# How to compute the size of a polygon moving along non-circular guide rails

I am trying to compute the optimal (a non-optimal solution would also work) size of carts that can move along guiding rails. Each cart is connected to the rails at two points. Each cart is also connected to the cart in front of it and the one behind it. It is a closed system. The carts can be connected directly as in picture (a) or there can be some arbitrary but constant size connector between them. In the first step I am just looking for a solution that works mechanically. It would also be great to know what is the best arrangement that allows the least force.

In picture (a) I can find many solutions. However, my guiding rails have to have a shape as in (c). I can change the rails a little bit but they will have to have inside angles (red) and outside angles (yellow) as depicted.

I tried experimentally to find a working solution through simulation, but is there an analytic way to compute which size of carts could move along these rails?

I don't have a kinematics background and don't know what it is that I am looking for. Any pointer towards keywords/ papers where I can start my research would be very much appreciated. So far I am stuck at "Polygon Effect".

• I think that if you try to work this out for a square track with radiused corners you will find a fundamental problem as the total path length changes with angle. Have a look at how chains run around chain wheels for some inspiration. Also chain conveyors or "table-top conveyer" (TTC) may yield some ideas. Aug 4, 2019 at 12:15
• This is exactly a chain with very long links - arrange your track in the same way. Set the bends to have a circumference that divides into a whole number of links, and add in a 'tensioner', whereby you can adjust the length of two straight bits of track to change the overall length Aug 4, 2019 at 16:53
• @JonathanRSwift Thanks for your suggestion. I like the idea of using tensioners to adjust the tracks. That could be a solution for me. Can you recommend literature by any chance? Aug 5, 2019 at 20:21
• @Transistor thanks for the inspiration, that's very helpful. Aug 5, 2019 at 20:22

There's no uniform radius of curvature with your shape (c) so you can't have a closed series of rigid linkages that's capable of moving around the track. You can use a flexible link or cart as a tensioner, as Jonathan R Swift mentioned, but otherwise it's not possible.

:EDIT: - Consider this example of three carts/points on the track. In your (a) and (b) case, the radius of curvature is constant, so the arc length $$s = r\theta$$ is constant, so the length of the chord that spans the endpoints of the arc is also constant.

In case (c), you can see that there's no way for the bottom-right point to move down the right "arm" of the track because the bottom linkage, if it's rigid, isn't going to allow that. Similarly, the right linkage between the bottom-right point and the top point isn't going to expand and let the bottom-right point move down, either. The mechanism is jammed.

It might look like it's going to work if you make the links smaller but, as long as they are of some finite length $$\ell>0$$, they'll always be cutting a corner off and again, since the radius of curvature isn't constant, you'll find the length of track spanned when it's straight, for example, will be shorter than the length of track spanned when it's curved and so it'll still jam.

If you need to have the chain be rigid then the tensioner needs to be incorporated into the track/rail.

• Thanks for your answer. And you are probably right. Intuitively, I thought that it should work for two (different sized) half circles with straight bits in between, just like a bike with only one gear. Is that wrong? Because that example doesn’t have uniform radius. The “carts” in that example are much smaller, so I assumed that for cart size —> 0 there would also be a solution for (c). I wanted to avoid tensioners because the drive is supposed to hook directly into the linkages (also similar to a bike). When the gaps in between have different lengths, the whole system becomes irregular. Aug 5, 2019 at 20:08
• @danny - A bike chain uses a tensioner. For a bike chain, the links are all rigid but the track is variable. One "corner" of the chain track is a spring-loaded sprocket, which means that the chain is a rigid series of links but the track is deformable. Something's got to give. Aug 5, 2019 at 22:39
• A "fixie" doesn't have a tensioner - the only reason a geared bike does is to take up slack when changing gears Aug 6, 2019 at 5:51
• @JonathanRSwift - There's also no fixed track for a bike chain; even without a tensioner the chain is free to sag as necessary to make up for any path error. Either the chain or the track has to be flexible. Aug 6, 2019 at 18:02