Understanding the physics behind an elastic sphere

[JordanLC]

TL;DR Summary

Non engineer/scientist trying to gain an understanding of how internal pressure (from air, water, sand, etc) of an elastic sphere affects it's shape.

First off, I'm not a scientist or engineer and I apologize if I don't give a clear description of my question. I'm beginning a personal project and was hoping for some knowledge and assistance.
What I'm trying to achieve is to have a spherical object (it will be at least twice as wide as it is tall) that can maintain it's shape while also being able to rotate in any direction. If I were to put it on the ground, the top and bottom would be flattened into a circle and it's only a few inches tall, then for it to be moved along the ground, rotating in any direction while maintaining a flat surface and circular shape.
I began with a playground ball, to try and understand the relation of it's shape with it's internal pressure. Unfortunately the ball was too inelastic and as I let out air wouldn't flatten since the it couldn't stretch past it's diameter, if that makes sense.

So what kind of materials would I want to look at? I was thinking of using sand to fill it instead of air, so that if you were to step on it, it would retain its shape better.
How could I gain a better understanding of how the internal pressure will affect the shape?
Please ask me anything that could help me clarify what I mean for you.

 

[hutchphd]

Maybe you could start here:

https://en.wikipedia.org/wiki/Young–Laplace_equation
The salient point is that for a given surface tension the contained pressure is inversely related to radius of curvature. Does that get you started?

 

[JordanLC]

Yes that certainly does, perhaps exactly what I was hoping to find, thank you!

As an update, I thought perhaps to try liquid latex, at the very least as a test material.

 

[some bloke]

Are you essentially looking to create something similar to an omnidirectional caterpillar track? IE a flattened ball which "rolls" along in any direction, maintaining a flat upper surface?

If you're wanting something physical to analyse, I would try using some balloons. I know they aren't spherical, but you can fill them with sand, water, oil etc. and then sit them on a flat surface to see what sort of results you can achieve. You can observe the difference in shape and roll-ability.

I believe that a highly elastic rubber (EG latex) would be a good option for your designs. If I was going to try to make a perfect sphere of rubber filled with water, I would look at freezing a sphere of water using a mould, making 6 half-spheres of rubber and then putting 2 of the half spheres on the ball (to make a whole sphere), paint them with liquid latex and then put the other 2 over the top, offset by 90°, paint again and then put the last 2 over, offset by 90° in the other axis (so there is no location where the water can escape). Put it in a support whilst the water melts and the latex dries, leave for a couple of days I guess. I would expect this to result in an elastic sphere full of water.

If you could get a ball of ice the right size, you could cut the nozzles off of 2 balloons and then put them over the ball, as above. As long as you have to stretch the balloon, I expect the tension will hols the water inside (I'd still use latex glue though).

Hope this helps!

 

[JordanLC]

Thanks for the ideas! I was actually just looking at balloons seeing if I could find some that are made to be spherical. If I can find that I can create a mold to do the latex.
Would you happen to know if there are different sorts of latex? I was trying to but everything I was searching was just discussing brands for making masks.
That's a clever idea, I'll definitely have to keep that in mind. I was also thinking of finding a some sort of water toy with a big rubber hole (the ones that you can pull out, open up then fill it with air, can't think of a better name ha) that I can pour sand into.
The nice thing about the latex is I'll be able to easily test different thicknesses by letting them sit longer.
I'll let you know what I've found, maybe you'll make connections I don't!

Thanks again!

 

[some bloke]

Thinking about it, as a balloon is already predisposed to become a sphere, you wouldn't need a perfect one. you could create 2 rough half-spheres of ice using a bowl twice, or 2 of the same size bowls, then stick them together into a sphere by wetting and re-freezing them together. Then stretch 2 balloons over it from opposite directions, glue one over the other and then let it thaw. voila - one elastic rubber ball full of water.

You could simply add more balloons to achieve a thicker layer, for your initial experiments. Remember to change the angle you fit them, otherwise the overlap will be disproportionately thick.

I don't know if you could use a sand/water mix to do the same and achieve mostly sand inside. I can't think of any other way to do it with just sand.

 

[DaveC426913]

Thinking about it, as a balloon is already predisposed to become a sphere, you wouldn't need a perfect one. you could create 2 rough half-spheres of ice using a bowl twice, or 2 of the same size bowls, then stick them together into a sphere by wetting and re-freezing them together. Then stretch 2 balloons over it from opposite directions, glue one over the other and then let it thaw. voila - one elastic rubber ball full of water.

Call me dense but what does the above accomplish that isn't just as easily accomplished by simply filling a balloon with water?

 

[some bloke]

Call me dense but what does the above accomplish that isn't just as easily accomplished by simply filling a balloon with water?

My aim was to create a (roughly) perfect sphere without a nozzle sticking out of the side. balloons are also usually teardrop shaped to some extent. It was mainly to see if I could think of a way to do it!

 

[JordanLC]

Call me dense but what does the above accomplish that isn't just as easily accomplished by simply filling a balloon with water?

Water would work in creating the shape I imagine, and will work for testing, but the overall goal will require a material that can be withstand a considerable amount of pressure being placed on top sometimes more pointed than spread out. For that I intend to use sand.
With the smaller balloon for testing I want to get an idea of how a sphere of latex filled to a certain volume will spread when laying on a flat surface with and without pressure

 

[some bloke]

can you elaborate on your end goal? mainly on what the flattened sphere will be doing? I don't think that many elastic materials will respond well to a point being pressed into them, sand or not. If anything the sand will make it more likely to burst.

 

[Baluncore]

I do not think a flattened sphere can be moved, beyond the stretch in the skin or envelope. Consider a vertical plane aligned in the direction of travel. The length of the intersection of that plane with the skin or envelope will be a function of the distance from the centre line. A shear in the surface must therefore accumulate with any relative movement of the upper and lower surfaces. The only way to relieve that shear is to have an envelope material that slides against both the upper and lower external surfaces as they move relative to each other.
Filling the envelope with sand will increase internal friction and so also prevent movement.
How much movement does the OP require?

 

[JordanLC]

So the end goal is to have an omnidirectional non motorized treadmill of sorts. My idea is to have two half spheres lined with rollers, either ball bearing or simple rollers. One of the halves will have a portion cut out from the bottom to leave a space for the sphere.

Filling the envelope with sand will increase internal friction and so also prevent movement.

That's what I was hoping for, which would perhaps allow me to easily set different levels of friction for the sphere.

I don't think that many elastic materials will respond well to a point being pressed into them, sand or not. If anything the sand will make it more likely to burst.

This is why I thought of latex. I've never worked with it, but I figure if I make a sphere mold then I can test different levels of thickness with the latex to find a point where it will deform enough to maintain it's shape without bursting, but not so thick that it it will fold on itself, or I believe shear(I believe I know what they mean by shear at least) as Baluncore put it.

 

[JBA]

My idea is to have two half spheres lined with rollers, either ball bearing or simple rollers. One of the halves will have a portion cut out from the bottom to leave a space for the sphere.

Can you post a sketch of your planned "treadmill" assembly (at least in the region of the spheres) so we can better understand all of the elements that will affect the spheres?

That's what I was hoping for, which would perhaps allow me to easily set different levels of friction for the sphere.

Are you planning to have the spheres create some type of rolling resistance?

 

[JordanLC]

Can you post a sketch of your planned "treadmill" assembly (at least in the region of the spheres) so we can better understand all of the elements that will affect the spheres?

This is the general mockup with roller ball bearings.
It's not multiple spheres, but one large one that will set encased by the roller bearings.

 

[JBA]

Now I understand your above description. Can I assume the part of the sphere exposed thru the hole in the top is part of the running surface of the "treadmill"?

 

[JordanLC]

Now I understand your above description. Can I assume the part of the sphere exposed thru the hole in the top is part of the running surface of the "treadmill"?

Yessir

 

[JBA]

In that case the material for the sphere will need to be from a very durable material that can resist shoe sole abrasion as well as the shoe impact forces of the user; and be stiff enough not to deform excessively into the spaces between the bearings under load because that might add an unacceptable amount rolling resistance for the sphere.

Just as a general thought, the amount of sphere filling that works for a 140 lb user might not work for a 220 lb user because of excess flattening to the point that it drags on the edges of the top cover; or visa versa, if it worked for the heavier user it might not flatten enough to make a suitable running surface for the lighter user.

 

[JordanLC]

In that case the material for the sphere will need to be from a very durable material that can resist shoe sole abrasion as well as the shoe impact forces of the user; and be stiff enough not to deform excessively into the spaces between the bearings under load because that might add an unacceptable amount rolling resistance for the sphere.

Just as a general thought, the amount of sphere filling that works for a 140 lb user might not work for a 220 lb user because of excess flattening to the point that it drags on the edges of the top cover; or visa versa, if it worked for the heavier user it might not flatten enough to make a suitable running surface for the lighter user.

Both good points.
But for now I'm just trying to get a proof of concept. Get an idea of how different pressures will affect the bearings and the sphere's ability to rotate and such.

 

[JBA]

The thing that worries me is your focus on pressure rather than fill. Assuming you start with a sphere, lacking any eternal load on the top of the sphere, the amount of flattening will depend upon the filling weight across the bottom of the sphere so you will have to start with a sphere with a radius a bit less than that of your ball bearing cavity of the sphere holder so that when filled to the proper level the sphere to deform to the point that the top segment of the sphere will be flat, or nearly so, and even with (or maybe slightly above) the top surface of your holder.
Also, the filler used must be an incompressible or nearly incompressible material, i.e. like sand as has been already been discussed, because that upper flat region of the sphere intended for the user to step-on must support their foot weight without significant deformation to be safe and stable for the user. Any "pressure" placed in the sphere will cause the top exposed region to try and reshape itself into a spherical soft "bump" any time the foot load is removed and that is not what you want on your treadmill.

 

[some bloke]

If you need as little deformation on the surface as possible, an internal structure for the sphere may be more useful than a fluid. you could have a standard omnidirectional treadmill in the centre to maintain a flat surface, and use braking within this to give a variable resistance to movement. It could contain a battery and utilise wireless charging to keep it alive.

If you're after an omnidirectional treadmill, for Virtual Reality style usage, they already exist - they're essentially a treadmill made of treadmills.

 

[JordanLC]

If you're after an omnidirectional treadmill, for Virtual Reality style usage, they already exist - they're essentially a treadmill made of treadmills.

There are a few companies that make Omnidirectional treadmills, but none are on the consumer market, and all except for one are simply frictionless bowls. From what I've heard they aren't very satisfying to use. The treadmills made of treadmills is giant and unwieldy, and watching the video you see it has to predict your movement, it just seems over-complicated.
Honestly, the lack of VR treadmills is what inspired my idea. Locomotion is the one thing I feel is holding back the VR industry, and the only thing that's kept me from getting a headset( I have bad motion sickness ) as well as playing Fallout 4 haha
I'm hoping that my idea will be simple and effective enough it could be cheaply made and really help bolster the VR industry, if it works like it does in my head (ie perfectly). We shall see though. Wednesday I'll get my roller bearings and sand and begin testing how latex works with it.

If you need as little deformation on the surface as possible, an internal structure for the sphere may be more useful than a fluid. you could have a standard omnidirectional treadmill in the centre to maintain a flat surface, and use braking within this to give a variable resistance to movement. It could contain a battery and utilise wireless charging to keep it alive.

I'm not sure what you mean by internal braking. While I'd like to keep it as simple and cheap as possible, I'm curious how that would work

 

[some bloke]

I can't upload a sketch at the moment, but I'll try to explain it as well as I can!

create something which will sit inside your stretched rubber sphere. The device would have ball bearings on the outside, which will line up with the gaps between the ball bearings of the outer shell. The top of the device will be flat, with smaller rollers to stop it from being bumpy. Wrap the sphere around it and then fit it into the outer shell. The rubber will move between the inner & outer bearings. the internal part will be held in place by the location of the outer bearings, forming 3-point holds on the inner bearings (preventing them from moving in any direction) with the rubber membrane in between. The sphere will roll around the inner, inside the outer, and with a flat top for walking on.

I'm not sure what the difference would be between an omnidirectional treadmill and your design, so far as the necessitating prediction of the users movements - both result in a flat surface which can move in any direction, so surely any requirements an Omni-treadmill has would also be required to make yours work? Otherwise you could simply have an Omni-treadmill without the motors to achieve a freely-moving omnidirectional surface?

I do see how your design would be much more compact than an Omni-treadmill, however, with many less motors etc.

 

[JordanLC]

The device would have ball bearings on the outside, which will line up with the gaps between the ball bearings of the outer shell.

Ahh that's pretty clever! The only problem I could see is the ball bearings clamping the material between each other as weight is put on top.
The other problem is, at least at the consumer level, ball bearings are pretty expensive. I was thinking about rollers, but they aren't much less expensive and I'm unsure how the friction of rollers not parallel to the movement would affect the movement.

For anyone curious what we're talking about Infinadeck.

Otherwise you could simply have an Omni-treadmill without the motors to achieve a freely-moving omnidirectional surface?

Yeah, I'm trying to stay as simple and small as possible (not to mention cost effective). I'm even going to try forgoing a bowl shape and see if I can do a completely flat bottom, with two rows of rollers lining the top and bottom all around the sides.
While ideally I could become a millionaire (ha-ha), I want vr treadmills to boom so that developers can make games in which you can walk around. There are already some cool VR mmo's out there even without a walking platform!

 

[some bloke]

You will also need to consider how you capture the movement, which will have to translate to a walking motion in the game - if it's all free-rolling rollers, it might not be easy. Perhaps using old fashioned mouse rollers for a prototype would be a good idea? they are rubberised, already mass produced and designed to capture movement. You could likely get a box full of them for the prototype much cheaper than bearings.

 

[JordanLC]

I do have some ideas on that, one that's ridiculously simple though I'm not sure it's technically possible due to it being patented. I'm going to have to dig deeper into how that works.

 

[JordanLC]

So far the only issue seems to be the balloon latex catches on the metal I was wrong, it's actually getting caught between the bearing and it's metal encasing ( when I put pressure down) of the roller bearings. It does stick to the metal exterior but only on the exterior rollers. If the latex can't expand to hang over the roller then it won't be an issue.
I ordered some metal bearing balls to test as well, though that would require a special metal bowl to allow them to roll in place, or maybe something to coat the ball and metal like surfboard wax (we used that on our sliding metal door when I was a kid).
First I used just a regular balloon, filled it with sand and then blew some air into it. It was expanding too much so I doubled up on the balloons.
Filling the double balloon with 1/3 and 2/3 air seems to be a good combination so far. The latex can still expand slightly and the sand has enough room to roll over itself and stay on the bottom.
I honestly won't be able to properly test this until I have a bigger sphere to work with. Unfortunately that will require a lot more bearings, which are fairly expensive in mass.

 

[JordanLC]

So for anyone curious, ball bearings by themselves work fantastically, far better than the roller ball bearings. They rotate much more freely, and can handle weight much better.
I'm moving on to larger scale testing, large enough for walking with full weight, albeit with small steps.
The problem with ball bearings is trying to keep them in one spot. I figure once I can fill the surface with ball bearings so they don't have space to move it should be fine.
The other problem will be to keep the sphere in a central spot.
Eventually I'll make some sort of encasing like the roller ball bearings have, where it just comes over the edge of where the sphere expands outwards to the point where it keeps it from moving laterally.

Anyone have any knowledge on a material that won't provide the sphere with friction, or at least very little?