:2.15 10.00 15> 38x17 58.12 7.37% 182.58 13.45 15' 2.58" 347.02 1:2.24 10.37 16> 48x20 62.40 5.56% 196.04 10.89 16' 4.04" 323.21 1:2.40 11.14 17> 38x15 65.87 11.46% 206.93 23.70 17' 2.93" 306.20 1:2.53 11.76 18> 48x17 73.41 3.53% 230.63 8.13 19' 2.63" 274.73 1:2.82 13.10 19> 38x13 76.00 9.47% 238.76 22.62 19'10.76" 265.37 1:2.92 13.57 20> 48x15 83.20 15.38% 261.38 40.21 21' 9.38" 242.41 1:3.20 14.85 21> 48x13 96.00 0.00% 301.59 0.00 25' 1.59" 210.08 1:3.69 17.14  SP CRxFW GI GIdf DI diff DF PRPM RT MPH Here's what it all means, and how it's calculated. Try to figure out the formulas!
(CR=front chainring teeth, FW=rear freewheel teeth, WD=wheel diameter.)
SP Shifting Pattern from 1 (low, for climbing) to 21 (high, for cruising). CRxFW Each ChainRing x FreeWheel combination (front x rear) GI Gear Inches is the traditional way to measure bike gears, calculated as Gear Ratio x Wheel Diameter. It's a biker's way to compare gears combinations on bikes with different size wheels. It's also the size of the front wheel on an equivalent old fashioned PennyFarthing or highwheeler bike. (Does this have something to do with torque and moment arm and mechanical advantage?)
GI = WD x (CR / FW) GIdf Percent increase between this GI value and the next GI value. DI Distance traveled in inches with a single pedal rotation
DI = GI x pi diff Difference in inches between this DI value and the next DI value. DF Distance traveled in feet with a single pedal rotation
DF = DI / 12 PRPM Pedal Rotations Per Mile
PRPM = 5280 / DF RT Gear Ratio
RT = CR / FW MPH Miles Per Hour in this gear with a cadence of 60 rpm. Cadence is a bikers' term for how fast you're pumping. The ideal is to always keep peddling at the same rate and use the gears to adjust the terrain. My own cadence is far from ideal!
MPH = (DI x 3600) / (12 x 5280) Really, all these numbers just show the same basic gear ratio data in different ways. In my 12th gear (for example), I'm on the middle 38tooth cog of my front chainwheel and the middle 20tooth cog of my rear freewheel. The gear ratio is 38 / 20 = 1.90. When I pedal around once, the chain will move around 38 teeth, so my rear 20tooth cog goes around almost twice, rotating the rear wheel 1.90 times. I'll travel about 155 inches on my 26 inch wheels, or almost 13 feet with every rotation of my pedals. I'll have to pedal 408 times to go a mile in this gear, and I'll go about 9 miles per hour if I keep pedalling at that cadence of one turn per second (or 60 rpm).
That's mostly trivia for hardcore bikers. More important is that this gear is less than 2% different from my next lower gear, but almost 10% different from my next higher gear. It's not worth shifting down for such a small advantage, but it will make a big difference if I shift up.
My truck has a 5 speed stick shift, with the simple shifting pattern written on the shifter knob. It's more complicated with my bike. Here is my bike's shifting pattern, with the gear cogs numbered 13 and 17 from insidetooutside like they are on my shifters:
My Bike's
Shifting ChartSmall
1  28Medium
2  38Large
3  481  30 1 4 7 2  26 2 6 10 3  23 3 9 13 4  20 5 12 16 5  17 8 15 18 6  15 11 17 20 7  13 14 19 21 I expected my bikes' shifting pattern to be complicated, but not this complicated! Except for the extremes, all successive shifts involve using both front and rear shifters in a complex pattern. But ten of my shifts give less than a 5% advantage, and several combinations are virtually identical. And I shouldn't use the innerouter combinations like 2813 and 4830 that can grind down my cogs and chain. If I had more gear combinations, there would be even more repeats. More gears are not better unless they are the right gears!
It's interesting that my gear in the middle has a ratio of
{0.93 + (3.69  0.93) /2} = 2.31 , which is between my 15th and 16th gear combinations, near the high end. Sure, I need more control when climbing hills.Usually I think of my front gears as 1low (for hills), 2medium (for flats), and 3high (for downhill). Then I use the rear shifter to adjust until I run out of room and need to change the front. Sometimes on steep hills I wish I had a lower front gear, but by then it's easier to walk and push. I tried shifting both gears at once in the mathematical order, but I can't do it very well so it's not worth it for me. In fact, racing bikes have (or used to have) the shifters down on the bike frame which would make it impossible to shift both front and rear gears at once. Maybe that's why racing bikes usually only have two front chainwheel gears. Road racers have fewer gear combinations, but they can shift fast in a meaningful pattern to the right gear using just one shift at a time in a crossover pattern like this:
Racing Bike
Shifting ChartSmall Large Large 1  2 2  3 3  4 4 5 5  6 6  7 Small  8 Competitive bikers must spend a lot of time thinking about the ideal gear combinations for their style of riding. I recieved this email from Cynthia about a conversation with a recent inductee into the Mountain Bike Hall of Fame in Crested Butte, Colorado:
I asked him how consciously and constantly serious bike riders think about their gear ratios, and he said that at every moment, the cyclist knows precisely what gear he is in. Precisely. They even use certain ratios as part of their strategies and sometimes try to shift very subtly so that the opponent doesn't know. A guy will say, "I went up that hill in a 42x19," and a knowledgeable cyclist will understand a hundred things about the climb and the way it was handled just by those numbers. (Gosh, I never realized this!) He said for example, someone will say, "I did that descent in a 53x13," and that instantly tells you it must have been a very steep hill, maybe a tailwind also, and definitely the guy was moving fast. It's like a whole numerical language I never understood. And they skip around, playing the gears like a musical instrument, intuitively knowing exactly what they are doing. Now I understand [his] frustration with me when we ride together  he will take one look and tell me I'm in a very stupid gear. It also helps me to understand something he told me about bike racing: Assuming you have the right fitness and body for it, "It's 90% between the ears, and 10% being able to withstand pain a little bit longer than the next guy."Fortunately, the rest of us can go for a bike ride in the country on a beautiful spring day and usually find a gear that is good enough!Here are some links for further Web research on bike gears:
 The real stumper is for you to figure out your own bike gears. I used Barry Masterson's handy Gear Inch & Shifting Pattern Calculator. There's another JAVA calculator here, but it didn't allow my gears. You can find shareware bike gear calculator software like this with a Web search. I haven't tried it.
 I found help for this stumper at UtahMountainBiking.com, Chris'z Corner Tech Tips, Ken Kifer's Cycling Cadence and Bicycle Gearing, and Hardcore Bicycle Science. I'm sure there are many more relevant sites.
 The San Francisco Exploratorium has a site about the Science of Cycling, including a page of links. We visited The Exploratorium on our recent Dunn Middle School trip to San Francisco. I'm impressed that it's still a fine institution without a corporate presence.
 There is an interesting math puzzle here to find the most useful combination of chainwheel and freewheel cogs that give an advantage of 10% or so per shift. This kind of math is called Linear Programming and the main strategy is the simplex algorithm. Here are links to an FAQ and a tutorial.
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