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radius of curvature of gouge size


Seth_Leigh

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Going back to this post on Roger Hargrave's gouging method for creating the edge thickness and channel at the same time using gouges, I'd like to make two gouges, a 10mm wide #7 and an 8mm wide #8 sweep.

How does one calculate what radius of circle to curve the gouge to for it to be a #7 or a #8 sweep?

My current plan is to take some O1 toolsteel I have in strip form, take a 7" long piece of it 10mm wide and beat the last two inches of it over a piece of galvanized steel pipe to form it into the proper curve for the #7 gouge, and do the same with an 8mm wide strip on a slightly tighter piece of pipe for the #8 gouge. But I don't know what diameter of pipe would most closely match these gauges.

Can anyone give advice on how to calculate this radius? Or, can anyone offer advice on perhaps a better way to make these two gouges?

I plan on using a #10 coffee can with some charcoal in it blown with a hair dryer to heat the steel to red hot for quenching in some vegetable oil after it's formed into a gouge.

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""How does one calculate what radius of circle to curve the gouge to for it to be a #7 or a #8 sweep? ""

I do not know how they define sweep, but I can give you a worksheet to relate circle radius to chord length to "sagita". The Sagita is the distance from the center of the chord to the circle. This is a well known term from the pages of the telescope (mirror) makers' book. The derivation involves two similar right triangles and Archimedes formula A^2 + B^2 = C^2 One gets a simple algebraic equation to solve.

I will check to see if your e-mail is on your bio and send an old Lotus Symphony spreadseet. It can be imported to Excell and the formulas should work fine. If not, let me know. I have Excell and can take a look-over. I don't like Excell....... Symphony is about 1.5M of DOS and does everything Excell does on 50 except give you a lot of templates for stuff you will never use. (The original DOS Excell was a rip-off of Lotus. and with a spreadsheet half the size.)

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Seth- I arrived at the following by very unscientific methods (sorry jm), but the EXACT curve or sweep is not critical anyway. You can calculate the diameter of a pipe (form) for a particular gouge as follows: Multiply the desired gouge width by: For 6 sweep by 1.857; For 7 sweep by 1.667; For 8 sweep by 1.143; For 9 sweep by 1.048; For 11 sweep by 1.000 Example: For a 12mm gouge with a 7 sweep, 12 x 1.667= 20.004mm diameter pipe form. Hope this helps. Ron.

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Seth, when I made the two channel gouges several years ago, I didn't know what were those

sweep numbers either. Use 2 or 3 cross-arch curves and draw the cross-sectional diagrams,

one within the C and another at upper or lower bout. You can actually see the width and

the radius for the gouge. Since it cuts to the very edge of the plate, the edge is quite thick

to start with (6mm for the bout, 6.5 for the corners and C). The problem of this method is

in the carving of C channel of back, particularly if it is highly flamed. I use a smaller

and more curve gouge to remove as much wood as possible, then use the right gouge to

"clean" it.

I don't use RH method any more. I have some questions about this method: The thickness A in

Fig.8 is different for C and for upper and lower bouts as shown in Sacconi's book. The

thickness of C-edge is not constant. If the channels are carved when the plates are not

glued to the ribs, you can not account for the irregular graduations near the edges. If it

is carved after assembly, it is difficult to do.

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Quote:

Seth- I arrived at the following by very unscientific methods (sorry jm), but the EXACT curve or sweep is not critical anyway. You can calculate the diameter of a pipe (form) for a particular gouge as follows: Multiply the desired gouge width by: For 6 sweep by 1.857; For 7 sweep by 1.667; For 8 sweep by 1.143; For 9 sweep by 1.048; For 11 sweep by 1.000 Example: For a 12mm gouge with a 7 sweep, 12 x 1.667= 20.004mm diameter pipe form. Hope this helps. Ron.


Ron1, thats a useful set of formulae for us unscientific types, I'll print that off and keep it for future reference.

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John- the numbers aren't completely arbitrary, but neither are they entirely mathematically accurate. They should suffice though- I just figured my simplistic approach might be used more easily, even if not quite as accurately as a "real" formula.

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Thanks for this information......... I was hoping they were real results that could be duplicated easily in a calculation. You can see the advantage. You could order a given sweep and width and know what to expect in the mails. Actually, it was hardly "scientific." Most kids who passed ordinary algebra and geometry in high school could do it. Not even trigonometry etc.

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The problem here is we want to know how the sweep numbers are defined. I suggest (or guess) an arbitrary formula: sweep number n=(d + 10h)/d, where d is the half cord length and h, the sagita. For a chisel, h=0 and n=1. For a half circle, d=h=r, n=11. For 8mm number 8 sweep, the above formula gives h=2.8mm. Then the radius r=(h^2+d^2)/2h=4.2mm. Similarly, 10mm #7, r=5.7mm.

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O.K., one of you mathematical geniouses will have to make the calculations, but I really feel they should then be put forth in the manner I showed my "close enough" figures. Then us 'average Joes' will be able to use them. It's been over half a century since this kid passed algebra. Honestly, I thought Saggita was a small town in Michigan. And I bet most of the other readers of the tread did too.

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Wouldn't it be easier to get a circle template from a drafting shop and match the curves to a sweep chart from one of the tool maker's? That would eliminate the math.

If it is helpful, I would be willing to do a chart of sweeps vs. radius of curvature, but I have to get ready for a trip. I will try to do it before I go, and post it soon, but I might not get it until next week. Maybe someone else is working it out and will have it before me.

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"O.K., one of you mathematical geniouses will have to make the calculations, but I really feel they should then be put forth in the manner I showed my "close enough" figures. Then us 'average Joes' will be able to use them. It's been over half a century since this kid passed algebra."

Exactly

"Honestly, I thought Saggita was a small town in Michigan. And I bet most of the other readers of the tread did too."

Being an amateur telescope maker many years ago I can enlighten you on the mysteries of "Sagita".

The word is actually spelt "Sagitta" and is, simply put, the depth of curvature in a spherical mirror. The formula for all but a small minority of mirrors is equally simple, being: Sagitta equals r2/2R, where r2 is the radius of the mirror and R is the radius of the curvature ground into it (focal length).

The limitatios of the fonts here don't allow me to show the exact formula for short focus mirrors, compound telescopes and eyepieces, but it is the same whether derived algebraically, geometrically or from the Cartesian equation of the circle.

There is also an alternative formula for S, derived by expanding r2/2R, thus: S=r2/2R + r4/8R3 + r6/16/R5......, this alternative formula has an advantage for those of us who have forgotten how to do sqare root, and its first two terms will give close enough approximation to exactness for any mirror.

However, without having to use sundry formulas, if any on this forum have a set of gouges, a practial way to work out the #sweep diameters you require, would be to draw round templates that fit neatly in the concave curve and then measure that diameter. IMHO That's all that you and the rest of us require for making our own suitably shaped gouges. It would be interesting if a member would take up that challenge to see how closely the results compare with those from your own simple formula.

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Re the final formula from the link in your earlier post:

"

d^2 + h^2

r = ---------

2h

So the radius is just the sum of the squares of the height and half the

length, divided by twice the height."

I thought I'd run the formula from the link you gave, past a set of figures and it works fine.

ie if you have a mirror 6" in diam with a focal length (radius) of 56":

r2/2R = 9" divided by 112" = 0.O803571"

then using your formula in reverse:

9" + 0.0064572" = 9.0064572" divided by 0.1607142 = 56" radius

So if a gouge has a width of 12mm and a concave depth of 3mm, that formula reads in laymans language:

6mm squared plus 3mm squared, with that total divided by 3mm times two, equals the radius of concavity, thus:

36mm + 9mm = 45mm divided by 6mm = 7.5mm radius, or a forming tool required 15mm in diameter.

Is this calculation correct?

Note: On reflection the width measurement used in the calculation should probably be the internal width of the gouge, not the external width, for the correct diameter of the forming tool to be determined.

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Thanks for expanding on what I posted, Alex. Your expanation is good. Too often we look for a "formula" when we really just need a measurement. I prefer the comparison method to the formula, comparing the gouge to a circle of known diameter.

The real genius is not so much in the math, but in the art that goes into instrument making, or into tool making. Compare the prices of the math books containing all the neat equations to the great instrument books. The math books gather dust, while the instrument books just zoom in price.

I hope everyone realizes that each good gouge may have a very slightly different curve, since they are made by hand.

There is another sweep measurement system by Hans Karlsson. It is based on the curve of a circle designated by the radius measured in millimeters. You can see his tools at http://www.countryworkshops.org/cwbulletin.pdf .

Also,

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Dear Froggie,

. "" . I prefer the comparison method to the formula, comparing the gouge to a circle of known diameter.

The real genius is not so much in the math, but in the art that goes into instrument making, or into tool making. Compare the prices of the math books containing all the neat equations to the great instrument books. The math books gather dust, while the instrument books just zoom in price. ""

So let's hear your dissertaion on "Genius." So far as making fiddles is concerned, I have had various interestting experiences. The country's finest maker from the standpoint of woodwork (guess who) has commented to me on things that I did slightly in a clumsy manner. "This curve might be changed to go just so......" If sweep numbers are approximations, then I suppose "close enough" is not good enough.. Other comments:

A.) You said this before, about comparing to circles of known radius..... Problem is that the sweep number does not correspond to a radius alone, it is a radius in comparison to the width of the gauge and the "sagitta".

B.) Notice that all the blue collar jobs are going abroad. The real genius is not in the math? This is hardly math, it is simple high school stuff. By instruments, do you mean musical instruments or instruments such as used by toolmakers and machinists (all working overseas, of course.) ?

C.) Thanks for the URL, I worked it out years ago, and offered an Excell (actually Lotus) spreadsheet which would find any of the three in terms of the other two. I sent it to one person who did not have Excell. I will send it to anyone else who writes me an e-mail. But I am not so sure it is going to work for sweep numbers. I still do not know how much these are arbitrary or correspond to a true ratio of d/h.

You are right about one thing, the real significance of the sweep number does not mean much. I have a few pretty flat ones of various widths and some more curved ones. That is all I need.

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Dear RON1,

Saginaw is in Michigan......... I see that you are retired, from what I do not know.

If you have grandkids ever, don't talk to them this way. Being an average joe will get them no place. Blue collar work will not exist for them. Encourage their math and don't make light of it as some kind of egghead excercise that plays second fiddle to true Pragmatism. The pragmatic work has already flown overseas.

The one thing the Westen Yankee culture has (more lacking in Asia) is creativity and insightful thinking. But this is changing fast too.

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Dear David,

Do you have a derivation that would come close to your guess? I think you are the only one who approached the problem in the right spirit.

Alex, as a telescope person, gave an expansion of sagitta in terms of Radius of curvature of the mirror and the mirror blank radius. I believe this follows directly from the normal sketch:

h = R - SQRT (R^2 - d^2) I did not do the expansion myself for a check, but I suppose a standard Taylor series would work fine.

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Hi Seth,

I made a set of gouges in 1982. I still use some of those gouges although I've bought both German and Japanese gouges since.

Here's how I did it: I bought bar stock of unhardened 01 steel. I bought a 1725rpm electric motor (often obtainable for free)and mounted a carborundum wheel that was as wide as the gouge I was working on. I then went to a woodworking catalogue that had the gouge sweeps printed on the page and cut out the sweeps I wanted. I then made a reverse template from them. The idea was to round the edge of the carborundum wheel to the same radius as the template. I then ground down the bar steel to the shape I wanted(convex wheel= concave gouge)-easy with unhardened steel. I then hardened the steel with a propane torch and tempered it in a toaster over(the bar steel came with a graph showing what temperature and how long to hold it to get the Rockwell rating you want)I quenched the steel in cheap olive oil-gave it a nice golden yellow color on the shank.

If I were you I wouldn't worry about making the guge a standard radius but rather work out the curve you're actually going to carve, either from a cross arch template or by making paper patterns following the surface of an instrument you like. If I were doing this fun excercise again I would attemp to do a cryogenic tempering of the gouge by dipping it in dry ice after the usual tempering process-you'll have to read up on cryogenic tempering because I haven't done the research for that. It took me about 6 hours to make a gouge with the handle I used brass plumbing joints to make the ferrule.

Good luck

Oded Kishony

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John,

I appologize if I stepped on your toes with my tongue-in-cheek comment about math books vs. music books. I was just making a joke. I did not mean that there is no genius in math. I just felt cheated way back when I tried to sell my college calc and diff-eq books and they said they would not buy them back.

Correct me if I am wrong, but I don't think that sweep is a circular section.

I do think you can approximate sweep with a circle, and yes, that is close enough for me. There must be a little tolerance there, as I don't see too many fractional sweep gouges.

I do appreciate the "partial credit" for pointing out the website

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"" I just felt cheated way back when I tried to sell my college calc and diff-eq books and they said they would not buy them back.

Correct me if I am wrong, but I don't think that sweep is a circular section.

I do think you can approximate sweep with a circle, and yes, that is close enough for me. There must be a little tolerance there, as I don't see too many fractional sweep gouges.""

Thsnks, and I just get irritated at times, not angry. You are the first to point out that most of these curves and not always circular. That is an absolute fact.

Sorry about the math books...... funny thing is, the math stays the same but fads in teaching tend to change. Also the pictures get more appealing, which makes a student want to buy the new book. I have boxes of such books myself, many of the classic texts on Quantum Mechanics. Partial differential equations is about as far as the classic QM physics goes.

I reacted as I did because there really are posters who seem to want the recipes and methods and nothing else. That is fine, but a working person in the field cannot afford to be too rigid with respect to science. (Unless he is Nigogosian ha ha !!) That was also tongue in cheek.

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Quote:

Dear RON1,

Saginaw is in Michigan......... I see that you are retired, from what I do not know.

If you have grandkids ever, don't talk to them this way. Being an average joe will get them no place. Blue collar work will not exist for them. Encourage their math and don't make light of it as some kind of egghead excercise that plays second fiddle to true Pragmatism. The pragmatic work has already flown overseas.

The one thing the Westen Yankee culture has (more lacking in Asia) is creativity and insightful thinking. But this is changing fast too.


John, I agree with your comments to Ron, that in our modern world, unless one has Certificates or Diplomas to prove their level of education, it will become evermore difficult to obtain work, other than blue collar.

However, education levels do not make a person any more intelligent than an uneducated person, just smarter in those fields studied.

This Thesis Basic Laws Of Stupid People is worth reading, even if just for a few laughts if you don't agree with it, as it claims that the same percentage of people are stupid (or intelligent) regardless of education or status in life.

One paragraph reads:

"Whenever I analyzed the blue-collar workers I found that the fraction å of them were stupid. As å's value was higher than I expected (First Law), paying my tribute to fashion I thought at first that segregation, poverty, lack of education were to be blamed. But moving up the social ladder I found that the same ratio was prevalent among the white-collar employees and among the students. More impressive still were the results among the professors. Whether I considered a large university or a small college, a famous institution or an obscure one, I found that the same fraction å of the professors are stupid. So bewildered was I by the results, that I made a special point to extend my research to a specially selected group, to a real elite, the Nobel laureates. The result confirmed Nature's supreme powers: å fraction of the Nobel laureates are stupid."

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Alex- thanks for the enlightening & understandable description of sagitta, etc. I agree the simple way to determine the diameter of a form would be to construct a circle on paper using the desired "sweep" arc. As a matter of fact, that's what I did to arrive at my "close enough" multipliers. I thought it should be simple & useable. The problem with the mathematical formula approach is that all reference to "sweep" is lost, which is the common term used when luthiers refer to gouges. Who ever heard of using a 3mm sagitta gouge? I'm trying to be serious here- however I do appreciate that these "easy multipliers" can & should be verified & adjusted for accuracy mathematically. I would hope someone with more of an aptitude for working with math formulas would take the time to do this- it would be a useful tool which would eliminate a lot of re-figuring whenever anyone wanted to make a gouge.

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