David Kerber wrote:
> In article <
[email protected]>,
[email protected] says...
>
> ...
>
>
>>>Instead of trying this on a normal wheel with lots of spokes, think about a 4-spoke wheel, with
>>>the wheel turned so that one of them is straignt up and down. If they are pre-tensioned to
>>>200lbf, and you then set 200lbs of bike and rider on them, the tension in the top spoke MUST
>>>increase by 200lbf total (front and rear combined). It will likely be more in the rear than the
>>>front, but it the combined increase must equal the total weight of the load. A simple statics
>>>calculation will show you that. The interesting part of this calculation is that if the spokes
>>>don't stretch and the wheel doesn't distort, the tension in the bottom and side-facing spokes
>>>doesn't change at all.
>>
>>Holy moly, you've invented anti-gravity!
>>
>>Firstly, you've started with an incorrect assumption, that the rim is infinitely stiff as compared
>>to the spokes. In
>
>
> That was only a simplifying assumption for illustration purposes. I am well aware that it is not
> the case in real life.
>
>
>>actuality, the radial stiffness of the rim is much less than the radial stiffness of the spokes.
>>So when you apply a load to the wheel, the bottom of the rim will easily bend inward allowing the
>>bottom spokes to absorb the load, and very little of the load will be allowed to be transferred to
>>the top of the wheel.
>
>
> SOME of it must be transferred, because the total vertical load on the axle must increast to equal
> the total weight of the load. That difference is spread out among lots of spokes, so the change in
> tension in any one spoke is probably only a few pounds. That's why you don't hear it.
>
>
>
>>But even if the rim were infinitely stiff, you've still got it all wrong. I think you'll agree
>>that as you tension the spokes, they will elastically stretch in proportion. According to you, the
>>top spokes increase tension, but the bottom spokes don't. Therefore the top spokes will stretch
>
>
> I didn't say they didn't stretch; I said that *If you make the simplifying assumption that nothing
> deformed* then the tension in the bottom wouldn't change. That is obviously not a real-life
> assumption. However, I still maintain the the tension in the top spokes MUST increase to handle
> the added load. Draw a simple free body diagram with a 4-spoke wheel and work out the
> calculations. As you said below, if you take the real life case in which the spokes and wheel
> deform under load, then the increase in tension in the upper is smaller, because it is partly
> counter-acted by the decrease in the lowers, but it CANNOT fully cancel out, or there would be
> nothing to support the added load on the axle.
>
>
>>Before you get yourself further tripped up in half thought-out theory, I really suggest you try
>>the spoke plucking experiment mentioned above. It can be very illuminating.
>
>
> It wouldn't surprise me a bit if you couldn't tell the difference in tension by the sound of a
> plucked spoke, especially in a high spoke- count wheel (because the load is distributed over so
> many spokes), but it MUST be there, and with a sensitive enough tensionometer, you could measure
> it. Do the calculations!!!
The calculations will depend on what is assumed about the relative stiffness of the spokes vs. the
rim. If we make the assumption that the rim is infinitely stiff and the spokes are elastic (as Mark
pointed out this is a very poor assumption for a real wheel), then your 4-spoke wheel looks like a
rigid hoop connected to the hub with strong 'rubber-band' spokes. The rubber bands each have a
pre-load tension of 200 lbs-f, so the hub feels a pull of 200 lbs downward from the bottom rubber
band, a force of 200 lbs upward from the top rubber band, and cancelling 200 lb forces forward and
backward by the two side rubber bands. Now the hub axle is pushed down by the weight of rider + bike
in the amount of 100 lbs-f. With a rigid hoop, the rubber band on the bottom will get shorter and
have a reduced tension while the rubber band on the top will be stretched by an equal amount and
have an increased tension. The two side rubber bands will only have an insignificant increase in
their lengths, so they'll have very slight tension increases which we'll ignore. If the bottom and
top rubber bands have the same spring constant, the reduction of tension from the bottom one will be
equal to the increase of tension in the top and the sum of these two effects must balance the
downward force on the hub of 100 lbs-f. So the bottom spoke sees a reduction of its preload from 200
lbs-f to 150 lbs-f, the top spoke sees an increase from 200 lbs-f to 250 lbs-f and the net force on
the hub is balanced as expected: 250 up - 150 down - 100 down from rider+bike = 0 net force.
The other extreme is that the rim is very flexible and the spokes are more rigid. In this case when
the wheel is pushed down by the weight of bike+rider, the bottom of the rim deforms while the rest
of the rim maintains its shape. In this case the bottom spoke sees a decrease in tension but the top
and side spokes will have no change. If the force from the weight of bike+rider is 100 lbs, then the
tension in the bottom spoke must decrease by 100 lbs-f to balance it and therefore change from the
initial preload value of 200 to 100 lbs-f while the tension in the top spoke remains at 200 lbs-f.
The net forces on the hub are still properly balanced: 200 up - 100 down - 100 down from bike+rider
= 0 net force.
In the case of my actual bicycle wheels with a lightweight rim and 36 14 gauge spokes almost all of
the radial rigidity comes from the spokes and very little from the rim (the rim alone would
immediately collapse under my weight). Therefore it closely matches the second scenario and I'd
expect almost no change in the tension of the top spokes when the wheel is loaded.
