Producing Concave Surfaces With A Vertical Mill

Not to worry, you didn't know. We got way off topic and yours was a funny yet tactful post :)
 
Billh50 said:
There is a formula for cutting a large radius such as 32" in a 3" wide part with a 6" cutter as an example. The formula tells you how much to tip the head of the mill. I would assume if you used that formula and put your cutter at it's lowest point on center of a round piece it would create a concave the same way when turning the part in a rotary table.
The formula is this. Divide 1/2 radius of the cutter by the radius to cut. The answer will be the Tangent of the angle to tip the head of the mill.

What you are really cutting is the bottom of a parabola.
Click to expand...
Do not believe that is correct, any mathematic proof ?
 
I worked on this topic for several months off and on last year. I had a client that wanted a spherical bearing with a radius of 12". Came to the same conclusion. I did not have equipment or tooling that would accomplish task. (surfaces must mate.) But many surfaces can be made to approximate another as I have done with a parabola and a circular arc thousands of time in my checkered and dissipated career. A 90 deg circular path must be rotated to achieve a spherical segment. its geometry.
 
I used this technique 30 years ago to put a raised arc along a plate. Having it be higher in the center means cutting on the upside of the tool path so for a 6" long plate the flycutter needed more than a 6 inch radius. Otherwise it would start cutting a dish on the backside. I always assumed it cut an ellipse because it would put a smaller radius at the edges if you made the swing radius only slightly bigger than the width of the part. This trick is often used with table saws by tilting the blade and pushing the wood across sideways to make a shallow depression, like a serving dish. The part I designed was a slightly curved backing plate for a silicone rubber pad. It was used for making microfiche (film) duplicates by pressing the original and film together and flashing light to cast a shadow of the master onto the film. The curved foam made line contact down the center and squeegeed any air bubbles as it compressed.
 
The best way I can explain it is this. Hold a round plate horizontal. Now slowly turn it toward you and look at the bottom half of the rim as you turn it. It is slowly becoming becoming a parabola with narrower legs and smaller radius at bottom. Now if you are just raising the table into the cutter as you cut across the surface you are creating a parabola which if you go past center of cutter it will have straight sides. But by using just the lower part of a large cutter on a small width part you will be using the very bottom of that parabola, which even though not a perfect radius, is so close to one that it would be very hard to tell the difference.
 
Look up how spherical lens surfaces are generated. This is essentially the same thing. The lens people use a specialized tool and diamond cutters, but they make spherical lenses and other more exotic surfaces with a similar setup. The key is the cutter must sweep through the center of rotation as it has been explained to me.

Google "generating spherical surface"
 
I can't believe I'm wading into this, but here goes.

There is no Parabola. There is no part of a parabola. There is nothing parabolic about any of it.

With two rotating axes, there can only be circular (spherical) and ellipse (ellipsoidal) surfaces produced. There cannot even be radii produced larger than the sweep of the cutter. The cannot be any hyberbolics.

What is produced is an ellipsoidal (swept 3D ellipse) that can be used to *approximate* a spherical surface. Close enough for even aircraft work (that is typically .03 at best. Careful probing on your local CMM will reveal this. Yes, I've seen the charts and "formulas", but the geometry is not up for opinion.

This is because all of these fall into a class of curve known as 'conics'. Someone posted a cone that had been cut up. That's why the cone matters.

Analytically, these curves all follow what is known as the "2nd degree general equation, or <deep breath> Ax^2+By^2 +Cxy+Dx+Ey+F=0
So when you have, say, A,B,and F coefficients, you have a circle. (where F = radius^2).
Mathematically, without a distinct Cxy component (and a particular proportion at that) no parabola can exist. If you're working with circles, like rotating things are, no Cxy component enters into the equation. Even if they're eccentrically rotated, the resultant intersection is still just elliptical.
So, returning to the cone, we can see that when the cone is sliced exactly perpendicular to the axis, a circle results. Only then. When sliced exactly perpendicular to the base, a parabola results. Only then. All the infinite other slicings that are not normal to base or axis are ellipse and hyperbola.
I got into this both as an Engineer and as a Lofter before that. I lofted portions of some small planes you've heard of. Everything subsonic, including airfoils and propellers, are lofted to 2nd degree equations or combinations thereof. As an Engineer, i investigated 'best-fit conics' for surveillance algorithms. (There are no true circles in nature, so finding one with your UAV means you found something manmade... like a truck.)
In defense of anyone i might have offended, the word "parabolic" is like "aircraft aluminum" or "surgical steel" and is thrown around with reckless abandon.. often by folks that really ought to know better but choose to go with the flow instead of correcting a roomful of people (BT;DT). So it's no one's shortcoming if they haven't ventured deep into GeekLand to sort all this out.
Also, very often the differences between these geometric distinctions are microscopic. Indeed, unless your cutter is *perfect*, which none can be, you're cutting an ellipse, anyway.
All i'm offering is analytical theory. Not how to cut stuff.

Wrat
 
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