Thursday, January 26, 2012

Experiments with measuring bike geometry

The current weather in Montreal has kept me indoors more than I wanted. Inspired by an article in the current issue of Bicycle Quarterly, I decided to measure the geometry of my Gunnar Roadie.The article in BQ describes how to measure bike geometry from a photograph, and an older article gave instructions how to measure directly on the bike. As a quick summary: measuring on the bike works better for almost all dimensions but trail and fork rake. Trail and fork rake can be measured on a picture, but it's hard to get them accurately.

Measuring angles

For measuring angles, using a digital angle finder or an angle finder app for your smartphone is the best option. According to the interwebs, the angle finder apps are pretty accurate. The angles you need are that of the seat post, the head tube and the top tube. The top tube angle is only important for calculating the effective top tube length, as described below. So if you plan on measuring the top tube length directly, you can skip that step. For most bikes, seat tube and top tube angles will be somewhere between 70 and 75 degrees. Top tube angles used to be 0 on most bikes, but after the sweeping success of compact geometry with sloping top tubes, there now is a large range of angles. In order to get good measurements, measure multiple times and make sure that your angle finder is flush with the frame. Many frames have butted tubing that will be thicker near the intersections of the tubes and you should thus measure in the middle of the tubes.

Measuring lengths

The seat tube is probably the easiest thing to measure. The cranks are somewhat in the way of putting a ruler on there, but it's still easy to get a good measurement. Just make sure to note which measurement you're taking: either center (of bottom bracket) to center (of top tube intersecting the seat tube) or center to top (of seat tube). In addition, you will probably already know this number, as this still is the most common way in which manufacturers size their frames.

The top tube is a little bit more tricky, at least if it is a sloping top tube. In order to measure the effective top tube length you have to put a level on your ruler and then measure the distance from the center of the head tube to the intersection with the seat tube (or the seat post as its extension). I find this pretty difficult to do properly, and fortunately there is another way by employing the powers of trigonometry: You'll need the length of the actual seat tube and the angles of the seat and top tube. Using the laws of sines, you can calculate the effective top tube length x as follows:

$x=\genfrac{}{}{0.1ex}{}{\mathrm{sin}\gamma \cdot a}{\mathrm{sin}\beta }=\genfrac{}{}{0.1ex}{}{\mathrm{sin}\left(180-\alpha -\beta \right)\cdot a}{\mathrm{sin}\beta }$

Another relevant dimension is the bottom bracket drop, that is the distance from a line between the two wheel dropouts and the center of the bottom bracket. Jan Heine suggests tying a string around the front and rear quick release skewers and then using a ruler to measure the distance from the string to the bottom bracket. You have to correct that measurement with the radius of the skewers.

Next is the front-center distance, i.e. the distance from the front wheel axle to the bottom bracket. The problem is that the fork has to be completely straight in order to get an accurate measurement--something not easily to achieved. Jan Heine suggests measuring on both sides and then making sure to get identical readings, but I found that rather difficult.

Now we get to the tricky part: measuring fork rake and trail. I won't go into the discussions if high trail or low trail are TEHBESTTHING(tm) and instead only show how to best measure trail. Measuring trail directly is almost impossible. Trail is defined as the distance between where the front tire touches the ground and the extension of the steerer axis, as shown in the following figure.
 Fork rake r and trail t
Projecting the head tube and the vertical axis through the hub onto the ground accurately just doesn't work properly. [1] Fortunately, you can use the fork rake and wheel diameter to calculate the trail. Measuring the fork rake is easier than measuring trail, but it's still tricky. Bicycle Quarterly use a custom made tool or they measure rake from a digital image. If you only have access to the image of a bike, it's even possible to get all of the above measurements from the image; however, I had a hard time producing a distortion-free image with the lenses at my disposal.

First, take a picture of the front part of the bike. Using your image editor's (I'm using Photoshop) measuring tool, rotate the bike so that the head tube angle matches the value you got from the angle finder. Measure some other tubes, too, to see if and where there is distortion. In addition, draw a circle and see if it is well aligned with the rim. If that shows too much distortion, take another picture and repeat the process. Now you will determine the scale of your image. Draw a vertical line through the hub and measure the distance from the center of the hub to the outer rim. It doesn't really matter which unit of measurement your software uses. On your bike, this distance is a known variable, and dividing the real value through the measured value will give you the scale of your image. With my bike, I measured 1.729 in the image and by dividing 316 (622mm bead seat diameter plus 10mm, then divided by 2) by 1.729 I got a scale of 1:182.8. It is a good idea to repeat the measurement a couple of times on different sections of the rim to once again control for distortion. I measured 1.729, 1.758, 1.651, 1.524, showing that there is still a good deal of distortion. One might be tempted to just take the average of these measurements, but since distortion is not random, this is not advisable. It is preferable to keep taking more pictures, with different lenses and at different angles to achieve a minimum of distortion. For now, we'll just assume that the 1:182.8 scale is correct.

Now we can actually measure the rake: Draw a thin line through the steerer axis, extending it all the way to the end of the fork blades. The draw a line parallel to the steerer axis through the hub. Parallel lines are easiest to get with copying and pasting the first line or drawing a rectangle. Measure the distance between the two parallels. In my case, this was 0.240, yielding a rake of 0.240 * 182.8mm = 43.9 mm. Common values for fork rake are between 40mm and 75mm, with most forks being in the lower range of this spectrum.

Using fork rake, head tube angle, and wheel/tire size you can now use the excellent online trail calculator to get your bike's trail. For Gunnar that would be 54mm trail with a 25-622 tire. Since small measuring errors in any of the measurements can throw the calculation off easily it's hard to say how accurate this value is. It would put Gunnar in the medium trail range (Heine gives 25-80mm as the common range).

To conclude, measuring a bicycle's geometry, either on the bike or from a picture, can be tricky. For many bikes you will be able to get the correct values from the manufacturer (please note, though, that most likely there will be minor aberrations from the values on your frame as artefacts of the manufacturing process). If you want to measure your own frames it might be a good idea to start with a bike with known dimensions to give you an idea of how good your own measurements are.

[1] I wonder if one could use a laser pointer attached to a concave piece of metal that matches the head tube's curvature. I've ordered a laser pointer and tinker around with it a bit.