An Off The Wall Use For CAD

[Nigel Graham 2Participant @nigelgraham2]

I’ve sometime said I find TurboCAD Deluxe in 2D mode useful for geometrical constructions: most recently for designing a disc valve giveing constant flow over a range of rotation angles, for a combined vacuum- and mechanical- brake control on a loco driving truck. From that I drew the actual parts in Alibre Atom.

This is an oddity though.

I’ve a friend who lards his Christmas cards with mathematical puzzles, giving the answer next Christmas. A keen runner, most have been about runners racing up and down hills. I thought solving them requires deriving simultaneous equations far too hard for me; but he shows some other method simpler but not at all obvious.

Then this last Christmas: “Runners have to vault over three identical logs, in a cross-country race” – see diagram,, showing the logs, of 32.26cm dia, stacked horizontally and parallel.

Now forget the athletes….

Calculate:

1) the cross-sectional area of the central cavity, and

2) the maximum diameter of a cylinder that will fit the cavity, touching each log without displacing it.

A rough sketch showed me the geometry hence how to calculate it, but as well as other errors I needed several attempts to calculate the area of the triangle that it gives, with its apices on the log centres. It height as in (half-base X height) stubbornly resisted several methods but eventually yielded to Pythagoras.

The area wanted is the triangle’s area minus the total area of the three sectors it delineates.

So that was Question 1) done.

Next, how the heck can you work out that inscribed diameter? Only by a CAD drawing, from which I found I had guessed correctly the pipe’s axis is on the triangle’s bisectors. The dimension tool gave the radius and I proved it by drawing the circle.

Trevor wanted the answer only to whole numbers. Here is the solution: 1) 41cm^2, and 2) 5cm.

[BazyleParticipant @bazyle]

Well the second part is easy for a Muddle Engineer. Take 3 bars out of your stock and stack them in the triangle. Then check the hole in the middle with your set of drills until you fid one that fits and calculate the answer from the ratios.

[blowlampParticipant @blowlamp]

Nigel.

I find the cross sectional area to be closer to 42cm2.

 

Martin.

 

[Kiwi BlokeParticipant @kiwibloke62605]

Saw this as I sat down to lunch. 2nd question doesn’t need millions of transistors beavering away, driving CAD, just basic trig and Pythagoras. The key is to consider the triangle formed by the centres of the two lower logs and the centre point of the inscribed circle. Its ‘height’ is found by log radius X tan 30. Plug that into Pythag to find hypotenuse, then subtract the log’s radius, and double the result. OK, I admit I had to employ quite a lot of transistors in a scientific calculator… No doubt there’s a quicker way – I’ll leave that to proper mathemagicians to reveal.

[DC31kParticipant @dc31k]

In some areas of study, the answer to Life, the Universe and Everything is 42.

I wonder if the slightly odd log diameter was chosen with this in mind.

Perhaps send a reciprocal question to your friend, asking for the diameter that would make the area exactly 42.

Is the sender a dress maker or Blue Peter presenter? I do not understand the cm things.

 

 

[Michael GilliganParticipant @michaelgilligan61133]

Some people do this stuff for very practical reasons:

http://pkgsolutions.co.uk/boxcomp/stagger.html

MichaelG.

.

Edit: … as, of course do Bees

[Robert Atkinson 2Participant @robertatkinson2]

There are other uses for this. One I was involved with was very accurately spacing a grid of pins. These were used for printing arrays of samples on microscope slides. Two layers of ceramic ball bearings were used with the ball diameter setting the pitch and the gap setting the pin diameter. The company patented it. A couple of links:

https://worldwide.espacenet.com/patent/search/family/025531577/publication/US2003086827A1?q=pn%3DUS2003086827A1

https://worldwide.espacenet.com/patent/search/family/025400237/publication/US2003002962A1?q=pn%3DUS2003002962A1

https://arrayit.com/microarray-pins-genetix-technology.html

Robert.

[Nigel Graham 2Participant @nigelgraham2]

My friend promptly replied with the proper answer (42) – with Hitchhikers’ Guide reference.

So DC31k, you may be right about the set diameters!

No, he’s not a dress-maker or Blue Peter presenter, but I think he was an IT specialist when working, and amateur geologist as retirement hobby. Indeed he gained his PhD in just that! Another friend – builder by trade so accustomed professionally to metres and millimetres – was telling me this afternoon about his frustration when his daughter (now adult) keep using measurements like “7.3cm” instead of “73mm”. It is because schools will insist on teaching centimetres.

Since I was so close I must have used the (or ‘a’) correct method but either mis-typed one of several 7-digit numbers into the calculator, or rounded the answer the wrong way.

 

KiwiBloke –

I’d assumed using trig to solve the triangle linking the centres, and that kept going wrong. Then realised I already had the values enabling me to use Pythagoras instead.

That cracked Q.1 (the area).

Q.2 was far harder. I could not see any possible way to solve it numerically – not the arithmetic itself, but which arithmetic to use.

I think that is the key to being able to use mathematics. It’s not just knowing standard formulae and rules; but being able to examine the problem and see which of them to apply when no method is obvious.

I have seen scientific reports stuffed with the most terrifying maths derived from measurements in the physics and engineering experiments they describe; and wonder how on Earth the author even knows how to use the tables of physical readings to discover the equations giving them.

I recall one in particular. It examined the vibration properties of certain materials when subjected to an impact.

The Report? Part-typed, part hand-written, some twenty pages of Extreme Mathematics, including page-width expressions prefixed with several definite-integral signs, resembling a stylised swannery.

The Apparatus? Calibrated accelerometers… and an ordinary claw-hammer screwed to a wooden metre-rule suspended on a cup-hook, released by hand to swing a set distance against the sample.

 

Michael –

I like that! If I ever needed pack a lot of cylinders into a box, I’d need ready-made calculators. Or stack some between angle-plates and measure the stack. Or solve it graphically like the three logs. I could not derive the formulae.

I suppose the principle might be used for designing, e.g. a tube-plate or a “Rosebud” grate, with the circles enlarged to take in the desired spacing.

 

Robert –

I studied the patent applications carefully but could not really see how the spacing is made, using bearings. The drawing appeared to show the pins slide in blocks drilled with matrices of holes.

Nevertheless those, and the accompanying publicity material, are impressive in their very high-order engineering indeed.

I have seen something perhaps related in principle, a vernier-height gauge that had a “coarse” setting in, say, 10mm increments by the vernier assembly having a finger that went between ball-bearings of that diameter in a slotted, vertical, cylindrical colum.

 

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