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From: I.N. Galidakis on 16 Jul 2010 01:52
Here's a "funny" engineering problem I encountered.

A certain "device" I am designing, needs to have internally (at least) two gears
with ratio r to transmit torque and rotational movement. The mathematical design
predicts that the system signal of the device will be periodic if r\in Q with
period p=n/GCD(m,n), where r=m/n (on the visual plane's Complex unit circle)
whereas it will be chaotic (and dense on the visual plane's Complex unit circle)
if r\in R\Q.

Without thinking much, I embarked to give instructions to my (very expensive)
engineer for an r\in Q for the two gears, as it should be. And now that I am
thinking about it, because of manufacturing error tolerances, there's virtually
no guarantee that the constructed gears will have the desired r.

I could give an r=1/2 for the two gears, but because of manufacturing error
tolerances, r could be _any_ irrational very close to 0.5.

In general for any such r, there are *exactly* 2 manufacturing possibilities:
Either r\in Q, as it should be, or r\in R\Q. In the second case I am screwed and
the final construct will perform wrong.

In fact, because the Lebesgue measure of the rationals in [r-eps,r+eps] (where
eps is the manufacturing error tolerance) is 0, whereas that of the irrationals
is 2*eps, it would appear that it is _much_ more likely for the manufacturing
process to produce an irrational r, than a rational one.

Isn't math wonderful? It analyzes _completely_ the behavior of the final system
(for any r\in R) and predicts that the final device which will come to my hands
will be completely WRONG.

New Murphy's Law:

"You can use mathematics to completely analyze the behavior of *any* mechanical
device, but the same math will show that the device WON'T work" --- Addendum to
the Physics Law: If it DOESN'T work, it's Physics.

:-)
--
I.

From: Tim Little on 16 Jul 2010 04:45
On 2010-07-16, I.N. Galidakis <morpheus(a)olympus.mons> wrote:
> Without thinking much, I embarked to give instructions to my (very
> expensive) engineer for an r\in Q for the two gears, as it should
> be. And now that I am thinking about it, because of manufacturing
> error tolerances, there's virtually no guarantee that the
> constructed gears will have the desired r.

How do you get an irrational ratio with gears? Belts, I could
understand. But gears have integral numbers of teeth.


- Tim
From: I.N. Galidakis on 16 Jul 2010 15:41
Tim Little wrote:
> On 2010-07-16, I.N. Galidakis <morpheus(a)olympus.mons> wrote:
>> Without thinking much, I embarked to give instructions to my (very
>> expensive) engineer for an r\in Q for the two gears, as it should
>> be. And now that I am thinking about it, because of manufacturing
>> error tolerances, there's virtually no guarantee that the
>> constructed gears will have the desired r.
>
> How do you get an irrational ratio with gears? Belts, I could
> understand. But gears have integral numbers of teeth.

The device I am designing doesn't _have_ to use gears. It could use wheels with
or without conveyor belts OR gears.

This incompetent retard, "Porky", actually could have given me this information
about gears *politely* (like you have in the form of a question) and I would
have thanked him for it, because it would've simplified my design enormously,
but instead, he chose to insult.

I think he has a grudge with me or something, after I commented on some bullshit
he wrote on numerical analysis, a while ago.

But never mind all that. Problem solved :-)

> - Tim
--
I.

From: Inv Alid on 16 Jul 2010 21:00

"I.N. Galidakis" <morpheus(a)olympus.mons> wrote in message
news:1279259554.423069(a)athprx04...
> Here's a "funny" engineering problem I encountered.
>
> A certain "device" I am designing, needs to have internally (at least) two
> gears with ratio r to transmit torque and rotational movement. The
> mathematical design

is trivial.


> Without thinking much, I embarked to give instructions to my (very
> expensive) engineer for an r\in Q for the two gears, as it should be. And
> now that I am thinking about it, because of manufacturing error
> tolerances, there's virtually no guarantee that the constructed gears will
> have the desired r.

nope, this is trivial.

how many significant digits is r specified to, and what is the acceptable
error band ?

>
> I could give an r=1/2 for the two gears, but because of manufacturing
> error tolerances, r could be _any_ irrational very close to 0.5.

which is completly acceptable, as your specification is NOT
0.50000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

<snip rest>


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