Hey, I’m new here, and an aspiring mathematician. I recently found myself on the issue of wheel shape, specifically how the perimeter of a shape correlates to the distance it travels over one rotation. So (in theory of course) a triangle and a circle with the same perimeter would travel the same distance per rotation.
I wanted to ask you all, the ones building bots and whatnot, if any of you have tried non-circular wheels. This doesn’t necessarily mean using triangular wheels or square wheels, but something a bit closer to the round shape.
The reason I bring this up is that I recently noticed that as you increase the number of sides on a shape with a constant perimeter, the total area increases. When applying the third dimension, this means higher volume, thus mass. While it isn’t much, changing from circular to decagonal wheels (theoretically of course) would cause the wheels to weigh less. Have any of you tried anything similar, or considered/disproved such? I have included a graph to demonstrate the correlation between this, with x representing the perimeter, and y being the area of the shape. Included are 3 sided to 12 sided (All equilateral, further math needs to be done on irregular shapes), followed by the standard circular. https://www.desmos.com/calculator/rmzwn55efr
Furthermore, since these wheels pivot from point to point, they touch the ground a bit less than a circular wheel would. What kind of impact would this lessened grip have on your bots? Positive or negative?
Sorry that I’m throwing this out there, but engineers/robotics enthusiasts such as yourselves seem like the prime people to ask/converse about this with. Any commentary would be loved!
So from my minimal knowledge on wheel shape for I generally only use round wheels, “why reinvent the wheel:p.” I urge you to watch this mythbusters video! They did square wheels, but it would be a similar effect with the other non round wheels.
They would have a lessoned grip on the wheels, depending on what part they were touching, and would make turning the robot really difficult, especially when driving, making for a VERY rough and unpredictable ride. Plus this would put a huge strain on the motors when starting in a flat position, for it would have to turn it to gain speed and to get over any corners.
Your question is a very good one and it is evidence of someone constantly pushing the edge of what could/should be done during the design of a machine.
You are right that there could be benefits of weight reduction by going to a different shape of wheel. For the purposes of this post, let’s consider 3D prisms of basic shapes and not any fancy 3D geometries that could really complicate things.
In my humble opinion, there are four main reasons that cause people to use cylindrical wheels instead of other 3D prisms:
We’ve always used cylindrical wheels - This is not a great reason, but most people don’t question the status quo.
Predictable height from wheel axis - The center of a cylindrical wheel will always be at a height equal to the radius of the circle. A polygonal prism will have a height that fluctuates as it rolls, meaning each corner of the machine will raise and lower. This can change the angle at which the frame sits, causing mechanisms on the machine to be set at incorrect angles.
Reduced vibration - If you imagine each wheel raising and lowering (see #2), now consider a robot driving quickly across the floor. This would now create vibration throughout the machine. With a few designed exceptions, vibrations are generally a bad thing for mechanical equipment. Bolts can rattle loose, parts can deform, and wear can show up on a number of different ares of the machine.
It’s easier to shed weight elsewhere - My best estimate for the weight you would be saving is around one pound. It’s generally pretty easy to find that kind of weight elsewhere in the machine. You could design components out of less-dense materials, for example.
With all of that said, it’s very possible that I am overlooking another reason to utilize non-cylindrical wheels, besides weight. Keep asking questions like these and one day, you’ll find yourself redefining the status quo.
I think the OP is saying not a standard triangle but a shape of equal width like jmadigan posted which would mean it would have a constant height from the ground and it wouldn’t have vibration problems.
With that design, you must consider that the guards and elaborate suspension systems above each wheel would be required to maintain a constant height and eliminate vibration. If weight savings is the end goal, I think we’d be heading in the wrong direction.
I’m not convinced that this is the case. Even if the machine is remaining at a constant height (like the rider on the bike with triangle wheels in the video), the suspension is still moving up and down, and at constant angular speed of the wheels, the suspension system is moving with some constant frequency (changing angular speed = changing frequency). And, unless your suspension and wheels have no mass, their motion changes the center of mass of the machine with time, and introduces oscillating reaction forces and torques in the joints mating to the suspension due to the accelerations of the mass centers of the various components. So, in reality, you are still introducing some vibration into the system as long as these oscillations are occurring. Plus, the suspension itself IS vibrating, so it does still face all the problems of machine vibration.
It is important to note also that cyclic loading, though not initially apparent, has a serious impact on overall strength of most materials. This type of failure is called fatigue failure. It is, for example, why airplane cabins are only rated to a certain number of pressure cycles (airplane cabins being pressurized are just basically aluminum tanks being deformed). By introducing any sort of changing load in a system, you need to start taking into account fatigue, and there is an entire field of solid mechanics devoted just to understanding fatigue modes and how they happen. Some cyclic loads are unavoidable, like a rotating shaft, where the load is applied in a constant direction (geartrain, etc), so as the shaft turns, the load on a specific point changes in time.
These fatigue modes can occur very rapidly depending on the speed of the system, but they are not the only problem. As someone else said, issues of wear and nuts vibrating loose are some other contributors to the list of reasons to not introduce undue vibration into your system.
Please consider the power input for your various choices. Turning a triangle from a rest position is likely to brownout your electrical system.
I can also say with some certainty that while mathematically triangle and circle should move the same distance, in practice they will not since all surfaces will not mate with the floor during all movement. As such, any feedback based calculations will be inaccurate.
Curves of constant width are cool, but keep in mind that they aren’t constant radius, meaning that the distance from your drive shaft to the floor is going to change as you move. This might be saying the obvious, but by definition the only 2D shape that has constant radius is a circle. The bicycle that a couple people linked to only works because the wheel rubs against special fenders on the frame, which is hardly practical for most designs.
That said, some teams have used slightly non-circular wheels for other purposes. Way back in 2003 (Stack Attack), I remember that some teams machined their own wheels that could grip the wire mesh of the ramp. In 2004 (FIRST Frenzy), there were several clever wheel designs created to help teams drive up the steps (a 6" jump) to hang on the bar.
I wonder if non-circular wheels could have been used to an advantage in 2011 (Breakaway), 2016 (Stronghold) or other games which involved a playing field that wasn’t level or had barriers to cross?
I know we didn’t want to complicate things in 3D, but here is an awesome video on circular wobblers with all the math included. I seem to remember a video somewhere that showed this being done with ellipses that negated the wobble effect while still rolling true. Bonus points if someone can produce a picture or a video of the elliptical version (if it even exists, I have an eccentric imagination (sorry bad pun, had to stick it in there )).