How to build a fixie

Last week I built a fixie.  Here's how I did it.

First I went on Craig's List and looked for 1970s-ish steel bicycles that would make good fixies.  I tried a couple of sellers before settling on the orange Schwinn Varsity, which I bought for $120.  This is what the bike looked like:

I took of the wheels, the chain, the crankset, the rear brake, the front and rear derailleurs, and all the shifters.

From eighthinch.com I purchased an Amelia wheelset with a 16-tooth cog on the rear wheel, which cost $109.50, including shipping.  I also purchased a 44-tooth chainring for a one-piece crankset and an eighth-inch-wide single-speed chain from harriscyclery.net for $31.40, including shipping.  From a local bike shop I bought white Fizik bar tape and a new brake cable (I already had on hand fresh brake pads), which cost about $24.  In total I spent about $285 on the bike and parts.

The first step in reassembling the bike was to install the wheels.  The cog on the rear wheel has right-handed threads, so when you pedal forward, the cog gets tighter.  But on top of the cog goes a lockring that is reverse-threaded, so when you push back on the pedals to stop, you don't unscrew the cog.  The rear stays had to be squeezed together a little bit due to the lack of a freewheel.  Next I cleaned out the bottom bracket, composed of the cups and ball bearings, which was full of grime and grease.  The one-piece crank, which is pretty unique to this old style of bike, looks like this:

I greased up the cups with white lightning grease and put the bottom bracket together, ball bearings and all, this time substituting a single 44-tooth chainring for the double that the bike originally came with.  That extra hole you see that looks kind of out of place fits into the drive pin located on the one-piece crank.

Next it was time to install the single-speed chain.  Getting the chain sized appropriately is important because a fixie has no derailleur to take up the extra slack.  Fortunately the old Schwinn had horizontal drop-outs (the C-shaped indentations in the frame where the rear axle sits), which allow some wiggle room if the chain is not perfectly taut.  I erred on making the chain a bit long, so that I could just move the rear axle back in the drop-outs to make it taut.

Once I had the chain on, the bike was ready to ride, but not necessarily safely.  I installed new brake pads and a new brake cable on the front.  I wrapped the handle bars with white grip tape to make it look flashy.  And unfortunately, the bolt holding the handlebars to the stem was missing, which I didn't know about when I bought the bike, causing the handlebars to wiggle to and fro while you were riding.  After going to multiple hardware stores and multiple bike shops, I still was not able to find a bolt that fit the threads.  I think Schwinn literally invented their own bolt size!  So I used a bolt that was a little too small and tightened it down hard with a nut, which seems to work for now.  Finally, I took off a piece of metal that used to be holding shifters by removing the stem temporarily from the headset.

And that was it!  It's going to take a bit of practice to get used to riding fixed gear.  I didn't realize I coast fairly often, mostly when riding around the city.  With the fixie, you have to keep pedaling all the way up to the red light!  I haven't gone up any giant hills yet, but fortunately they don't exist in Michigan.

  

Gear Ratio Math

This weekend I purchased a 1970s Schwinn Varsity, which this article calls "the single most significant American bicycle."  I am in the process of converting it into a fixed gear bicycle, and I'll have photos and a description of that process soon.  Today's post, however, is on some math with fixed gear bicycles.  On fixed gear bikes, there is one speed, one chainring, and one cog.  What makes a fixed gear different from a generic single speed is that the drivetrain is connected directly to the hub of the rear wheel without a winch mechanism permitting free turning of the wheel while the pedals remain stationary.  You have to pedal continuously on a fixed gear bike as long as you're moving forward!

One of the most important decisions when making a fixed gear bike is choosing which gear you want to be stuck in forever.  I chose to make this decision based on my desired leg revolutions per minute and desired speed.  Since I'll be using the bike for getting around town and maybe a few longer rides, I want to be going about 18 mph when I'm pedaling at 90 revolutions per minute.  Now, how do I figure out what gear ratio to use?

18 mph = 28,962 meters per hour = 482.7 meters per minute

The circumference of my bike wheel (700x23c size) will be 2.09858 meters.  This means the rear wheel will need to turn 230.013 turns per minute.  I want my pedals to be turning at 90 per minute.  This means the gear ratio I'll need is 230.013/90 = 2.5557.  Of course, chainrings and cogs come in limited sizes, and to minimize wear on the chain both should have an even number of teeth.  I've decided on a 42x16 combination, which is actually a gear ratio of 42/16 = 2.625, corresponding to a speed of 18.5 mph at 90 rpm.

Now that we've done some basic arithmetic, how about a little number theory?  As I was reading up on fixed gear bikes, there was much discussion on "skid patches."  Some die-hards believe that adding a break to a fixed gear ruins its sleekness, and so they rely on skidding to stop their bike.  Basically they lean forward to take weight off the back wheel, lock their legs, and then skid to stop.  Since most people skid with their pedals in the same position every time, this causes skid patches to form on the tire at certain locations.  The gear ratio determines how many skid patches you end up with.  For example, if your gear ratio is 45x15, then every pedal stroke corresponds to 3 whole wheel revolutions, and you will have one skid patch.  If your gear ratio is 42x16, as I plan mine to be, you will have 8 skid patches.  How do you determine how many skid patches you'll have?

Theorem.  Let a/b be the reduced gear ratio, meaning that a and b are relatively prime integers (In the above example, 42x16 reduces to 21/8).  Then there are b skid patches.

Proof. First we show that there cannot be more than b skid patches.  If a/b is the reduced gear ratio, then for every one pedal revolution there are a/b revolutions of the wheel.  So for every b pedal revolutions there are a wheel revolutions.  Thus, b pedal revolutions returns us to the original starting location on the wheel, since a is an integer.  So there can't be more than b skid patches.

Now we show that there can't be fewer than b skid patches.  Let's assume that there are fewer than b skid patches.  Then there exist two different numbers of pedal revolutions that correspond to the same location on the wheel.  Let's call these two different numbers of pedal revolutions m and n, with 0<n<m<b.  If m and n pedal revolutions get us to the same spot on the wheel, then their difference, m-n pedal revolutions will get us to the same spot on the wheel.  Now m-n pedal revolutions is equal to (m-n)(a/b) wheel revolutions, which must be an integer in our case.  This means that b divides m-n or b divides a.  But b can't divide m-n because m and n are both less than b, and b can't divide a because b and a are relatively prime.  This is a contradiction, so after every pedal revolution [0,1,2...b-1] we must be at a different location on the wheel, and there must not be fewer than b skid patches.

There are no more or no fewer than b skid patches, so there must be exactly b skid patches.

Some fixed gear skidders are ambidextrous, and can skid with either their left or their right leg forward.

Theorem.  Consider the case of an ambidextrous skidder.  If the reduced gear ratio a/b has an even numerator, then there are b skid patches.  If the reduced gear ratio has an odd numerator, then there are 2b skid patches.

Proof.  Ambidextrous skidders can skid every half pedal revolution, which is equivalent to a situation where we have a single-side skidder using a front chainring half as large.  The gear ratio for an ambidextrous skidder, then, is effectively (a/2)/b.  If a is even, then this can simplify such that a and b remain integers, and as above there are b skid patches.  If a is odd, the gear ratio could only be simplified to a/2b and according to the theorem above we have 2b skid patches.  

Liquid Nitrogen and Helium

This evening we made liquid nitrogen ice cream.  We used a very simple recipe--one quart of heavy cream, one pint of milk, 3 tablespoons of vanilla, and 1/2 cup sugar.  Then we added about half a gallon of liquid N2 to bring the mix down to 77 Kelvin (-321 degrees F).  The ice cream was quite smooth and delicious.

Ned and Liesel enjoying liquid nitrogen ice cream.

I was curious about some properties of extremely cold matter that I remember learning in high school and decided to read up on it.  Other than being extremely cold, nothing particularly exciting happens around the temperature at which nitrogen is liquid.  Liquid helium, however, is considerably colder (boiling point of 4.2 K), and at these temperatures we start to see very interesting quantum mechanical phenomena.  For example, if you cool liquid helium a couple degrees below its boiling point, it becomes a superfluid, with bizarre properties.  Atoms of most elements settle into a solid at cold enough temperatures due to intermolecular interactions.  Helium, however, is so light and has such weak intermolecular interactions that even at absolute zero it remains a liquid.  Since helium stays liquid near absolute zero, it can transform into a Bose-Einstein condensate, in which all the atoms of the liquid move in unison with each other.  Superfluid helium can leak through its container, finding openings between the container molecules that ordinary fluids would not be able to penetrate.  Superfluids also have zero viscosity, which means they can climb up the walls of their container, seemingly defying gravity, and spill over the edge, eventually emptying the container.


More practically, liquid helium is used for cooling superconducting magnets used in NMR and MRI machines.  Superconducting magnets are made from superconducting wire, which when cooled below a critical temperature, exhibit zero electrical resistance.  Electrical current will flow forever in a loop of superconducting material.


Really cool video on superconductors.

As you might expect, it is pretty simple to make liquid nitrogen, since it makes up 80% of air.  All you have to do is compress air, allow it to radiate off its heat, and then let it expand, which will cause the gas to get extremely cold.  This process is repeated several times until liquid nitrogen is achieved.