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Algebra/calculus Problem


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Ok, what I'm trying to do is develop an equation that will let me calculate the y coordinates of the tip of a ball nose endmill and any point of the x travel. This way I can mill a perfect radius on the fretboard and repeat that with the frets installed.

Heres a simplified picture of the problem.

problem.jpg

The equation of a circle is x^2+y^2=r^2 In this case r=7.25 (fender)

Now that is the easy part. The problem is that as x grows, the point of contact moves from the tip of the cutter toward the x axis so if the tip is calculated using the above equation, one side will dig in.

What im trying to calculate is an equation to describe the little extra that has to be added to the y component to keep the radius true.

Anyone have any ideas?

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I have no idea how you program or use these things, but I'm guessing that normally you would write an equation to control the position of the tip (the bit directly below the centre) of the ball thing... would it be possible to set it so that your equation controlled the position of the centre of the ball instead?

Then you could just set the radius 5/8" larger than the one you wanted.

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I have no idea how you program or use these things, but I'm guessing that normally you would write an equation to control the position of the tip (the bit directly below the centre) of the ball thing... would it be possible to set it so that your equation controlled the position of the centre of the ball instead?

Then you could just set the radius 5/8" larger than the one you wanted.

Its not a cnc, although what you mentioned is called an offset and its exactly how it would be done in a program.

THis is a manual mill im working with so im doing the cnc work manually. It seems like a lengthy process but with a digital readout its not so bad.

You bring up an excellent point tho...now how to work it that way is the question.

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Otherwise I THINK the equation:

(x^2) + (y+r)^2 = (R+r)^2

Where R is the intended fretboard radius and r will be the radius of the cutting bit (5/8)

Might do it, but check before you take my word for it.

(Thats the same sort of idea as controlling the position of the centre of the ball)

Its hard to explain what I'm thinking without a diagram, but that should sort of 'indirectly' control the position of the centre of the ball by controlling the position of the tip...

Sort of the same as the offset idea.

I wouldnt understand what I just wrote if I were you reading my post... this really needs a diagram I think.

(This is good revision for my upcoming maths exams!)

Edited by Ben
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I'm not so sure, I just drew this a diagram in A9CAD (poor man's autoCAD).

paths3yv.jpg

I know the diagrams a bit convoluted, but it seems to show that the path of the tip of the bit is the same as the path of the centre, but 'r' units lower in the Y-axis, and that the path of the centre is the same as the circumference of your desired radius fretboard radius + the radius of the cutting bit... hence the equation.

Obviously you are only cutting a small section of the top of that circle for your fretboard radius, but I drew the whole circle to try to illustrate the point.

I can email you the original above diagram in DWG format if you like.

I should probably do my own maths work now, I hope I helped you somehow.

Ben

EDIT, I had to shrink the image because it was too big (640x640), am I right in thinking 640x480 is the allowed limit?

Edited by Ben
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