wheel puzzle

Unicycles and Equipment
1

I was thinking about this the other day and it fooled me for a minute. If you like to be fooled, read on:

When you are standing or sitting on your unicycle you apply a force down on the hub. The spokes transmit this force to the rim. So the rim sees a downward force. Your tube is filled with air that acts against the rim. The pressure is THE SAME EVERYWHERE. Meaning that it doesn’t push harder on the bottom than the top. To be static it would have to push up as hard as your weight pushes down. See if you can figure out this seeming contradiction. If you get it, don’t post the answer right away, so that it bothers some people. I’m not sure if I made this clear?
-gauss

2

Uh…

3

Re: wheel puzzle

Nope, I dont think you did. I do understand what you are saying, however, I have no clue what you are asking. What is the question that you wish us to post the answer to?
-David Kaplan

4

Re: wheel puzzle

On Mon, 13 May 2002 15:09:24 -0500,
UniDak <UniDak.4lyjm@timelimit.unicyclist.com> wrote:

> gauss wrote:
> > * I’m not sure if I made this clear? *
>
> Nope, I dont think you did. I do understand what you are saying,
> however, I have no clue what you are asking. What is the question that
> you wish us to post the answer to?

What he’s saying (if I’ve got it right) is something like this:

When you are on the unicycle, your weight is applied to the hub (either
through the cranks or the frame via the bearings, but that’s incidental).
Your weight is a downwards force. Your weight is conducted from teh hub
through the spokes to the rim. The rim therefore resists your weight.

However, the rim is not touching the ground. The rim is held off the
ground, supported by the air in the tyre. Since the air is fluid with
very low viscosity, and there is only a single chamber, the pressure
everywhere within it must be the same. But if the pressure is everywhere
the same, how can it resist the downward force? The air must be pushing
upwards on teh rim to hold it off the ground, but it must have the same
pressure in teh top of the tyre pushing down just as hard. This is his
conundrum.

It might be easier to visualise by imagining a tyre in teh which the air
is in dozens of separately inflated and sealed pockets distributed around
teh rim. If you looked at this, you’d say the bottom pocket compresses,
so teh pressure in it increases, so that’s what holds the rim up. But
that doesn’t work in a normal tyre - if you compress the air at the
bottom, it just moves round a bit.

So the question is, where is the force that holds the rim off the ground?

The question (or rather the answer) is similar to one you can introduce
earlier in the load-path - does the hub hang from teh spokes above it, or
does it push on teh spokes below? If it hangs, why can’t you stand on an
unlaced wheel rim without knackering it? If it stands, why can’t you
stand on teh ends of a few spokes without them buckling?

regards, Ian SMith

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5

Well said!

6

Re: wheel puzzle

On 13 May 2002 20:53:50 GMT, Ian Smith <ian@achrn.demon.co.uk> wrote:

>The question (or rather the answer) is similar to one you can introduce
>earlier in the load-path - does the hub hang from teh spokes above it, or
>does it push on teh spokes below? If it hangs, why can’t you stand on an
>unlaced wheel rim without knackering it? If it stands, why can’t you
>stand on teh ends of a few spokes without them buckling?

I think it hangs (mainly). But if you would stand on an unlaced wheel
you apply localised force, whereas the hub hangs also on spokes that
are at an angle. In addition, and this may be even more important, for
the wheel to knacker, it initially would become sort-of-oval-shaped in
that the horizontal wheel diameter grows. But this is prevented by the
spokes that are horizontal at that moment in time - ergo: no
knackering.

Gauss’s puzzle is a good one. I have an answer but I’m not sure if it
is right. This makes me think (not that I would want to do that), is
there an online equivalent to giving off a sealed envelope?

Klaas Bil

“To trigger/fool/saturate/overload Echelon, the following has been picked automagically from a database:”
“gorilla, ~, 1080H”

7

Aagh, dont be daft lad. I did understand the explaination. It was the part about “toes it hang, or does it stand” that i was missing in the passage. I understand it, but i was not able to figure out that that was the question at hand.
CAUTION ANSWER BELOW if you want to ponder this a bit more,dont read ahead**

The answer is that it hangs primarily. Infact, I believe that it was Spinergy who came3 out with a type of flexable spoke. There is no way the hub could stand with full pressure over it if a rim was laced only from the bottom half. Though it may stand longer if only laced from the top, it would still eventually turn to rubbish. There need to be all sorts of forces acting on the rim. Harper or someone could give you the technical parts, but my answer is that IT HANGS.
-David Kaplan

8

Damn it KLAAS! I hate you! As I was writing my response, you sneaked by, and posted yours first. Here I am thinking that I am so smart, and then I am brought to my post, and I see yours laughing at me. Saying “Haha I got here first” Now everyone is going to think I just copied your response, and that I am dumb. I AM NOT A CHEATER!!!:frowning: :frowning: :frowning:
-David Kaplan

9

Re: wheel puzzle

How about if you sat on your unicycle and had a friend “pluck” the spokes above
and below your hub. My understanding from reading bicycle crap is that you will
find the spokes under your hub to be carrying the load.

But what do they know…

David Maxfield
Bainbridge Island, WA

10

Ok. There are two separate puzzles here. One is that how do the spokes work (puzzles submitted by Ian). The other was the original one that I posted and Ian clarified, followed by Klaas Teasing everyone.
-gauss

11

Re: wheel puzzle

Spoiler - maybe…
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The pressure pushes equally in all directions. Therefore it pushes on the
tyre as much as on the rim. At all points around the wheel the pressure
in the tube acts to (a) compress the rim and (b) push out the tyre.

The force compressing the rim has no net effect - increasing the pressure
will simply put the rim under greater compression: it will not counter
any external directional force applied to the wheel.

The force pushing out the tyre only has an effect where the tyre
is in contact with the ground (where it is equal and opposite to the force
exerted by the ground). I believe this is what you have overlooked.

When you mount the unicyle the rim is lowered, deforming the shape of the
air
chamber and reducing its volume, and thereby increasing its pressure. The
pressure increases until the downward force applied to the part of the tyre
touching the ground once again equals the downward force of the unicycle
on the air chamber.

When the pressure in the tyre is high, the required deformation is small.
When it is low, you may bottom out before balancing the pressure against
the additional weight.

Arnold the Aardvark

12

THE ANSWER
(I think)
Hey, nice! Just one thing to add. your answer dodged the idea of how the rim can be balanced. The original question was about how the pressure is the same all over the rim, yet you can somehow apply a static downward force to the rim. The answer, I think, Is that the force is transmitted as a traction force between the tire and rim. I was carful in the original post to use the word tube instead of tire. The tube stretches the sidwall of the tire. The spot on the ground has its “stretch” relieved. As the bottom stretch is relieved, the top stretch of the tire applies an upward traction force on the rim to counter your pushing down. Hope you all had fun with this.
-gauss

13

RE: wheel puzzle

> a downward force. Your tube is filled with air that acts against the
> rim. The pressure is THE SAME EVERYWHERE.

Are you saying that the air pressure is the same everywhere in the tire? Yes
I think that’s true. But it ends there. The tire itself reacts differently
at the bottom than around the rest of it. So?

JF

14

Re: wheel puzzle

On Tue, 14 May 2002 08:28:01 -0500,
gauss <gauss.4nago@timelimit.unicyclist.com> wrote:
>
> THE ANSWER
> (I think)

I think gauss is quoting someone here, but I’ve not seen what he’s
quoting, however I agree with his explanation:

> force is transmitted as a traction force between the tire and rim. I
> was carful in the original post to use the word tube instead of tire.
> The tube stretches the sidwall of the tire. The spot on the ground has
> its “stretch” relieved. As the bottom stretch is relieved, the top
> stretch of the tire applies an upward traction force on the rim to
> counter your pushing down. Hope you all had fun with this.

I’d phrase it slightly differently (and avoid the word traction, which
generally means a force in an altogether different direction when
referring to wheels) - the pressure in the tyre prestresses the sidewall,
so the tyre carcase is everywhere pulling the rim outwards (away from the
hub). At the point of contact with the ground, the tyre is pushed
inwards, thus relieving the prestress in teh carcase wall.

Therefor, the tyre pushes up on teh rim, by locally reducing the amount by
which it pulls down. It’s the same basic answer as gauss’s.

Incidentally, I believe any talk of changes in pressure is a complete red
herring - I don’t believe the pressure change in a loaded wheel is
relevant. My reasoning is that at the pressures we are dealing with, air
is an elastic gas (it’s elastic up to many, many times these pressures).
So since we have a constant mass of air within the tyre, the pressure will
be inversely proportional to the volume (PV will be constant). When you
load the wheel and the tyre deforms, it moves in such a tiny amount I
don’t believe the change in volume (and hence change in pressure) is
enough to affect anything. Furthermore, although the tyre tread moves in
towards the hub, the sidewalls bulge, thus the volume change is really
really tiny.

regards, Ian SMith

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15

Re: wheel puzzle

On Mon, 13 May 2002 17:37:52 -0500,
UniDak <UniDak.4m59b@timelimit.unicyclist.com> wrote:

> The answer is that it hangs primarily.

No, structurally it stands.

> Infact, I believe that it was
> Spinergy who came3 out with a type of flexable spoke.

Irrelevant - spokes are flexible. At least they’re fairly flexible,
and since they have pivoting connections at hub and at rim no bending can
be conducted to them, so they may as well be infinitely flexible (ie, bits
of string).

> There is no way
> the hub could stand with full pressure over it if a rim was laced only
> from the bottom half. Though it may stand longer if only laced from the
> top, it would still eventually turn to rubbish.

No, you can’t lace a wheel with spokes round only half the rim. Yopu
don’t actually get as far as doing anything with it - the rim would not
remain cuircular, the hub would not be central, and you wouldn’t have
anything useable as a wheel.

> There need to be all
> sorts of forces acting on the rim. Harper or someone could give you the
> technical parts, but my answer is that IT HANGS.

As I say up above, it stands. It stands on teh reduced tension in the
lower spokes. It has been demonstrated both with instrumented wheels
under test and with finite element analysis that teh tension in the lower
spokes reduces, while the tension in the side and upper spokes stays
pretty much the same as in an unladen wheel. This is why you should keep
your spokes tight - the wheel only works while the tension in the spokes
is greater than the maximum force you’re going to apply to the rim. If
you load the rim enough to make a spoke go slack then suddenly you’re
dependant upon the bending stiffness of the rim, and wheel failure is
normally not far away.

If anyone is still interested teh definitive text on teh subject is
probably “the Bicycle Wheel” by Jobst Brandt. It’s ISBN 0-9607236-4-1,
and is about 150 pages of book talking about nothing but bicycle wheels,
covering both the theory and the practicalities of building and
maintaining wheels.

regards, Ian SMith

|\ /| no .sig
|o o|
|/ |

16

Re: wheel puzzle

On Tue, 14 May 2002 08:28:01 -0500,
gauss <gauss.4nago@timelimit.unicyclist.com> wrote:
>
> THE ANSWER
> (I think)

I think gauss is quoting someone here, but I’ve not seen what he’s
quoting, however I agree with his explanation:

> force is transmitted as a traction force between the tire and rim. I
> was carful in the original post to use the word tube instead of tire.
> The tube stretches the sidwall of the tire. The spot on the ground has
> its “stretch” relieved. As the bottom stretch is relieved, the top
> stretch of the tire applies an upward traction force on the rim to
> counter your pushing down. Hope you all had fun with this.

I’d phrase it slightly differently (and avoid the word traction, which
generally means a force in an altogether different direction when
referring to wheels) - the pressure in the tyre prestresses the sidewall,
so the tyre carcase is everywhere pulling the rim outwards (away from the
hub). At the point of contact with the ground, the tyre is pushed
inwards, thus relieving the prestress in teh carcase wall.

Therefor, the tyre pushes up on teh rim, by locally reducing the amount by
which it pulls down. It’s the same basic answer as gauss’s.

Incidentally, I believe any talk of changes in pressure is a complete red
herring - I don’t believe the pressure change in a loaded wheel is
relevant. My reasoning is that at the pressures we are dealing with, air
is an elastic gas (it’s elastic up to many, many times these pressures).
So since we have a constant mass of air within the tyre, the pressure will
be inversely proportional to the volume (PV will be constant). When you
load the wheel and the tyre deforms, it moves in such a tiny amount I
don’t believe the change in volume (and hence change in pressure) is
enough to affect anything. Furthermore, although the tyre tread moves in
towards the hub, the sidewalls bulge, thus the volume change is really
really tiny.

regards, Ian SMith

|\ /| no .sig
|o o|
|/ |

17

Re: wheel puzzle

On Mon, 13 May 2002 17:37:52 -0500,
UniDak <UniDak.4m59b@timelimit.unicyclist.com> wrote:

> The answer is that it hangs primarily.

No, structurally it stands.

> Infact, I believe that it was
> Spinergy who came3 out with a type of flexable spoke.

Irrelevant - spokes are flexible. At least they’re fairly flexible,
and since they have pivoting connections at hub and at rim no bending can
be conducted to them, so they may as well be infinitely flexible (ie, bits
of string).

> There is no way
> the hub could stand with full pressure over it if a rim was laced only
> from the bottom half. Though it may stand longer if only laced from the
> top, it would still eventually turn to rubbish.

No, you can’t lace a wheel with spokes round only half the rim. Yopu
don’t actually get as far as doing anything with it - the rim would not
remain cuircular, the hub would not be central, and you wouldn’t have
anything useable as a wheel.

> There need to be all
> sorts of forces acting on the rim. Harper or someone could give you the
> technical parts, but my answer is that IT HANGS.

As I say up above, it stands. It stands on teh reduced tension in the
lower spokes. It has been demonstrated both with instrumented wheels
under test and with finite element analysis that teh tension in the lower
spokes reduces, while the tension in the side and upper spokes stays
pretty much the same as in an unladen wheel. This is why you should keep
your spokes tight - the wheel only works while the tension in the spokes
is greater than the maximum force you’re going to apply to the rim. If
you load the rim enough to make a spoke go slack then suddenly you’re
dependant upon the bending stiffness of the rim, and wheel failure is
normally not far away.

If anyone is still interested teh definitive text on teh subject is
probably “the Bicycle Wheel” by Jobst Brandt. It’s ISBN 0-9607236-4-1,
and is about 150 pages of book talking about nothing but bicycle wheels,
covering both the theory and the practicalities of building and
maintaining wheels.

regards, Ian SMith

|\ /| no .sig
|o o|
|/ |

18

Re: wheel puzzle

Wait a minute. Gauss asked not to post a solution to his tube pressure
puzzle. I mentioned I had a solution but I didn’t spell it out.

Ian (besides clarifying Gauss’s puzzle nicely) asked the question
about spokes. I said the hub hangs, and I answered the wheel
knackering question associated with it. But, really, I believe you
that your answer was original too.

OK? :-)?

Klaas Bil

On Mon, 13 May 2002 17:40:49 -0500, UniDak
<UniDak.4m5hn@timelimit.unicyclist.com> wrote:

>
>Damn it KLAAS! I hate you! As I was writing my response, you sneaked by,
>and posted yours first. Here I am thinking that I am so smart, and then
>I am brought to my post, and I see yours laughing at me. Saying “Haha I
>got here first” Now everyone is going to think I just copied your
>response, and that I am dumb. I AM NOT A CHEATER!!!:frowning: :frowning: :frowning:
>-David Kaplan
>
>
>–
>UniDak - David Kaplan
>
>And thats all I have to say about that. -Forrest Gump
>------------------------------------------------------------------------
>UniDak’s Profile: http://www.unicyclist.com/profile/311
>View this thread: http://www.unicyclist.com/thread/18063
>

–
“To trigger/fool/saturate/overload Echelon, the following has been picked automagically from a database:”
“Li Hongzhi, CFD, Lexis-Nexis”

19

Re: wheel puzzle

On 13 May 2002 23:32:46 GMT, maxfieldd@aol.com (Maxfield D) wrote:

>How about if you sat on your unicycle and had a friend “pluck” the spokes above
>and below your hub. My understanding from reading bicycle crap is that you will
>find the spokes under your hub to be carrying the load.
>
>But what do they know…

Not much I think! How can the spokes under your hub be under tension?
They would drag the hub down, only accellerated by the rider’s weight.
On the contrary, the rider’s weight has to be compensated by tension
in spokes above the hub (or at least that’s what I think).

Somebody do the friend plucks spokes gag and tell us.

Klaas Bil

“To trigger/fool/saturate/overload Echelon, the following has been picked automagically from a database:”
“Li Hongzhi, CFD, Lexis-Nexis”

20

In fact all the spokes should always be in tension. The reason a wheel works is because you have this residual stress in the wheel all the time. If you load the bottom hard enough to relieve this tension on the bottom, then as Ian pointed out your not far from loosing some money. I guess answering the puzzle was fair enough. It had been up for a few days. Finally, the person I quoted was myself. Actually I’ve never read anything on the subject, but have learned a thing or two school that helped me figure out what was going on. Thanks, Ian, for also clarifying my answer. That was nicely explained. Do you know where I could score one of those books. I looked on amazon, but it diddn’t seem to have any.
-gauss