Member postings for Michael Gilligan

Thanks, John

… complete with curlicues if we zoom out a bit:

**LINK**

https://www.proofwiki.org/wiki/Definition:Trochoid/Prolate

MichaelG.

You are welcome to a copy of my QCAD file if you want to investigate

… I certainly won’t be doing so

MichaelG.

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Edit: __ for clarity: I have established to my satisfaction that the ‘extended rod’ curves are NOT cycloids

It doesn’t matter in the slightest  to me how close they might come dimensionally in any specific implementation.

Not a Cycloid = Not a Cycloid … that’s all there is to it.

Edited By Michael Gilligan on 10/09/2023 18:52:16

Thanks, Dave

Definitely yellow in my Album

**LINK**

https://www.model-engineer.co.uk/sites/7/images/member_albums/79913/927884.jpg

… I did swap the picture : perhaps things are out of synch. now

Edited By Michael Gilligan on 10/09/2023 17:07:46

Edited By SillyOldDuffer on 10/09/2023 17:11:40

With thanks to Christiaan Huygens for discovering the Cycloid

… and a nod of respect to all those who participated in the recent, somewhat lively, discussion.

Here is a screen-grab from a little exercise I have just done in QCAD

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First I created a ‘wheel’ with 96 spokes, then attached a green marker at one point on the circumference, and yellow markers at three more locations on a radial extension to that spoke. [thus mimicking what dave8 described]

Then drew the circumference of the wheel, plus a ‘road’ of equal length on another layer, and hid the original wheel [to avoid cluttering the image]

Grouping the four markers, and then copying that group ‘displaced & rotated’ to the other 95 positions produced four elegant curves … one of which is a Cycloid; but I know not what to call the others.

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MichaelG.

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Discuss or ignore as you wish … My own curiosity is satisfied

MichaelG.

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Note: __ With hindsight, I don’t think creating the ‘spokes’ served any useful purpose … except to keep my mind focussed on what I was doing.

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Edit: __ Apologies, I seem to have red dots where there should be yellow

… can’t work-out what happened there  but I will try to fix it.

Edited By Michael Gilligan on 10/09/2023 17:02:05

This discussion seems to have taken an unpleasant change in style

… I now regret that I unintentionally revived it at the end of page 4

MichaelG.

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… PIN should of course be capitalised, which they failed to do in either language version

But what really amused me was the English-only device.

MichaelG.

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I hesitate to write this … but here goes:

I think it is ‘intuitively obvious’ from Huygens

But, of course, that intuition only applies to a simple pendulum with flexible string.

MichaelG.

This is the “dual language “ labelling on the ticket collection machine at Prestayn Railway Station:

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MichaelG.

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Next big news is the introduction of 20mph speed limits, from 17 September 2023

… You have been warned !!

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A number or a house name

My present PostCode currently covers only a group of dwellings that all have names.

MichaelG.

I don’t think this does anything to resolve the issue raised by dave8

[regarding a small generating circle with a long rod appended radially]
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But it’s a neat interactive demonstration, with a simple explanation of the mathematics

… so I commend it to all readers

**LINK**

https://www.geogebra.org/m/QeQ9aA5e

MichaelG.

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P.S. __ @dave8 … Please do correct me if I have misrepresented you !!

Tony, Duncan, Mike, Dave

Thanks for your comments/observations

Just to add a personal ‘position statement’ … I am well aware that other avenues have already led to more accurate pendulums, but I have long been interested in Huygens’ approach.

The mathematical proofs, I regret to say, cause me to glaze-over … so the simple geometrical exercise is much more to my taste.

The other difficulty, of course, is my trivially small command of Latin … There is no way that I could translate Huygens’ text, so I am reliant upon Ian Bruce for the English and Huygens for the pictures !

… The best I can manage is a “sanity check” on a few individual words from Bruce’s translation, but the grammar of ‘Scientific Latin’ can be full of subtleties beyond my comprehension.

I look forward to any further insights.

MichaelG.

Right … This is the patent that I should have linked : **LINK**

https://worldwide.espacenet.com/patent/search/family/010123251/publication/GB1215583A?q=dickson%20donald%20john

I had both of them, and evidently selected the wrong one

It doesn’t say much about the manufacture of the tool-holders, but it does put them in their original context, and is interesting for that.

MichaelG.

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There may be more, but I’ve never found it/them

The translation is obviously into spoken English, but the important words appear to correspond nicely to the Latin.

So the big problem is:

with the diameter of the base equal to half the length of the pendulum

MichaelG.

Ian Bruce:

Upon a flat table the rule AB is fixed, half a finger wide. Then there is present a cylinder C D E truly with the diameter of the base equal to half the length of the pendulum; and F G H E is a small bandage, or better a thin metal foil, fixed to the rule at the point F of this, and to some point of the circumference of the cylinder E, thus in order that it shall be partially wrapped around the cylinder alongside the rule A B. Moreover a sharp iron point DI shall be fixed to the cylinder, projecting a little below the base, and thus in order that the circumference of the cylinder corresponds exactly to it. [p. 11]

Thus with these in place, if the cylinder is rotated along the rule A B, with the metal foil in between as far as FG, and with that always extended as far as possible, the sharp point I applied below on the plane table will describe the curved line KI, which is called the cycloid. Truly the circle CDE is the originator of the curve, applied to the base of the cylinder. Since if now we apply the plate KL to the rule A B ; that first part of the cycloid KI is traced out, then we invert the plate, and in the facing surface the similar curve KM is inscribed, arising from the same point K. Then we will have formed the figure MKI, following these lines carefully for the figure of the plates it is necessary to adapt, through the gap between which the pendulum is hanging. Moreover the small portions of the arc KM, KI are sufficient for the use of the clocks ; with the remainder of the curve not to be used in the future, and to which the string of the pendulum does not have access. Truly, in order that the wonderful nature of the line and the effect may be understood better, the whole semicycloids KM and KI, here seen to be expressed by another diagram, between which the pendulum KNP is suspended and moving, [of length] twice the diameter of the generating circle, and the oscillations of any amplitude, as far as the largest of all through the arc MPI will be made in the same times : and thus, so that the centre P of a sphere hung on, is always moving to and fro on the line M P I which is part of a whole cycloid. I do not know of any other line with this conspicuous quality, except for this given line, as truly it describes its own evolute. Moreover these things which have touched on, concerning the descent of the weight and the evolute of the curve, we will go through and each will be explained in what follows. [p. 12]

Huygens:

Super tabula plana affigatur regula A B, semidigiti crassitudine. Deinde fiat cylindrus C D E eadem illa altitudine, diametrum vero baseos, dimidiae perpendiculi longitudini, aequalem habens; sitque F G H E fasciola, seu potius bractea tenuis, affixa regulae in huic F, cylindro vero in circumferentiae puncto aliquo E, ita ut partim huic circumvoluta sit, partim extendatur juxta latus regulae A B. Cylindro autem infixa sit ferrea cuspis D I, pauxillum ultra basis inferiorem prominens, atque ita ut circumferentiae ejus exacte respondeat. [p. 11]

His ita se habentibus, si cylindrus secundum regulum A B volvatur, bracteolae tantum FG crassitudine intercedente, eaque semper quantum potest extensa, describet cuspis I subjecto tabulae plano lineam curvam KI, quae Cyclois vocatur. Circulus vero genitor erit CDE, cylindri adhibiti basis. Quod si jam laminam KL ad regulam A B applicuerimus; exarata primum ea cycloidis portione K I, invertemus deinde ipsam, & in superficie adversa similem lineam K M, ab eodem puncto K egredientem, incidemus. Tum figuram MKI, accurate secundum, lineas istas, efformabimus, cui figurae lamellarum interstitium aptari oportet, inter quas per pendiculum suspenditur. Sufficiunt autem ad horologiorum usum portiones exiguae arcuum K M, K I; reliquo flexu inutili futuro, ad quem perpendiculi filum accedere non potest. Verum, ut mirabilis lineae natura atque effectus plenius intelligantur, integras semicycloides KM, KI, alio schemate hic exprimere visum fuit, inter quas suspensum agitatumque Pendulum KNP, diametri circuli genitoris duplum, cujuscunque amplitudinis oscillationes, usque ad maximam omnium per arcum MPI, iisdem temporibus confecturum sit: atque ita, ut appensae spherae P centrum, in linea M P I, quae & ipsa cyclois integra est, semper versetur. Quae proprietas insignis nescio an alii praeter hanc lineae data sit, ut nempe se ipsam sui evolutione describat. Haec autem quae dicta sunt, in sequentibus, ubi de descensu gravium, deque evolutione curvarum agemus, singula demonstrabuntur. [p. 12]

Tony

As you have so obviously got a grip on all of this … Would you please do me the favour of reading Huygens’ description of his layout process; and then explain exactly the result what he is doing ?

Something in it doesn’t quite ring true

The English translation of text is on the page that I posted

But, for everyone’s convenience, I will post the transcription of the Latin and IanBruce’s translation shortly.

It all seems consistent in itself, but …

MichaelG.