Thread: The Secret to The Secret to Cambering Bevel Up Plane Blades

Originally Posted by pmcgee

I think I can tie where Josh got us to to my post regarding triangles and rectangles being mushed around.

Attachment 251958

Imagine in the upper block there are two diagrams superimposed. One a rectangular prism (P), and the other a rectangular prism with part of the front face swiped off it vertically (PC).

As I said regarding the two-dimensional analogy, if P were transformed back at an angle (phi), maintaining the same height, the volume is left unchanged. You can easily verify that if you cut off the new triangular-prism-shaped section that appears at the back and glue it onto the now-sloped front you get back the original prism P.

And the same applies to the (vertically) cambered prism PC.

So consequently, for any given Depth of Camber d, the volume removed to put that depth of camber on any angle of bevel phi is always the same.

But - and this is where it ties back to Josh - for different bed angles theta, you will need a different depth of camber if you want to maintain the same width of shaving.
Alternatively you can put on the same depth of camber at a steeper bed angle and accept a narrower shaving.

It took me quite a while to get the illustration ... I wanted to squash the top prism, but settled for making a second angled one. I put the above in bold for my benefit because I've just twigged that the bed angle is coming into play and it seems to me at the moment that the bevel angle is not ... but the brain is now basically numb, so I'm gunna let that sink in for a while.

I will point out that Josh's formula ... only involves the following terms ...

Bed angle, Radius of curvature of the camber, and Width of the blade.

Cheers,
Paul

Paul,
I have looked at it from a few ways and if the primary bevel is established then the materiel removed for a given camber is constant for all bevel angles since the volume of the camber is the product of the (area of the bounding rectangle - area of the segment) and the thickness of the blade, nothing to do with the bevel angle. This is true because the bevel of the cambered bevel is parallel to the bevel of the straight blade and hence any section can always be made into a rectangle.
I have also come to the conclusion that Professor must have made an error in his calculation for the materiel removed, I think he calculated the volume remaining?

I will show my workings although it will be via an attachment at a latter stage when the grinding is less intense.

I have also done the calculations (not approximations) for some variations of the bedding angle and bevel angle starting with a square piece of steel
the following are for a 50mm wide blade x 3mm thick with 1.59mm shaving. I verified numerically as well. 33 and 48 degree bevels for BD done for comparison to BU. Radius for camber at bed angle 12=45, 45=140, 50=152, 55=162, 60=171

1. BU bedded at 12 degrees 25 degree bevel: Material removed for straight bevel 482, Material removed for camber bevel 368. Total = 850 cubic mm
2. BU bedded at 12 degrees 33 degree bevel: Material removed for straight bevel 346, Material removed for camber bevel 368. Total = 714 cubic mm
3. BU bedded at 12 degrees 48 degree bevel: Material removed for straight bevel 202, Material removed for camber bevel 368. Total = 570 cubic mm

4. BD bedded at 45 degrees 25 degree bevel: Material removed for straight bevel 482, Material removed for camber bevel 112. Total = 594 cubic mm
5. BD bedded at 45 degrees 33 degree bevel: Material removed for straight bevel 346, Material removed for camber bevel 112. Total = 458 cubic mm
6. BD bedded at 45 degrees 48 degree bevel: Material removed for straight bevel 202, Material removed for camber bevel 112. Total = 314 cubic mm

7. BD bedded at 50 degrees 25 degree bevel: Material removed for straight bevel 482, Material removed for camber bevel 103. Total = 585 cubic mm
8. BD bedded at 50 degrees 33 degree bevel: Material removed for straight bevel 346, Material removed for camber bevel 103. Total = 449 cubic mm
9. BD bedded at 50 degrees 48 degree bevel: Material removed for straight bevel 202, Material removed for camber bevel 103. Total = 305 cubic mm

10. BD bedded at 55 degrees 25 degree bevel: Material removed for straight bevel 482, Material removed for camber bevel 97. Total = 579 cubic mm
11. BD bedded at 55 degrees 33 degree bevel: Material removed for straight bevel 346, Material removed for camber bevel 97. Total = 443 cubic mm
12. BD bedded at 55 degrees 48 degree bevel: Material removed for straight bevel 202, Material removed for camber bevel 97. Total = 299 cubic mm

13. BD bedded at 60 degrees 25 degree bevel: Material removed for straight bevel 482, Material removed for camber bevel 91. Total = 573 cubic mm
14. BD bedded at 60 degrees 33 degree bevel: Material removed for straight bevel 346, Material removed for camber bevel 91. Total = 437 cubic mm
15. BD bedded at 60 degrees 48 degree bevel: Material removed for straight bevel 202, Material removed for camber bevel 91. Total = 293 cubic mm