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Jan, yes that's a classic beauty.  Related to that: https://www.youtube.com/watch?v=ovJcsL7vyrk

Jan, I'd like to see that.

cbwx34, give this a shot: https://www.youtube.com/playlist?list=PLiaHhY2iBX9g6KIvZ_703G3KJXapKkNaF

Rich, I was learning about complex numbers and this was a good exercise to try them out.  I already knew that one could convert trig functions to Z-plane, but then the realization came that the complex plane representation was both simpler than the sine/cosine form and easier to work with. 

[i]

Your formulas are for me rare demonstration how complex numbers may be useful. Until now I used complex numbers only to solve quadratic equations or describing the phase offset between current and voltage.

That great because I put this together after learning more about complex numbers myself.  The cyclic nature of the powers of i that works for phase also works for angles.

[y]

Jan, after a little more work I think the y formula reduced to the expression attached, which I'll edit into my original post.  Please check it and report any errors.

[P]

For cbwx34 and others who are not yet fans of the complex plane there is another equally simple if more verbose way to express this, and that is using vectors.  We'll need to define a two argument function P that takes (magnitude, angle) and returns Cartesian components (x, y).  (I avoid the standard r/theta to avoid confusion with their earlier use.)

P(m, a) = ( m cos(a), m sin(a) )

  Then we can write:



|| (x, y) || is the norm, i.e. distance, i.e. sqrt( |x|^2 + |y|^2 ).  All angles are in radians.

What this represents is defining the contact point on the wheel as (0, R) and then walking (d-r, theta) and (r+t, theta-90°) from that point for the composite tool depth and rest offset respectively, giving us the coordinates of center of the tool rest relative to the origin at the center of the wheel.

[i]

I took this as an excuse to play with complex numbers, so if you don't like math you're not going to like this.  Nevertheless I think the form produced has an elegance that the equivalent sine/cosine expressions lack.  i is the imaginary number sqrt(-1).  e is Euler's number.

Plugging the numbers you gave:

theta = 20 ° (in radians)
d = 140mm
R = 125mm
t = 6mm
r = 6mm

I get 74.82mm for the gap.

[x]

This has been done by others, and I don't even own a Tormek.  However it was a good exercise to derive the formulas myself and I'd like to share the results that I find pleasing: the formula for the gap (green line), which does not use x or y, and the formula for y given x.

gap   =   

y   =   

Thank you; there is indeed much more to abrasive performance than median particle size.

Thanks!

I registered to use the search function but I was unable to find the information I am seeking.

Have micron grades for the Tormek diamond wheels been published?