wheel puzzle

Unicycles and Equipment
21

RE: wheel puzzle

> 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).

Pardon me if I come in sounding too un-engineer-like.

If you guys want to argue about spokes, why not establish the basics with an
old Semcycle wheel, the kind with radial spokes. If you can agree on that
one, then work your way up to what we all ride now, which has spokes going
in every direction. Why do they do that? Because it makes the wheel more
solid.

Radial wheels are very common these days on high-end bikes, especially
racing bikes, but only on the front wheel. Sometimes they’re used on one
side of the back wheel, but let’s ignore that for now. The front wheel hits
plenty of bumps, but the reason radial works there is because there is no
driving force applied to that wheel.

Radial spokes are weak in the direction of rotation. In other words, if you
took an old Semcycle and faced it toward a wall, you could press down on the
forward pedal and literally see the hub rotating back and forth as you
pressed on the pedal. This setup required constant attention to spoke
tightness, and eventually led to broken spokes.

A cross-spoked wheel has tension in all sorts of directions. When load is
applied, in the form of weight bearing down on the axle, a whole bunch of
the spokes, facing in all different directions, transfer this load to the
rim. They’re not on the top or bottom, they’re probably mostly on the sides.

So if the original question was which spokes take up the tension, I think
the most accurate answer would be “most of them” on a cross-spoked wheel.

If the question was whether the pressure in the tire stays the same with or
without load, the answer is yes (with possible micro variations that don’t
affect anything a rider can detect).

If the question is why does the tire compress if the air pressure doesn’t
change, it’s because you may need to lose a few pounds :slight_smile:

Stay on top,
John Foss, the Uni-Cyclone

not a scientist, just a unicyclist

“Everything I know about physics I learned from riding a unicycle (and
watching “professional” wrestling).”

22

Re: wheel puzzle

On Wed, 15 May 2002 09:36:58 -0700,
John Foss <john_foss@asinet.com> wrote:

> Radial spokes are weak in the direction of rotation. In other words, if you
> took an old Semcycle and faced it toward a wall, you could press down on the
> forward pedal and literally see the hub rotating back and forth as you
> pressed on the pedal. This setup required constant attention to spoke
> tightness, and eventually led to broken spokes.
>
> A cross-spoked wheel has tension in all sorts of directions. When load is
> applied, in the form of weight bearing down on the axle, a whole bunch of
> the spokes, facing in all different directions, transfer this load to the
> rim. They’re not on the top or bottom, they’re probably mostly on the sides.

The bottom spokes at the rim are at the side of the hub (normally,
assuming you’ve got the maximum possible number of crossings, but most
wheels do), but I think this is what everyone has meant by ‘bottom spokes’
in the discussion - it’s what I’ve been calling bottom spokes.

In addition to the issues of torque transmittal, most hubs are made
assuming tangential spoking at the hub. That means the amount of metal
between the spoke hole and the edge of the hub is too small to take much
radial spoke tension - it is possible to tighten radial spokes enough to
tear them out of normal hubs. With spokes tangential at teh hub the
tension is not pulling through the thin bit of metal, but rather would
have to tear through a sizeable chunk of hub to pull free.

> So if the original question was which spokes take up the tension, I think
> the most accurate answer would be “most of them” on a cross-spoked wheel.

No, it’s still the spokes attached near the bottom of the rim that carry
the load imposed on teh wheel, which they do by reducing their tension.

> not a scientist, just a unicyclist
>
> “Everything I know about physics I learned from riding a unicycle (and
> watching “professional” wrestling).”

Well, it was a pretty good description of torque transfer problems with
radial spoking, none the less. I’m not sure what part of it was
contributed by the wrestling.

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23

Re: wheel puzzle

On Tue, 14 May 2002 21:49:30 -0500,
gauss <gauss.4obqa@timelimit.unicyclist.com> wrote:

> In fact all the spokes should always be in tension.

Absolutely.

> 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.

No, sorry - I’ve had mine about 10 years, I guess maybe it’s out of print
now. It was published by “Avocet Inc., Menlo Park 94026” if that helps.
Library of Congress Catalog Card number 81-69715
ISBN 0-9607236-4-1

regards, Ian SMith

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24

I bought the 3rd edition of Jobst Brandt’s book just a couple of months ago; I think it was amazon.com. Yes - here it is (http://www.amazon.com/exec/obidos/ASIN/0960723668/qid=1021500013/sr=8-1/ref=sr_8_1/102-5593326-6372103). It is very illuminating; I now see cycle wheels completely differently. It is WORTH it for the understanding.

Basically the cycle sits on the rim; the spokes keep the rim in shape. Just like a bridge. All the spokes on the wheel are in tension all the time. They pull the rim towards the hub equally all the time, thus keeping the rim circular. The rim doesn’t collapse when you sit on your cycle because the spokes are keeping the rim circular much harder than your weight is trying to flatten it.

To say that the “bottom spokes carry the load by reducing their tension” is missing the point. To say that the wheel’s reaction to load is that the bottom spokes’ tension is lower is more accurate because it says that the entire wheel is carrying the load.

Without the spokes, the rim deforms. Forces greater than the spokes can handle will deform the rim. Properly tensioned spokes support the rim properly. Untensioned spokes do not, and wheel failure is the result. Stiffer, stronger rims demand less of the spokes, and transfer the stress tangentially to more spokes more effectively, making a stronger combination.

When a rider sits on the cycle, the rim deforms slightly inwards at the point of contact with the ground and the spokes in that area have decreased tension. Where does that stress go? It is transferred to the rest of the wheel through the rim. Accordingly, the rest of the rim increases slightly in diameter and the spokes in that area have more tension on them. See Brandt 3rd edition pp 19. So it is the entire wheel that responds. Whether or not one will be able to hear this by plucking the spokes I don’t know. By careful note-taking and the use of a tensiometer, which I don’t have, one should be able to see the difference, though. Be sure to examine all the spokes, not just one or two, and use a sufficiently large load that will not move around during the data-taking.

The pressurized tube/tire does compress the rim inwards as well (see 3rd edition pp 14). So it doesn’t prestress the rim outwards, as suggested in an earlier post. In fact, it does the opposite.

The tire pressurizes somewhat when you sit or hop on it because the strong elastic walls of the tube, tire, and rim do not allow it to expand sufficiently to keep the pressure constant. It is both this pressurization and the rubber elastic expansion which allows a prehop to help one attain greater height – the stored energy is released as upwards kinetic energy. Brandt, however, is silent about the relative amount of air compression.

25

Re: wheel puzzle

On Wed, 15 May 2002 00:32:04 GMT,
klaasbil_remove_the_spamkiller_@xs4all.nl (Klaas Bil) wrote:

>How can the spokes under your hub be under tension?

I need to qualify this statement. All spokes are under tension at all
times, but the spokes below the hub are under lower tension.

Klaas Bil

“Seems that quotes are quite fashionable these days” - Bruce Edwards

“To trigger/fool/saturate/overload Echelon, the following has been picked automagically from a database:”
“ie.org, Blenheim, WWSP”

26

First of all a number of points here have already been said (in different ways) but I wrote this off line so here is my point of view.

Many years ago (not that many though) I pondered about this and these are the conclusions I came to. They’re not based on any experiment or study of wheels, I was just extrapolating from my own knowledge.

Does a hub hang or stand on the rim?

How could it stand? First all the tension in the spokes below the hub would have to be released for them to take the compression force necessary for the hub to stand on the rim through the spokes. Then what is stopping the nipples pushing through into the inner tube? As far as I’m concerned standing is a nonstarter.

So it must hang then? Well at first yes. The key to this is distribution and translation of directional forces. As the mass of your body is transferred through the seat and pedals to the hub a downward force is applied to the spokes above and below the hub. As all the spokes in the wheel are under equal tension (hopefully) the tension in the spokes below the hub are reduced and in the spokes above, it is increased. This has the net effect of trying to deform the wheel into an ellipse; the vertical force trying to squash the rim has been translated through the rim into a horizontal force trying to pull the rim away from the hub. This increases the tension in the spokes either side of the hub therefore distributing the force. I would guess about 3/4 of the spokes are under increased tension and the other 1/4 has reduced tension (don’t quote me on that). Without this distribution of force those flimsy spokes would just stretch, lose their tension, and allow the wheel to deform. You probably know a 48-spoke wheel is stronger than a 36-spoke wheel (all other things being equal), if you divide the force between 48 spokes each one has less force applied than 36 spokes (also the distance between the holes in the rim is less = better distribution), logical really.

As unicyclists all our weight is distributed to one wheel and the myriad of the forces applied to the hub are unique and complex and I have only described the wheel with static force applied. Of course if a wheel were only to be used statically it wouldn’t need to be as strong as they are.

The original question in this thread was regarding the tire. As far as I was aware the pressure (more correctly the pressure difference) imparts a force on the inside the tire to give it strength and rigidity to resist deformation. The forces are therefore transferred through the tire (to the rim) in a similar way as they are in the wheel (think of a cross-section of the tire) only pressure difference replaces spokes. If the tire were elastic like the inner tube this wouldn’t work so tire manufacturers use nylon and other materials to stop them from stretching excessively and to give them some internal rigidity (to stop tire pinching for example) whilst still allowing them to deform in a controlled way to absorb shocks (this goes into wave mechanics which is way beyond this post, or me). As the tire volume changes very little when compressed any change in pressure would only be slight and make little, if any, difference.

Arches have been used for thousands of years as a means of distributing a force and the wheel is just two arches. Incidentally a suspension Bridge works on the same principle only in reverse (the towers are pulled together).

Here is a small experiment for you to try, first you will need a piece of A4/letter paper, one CD (out of its case) and two video cassettes (or books of similar size). First place the two video cassettes about six inches apart then place the piece of paper bridging between the two cassettes and finally place the CD in the middle of the paper. As you would expect the paper bends and cannot support the weight of the CD. Now take the piece of paper and curve it and place each end in between the two cassettes so you have an arch. Now place the CD on top of the arch and the piece of paper can now support the weight of the CD.

A lot is made of the material used in construction but the design is of equal if not greater importance and the spoked cycle wheel is a very good demonstration. They are light, not too rigid (unlike solid wheels), and very strong for the amount of material used.

If you’re still reading this far thanks :slight_smile:

Cheers, Gary

27

Re: wheel puzzle

On Wed, 15 May 2002 17:27:16 -0500,
U-Turn <U-Turn.4pu4m@timelimit.unicyclist.com> wrote:

> The pressurized tube/tire does compress the rim inwards as well (see 3rd
> edition pp 14). So it doesn’t prestress the rim outwards, as suggested
> in an earlier post. In fact, it does the opposite.

No-one said it did. I said the pressure pre-stresses the tyre sidewalls,
which is true, but no-one has yet made any comment on the effect of the
tyre pressure on the rim.

> The tire pressurizes somewhat when you sit or hop on it because the
> strong elastic walls of the tube, tire, and rim do not allow it to
> expand sufficiently to keep the pressure constant. It is both this
> pressurization and the rubber elastic expansion which allows a prehop to
> help one attain greater height – the stored energy is released as
> upwards kinetic energy. Brandt, however, is silent about the relative
> amount of air compression.

No, there I believe you are wrong. The tyre does not stretch
significantly, because of the threads within the carcase. The rubber of
the tyre cannot undergo much elastic deformation because the threads are
relatively inelastic.

regards, Ian SMith

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28

Re: wheel puzzle

On Thu, 16 May 2002 05:49:03 -0500,
unicus <unicus.4qsma@timelimit.unicyclist.com> wrote:

> How could it stand? First all the tension in the spokes below the hub
> would have to be released for them to take the compression force
> necessary for the hub to stand on the rim through the spokes.

The structural effect is predominantly an increase in the upward force
applied to the hub by the spokes due to a reduction in the tension in teh
spokes below.

> spokes in the wheel are under equal tension (hopefully) the tension in
> the spokes below the hub are reduced and in the spokes above, it is
> increased.

But the point is that the tension in the spokes below reduces
significantly, while the tension in the spokes above and to the sides
hardly changes. Hence the assertion that it predominantly stands on teh
lower spokes.

> reduced tension (don’t quote me on that). Without this distribution of
> force those flimsy spokes would just stretch, lose their tension, and

They’d only lose their tension if they were loaded beyond their yield
stress. That’ll be at something like 150kg per spoke (actually, that’s a
guess, but I reckon it’s not a bad one). I think you’ve already
completely unloaded the lower spokes and buckled the wheel before this
happens.

> to resist deformation. The forces are therefore transferred through the
> tire (to the rim) in a similar way as they are in the wheel (think of a
> cross-section of the tire) only pressure difference replaces spokes.

Not sure what you mean by pressure difference here.

> Incidentally a suspension
> Bridge works on the same principle only in reverse (the towers are
> pulled together).

Well, they rely on tension in slender elements (the hangers), but the
hangers are not prestressed against each other, or against the towers, so
I’d say the similarity is a bit tenuous. (I don’t really agree with teh
arch comparison either, but that’s straying ever further off topic).

regards, Ian SMith

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29

That is what I was responding to… the tire doesn’t pull the rim outwards from the hub, it pushes it inwards.

I didn’t say to what degree it stretches. I said that there are two effects: the air pressurizes and the tube/tire stretches. In fact, I said that I didn’t know because Jobst Brandt doesn’t address it. However, it is true that the two effects together store a significant amount of energy that is retrievable in a pre-hop.

This area, like the electrons/holes idea in electronics, is really a matter of semantics, semantics that are very useful to an engineer but get in the way of lay discussions. The wheel supports the cyclist. To say “hangs” or even “stands” is an oversimplification that happily (for me) Jobst Brandt’s wonderful book eliminated.

What’s off-topic about it? This is a great discussion and, I hope, interesting to people who haven’t yet seen the light and ordered Mr. Brandt’s book. :smiley:

I personally feel the arch comparison is excellent, and really shows up the genius of the wheel. Think of a stone arch bridge with someone standing on it. It is a lot like the bottom of the wheel (upside down of course). The rim is the set of stones, gravity is the spokes, the person is the force exerted on the wheel by the ground. The wedge shapes of the stone are like the finite elements of the rim, trying to go down (the stone, because of gravity) or towards the hub (the wheel, because of the tensioned spokes). But they can’t go down because they also press outwards, and there is pressure back! In the stones’ case, the ground at the ends of the arch pushes the end stones towards each other, which press inward on the next stones, etc. In the wheel’s case – here is the true genius – the rim elements push outwards but can’t move because of their adjoining elements, which press outwards along the rim, all the way around the circle infinitely. The elements stay in place because the spokes are pulling on them… It is pure magic! Building a wheel, round, and round, tensioning each pass, illustrates the ideas, the structure of forces that are the true substance of the wheel. The wheel is an infinite arch.

Another glorious idea, to me anyway, is that the wheel rolls, and even though it is moving laterally and rotationally, the force is still on the top of the arch, the top of the bridge.

So if this ramble made any sense, you may see that the spokes are not really the focus, the rim is. The spokes help the rim keep its shape. The hub allows the spokes to do that. The cyclist’s weight/downwards force on the hub affects the amount of the bridge’s gravitational pull. If the force of the cyclist is too great, gravity becomes zero (the “bottom” spokes lose their tension), the stones float a little (the rim loses most of its shape-holding compression), and the strong side forces accompanying that downward force blast the stones sideways, destroying the bridge.

If this is incomprehensible or difficult to understand, it’s due to my writing, not the ideas I’m trying to express and I’d be glad to elaborate on part or whole if anyone would like.

The analogy is partly difficult because of the upside-downish comparison. The bridge has a person on top, gravity pulls the stones downward, the stones push outwards as they try to fall and actually hold each other up. The wheel has the ground on the bottom, pushing up, the spokes pull the rim upwards, the rim pushes outwards as it tries to flatten but can’t because of the rest of the wheel. When the cyclist puts force on the hub, lessening the amount of force in the “bottom” spokes, it is like reducing the force of the gravity pulling down the stones of the bridge.

30

Re: wheel puzzle

On Thu, 16 May 2002 22:26:04 -0500,
U-Turn <U-Turn.4s2om@timelimit.unicyclist.com> wrote:
>
> > the pressure in the tyre prestresses the sidewall,
> > so the tyre carcase is everywhere pulling the rim outwards (away from
> > the hub).

> That is what I was responding to… the tire doesn’t pull the rim
> outwards from the hub, it pushes it inwards.

No, the tyre PRESSURE may push inwards, but the tyre CARCASE is pulling
outwards. The tyre carcase can’t possibly be pushing in on teh rim - its
far too thin and would just buckle. It’d be like trying to push something
with a piece of string - you can’t do it. The tyre sidewall is under
tension, so it can only be pulling anything to which it is attached.

> > No, there I believe you are wrong. The tyre does not stretch
> > significantly, because of the threads within the carcase.

> I didn’t say to what degree it stretches. I said that there are two
> effects: the air pressurizes and the tube/tire stretches. In fact, I
> said that I didn’t know because Jobst Brandt doesn’t address it.
> However, it is true that the two effects together store a significant
> amount of energy that is retrievable in a pre-hop.

Yes you did say to what degree - you said it does so sufficiently to store
useful amounts of elastic energy. You’ve just said so again. I said the
tyre does not stretch significantly, that is, any effect of the stretching
is insignificant, ie not significant.

> > I don’t really agree with teh
> > arch comparison either, but that’s straying ever further off topic).
>
> I personally feel the arch comparison is excellent, and really shows up
> the genius of the wheel.

OK, I’ll go through your elaboration of the comparison, explaining why I
don’t like it. I still thing it’s heading off topic though.

> Think of a stone arch bridge with someone
> standing on it. It is a lot like the bottom of the wheel (upside down of
> course). The rim is the set of stones, gravity is the spokes,

Gravity acts down, the spokes act radially. Thus your loading pattern is
different - sufficiently different to completely alter the profile of an
load path, which in an arch under uniform loading is parabolic, and in a
rim loaded only by spoke pretension will be substantially circular. In
the arch the load at one point is radial, but over the majority of the
arch a significant component of teh imposed load is circumferential.

Furthermore, on most arch bridges self weight is the predominant load
effect and imposed loads are a relatively small component. This is the
reverse of teh case of the bicycle wheel.

Furthermore, in a bicycle wheel the spoke loads are equal. In arch
bridges the fill between the spandrel walls is deeper over the springings
than over the crown, so the applied load varies across the arch.
(Actually, this is not universally true - there are a few ancient chinese
masonry arches with a uniform fill, and a very few european arches.)

> the person is the force exerted on the wheel by the ground.

> The wedge shapes of the stone are like the finite elements of the rim,

I really wouldn’t introduce teh term finite elements with respect to teh
stones of teh arch (incidentally the technical term for the stones is
voussoirs). It may be liable to cause confusion regarding ‘finite element
analysis’, and the voussoirs are not particularly good examples of the
elements in a finite element analysis.

> trying to go down
> (the stone, because of gravity) or towards the hub (the wheel, because
> of the tensioned spokes).

ie, in different directions.

> But they can’t go down because they also press
> outwards, and there is pressure back! In the stones’ case, the ground
> at the ends of the arch pushes the end stones towards each other,

Indeed, another difference - teh arch has an unbalanced lateral thrust
resisted by other components (the abutments) while the bicycle wheel is
complete within itself.

The other fundamental difference is that an arch operates happily made in
a material with no tensile, and hence no flexural, capacity. The rim of a
bicycle wheel is subject to bending, reliant on teh tensile performance of
the material from which it is made.

I also don’t like the arch-wheel analogy because teh wheel very
efficiently uses the material from which it is made - all of the material
sees stresses that are a significant proportion of the strength.
Conversely a typical stone arch may have a barrel that is hundreds of
millimetres thick, but a zone of thrust of only a few millimetres - so the
vast majority of teh material is unstressed.

Thus, I conclude that teh similarities that do apply are that both teh
arch and the bicycle wheel are curved, and that they don’t collapse when
an appropriate load is appled. You could make the same comparison to a
banana (though the load is somewhat smaller than either the arch or the
wheel case, the ratio (arch collapse load):(wheel collapse load) may well
be similar to that of (wheel collapse load):(banana squash load)).

regards, Ian SMith

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31

I was talking about the tube/tire combination, pressurized with air. But the tire by itself presses inwards; have you changed a bicycle tire? It is like a rubber band, whether the bead is steel, kevlar, or whatever.

This hasn’t said anything new. Do you have numbers to quantify “not significant”?

Those are the boundary conditions; which the wheel so ingeniously eliminates. As far as the direction of load, when you resolve the load into vector components (parallel and perpendicular to the element in question), the similarities become clear.

This point, the midpoint of the arch, is exactly the point on the unicycle wheel’s arch where the load exists, except for situations like straddling two rocks, or hopping on the wheel. When the rim deforms under radial load (see Brandt pp 19), the circumferential load is distributed around the wheel by the rim. This is one reason why the rim strength is important in a wheel.

Yes, but that is irrelevant to the analogy. Just because the wheel is more efficient than most arches doesn’t mean that it isn’t like an arch.

Again, this doesn’t mean that the wheel doesn’t work like an arch fundamentally does. Most arches, such as bridges, have to interface smoothly with a planar (linear in 2d) surface; the wheel does not.

Yes, that’s exactly my point – the wheel is an ingenious infinite arch. Those “unbalanced components”, instead of being transmitted to the ground, are zinged right around the wheel to balance themselves out.

I’m sure that I’ve made the case that there are more significant correlations.

As far as bananas go, they make great ice cream sundaes, but terrible arches! Apples, on the other hand, make great snacks but terrible verbs. :roll_eyes:

Enjoying the discussion…

32

Ian Smith writes:

So it hangs then. Therefore the spokes above the hub are at a higher tension (and to the sides) than below which although the hub is not purely hanging in my view it cannot be called standing.

So it stands then. Where does the energy go? A spoke is tensioned to about 80kg to 90kg (I read this somewhere but can’t remember where now), which on a 36-spoke wheel is 2880kg to 3240kg of tension in the wheel. When you sit on your uni the spokes below the hub reduce their tension, where does the energy go? When you get off your uni the spokes below the hub return to their original tension, where does the energy come from? The energy remains in the wheel as a structural unit and is just transferred around the rim and through the spokes to increase tension to appose reduction elsewhere.

Yes true. That is what I was trying to say, if the load wasn’t distributed through the wheel and instead through only a few spokes they wouldn’t be strong enough to cope and the wheel would deform.

I prefer to talk about the pressure difference rather than just pressure and here I was referring to the pre-stress in the tire.

I was referring to the translation of a downward force to a horizontal force through the large curved cables between the towers (which is then translated into a downward force through the towers by the cables anchored either side of the bridge). Curves, arches and wheels are stronger than merely the material they are made from.

I once tacoed (?) a rear bike wheel (I was young and higher = better) and there was hardly any tension left in the spokes but after moving the brake out of the way my friend (it wasn’t my bike) manage to ride it home (the tire was ok surprisingly but the wheel was binned, I had to pay for it though).

Anyway we could debate this for ages, and others probably will/are, but I’m a little unhappy at the moment so I won’t be, sorry.

This was an interesting discussion and if you want to know why I’m unhappy look at my post “Help me, suggestions please” (it has nothing to do with this thread).

Gary

33

Re: wheel puzzle

On Fri, 17 May 2002 20:22:45 -0500,
unicus <unicus.4trkb@timelimit.unicyclist.com> wrote:
> Ian Smith writes:

> > But the point is that the tension in the spokes below reduces
> > significantly, while the tension in the spokes above and to the
> > sides hardly changes. Hence the assertion that it predominantly
> > stands on teh lower spokes.

> So it stands then. Where does the energy go? A spoke is tensioned to
> about 80kg to 90kg (I read this somewhere but can’t remember where now),

Sounds about right to me, since I guessed teh elastic limit for s spoke to
be in the 150-200kg range.

> which on a 36-spoke wheel is 2880kg to 3240kg of tension in the wheel.
> When you sit on your uni the spokes below the hub reduce their tension,
> where does the energy go?

I’m not sure energy balance is teh right way to look at it. I’d consider
a work balance. You just sat on it. You applied a force to the hub which
moved a distance. That force times that distance was balanced by the sum
of the force in teh spokes times the distance they shortened (or reduced
their elongation, if you prefer). Obviously, if you really wanted to do
that calculation you’d need an integration since teh tension in teh spoke
is not uniform as the spoke deforms.

> When you get off your uni the spokes below the
> hub return to their original tension, where does the energy come from?

Bending stiffness of the rim? As I said, I’m not convinced that trying
tro track the energy does anything interesting. It’s not so easy to see
where energy ‘goes’ because there are all sorts of nasty lossy things
going on in the vicinity - hysteresis in the tyre rubber, viscosity of the
air in teh tube and so on. While some of these I’d be happy to rule out
on just a moments thought (eg viscosity of teh air), the others are not so
obvious.

> > Well, they rely on tension in slender elements (the hangers), but
> > the hangers are not pre-stressed against each other, or against the
> > towers, so I’d say the similarity is a bit tenuous.

> I was referring to the translation of a downward force to a horizontal
> force through the large curved cables between the towers (which is then
> translated into a downward force through the towers by the cables
> anchored either side of the bridge). Curves, arches and wheels are
> stronger than merely the material they are made from.

OK, that’s fair enough. I was maybe looking too closely and trying to
draw more precise comparisons.

regards, Ian SMith

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34

Re: wheel puzzle

On Fri, 17 May 2002 18:09:07 -0500,
U-Turn <U-Turn.4tljn@timelimit.unicyclist.com> wrote:
>
> > Yes you did say to what degree - you said it does so sufficiently to
> > store useful amounts of elastic energy. You’ve just said so again. I
> > said the tyre does not stretch significantly, that is, any effect of
> > the stretching is insignificant, ie not significant.

> This hasn’t said anything new. Do you have numbers to quantify “not
> significant”?

No, but we’re not talking numbers. If you think demanding numbers
rubbishes an argument then I can demand yours too if you like - how many
joules of energy (precisely) does this tyre stretch store for use in your
pre-hop? However, the discussion was qualitative, not quantitative, and
such demands take us nowhere.

I remain of teh opinion that teh stretch in a carcase is insignificant,
and that insignificant energy is stored by elongation, because the tyre
contains inelastic threads. The whole point of teh threads is to stop the
carcase stretching.

> > Gravity acts down, the spokes act radially. Thus your loading pattern
> > is different - sufficiently different to completely alter the profile

> As far as the direction of load, when you resolve the load
> into vector components (parallel and perpendicular to the element in
> question), the similarities become clear.

No, the differences become clear, because in an arch, over much of teh
arch a very large component is circumferential - something that a wheel
has hardly any of (only a relatively small point load at teh ground, if
the rider is braking or accelerating).

> > Furthermore, on most arch bridges self weight is the predominant load
> > effect and imposed loads are a relatively small component. This is the
> > reverse of teh case of the bicycle wheel.

> Yes, but that is irrelevant to the analogy. Just because the wheel is
> more efficient than most arches doesn’t mean that it isn’t like an
> arch.

Hang on - you’re saying that because they work in different ways, under
diufferent loadings and different load patterns, it doesn’t mean they’re
not like each other?

The point is not one of efficiency, it’s one of load patterns. The arch
works because most of the imposed load is self-weight, so you build the
arch to accomodate the line of thrust of teh self weight. It then remains
standing if you apply the required live load because teh live load is
fairly small, so teh thrust line doesn’t move much. I believe that this
is not true of a bicycle wheel.

> > Indeed, another difference - teh arch has an unbalanced lateral thrust
> > resisted by other components (the abutments) while the bicycle wheel
> > is complete within itself.

> Yes, that’s exactly my point – the wheel is an ingenious infinite
> arch.

That’s your point? Your point seemed to be that a wheel was just like an
arch, but now you agree with my observation that they are different?

> > Thus, I conclude that teh similarities that do apply are that both
> > teh arch and the bicycle wheel are curved, and that they don’t
> > collapse when an appropriate load is appled. You could make the same
> > comparison to a banana

> I’m sure that I’ve made the case that there are more significant
> correlations.

Good. I’m sure you haven’t.

regards, Ian SMith

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35

Re: wheel puzzle

I’ll try not to get too deep into the science and maths here but:

Spokes first of all:
the various tensions in the spokes always cancel each other out. Were this
not the case, then the hub would move, relative to the rim. Of course if it
moves even a fraction the tension in the spokes redistribute themselves to
maintain the balance. In an unloaded wheel the ideal is to have all spoke
tensions equal (with the wheel lying flat) The addition of a weight: YOU,
causes a RELATIVELY small change in the spoke tensions, those in the upper
half of the wheel seeing an increase, and due to a resultant small
distortion in the wheel, the lower spokes will also see a decrease in their
total tensions (measured using their vertical components of force for the
more scientific of us). The difference between upper and lower spoke
tensions now supports your weight.
Spokes are a one simple way of preventing too much distortion in a wheel rim
when it is loaded. They keep the wheel rim very nearly circular.

Imagine a wheel with spokes of elastic, and you may more easily grasp some
of this.

Tyres and pressure:
the air pressure at all points within the type is always uniform, but when
you load the wheel, the tyre distorts. (you may also see an increase in the
type pressure due to this deformation). lower the pressure in a tyre,
load it and you can easily see this and feel this. Newtons law says for
every action, there is an equal and opposite reaction, so the tyre, fights
back against the distorting force with an equal and opposite force, equal in
fact to your weight. (plus that of the unicycle to be accurate). If the
pressure in a tyre is say 50 pounds per square inch , and you (plus the uni)
weigh 150 pounds, the tyre will distort until 3 square inches of the tread
are flat on the ground. 3*50=150 (this simplistically ignores tyre
thickness and stiffness, but the principle is accurate).
Put more simply, sit on an object and it either holds you up by reacting
equally to you, or it collapses under your weight…
An inflated tyre is quite a strong structure, well able to transmit the
upward reaction force through to the rim, and thence via spokes to the hub,
frame and ultimately to your backside.

Here you might imagine the uni with just its inner tube and no tyre fitted.
It is easier (for me at least) to picture what happens.

Cheers

“gauss” <gauss.4knjy@timelimit.unicyclist.com> wrote in message
news:gauss.4knjy@timelimit.unicyclist.com…
>
> 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
>
>
> –
> gauss - memory fault (coredump)
> ------------------------------------------------------------------------
> gauss’s Profile: http://www.unicyclist.com/profile/651
> View this thread: http://www.unicyclist.com/thread/18063
>

36

Re: wheel puzzle

On Sun, 19 May 2002 17:40:54 +0100,
Naomi Sajeri <Naomi_Sajeri@hotmail.com> wrote:

> causes a RELATIVELY small change in the spoke tensions, those in the upper
> half of the wheel seeing an increase, and due to a resultant small
> distortion in the wheel, the lower spokes will also see a decrease in their
> total tensions (measured using their vertical components of force for the
> more scientific of us).

Nope, don’t like that explanation. I still believe the upper spokes see
hardly any change in tension. I’m not sure why you’re talking about
components of force - the force in a spoke is always purely axial in teh
spoke - it can be nothing else, since the spoke has pivotting connections
at each end. Well, actually there’s potentially a very small amount of
bending if your spokes are not a good fit to teh hub at their head, but
that’s not relevant at this point.

> pressure in a tyre is say 50 pounds per square inch , and you (plus the uni)
> weigh 150 pounds, the tyre will distort until 3 square inches of the tread
> are flat on the ground. 3*50=150 (this simplistically ignores tyre
> thickness and stiffness, but the principle is accurate).

I’m not sure I believe this either. It also ignores the effect of teh
prestressing of teh tyre carcase which I believe could be a significant
effect at higher pressures.

Consider, 50psi is about 3.5 kgf/cm2. That is 0.35 N/mm2. You can push
1 N/mm2 with your thumb. However, you can’t deflect a tyre at 50psi very
far with your thumb - the tyre is resisting at a lot more than the
pressure x contact area.

Incidentally, before someone queries my 1N/mm2, I just checked my
recollection. I can push something like 15 to 17 kg with my thumb,
according to my scales. 15kg is 150 N. With the help of some oil and a
bit of paper I determine that the contact area while I do this is
ellipticallish measuring 13mm x 15mm. I’ll say that’s equivalent to a
circle with 14mm diameter, or 7mm radius. pi times 7 squared = 3.14 x 49
= about 154 mm2. So I get 150 to 170 N on 154 mm2, or about 1 N/mm2.

> Here you might imagine the uni with just its inner tube and no tyre fitted.
> It is easier (for me at least) to picture what happens.

While an innertube only setup might behave as you describe, I don’t
believe that an inelastic tyre carcase will. If you imagine that, you’ll
picture what happens when you put just the innertube on, but you won’t
necesarily picture what happens when you’ve got a tyre too. I don’t know
anyone who rides on just an innertube - I’m not convinced it’d work, since
you can inflate innertubes to silly sizes (I’ve seen a photo of a
normal 27" tube going up to something like 10’ major diameter and 1’
minor diameter) at relatively low pressures. If you tried riding just an
innertube I think it wouldn’t hold you off teh ground, regardless of how
much air you put in. It’d be like standing on a kids balloon - either it
pops, or your foot gets to the ground while the balloon bulges all around
it.

regards, Ian SMith

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37

Time for me to bow out… thanks all for the interesting thread.
:sunglasses:

38

Re: wheel puzzle

“Ian Smith” <ian@achrn.demon.co.uk> wrote in message
news:slrnaedfq9.tl.ian@phlegethon.smithnet…
>>I’m not sure why you’re talking about
>> components of force - the force in a spoke is always purely axial in teh
>> spoke - it can be nothing else,

Yes indeed the force is along the spoke, but the same applies if you are
standing on a tightrope. Stand on a tightrope and it will deflect downwards
where you are standing, It will therefore stretch, and being stretched will
now be carrying greater tension. But if you resolve the tensional force
along the rope into a vertical and horizontal component, the vertical
components in the two sections of rope either side of you add up to just
enough to support your weight. The tension in the rope will be different,
and if the amount of “dip” in the rope is small, the tension in the rope
will be a lot greater than your weight. Simple physics/mechanics.
Get 2 people to hold a long rope taught, then ask a small kid to pull the
middle of it sideways…you wont be able to hold against it, you will be
pulled…at least until the two halves of the rope make a significant
angle at the kids hand…
The same force resolution applies to spokes in a wheel…

>>I still believe the upper spokes see hardly any change in tension.

If you consider a unicycle with only 4 spokes, top, bottom and two sides.
the two side spokes will carry no weight, and as such will not see a change
in tension. The upper spoke will be subjected to a stretching force as the
hub and your weight bears down on it. As the top spoke stretches, the hub
goes a little lower relative to the rim, the lower spoke gets shorter, and
its tension reduces. Your weingt now is equal to the differences in tension
between upper and lower spoke. All that adding more spokes does is to
complicate the mathematics.

>>I still believe the upper spokes see hardly any change in tension.

Imagine then a unicycle built with a very lightweight hub, and with all its
spokes made from elastic bands. Lightweight hub, so the bands hold the hub
roughly central. Now stand on the pedals, the top elastic bands will
clearly stretch. Yes? In order to stretch an elastic band you increase
the tension in it. Spokes are elastic too, they just stretch rather less
with the same tensional force applied. I think this demonstrates the upper
spokes get a tensional increase?
We use spokes rather than elastic bands for wheels because the distortion in
the wheel will be far less.

> …I’m not sure I believe this either. It also ignores the effect of
teh
> prestressing of teh tyre carcase which I believe could be a significant
> effect at higher pressures.

The prestressing of the tyre is of no relevance here. The structure of the
type has to cope with internal pressure from within due to the air contained
(which itself tensions the tyre casing), and also has to deal with the
stresses of road contact. It will not effect how much of the tyre lies flat
on the ground. There will be minor adjustments to be made due to tread
patterns, which really just result in a locally uneven distribution of
pressure, but they are not really important when considering the principles
involved, which are better described with a “bald tyre”.
It is not really easy to do your thumb test scientifically, because the tyre
has a level of stiffness that a balloon does not have, so the experiment
would not be scientifically rigorous.
But if you wet your tyre with your oil or paint, whatever, rested it on the
ground, and looked to see how large an area it marked, then do the same but
adding your weight to the uni, you will see a much larger footprint. The
area in contact with the ground increases proportionally to the weight
carried.

> While an innertube only setup might behave as you describe, I don’t
> believe that an inelastic tyre carcase will

But a tyre is NOT inelastic, it is made of elastic materials, mainly rubber.
As a point of interest just about everything in your uni is made from
elastic materials. The metal itself is elastic…including the spokes!
>

PS (I normally get paid to teach this sort of stuff…please send your
course fees to…)

Naomi

39

Re: wheel puzzle

On Sun, 19 May 2002 09:49:19 +0100,
Naomi Sajeri <Naomi_Sajeri@hotmail.com> wrote:

> “Ian Smith” <ian@achrn.demon.co.uk> wrote in message
> news:slrnaedfq9.tl.ian@phlegethon.smithnet…
[color=darkred]
> >> I’m not sure why you’re talking about
> >> components of force - the force in a spoke is always purely axial in teh
> >> spoke - it can be nothing else,[/color]

[snip somewhat garbled description of resolving vectors]

> The same force resolution applies to spokes in a wheel…

In what way? Or is this proof by assertion?

How about you stop trying to talk down to me and assume that I have an
engineering degree and some years of practical engineering experience - it
might help the clarity of your explanations.
[color=darkred]
> >>I still believe the upper spokes see hardly any change in tension.
>
> If you consider a unicycle with only 4 spokes, top, bottom and two sides.
> the two side spokes will carry no weight, and as such will not see a change
> in tension.[/color]

Only if you’re assuming the rim has not deformed, and the deflection of
the hub is small.

> The upper spoke will be subjected to a stretching force as the
> hub and your weight bears down on it.

And this will only be equal to teh effect at the bottom spoke if the rim
has infinite stiffness.

> As the top spoke stretches, the hub
> goes a little lower relative to the rim, the lower spoke gets shorter, and
> its tension reduces. Your weingt now is equal to the differences in tension
> between upper and lower spoke. All that adding more spokes does is to
> complicate the mathematics.

I wish you’d skip the baby-talk. I’m not disagreeing with the phenomenon
of elasticity. I’m disagreeing that the upper spokes change their
tension. Adding more spokes fundamentally affects the issue, because the
relative stiffness of rim and spokes is critical to teh distribution of
teh forces.

You seem to have assumed the rim is infinitely rigid. Why? It is being
loaded in flexure, and is generally of a less rigid material than teh
spokes, so why do you assume its behaviour is not contributing to the
behaviour of teh wheel?
[color=darkred]
> >>I still believe the upper spokes see hardly any change in tension.
>
> Imagine then a unicycle built with a very lightweight hub, and with all its
> spokes made from elastic bands. Lightweight hub, so the bands hold the hub
> roughly central. Now stand on the pedals, the top elastic bands will
> clearly stretch. Yes?[/color]

Doodums diddle-diddle do? Who’s a clever girl. Elastic bands!

I’ll try again - I UNDERSTAND THE RELATIONSHIP BETWEEN STRESS AND STRAIN.

LINEAR ELASTICITY IS NOT SOMETHING YOU NEED TO EXPLAIN TO ME.

Now, why do you insist on presenting a scenario in which teh rim is
dramatically stiffer in flexure than the spokes are in tension? Why do
you think this will provide any insights into the behaviour of a typical
unicycle wheel?

> In order to stretch an elastic band you increase the tension in it.

Wowee, who’d have thought it.

And when you apply a tension to one end of an elastic band the far end of
which is held by something that is a lot more flexible than the band, what
happens? Does the support move, perhaps? Does teh support move
sufficiently, perhaps, that the elastic band does not strain
significantly? If the band does not strain sugnificantly, is it possible
(perhaps, just a suggestion) that the stress in teh elastic band does not
change significantly? Perhaps?

> Spokes are elastic too, they just stretch rather less
> with the same tensional force applied. I think this demonstrates the upper
> spokes get a tensional increase?

Only for an infinitely stiff rim. If you have a source of these, I’m sure
teh muni chaps would be glad to hear of it. Otherwise, all this elastic
bands and baby-talk is irrelevant, since you’ve merely demonstrated that
an imaginary wheel that does not exist behaves in a particular way. This
tells us nothing about real wheels which do exist.

> > …I’m not sure I believe this either. It also ignores the effect of
> > teh prestressing of teh tyre carcase which I believe could be a
> > significant effect at higher pressures.

[skip irrelvant distraction of tread patterns]

> It is not really easy to do your thumb test scientifically, because the tyre
> has a level of stiffness that a balloon does not have, so the experiment
> would not be scientifically rigorous.

It doesn’t have to be - my thumb can exert three times the pressure you
maintain is the only resisting force, so even if I’m a bit out, it should
work. The fact that it comes nowhere near working demonstrates teh
fallacy of your argument.

> But if you wet your tyre with your oil or paint, whatever, rested it on the
> ground, and looked to see how large an area it marked, then do the same but
> adding your weight to the uni, you will see a much larger footprint. The
> area in contact with the ground increases proportionally to the weight
> carried.

So what? I think you’re arguing against what you’d like me to have said.
I don’t disagree that contact are increases with increasing load. How
about explaining why your description of the behaviour of tyres cannot be
replicated even with three times the pressure you say is necessary?

> > While an innertube only setup might behave as you describe, I don’t
> > believe that an inelastic tyre carcase will
>
> But a tyre is NOT inelastic, it is made of elastic materials, mainly rubber.

It is mainly made of relatively very elastic materials, but it contains
threads whose purpose are to prevent the elastic materials stretching.
At teh pressures we are concerned with, a tyre carcase is not
significantly elastic. Any (tiny, tiny) elongation of tyre wall does not
affect the behaviour of the tyre.

> As a point of interest just about everything in your uni is made from
> elastic materials. The metal itself is elastic…including the spokes!

You don’t say. So why does everything you’ve said about spokes assume an
infinitely stiff rim?

regards, Ian SMith

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40

Re: wheel puzzle

I have to say your posts make perfect sense to me Naomi.
Simple really: you put your weight on the hub, but the bike supports
your weight. the only way to transfer that weight from the tyre or rim to
the hub is via the spokes.
So as you say the forces in the spokes must change.
Maybe not as scientific as you but…