MEW 253: Workshop Techniques; Darren Conway

Bloody hell, Im lost

Ok, I tried to write something below but it gets technical very quickly and its not very well written (too lazy). So if you cannot handle wrong-prints please don't read the text below as it will only be an offense.

With that said let me see if I can shed some light on this issue. Normally when we are dealing with measurement uncertainty the error reported on a manufacturer sheet is a 95% confidence interval of the error. We are here assuming that the error is normally distributed and the error is 2 standard deviations from the mean.

Let us consider measurement X with uncertainty U i.e. we are seeing X+U in our instrument, where U is N(0,s), where s is the reported error of the instrument halved. Let f denote the function transforming our measurement to an angle, i.e. we are interested in f(X+U)-f(X). But notice that f is not linear and f(X+U)-f(X) is not normally distributed anymore, the author mentions "the linear part of the error" which I guess is referring to the fact that f is linearized as a function F and we can proceed in assuming that F(X+U)-F(X) is normally distributed (linearity). The author transforms the 2*Std(U) to 2*Std(F(X+U)-F(X)), i.e. two times the standard deviation.

Another important point here is that although the presence of one error affects the effect of another error so their transformed error (angle) are not independent anymore (they where independent to begin with). This can essentially be disregarded since the influence on angle is so small that considering them as independent is not a great error.

Now we come to the conclusion of this. Consider the four angular errors which according to the above can be regarded as independent, we wish to compute the standard deviation of this total error, so that we can compute a 95% confidence interval. Due to independence the sum of the variance is the variance of the sum, i.e. we can just add the squares together to get the variance of the sum.

To make this clearer we are considering measurements X1,X2,X3,X4 with uncertainties U1,U2,U3,U4, and denote Std(U1)=s1,...,Std(U4)=s4. We are also considering four functions f1,f2,f3,f4 that transforms a measurement to the angle (given the other values), to be clear f1(X1+U1,X2+U2,X3+U3,X4+U4), but we approximated this with F1(X1+U1,X2,X3,X4), (where F1 is a linear approximation of f1 around the point (X1,X2,X3,X4), in the following we will suppress the X2,X3,X4 depencence). Now what the author calculated was F1(X1+2*s1) which due to linearity is 2*Std(F1(X1+U1)), the same goes for F2,F3,F4. Now what we want to calculate is

2*Std(F1(X1+U1)+F2(X2+U2)+F3(X3+U3)+F4(X4+U4)) = 2*sqrt(Var(F1(X1+U1)+F2(X2+U2)+F3(X3+U3)+F4(X4+U4))

where Var indicates Variance (square of standard deviation). Now using independence we get

2*sqrt(Var(F1(X1+U1)+F2(X2+U2)+F3(X3+U3)+F4(X4+U4)) = 2*sqrt(Var(F1(X1+U1))+Var(F2(X2+U2))+Var(F3(X3+U3))+Var(F4(X4+U4)))

Which is then equivalent to what the author performed. So this means that a 95% confidence interval for the angle error is roughly 0.006 degrees when rounded off.

Quite sure somewhere is an error but the overall idea is sound, we are considering the calculated angular errors as standard deviations of independent random variables and thus squaring, summing and then taking the root gives the standard deviation of the sum.

Edited By Neil Wyatt on 12/04/2017 14:41:05

Neil,

Thanks for posting the table ... but is there any chance you could provide a link to it ?

... it currently runs under the Adverts.

MichaelG.

Thanks for taking the trouble Benny, I found that very helpful.

At school I spent most maths lessons staring out the window fantasizing about girls. Missing a out a few basic steps back then  doesn't half make simple mathematics difficult now.

Seriously though, compared with our international competitors, many of us Brits have a thoroughly bad attitude to maths. Smart people who get angry about misplaced apostrophes often brag about their incompetence with figures. Does anyone know why we despise maths?

Dave

Edited By SillyOldDuffer on 12/04/2017 10:41:19

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I think it has a lot to do with the way many of us were taught

1. As mentioned; Maths gets difficult when a step is missed; glossed-over, or misunderstood
2. There is rarely any discussion of practical uses to which the more complicated topics can be put.
3. We all seem to hit our own 'brick wall' of comprehension [blind spot] ... mine was Calculus.

I found calculus difficult; and couldn't see much relevance at the time ... so I 'zoned out'

Years later I came across several practical applications; and wished that Teachers had introduced the subject in that way.

MichaelG.

Maths never sunk in with me if I couldn't see a use for it ie algebra, equations, calculus. Basic maths and trigonometry I could envisage using, and I'd no problems with them. I notice MichaelG commented on same above

Roy

I 'got' calculus when I realised that you could produce a family of equations to show the position of something, the how fast it was moving, then how fast it was accelerating, then the rate at which its acceleration was changing.

Sadly, since then calculus has escaped.

Neil

.

Thanks, Neil.

MichaelG.

Many people did because, logically, you have to learn differential calculus (which, at that point, is a "yeah, so what?" topic) before you move on to integral calculus - which is lightbulb-popping and you find out such things as how to derive the π.r² formula for the area of a circle from first principles rather than "because teacher said so" (and incidentally, how to calculate orbital mechanics and a whole bunch of other arguably interesting stuff).

Then, once you get into integral calculus, differential calculus becomes a whole lot more relevant - and interesting - in its own right.

(Incidentally, I tried to put in π back there directly into the online editor using the alt-numeric-keypad method and it totally dumped the whole tab. Weird. I wonder if its an editor thing or a FF thing. I finally copy/pasted from Wordpad).

Edit: OK, so this editor doesn't like π (pi)

Edited By Bandersnatch on 12/04/2017 22:42:06

The main reason why students don't understand maths is because most teachers don't understand it either.

JA

This country's dislike of maths comes from the top, people who have degrees in Latin and Ancient Greek who think that qualifies them to run the country. They don't understand maths, so they disparage it.

I agree with comments above about teaching, when I explained differentiation to my daughter in the way I was taught so many years ago she got it straight away. Several GCSE papers she brought home as homework had questions which were fundamentally flawed. One I remember was to solve one equation which had 2 unknowns. The maths teacher would not accept that there were an infinite number of answers. If the teachers and examiners can't get it right what chance the pupils.

Edited By duncan webster on 12/04/2017 23:40:35

Hello

The mean square error in the table is simply the square root of the errors squared. Like many things in practical engineering, it is an approximation. In this case it is an approximation of the actual error. The purpose of this table was to illustrate that the angular error is tiny and that is all. If I was writing a math exam or an academic paper, I would have taken a different approach.

I intentionally avoided using or describing detailed statistical techniques in an article focused on practical workshop techniques. It wouldn't matter what error calculation was applied, the estimated error would be small. As it says in the article, the calculation is an estimate of errors, and that is sufficient to illustrate the point.

I am quite flattered that the article has been so closely scrutinized. At least it tells me that some found it worth the time to read. Forums like these are great for sharing knowledge, sounding out ideas and for providing constructive feedback. I hope you all enjoyed the remainder of the article.

Hi

It's early June and I finally received the magazine with my article. The formula submitted in the article is:

asin(d/D) -asin((d+e)/D)

Where:

e : the measurement error

D : the large radius of the taper

d : the small radius of the taper

The total "Mean Square Error" is simply the square root of the errors squared.

Darren

Duncan is right, but not only our leaders are so inhibited. The media is run by arts people so the science coverage is minimal. My problem at school, apart from maths, was Valency, in Chemistry. It appeared to be an arbitrary, meaningless number and I just didn't get it. Years later, when I discovered that it relates to the presence or absence of electrons in orbits around the atomic nucleus, it was, "Of course! Why didn't they say so!"

Geoff

I didn't take an interest in maths at school because I could never understand why anyone would want to work out how long it would take to fill a bath with the plug out!

Now if the teachers had come up with a practical application then I might have been more interested.

I learnt far more on my own after leaving school by working out the answers to real problems.

Brian

I'm kind of wondering why errors of 0.0001mm are being talked about on a setup that used a grubby bit of keysteel clamped to the lathe bed with a couple of off cuts of 2x2 and ply having just had a quick look at the pictures

One example of 'how long will it take to fill a bath with the plug out?' is the Titanic. The Naval Architect (Thomas Andrews) was on board when the ship hit the iceberg and calculated how long the ship would take to sink. He also pointed out that there weren't enough lifeboats for all the passengers and probably chose to go down with the ship.

I think most people had problems with the way maths was taught; much of the teaching was too abstract for my tastes. I have to say though that my smart mates had no trouble with that approach at all; perhaps the system was intended to identify high-flyers.

Dave

The error table relates to the precision of using a turned measured step and the top slide offset to measure the absolute angle of the taper set . The piece of grubby key steel and scrap wood is used for relative measurement of the top slide angle. Both methods are applicable to different applications.

Neil Wyatt:

"Sadly, since then calculus has escaped."

I have numerous books on "diff and int" which are looking for a good home and which would help you or any other interested member to recapture the rapture of maths.

Getting rid of my late parent's library is like trying to give away fivers in the street!