This is for a macro to be used in a cad/cam software, and it's purpose is to calculate the stepover distance of a ball nose cutter on a concave surface, based on the cusp height.
I found the attached pic in a google search, but the formula only yields approximate results – the “8” in the formula seems like it can’t be constant. For instance (I checked this by laying it out accurately in Autocad):
Given: Rsurface=.75, Rcutter=.25
When h=.0100”, the “8” needs to actually be 7.919639” to yield the correct S value of .17233295”. When h=.0075”, the “8” needs to actually be 7.939798” to yield the correct S value of .14943454”.
Again, my ‘correct’ S values were derived by laying the diagrams out in Autocad.
So bottom line; I am looking for a formula more accurate than the attached, to calculate the ‘S’ distance based on the cusp height ‘h’, with the Rsurface & Rcutter values being known (Rsurface=concave surface radius, Rcutter=insert radius). enter image description here
Solving the system of equations
$$ \left\{ \begin{array}{rcl} R (R-2 r)& =& (R-\delta ) (R-x-\delta) \\ d& =& 2R\sin(\theta)\\ r \csc (\theta )& =& (R-x-\delta) \csc (\psi ) \\ r \csc (\theta )& =& (R-r) \csc (\theta-\psi ) \\ \end{array} \right. $$
we get at
$$ d = \sqrt{\delta(2R-\delta)(2r-\delta)(2(R-r)-\delta)}\left(\frac{R}{(R-\delta)(R-r)}\right) $$
with
$$ d = \ \ \mbox{Sought distance}\\ R = \ \ \mbox{Larger circle radius}\\ r = \mbox{Cutter radius}\\ \delta = \ \ \mbox{Cusp height} $$
I hope this helps. Attached some results
$$ \left( \begin{array}{cccc} R & r & \delta & d\\ 0.75 & 0.25 & 0.0075 & 0.149435 \\ 0.75 & 0.25 & 0.01 & 0.172333 \\ \end{array} \right) $$