One gear…….. one idea

As i’ve mentioned in other posts, I ride a Condor Pista singlespeed (untill the Hardwood Single Speed is finished!). It’s my daily commuter into London, my weekend sportive ride, my tourer with paniers for long trips….. I think you get the drift. Like most other single speed and fixed gear riders, I always get asked by my geared friends why I make life so hard for myself by riding with only one gear. The short answer is that I don’t think I do. There are very few rides around London, including the North and South Downs and the Chilterns that are out of the range of one gear. I usually get dropped on the downhills as my legs don’t spin fast enough but uphill is never as bad as I think it should be. For most rides I ride 44t chainring with a 16t sprocket, or 73.2 gear inches, changing to a 15t or 17t sprocket very occasionally for particular rides. I reckon I can get up anything up to about 15% on 44×16 which means Ditchling Beacon in the South downs is just on the limit for me on my standard gearing. The monstrous Yorks Hill (http://www.catfordcc.co.uk/hillclimb/about.aspx?sm=21_1) in the North Downs at 24% remains well out of reach though, and certainly as the 9th hill at the end of the annual King of the Downs!

So, that brings me on to the point of this post. I was out on a ride in the Chilterns in February last year and Charlie had plotted a route that had a hill, at about 50 miles in, that he was confident would defeat me (unfortunately I can’t remember where it was). I was certainly up for the challenge. The sign at the bottom of the hill said 20% but I got out of the saddle and sprinted ahead of him. It wasn’t that bad and I was within metres of the top when it ramped up to 20%…… and defeated me. It was February though so I hadn’t done much training and wasn’t particularly fit. I’d get up it next time. Right? The next time I met the hill was in September. By this stage I had done a lot of cycling over the summer, several centuries, lots of hills and a tough triathlon in the Lake District. I was the fittest I’d been all year so I should have no problem with the last few metres……but I only made it about a metre further than before. Why?

This bothered me for a while as I simply couldn’t work out why I was so much fitter but didn’t get any further (and because Charlie teased me about it).I thought it must have something to do with another funny thing i’d noticed before. That is, when I get defeated by Yorks Hill at the KoTD every year, I have to get off and push my bike at about the same point as all the slightly less fit guys riding carbon fibre compact chainset bikes (2 chain rings rather than a triple chainset). Why do they have to get off when they’ve got sooooo many more gears than me?

So, when no one was looking, I turned to trusty old Excel to give me a hand! As with most engineering problems we’re going to have to make some assumptions and simplifications to demonstrate our point in an easy way and you’re going to have to remember a bit of GCSE maths as well : (  Here’s my model for me going up a hill on my bike. I’ve ignored fricton in the bike parts and air resistance (both minimal in this example)The total weight of me and the bike is W, there’s a reaction force between the road and each of the wheels and there’s a force driving me up the hill. At the point that I fall off because I can’t push any harder, I’m stationary, so all these forces balance as shown in the little force triangle I’ve drawn in the corner. Now, as the hill gets steeper, to keep the forces balanced R gets smaller and F gets bigger. How do we work out F?

If we know the angle of the hill and we know the weight of the rider and bike then:
F = Wsinθ
and R1 + R2 = Wcosθ The force is applied by me standing on the pedals with a proportion of my weight (xW) causing a tension in the chain. The chain then pulls on the rear sprocket which then turns the rear wheel causing a force F against the road surface.

There’s one more slightly trickier aspect in that you get maximum leverage on the cranks when they are horizontal and you get nothing when they are vertical. We’re going to use an average horizontal distance from the centre of the pedal to the centre of the bottom bracket as below.

I’m not going to bore you with the equation so you’ll have to take my word that the answer is that the average horizontal crank length = 0.637C or 63.7% of the full crank length.

Lets try an example on a 10% hill (θ = 5.7degrees) with my 44t (88.9mm radius) x16t (32.3mm radius) gearing, 700×23 tyres (334mm radius) and 170mm long cranks. That means the average crank lever arm is 0.637 x 170 = 108mm

The Force F = 0.0995W (using F = Wsinθ)

So the chain tension is 0.0995W x 334mm/32.3mm = 1.029W (the ratio of wheel radius to sprocket radius)

This force goes into the chain ring to be resisted by my weight on the pedals (xW). So, the force on the pedals to exactly balance F is:

1.029W x 88.9mm/108mm = 0.85W (ratio of chain ring to the average crank lever arm)

That is, 85% of my body weight is required on one pedal to balance on a 10% hill. More than this and i’ll go forwards, less than this and i’ll most likely fall on my arse! To speed this up I made a graph.

This shows that, using all the sizes and gearing above I can almost get up a 12% hill by standing up with all my weight on one pedal. Now, I know that I can actually get up about a 15% hill so the graph tells us this requires 126% weight on the pedals. I ride with cleats so I must be pulling upwards with my other leg by 26% of my bodyweight + the weight of the bike. That’s 26% of 85kg + 10kg which is 25kg per pedal stroke. Sounds like hard work and explains why I can’t keep it up for very long before my legs give in.

Ok, you’ve made it this far and you’re still waiting for the punch line. Right? How does this answer any of the questions that I started out with? I know that I can apply about 126% of my body weight down on the pedals as an absolute maximum to get up the last bit of a hill. But, using the graph again, to get up the elusive final part of that 20% hill I’d have to go up to 167% of my body weight. Let’s say everything up to 100% is possible for everyone, as that is essentially just like walking up stairs with all your bodyweight on one foot for every step. I call it the bit you get for free as you are just using your weight to it’s full extent. Anything above 100% requires pulling up with your other foot which requires cleats, technique and most importantly a very different set of muscles that you don’t use very much when you’re sat in the saddle for 95% of the ride. So, even though I got a lot fitter over the season I certainly wasn’t 2.6 times stronger/ fitter (I’m saying that going from 26% extra to 67% extra is an increase of 2.6x) and this shows me that realisticallyI’m never going to get up that 20% hill in my normal cruising gear. Once, I go past that 12% boundary where I need more than just my body weight, each small increase in steepness requires a significant increase in effort and strength.

Also, knowing my 126% limit, I can now work out a gear ratio that will let me get up my 20% nemisis. Using the calculation above with the same 44t chainring, I’d have to switch to a 21t sprocket from my usual 16t (and even that wouldn’t quite get me there but it’s the closest without changing the chainring). That’s a change of 31% on the gear ratio meaning that I’ll be spinning my legs 31% faster for the rest of the ride, just to get up the final part of one hill. No thanks! It also shows that when I switch between my 15/16/17t sprockets it wont really change the maximum gradient of hill I can get up by very much. It’ll just change my cadence over the ride and make each of the hills very slightly easier/ harder.

Finally, back to the guys on their carbon fibre compacts, walking up Yorks Hill. To get up a 24% hill in a 34t chainring with a 26t sprocket means 94% body weight on the pedals. That certainly means that you’ve got to get out of the saddle and, assuming they’re already pretty tired from the 90miles of hills that preceeded Yorks Hill in KoTD, that could well mean a premature walk just a few metres ahead of me whilst the smug triple chainsetted riders (on 30×26) are still ploughing on with a mere 83% body weight, which is still quite a lot of effort considering they have a gear ratio that’s more than twice as easy as mine.

So, the point is that as long as I’m happy to get out of the saddle from time to time when I get to hills, the 44×16 gear ratio that I like using really doesn’t limit me until I get to the steepest of hills, at which point geared riders also start to struggle because small increases in steepness past your 100% point require you to shift through the gears very quickly to maintain the same pressure on the pedals.

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4 thoughts on “One gear…….. one idea

  1. Very interesting! I was looking on the ‘net to see if there was a relationship between gear inches (or ratio) and body/bike weight that could replicate a given gradient while on the flat, so as to train for hills while living somewhere non-hilly (in this case NYC).
    BTW, I ride a ss/fixed flip flop hub for commuting with about the same ratio as you (46/16-17) on a 175mm crank, 700c, and system weight of about 102 kg. Sustainable cadence drops into the low 50s on 4% grade, just as an example of where my fitness is.
    Sorry to natter on, but great to have found your work on this.
    cheers

    • Hi xyxax
      I’ll start with the short answer. There isn’t a gear ratio that you can easily use without making your own gears as the commonly available chainring and sprocket sizes won’t get you close. Before I go into the long answer, let me introduce you to http://www.analyticcycling.com. I’m a big fan of this website but I’m sure it’s a bit of an aquired taste. You can use it to quickly give you the answer to your question above and plenty of others.

      Going from hill climbing to the flat has many differences but biggest of all is speed. On hills you use most of your power overcoming your own weight. On the flat you only lose about 10-15% of your power due to rolling resistance and the efficiency of the drive system. The rest is air resistance which the drag force is proportional to speed squared and the power required to overcome it is proportional to speed cubed.

      First lets work out what your power output is on the hills. Using the info above and http://www.analyticcycling.com/GearSpeedCadence_Page.html you’re travelling at 11.91mph (5.32m/s) at a cadence of 52. Then, using http://www.analyticcycling.com/ForcesPower_Page.html, with an frontal area of 0.6m2 for out of the saddle riding, Cd=0.5, Weight=102kg, Cr=0.004, Slop=0.04, V=5.32m/s, Cadence=52, Crank=175mm, pedal range=70deg. That gives you a power of 262W on the hills.

      Now, you want to train on the flat with the same power output. So stick the numbers into http://www.analyticcycling.com/ForcesSpeed_Page.html but this time using a frontal area of 0.5m2 as you’ll be in the racing position on the flat, a slope of 0 and power=262W. That gives you a flat training speed of 11.23m/s (25.1mph).

      Finally, what gear ratio do you need to ride at 25.1mph and a cadence of 52 on the flat? Back to http://www.analyticcycling.com/GearCadenceSpeed_Page.html. So, you need a 73/12 gear ratio riding at 25.1mph at a cadence of 52 to have an equivalent power/cadence on the flat. You can’t get a 73 chain ring so the next best thing would be to go as heavy as possible which is probably 53/12 (I’m not sure you can get a 12t freewheel but you can definitely get a 12t fixed track sprocket). To hit the same power you’ll still have to ride at 25.1mph but now at a cadence of 71 and this is about as close as you can get.

      Now you’ve seen the process you can go through your own scenarios on analytic cycling but there’s no real alternative to just hitting the hills!

      Cheers
      Nick

      • Hey Nick,

        Thanks for taking the time to respond so rapidly and thoroughly. As it turns out, I have the Analytic Cycling site bookmarked but haven’t delved much into it (and may not be smart enough to have used it to find an answer). In fact, I first started thinking about it while indoors on the rollers. I am 199cm tall and all too painfully aware of wind resistance. The rollers take that force out of the equation, but add the resistance of the drums (in this case 2.25 diameter), tire pressure, etc. Replicating a hill climb on the rollers would be great (it’s 25 miles to the nearest climb worthy of doing repeats) to help train when time is limited. The Kreitler’s have a chart of empirically obtained power numbers for different speeds on different sized drums (for a 70kg rider) that could be used with the Analytic site to adapt to my own specs. My “A event” this year is to finish a Vermont ride (330km north-to-south with 3500m climbing) which I finished just short of last year before time ran out. Nothing hellishly steep, just 10-12% in long ladder-like climbs. thanks again and great to meet your site.

        cheers

  2. I’ve never really trained on rollers so can’t confess to knowing much about how to tweak settings to get a particular resistance. If you knock out air resistance from the numbers above but still use 262W power, and a cadence of 52 then the only thing slowing you down is rolling resistance. This would need a rolling resistance coefficient of 0.05 (Compared with the rolling resistance of pumped up road tyres on the road of 0.004). So you need more than 10 times normal resistance. Do you have rollers with variable resistance or are they free running? If not then I doubt you can achieve enough resistance whatever you do unless you ride with your brakes on or apply tar to the rollers! You can probably get the right kind of feeling by increasing the resistance on a turbo trainer but you’ll just have to do that by feel though as Analytic Cycling doesn’t have a magic answer for that and I’m not about to start working it out!

    Cheers
    Nick

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