Elliptical Gears

          When I was in eight grades my first job was working on a conveyer belt at a metal and steel factory. I never asked why the boxes would slide slowly while we were placing the steel scraps in them and then speed up after they were full. I suppose that I thought some guy was watching the belt and keeping the speed irregular so that we could work more efficiently, but that would cost the factory extra money and the efficiency of the man watching the belt would not be very high. He may be more suited and more useful if he was cutting the steel or shaping the steel. Almost anything that he could do would have been better than sitting on a stool keeping the conveyer belt speed increasing and decreasing as the product moved along the line.

          After I started my Masters Degree I finally understood how that belt worked. It had to do with elliptical gears and the way they interact with each other that creates the varying speed. It’s fairly simple to understand if you’ve already read through the explanation of the ellipse in space, but the speed of an object traveling around an ellipse increases as it gets closer to the major axis (or the most extreme bend in the curve). Here’s a picture of two elliptical gears interacting that I created on GSP.

And here’s a link to these same gears that you can experiment with and watch the speed increase and decrease. Elliptical Gears

And here are the rules/directions to the construction of these two objects. It took me quite a while to come up with this, and there may be a quicker way to do it. If you find a quicker and more elegant way to do this then please do hesitate to email me at davidkdrew@yahoo.com.

Elliptical Gears

 

Construction on Geometer’s Sketchpad

 

            The following steps are a construction of how and hopefully why two gears that are congruent ellipses rotate and stay together. The idea behind the elliptical gears was created by a need to have an inconsistent gear speed. Such devices are used in wood and stone cutting. Anyway, here are the construction directions.

 

  1. Open GSP and start by placing a point anywhere on the plane, and label it point A. Construct two circles of variable size both with center A.
  2. On the smaller circle place a point B on the edge of the circle, and on the larger circle place a point D on the edge.
  3. Construct segments, in a thick line, AB and AD, and hide everything except points A, B, D, and those two segments.
  4. Make a dashed segment through B and D. Find the midpoint and call it M1. Make a perpendicular through M1 with segment BD.
  5. Construct a line through A and D and call the intersection of that line with the perpendicular line I.
  6. Select D and I in that order and select locus under the construct tab. It’s nicer to make the ellipse a solid thick line but not necessary.
  7. To avoid confusion let’s hide the lines, not the segments, that we’ve just made. And hide M1 as well.
  8. The trick is to make another ellipse congruent to the one that we’ve just created as well as lining up the intersection point. We can start by saying that point D is one of the foci of the second ellipse and the point I is where they will meet and stay together as the rotation occurs.
  9. Start by making a circle with center D and radius equal to segment AB. Next make a circle with center B and radius AD. Finally select points B and I in that order and construct a ray.
  10. The ray and the two circles all intersect at one point we’ll call C. Select the big circle and the ray and find their intersection. Now hide all that we’ve just created except for point C.
  11. Make a circle with center D and radius AD. Construct point E on the edge of the circle.
  12. Make segment CE and find the midpoint N. Make a perpendicular with CE through point N.
  13. Make a line through D and E and call the intersection with that line and the perpendicular from 12 point I2.
  14. Make a line through I2 and C.
  15. Select E and I2 in that order and select a locus again under construct. This will bring up another ellipse exactly congruent to the first one.
  16. Now hide everything except points A, B, C, D and I and the two ellipses.
  17. Select point B, go under Edit and choose Action Buttons, then go to Animation and create an animation of point B.
  18. Watch the point I go back and forth in non-constant speed.

 

 

          The last part of this presentation on conic sections and the ellipse is a quick investigation into bicycle cranks and gears. If you’ve ever ridden a bike enough to have to wash it you may discover something interesting. Last year, while I was cleaning my road bike I discovered that the front crank was wearing down or so it seemed. For those of you that don’t know the front crank is the big set of cranks closest to the handle bars. A road bike usually has two different cogs on the front crank with a certain number of teeth that grab the chain and propel the bike forward. There is also a front gear shifter that pulls the chain off one cog and moves it to the other cog depending if you want a harder or easier gear.  As I was cleaning my chain and gear cranks I discovered that the teeth on both the front cogs looked worn down so I took my bike up to Atlanta Cycle and asked them what the deal was. One of the mechanics blew my mind when he told me that nothing was wrong with my bike because it was built like that. He started throwing around some bike terminology that was boring a long winded but he did mention the word ellipse. He said that the front two cranks were built in an elliptical shape by the manufacturer, in the case of my bike it was the company Lemond Bicycles.

          Since I didn’t receive a full explanation of why the crank was elliptical I tried to make up my own answer. I came up with the theory that the gears were elliptical so that the power of your legs would be distributed more evenly through a regular pedal revolution. I thought that your leg power is weakest at the top a bottom of a pedal stroke so those places should be easier to turn over. It turns out that I was wrong, and according to another mechanic at Atlanta Cycling it all has to do with shifting gears.

          Here’s a diagram on GSP to show more detail of what I’m talking about.

          The front crank gets power from your legs and distributes it to the rear crank which turns the back wheel to propel the bike. Sorry for the terrible drawing, but it’s the best I could do with GSP. So back to the reason the front crank is elliptical: it’s shaped like that because of the gear shifters.

          Here’s a picture of my bike on the right and on the left is a close up of the front cogs and the shifting component.

 

          The picture on the right is called the front bracket. It consists of two cogs, pedals, a crank, some cable, and what is called a derailleur. The derailleur is the key and the reason for the elliptical design. When you shift from one gear to the next you may know that sometimes it takes longer to get into gear than other times. The reason is that the elliptical design helps the chain slide off the cog easier at a special point. And this special point is none other than the perigee for the crank and the right pedal.

          It makes perfect sense if you think about it for a minute. Let’s suppose that the crank were circular then the amount of force would be fairly strong in order to get the chain off a particular cog. But if the cogs had a weak point such as where the perigee is then the force would dramatically decrease. The perigee is the weak point because it is moving at a slightly faster speed than the back cogs so there is a small amount of time or a small margin of error that you could slow the front cog while not reducing the speed of the rear cogs. Therefore you can shift gears and slow the rotation of your legs for just a moment and not reduce the speed your bike is carrying you.

          Therefore we now know why the front cog of many road bikes is slightly elliptical.

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