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    Mechanics of the tensioned load-bearing wheel

    I remember reading in Jobst Brandt's "The Bicycle Wheel" that the tensioned (bicycle) wheel supports a load by standing on the bottom spoke, which loses part of its tension force in proportion to the load. Unfortunately, I couldn't remember all the details of it when trying to relate it to my friend, who thought the load was supported by "hanging" the hub off the rim and increasing the tension on the top spokes. Could anyone explain it to me better?

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    Senior Member DannoXYZ's Avatar
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    No, it doesn't hang from the top spokes, but from ALL of the ones not on the bottom. So let's say you have 100kg tension on all the spokes on a 32-hole rim. And there's 4 on the bottom that's unloaded from your weight. The round rim becomes a D-shape with the flat spot on the bottom. These four spokes lose their tension due to the rim getting closer to the hub. However, since the rim has a fixed circumference, the round part of the D has to expand in order to keep the same circumference (it's like pushing on one side of an inflated balloon, it expands on all the other sides evenly). This expansion on the rounded upper part increases the spoke-tension by an even amount on ALL the unloaded spokes.

    You can figure it out mathematically:

    50-kg load on wheel
    50kg/4 = 12.5kg tension unloaded on 4 spokes at the bottom (87.5kg tension)
    50kg/28 = 1.79kg tension increased for remaining 28-spokes above contact area (101.79kg tension)

    Of course, it's not so simple because spokes undergo tension-changes linaerly, not all at once. So the tension-decrease increases slightly as it nears the bottom and only fully at the very bottom. So the above scenario might have more tension relieved on the two at the very bottom and less on the two on the sides of those. But total tension-relieved is still the same total as the load.

    Basically it's a buoyancy equation where the load is redistributed evenly. Whatever tension is relieved off the bottom must be taken up the others.
    Last edited by DannoXYZ; 08-02-08 at 04:51 PM.

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    I had thought that while the bottom spokes support the load, the increase in tension in the other spokes can be attributed only to the maintenance of equilibrium and not to the load, thus the bottom spokes would remain (for argument's sake, as the tension changes are caused by outside forces) the only spokes subject to outside forces (which is the load), while the changes in tension for the other spokes are made by the forces already existing in the tensioned wheel. Yes/no?

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    Senior Member DannoXYZ's Avatar
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    Quote Originally Posted by nwcdr200 View Post
    the increase in tension in the other spokes can be attributed only to the maintenance of equilibrium and not to the load, thus the bottom spokes would remain (for argument's sake, as the tension changes are caused by outside forces) the only spokes subject to outside forces
    It's one and the same. What's "equilibrium" in this case? It's dealing with the extra forces from the load.

    Look at the directions of load. Draw a picture of a wheel and put a down-arrow at the hub. Then an up-arrow at the ground that balances the load. You can consider that the "outside force" of the weight-load actually starts at the hub "inside" the wheel. The result is that the hub and ground squeezes the spokes at the bottom together, thus relieving their tension.

    Tension and buoyancy can be difficult to grasp as it deals with distribution in a fluid-like manner. A wheel can be considered a suspension-bridge that's wrapped up around itself. If you push up underneath a suspension-bridge, the cables above it are actually relieved of their tension right?

    So it's actually the RIM that experiences the outside forces. The RIM then changes shape in response to load which causes the tension to change on ALL the spokes. In Jobst's book, he draws an exaggerated picture somewhere that shows the D-shaped rim. Compare the distance from the rim to the hub in the flattened D-part at the bottom versus the rounded parts above. So there is existing tension, which is what carries the load and that load is redistributed evenly.

    It's like a care tyre. The flattened part at the bottom squeezes up and reduces the volume at the bottom of the wheel. This increases the existing 32psi of pressure to say... 33psi. This extra 1psi is distributed ALL the way around the upper parts of the tyre. Basically the sidewall cords of a car-tyre loses tension, you can see that where it bulges out sideways, and all the remaining cords above take up that extra 1psi. You don't see a corresponding bulge above a tyre that corresponds to the bottom compression because the extra load is distributed evenly around the upper part of the tyre.
    Last edited by DannoXYZ; 08-02-08 at 04:58 PM.

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    Senior Member Mr. Fly's Avatar
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    Buy the book and read it?

    There's actually a very easy experiment you can do to demonstrate that the weight is supported by a loss in tension of the bottom spokes rather than an increase in tension on the top spokes.

    First, we should establish that spoke tension can be qualitatively measured by the tone the spoke produces when plucked. The plucking is best done to the middle of the supported ends. Higher tone equals higher tension, and vice-versa. We do not need to know what the exact tension is, only to establish that there is an increase or a loss of tension.

    Next, pluck the top and bottom spokes of an unladen wheel to establish a baseline tension/tone. Then get someone to apply a load to the wheel. Now, pluck the top and bottom spokes again. You should observe that the bottom spokes produce a lower-than-baseline tone but none of the top spokes offers any change in tone. On a very strong wheel (e.g., properly tensioned 36-spoked 559 wheel), you may need to apply a surprising amount of load to hear any effect.

    Have fun!

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    Senior Member DannoXYZ's Avatar
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    Quote Originally Posted by Mr. Fly View Post
    There's actually a very easy experiment you can do to demonstrate that the weight is supported by a loss in tension of the bottom spokes rather than an increase in tension on the top spokes.
    It's actually one and the same. The total loss of tension MUST be accompanied by an equivalent increase in tension. The only difference is that there's a lot fewer spokes losing tension than the ones increasing in tension. So you can easily measure the tension-loss in the 2-4 spokes at the bottom, but the much smaller increase in ALL of there 28-32 upper spokes is not as noticeable.

    But if you're comparing the 2 bottom spokes ONLY to the 2 top spokes, then yes, you'll see A LOT of loss in tension without much tension-increase on the top 2.

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    I went looking around the interweb and found this to support me:
    http://www.astounding.org.uk/ian/wheel/

    It has some finite element analyses near the bottom, which I found helpful, as well as this for his conclusion: "I conclude that it is perfectly reasonable to say that the hub stands on the lower spokes, and that it does not hang from the upper spokes. It is also wrong to say that the force distributes all around the rim and all the spokes contribute to holding up the hub - over a third of the spokes have an effect that pulls the hub down!"


    Mr. Fly - I actually remembered and performed that demonstration to my friend, but he remained unfortunately obstinate in his conceptions of the wheel.

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    So I happened to have a copy of "The Bicycle Wheel" on my coffee table, the chapter which actually covers this is a bit too long to quote, but I think y'all are talking around each other. The spokes on the top of the wheel actually do not change that much in tension, the D shape is caused by the detensioning of the lower spokes, wires under tension (spokes) behave like solid columns under load until all of the tension is used up. This means that you can assume that the tensioned spoke supports the hub the same as say a stick or a cement pillar would. So if a spoke is tensioned to 1000 lbF you can apply 1000 lbs of force downwards on the spoke before it again behaves as a wire as opposed to a solid column. The other spokes are necessary to maintain the tensioned equilibrium in the wheel and will change tensions slightly to maintain equilibrium, but the greatest tension change on the loaded wheel will be to the bottom spokes which are supporting the hub like solid columns.

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    Senior Member DannoXYZ's Avatar
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    The thing is, forces on a spoke is always in tension and the bottom spokes lose tension under load. That is, they experience lower amounts of force when load is applied. Columns experience an increase in force when they have load applied.

    And you're right, the upper spokes do not change tension as much as the few at the bottom, but they DO increase in tension when load is applied at the bottom; just a lot less because it's distributed. Another way to look at this is to consider that total-tension on the rim must remain constant.

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    You are right about columns... but in a statics analysis you can substitute a tensioned wire for a column. A tensioned wire loosing tension will behave exactly like a rigid column increasing in compression up until the point where all of the tension is used up.

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    Senior Member BCRider's Avatar
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    But if you see a decrease in the tesile load of spokes along the contact patch area this is due to a decease in their length.

    A fair way to look at a wheel is as an array of springs. Spokes are not totally rigid. They act like very high rate springs. Similarly the rim is flexible as well so you can consider it as a series of compression springs with very high rates that join the ends of the spokes with act as very high rate tesion springs. So if you reduce the tension loads of the spokes in the vicinity of the contact patch then you have reduced their length by a very small amount. But that means you have just moved the rim compared to hub in a way that places a higher tensile load on the spokes on the upper side of the wheel via the flexing of the rims "ring of compression springs". And since with something as flexible as a spoke the load can only be carried by an increase in tension then it's still fair to say that the bike and rider hangs from the upper half of the rim and upper spokes since that is where the tesion gain takes place.

    While I agree that you can look at tensioned spokes as a column until the tension dissapears this is a rather twisted and convoluted way of looking at tension loads in a wheel. It also suggests that the spokes pointed in directions other than straight up or straight down do nothing to support the wheel. But since we are talking about a situation where a far more accurate model is a series of very high rate springs constrained in a locked and balanced manner I really don't think the column model has much validity in the study of how a wheel carries its load.
    Model airplanes are cool too!.....

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    What's the point of talking about spokes as springs just to say that the hub hangs from the rim? Spokes are not springs, and while they are susceptible to lateral deflection, they do behave as a rigid structure under compressive forces, given that their tension is greater than the compressive load. There is a very simple aural test you can perform to confirm that the bottom spokes are the load-bearing ones, mentioned earlier by Mr. Fly. Pluck the bottom and top spokes to establish a baseline sound (make sure that the spokes you choose are on the same side, ie drive/non-drive), then, applying a compressive force to the wheel via the seat, pluck the top and bottom spokes again. The lower sound you will hear in the bottom spoke indicates that it is supporting the compressive load by compromising part of its tension; the lack of change in tone you will hear in the top spoke indicates that it is not bearing a load. You might say in response that there is no aural change detected because the force is distributed over the top third of the wheel, but seeing as that is only 10.66 spokes out of 32 that would be supporting the hanging load you should be able to hear a change if that were the case. If that test isn't enough for you, examine the site I posted earlier (http://www.astounding.org.uk/ian/wheel/), where you can find finite analyses of the forces on a loaded wheel which are fairly damning to the case of "hanging" a load.

    The column model is not only neither twisted nor convoluted but also exceedingly more accurate than the "high rate springs" model, and while it is true that every spoke changes in tension under load to maintain equilibrium within the wheel, it is only the bottom spokes that support the load. Try it for yourself - I am confident that your findings will match my own.

    If I am coming off as rude at all, I sincerely apologize and wish my comments to be taken in only the most humble and positive way; this thread has the subtle beginnings of a flame war.

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    Quote Originally Posted by nwcdr200 View Post
    I remember reading in Jobst Brandt's "The Bicycle Wheel" that the tensioned (bicycle) wheel supports a load by standing on the bottom spoke, which loses part of its tension force in proportion to the load.
    The term "standing on the bottom spoke[s]" is very misleading, especially when followed by "which lose part of their tension". It's not like a conestoga wagon wheel where the axle is compressing the lower spokes. In a bike wheel the spokes are all very tight and pulling the rim towards the hub. When the wheel is carrying a load, the bottom spokes are still under tension, but slightly less. They aren't being compressed.

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    Let's say God has my left arm and is pulling me up to Heaven, and the Devil has my right foot and is pulling me down to Hell. They're evenly balanced and I am stuck in the middle. Then somebody hands me the weight of my sins, and that makes me heavier. That should help the Devil pull me down, but being lazy, he relaxes a bit, and doesn't pull as hard, and I don't move. God's workload doesn't change. Would you say the Devil is now supporting the weight of my sins by not pulling down as hard? After all, God didn't have to pull any harder when that weight was added. It was the Devil who made the adjustment.

    That's what Brandt is saying about the bottom spokes. They are supporting the weight of the rider by not pulling down as hard as before. The upper spokes don't have to increase their tension (more than a trivial amount) to accomodate the load.

    .
    Attached Images Attached Images
    Last edited by cooker; 08-03-08 at 12:41 PM.

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    Senior Member DannoXYZ's Avatar
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    BC and cooker's got it down. Replace "spokes" with "strings" or "rubber bands" and the wheel will still operate exactly the same way. You can even lace up the wheel with a continuous line like a tennis-racquet and it would still work due to TENSION.

  16. #16
    Senior Member BCRider's Avatar
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    Quote Originally Posted by nwcdr200 View Post
    ....If I am coming off as rude at all, I sincerely apologize and wish my comments to be taken in only the most humble and positive way; this thread has the subtle beginnings of a flame war.
    Not taken that way at all. Debate can be described as "a friendly arugument". As long as the exchange is polite and we can all go and have a beer at the end of it and leave as friends it's all good.....

    From a purely mathematical point I'll accept the analysis described in that page. In fact it's an interesting and informative page from that aspect.

    The real issue seems to be the author going to a considerable effort to bring sematics into the issue and bend how he describes the summation of the forces to support his "standing on the lower spokes" concept.

    He brings in the definition of "standing" at one point and talks all around it and finally moves on leaving the reader with some vague impression that it applies to how a hub is supported by standing on the lower spokes. But our hub is only "standing" on the ground as much as a car is "standing" when supported by a suspension bridge. The road way is hanging from cables which are then supported by the end columns. Similarly our hub is hanging from the rim which then stands on the ground. But in both cases the hub loads and the car on the bridge are hanging from some element which then is used to transfer the loads to the ground by standing on it. The rim and tire in our case and the column in the bridge's.

    He brings up the issue of static tension in the spokes of the wheel and then dismisses this because the forces all balance. But we can't just dismiss the spokes preload. It's there and it's fundemental to how the wheel supports loads. We all talk about how spokes only fail when the tension drops close to zero and the metal has to move and work. So it is equally biased to dismiss the static spoke preload here as well. Yet this dismissal sets the stage for the descriptions to follow later in his conclusions.

    Yes his analysis shows a marked reduction in tensile loading of the contact area spokes. But to call it compressive is inaccurate. There is still tensile loads in the lower spokes, just less of it. But he set up the stage earlier by bringing up the preload tensions of the wheel and then dismissing them by saying they were balanced. But the wheel still knows the preloads are there and will see the changes in spoke tensions not as compression but as a reduction in tension. Semantics used again perhaps but it seems to be key to the idea of "standing on the lower spokes".

    The point remains that you can't "push a rope". The author has worked in a very nice circumlocution to support his idea of standing on the lower spokes versus hanging from the upper spokes. Yes the change in values in the majority of the spokes that undergo an increase may not seem all that grand but the point is that spokes can only support a load when tension is added and are less able to support a load when the tension is reduced. And the spokes with the increased tensions are along the sides and upper half of the wheel and those are the ones that support the load.

    Getting back to the plucking or light tapping of the spokes to indicate how all this plays out.... Sure, the lower spokes along the contact point are seeing the most reduction in tension and so you'll hear a marked tone decrease. Meanwhile the increase in the upper spokes that are supporting the shift in tensions is not noticable only because our ears and brain can't differentiate the small change in tone that each individual spoke sees. But it is there just as shown by the table in the analysis.

    Which all brings it back to the reality that the hub and load hangs from the upper spokes and the only "standing" going on is in the rim that is used to transfer the loads from the upper half of the rim to the ground just as the suspension bridge columns transfer the cable loads to the ground.


    Anyway, that's my take on the matter. Make of it what you will.
    Model airplanes are cool too!.....

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    Senior Member DannoXYZ's Avatar
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    Quote Originally Posted by cooker View Post
    Let's say God has my left arm and is pulling me up to Heaven, and the Devil has my right foot and is pulling me down to Hell. They're evenly balanced and I am stuck in the middle. Then somebody hands me the weight of my sins, and that makes me heavier. That should help the Devil pull me down, but being lazy, he relaxes a bit, and doesn't pull as hard, and I don't move. God's workload doesn't change. Would you say the Devil is now supporting the weight of my sins by not pulling down as hard? After all, God didn't have to pull any harder when that weight was added. It was the Devil who made the adjustment.

    That's what Brandt is saying about the bottom spokes. They are supporting the weight of the rider by not pulling down as hard as before. The upper spokes don't have to increase their tension (more than a trivial amount) to accomodate the load.

    .
    Souls, sins and spokes!! Hahahahahhaahh....

  18. #18
    Senior Member BCRider's Avatar
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    Quote Originally Posted by cooker View Post
    ...That's what Brandt is saying about the bottom spokes. They are supporting the weight of the rider by not pulling down as hard as before. The upper spokes don't have to increase their tension (more than a trivial amount) to accomodate the load.

    .
    If the part in red is from Brandt's book (I confess I have not read it or plan on getting it any time soon) then he has unfortunetly helped set the stage for this misconception. The upper spokes may only see a small change but with so many of them sharing the load this small change adds up.

    A single thread is easily snapped on its own but when woven into a belt it can support massive loads.
    Model airplanes are cool too!.....

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    Quote Originally Posted by BCRider View Post
    If the part in red is from Brandt's book (I confess I have not read it or plan on getting it any time soon) then he has unfortunetly helped set the stage for this misconception. The upper spokes may only see a small change but with so many of them sharing the load this small change adds up.
    Ah, but it adds up to much less than the weight of the rider on the fork. Before the rider got on, the upper spokes were pulling up to counteract the downward pull of the lower spokes. Now, the upper spokes are pulling up to counteract the weight of the rider and the downward pull of the lower spokes. However, the lower spokes aren't pulling down as hard. Their relaxation almost equals the weight added by the rider, so the upper spokes hardly experience a change in tension when a rider puts weight on the fork. That's the point Brandt is making. Just as God didn't have to pull harder when my sins were added, because the Devil stopped pulling down so hard, so the upper spokes don't have to work much harder to carry my weight, because the lower spokes are slacker and don't pull down as hard.
    Last edited by cooker; 08-04-08 at 09:31 PM.

  20. #20
    Senior Member BCRider's Avatar
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    You can't just add up the upper spokes that are in direct opposition with the lower spokes. The rim acts like an arch to distribute the forces and all the spokes on the upper and even some of the ones in the lower side all come into play sharing and constraining the various loads in the system. The true model of forces is far more complex then what is shown on that website.

    Some of the changes are directly due to the supporting of the weight and show up at the axle of the hub to support the outside load that started it all and some are locked into the wheel due to deformation loads. This second aspect is why the spokes just outside the deformation in the wheel diagram and table are shown as pulling outwards and downwards. They are reacting to the local teeter totter like levering bend in the rim that is induced by the localized rim deformation in the model. This actually makes those spokes see more tension even though they are on the lower side of the wheel. But that increase cancels out part of the decrease in the "compressed" spokes nearby so the total value for the reduced tension spokes needs to take that into account.
    Model airplanes are cool too!.....

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    Quote Originally Posted by BCRider View Post
    The true model of forces is far more complex then what is shown on that website.
    Ah, then tell us, what is the One True Model of forces in a loaded wheel? I fail to see how that diagram is wrong, given that it's on a computer program with inputs for spoke tensions and load weights. It's not just a drawing with of a wheel with a flat spot. Look at the table below it, there are the output tension values.

    As for the postulation that, under the hanging model, the distribution of forces among the top spokes would be too small to notice: first, I think we can agree on a wheel with 32 spokes, and further agree that .5 to .66 of those spokes would be supporting a hanging force on the hub (the spokes below the half line would be opposing the top spokes' effort, and the ones on the half line would be perpendicular to our vector and thus not supporting). I will say .66 of the spokes, because that makes more sense. That gives us 10.66 of the spokes that would support a hanging load. If the load was 100kg, then each spoke would be under ~10kg of additional force (completely discounting that the most vertical spokes would be under much more force). I fail to see how that would not be reflected in the finite analyses I linked, where deflections were multiplied x100, or even in an auditory response test. In my experience with wheelbuilding, tensiometers, and truing by sound, which is admittedly not extensive but certainly sufficient, you can tell a difference in sound for at least a change of 5kg.

    Finally, I'd like to make a semanticism in the terminology of forces on the bottom spokes. In this case, because the tension force exceeds the compressive force, it is correct to say that it is "in compression" while not "compressed," because it experiences a compressive force but not to failure.

  22. #22
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    For me - this is a great thread. Personally - I thumbed through the Brandt book, and when he appeared to describe spokes acting like columns and supporting compressive loads, put it down and bought Schraner. Now I understand more of what he was trying to say (but still don't think it he expressed himself clearly)

    I think the problem with the "hang" vs "supported" perspectives is that they are both trying to oversimplify what is happening and are poor terms to boot, since they seem to mean different things to different people.

    To the "hang" crowd - a spoke can not "support" any axial force like a column, since that involves compressive forces and the slender characteristic of a spoke means that it will fail in buckeling long before it carries any appreciable compression. Since the only element in compression is the rim, the hub is therefore hanging from the spokes.

    To the "not hanging" or "supporting" crowd - I'm guessing that they note on the finite element chart, that some of the spokes below the hub actually gain tension when the wheel is loaded, and would appear to be pulling down on the hub. The hub therefore can not be "hanging" from the spokes.

    Both sides are right as far as their perspectives go. The lower spokes are not carrying anything in the traditional sense of the word. Reducing the tension forces carried by a tension member, does not somehow turn it into a compresson member capable of supporting an axiel load. And yet - to say a hub is "hanging" when some of the spokes that increase under load are pointing down at a 45% angle (or more) doesn't seem quite right either.

    So where does that leave us ? Where we started - with the simple explanation that DannoXYZ provided and which was confirmed by Ian's bicycle wheel analysis of a 36 spoked wheel. An unloaded bicycle wheel starts out with all spokes in equal tension and the rim as a perfect circle. When the wheel is loaded through the hub, the rim deforms slightly at the bottom causing the lowest 15% or so of the spokes to loose tension, and the other 85% see an increase in tension due to restraining the hub from deforming into an oval.

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    Quote Originally Posted by cooker View Post
    Then somebody hands me the weight of my sins, and that makes me heavier. That should help the Devil pull me down, but being lazy, he relaxes a bit, and doesn't pull as hard, and I don't move. .
    I hate to break it to you - but the only reason the Devil isn't pullng as hard, is that the weight of your sins has moved you closer to hell. (remember - the rim has flattened out at the bottom. This strain at the rim decreased the distance from hub to bottom of rim and reduced the tension in the bottom most spokes)

  24. #24
    Prefers Cicero cooker's Avatar
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    Quote Originally Posted by bubbagrannygear View Post
    I hate to break it to you - but the only reason the Devil isn't pullng as hard, is that the weight of your sins has moved you closer to hell.
    Drat!
    But at least I'm not perceptibly farther from God

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    The guy in the website even says that he simplified things to make it easier to analyze. In the page where he explains the development of the model used he says what to analyse and what to skip is a bit of an art. But as for the data that is presented in the table I have not said I have any beef with that.

    My only real beef is with the human analysis portion of the data. Some aspects of the results are ignored, twisted or glossed over to further the idea of this "standing on the lower spokes".

    Just one such example of an area that wasn't really explained fully is the increase in tension in spokes near the bending points of the rim. A bicycle wheel is a locked system and that you can't look at any increase or decrease in any single or even a limited group of spokes in isolation. The whole wheel contributes to the job with various forces between spokes fighting each other and balancing out a lot of the tension changes in any individual or small group of spokes. Sort of like pushing your hands together super hard and then using your isometrically tensioned arms to reach out and move an object. It doesn't take much to move the object but the total forces involved in the balanced pressures between your arms and the amount needed to move the object are FAR higher than the amount needed to move the object. But the outside world only sees the effort needed to move the object and the locked isometric forces in your opposing arms are hidden other than to your arms.

    A full blown analysis of this would result in a hugely complex vector diagram or a ream of higher math equations. But I don't see any reference made to such a study of the findings. Instead the spoke loadings are only commented on in isolation or in small groupings. It's things like this that indicates to me that there's more to this model and data analysis that isn't being mentioned.

    But in the end it really is about not agreeing with the semantical idea of standing on the lower spokes.

    Instead I see it as the upper spokes have to take up the slack from the lower contact patch spokes that are no longer doing their job due to being under reduced tension. My own semantics? Perhaps. And in the end the whole idea of standing on or hanging from seems to come down to a case of semantics.

    Lets take a non wheel example to illustrate the point. A platform is attached to a crane lift line. A steady line from directly below is tensioned to hold the platform from swaying. There's a lot of tension in both the upper and lower lines. A man steps onto the platform.....

    The concept of "standing on the lower spokes" would suggest that the man is being supported by a decrease in the lower steady line tension.

    The "hanging from the upper spokes" suggests that the increase in tension in the line to the end of the crane boom is holding up the man.

    If I was the man on the platform I think I'd be happy that the line to the crane boom was there.....

    Using this compressive concept may be an interesting way to study the forces involved from a mathematical standpoint and certainly I ran across many similar examples of "relative observation point" convieniences when I studied physics in university. But those cases all recognised that it was a convienience for the sake of making the calculations easier rather than a real world reality. This whole standing on the lower spokes seems like such a convienience run amok to me.
    Model airplanes are cool too!.....

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