Re: [CAD_CAM_EDM_DRO] Re: Math for constant speed ellipse

> Todd,
>
> I took a crack at your problem because it is interesting. I'm not
> much of a mathematician so I analyzed it graphically using my trusty
> TI calculator. I think I have a possible solution.
>
> (1) Draw an ellipse whose X-axis dimension is 5 times the Y-axis. No
> special reason for 5, I just needed to pick a number.
>
> x = sqrt ( 25 - 25*y^2) would define the ellipse. (EQ 1)
>
> (2) Inscribe a circle with a raduis of 1 inside the ellipse. Sweep a
> projected radius at a constant angular rate (I picked 9 degrees with
> an initial offset of 4.5 degrees).
>
> (3) Take the sine for each increment. That will give you the value
> for "y", which you plug into EQ 1 to get the corresponding "x" on
> the elipse.

x = sqrt ( 25 - 25 * sin^2(theta) )

> (4) Calculate the segment lengths between each pair of x,y co-
> ordinates. First conclusion: The segment length (thus velocity)
> around x=0 is 5 times the velocity around y=0 for a constant angular
> rate. From this one can conclude that the angular rate around y=0
> should be 1/5 the rate around x=0. What is the relationship f(x) for
> this 5:1 range in velocity?

Ok. I plotted the segment lengths in Excel. The result is a non-linear
sawtooth.

> (5) Sum the segment lengths. Normalize the results by dividing the
> lengths by 5. Subtract each from "1". The resulting table now looks
> suspiciously like a sine table for every 9 degrees. It is in fact.

Maybe I'm not following correctly. The graph of this also looks like a non-
linear sawtooth.

> (6) Fooling with the calculator suggests the angular velocity for a
> constant path velocity should be:
>
> The relationship seems to be x/y -(x/y -1)*cos theta over the span
> of +/- pi/2. The result must range between 1 to 5 from +/- pi/4 and
> is the multiplier for "delta" summed to "theta" angle that sets the
> path velocity.

By x/y, do you mean the width of the ellipse divided by its height?

> Mariss
>
> None of (6) has been carefully checked.

Todd Fleming
flemingcnc.com