Blade Edge Parameters: ROH And All That Jazz (For Gnurds)

[tstop4me]

Pt. 1.  Reference Model.

In a recent thread concerning dance blades, the issue of depth of hollow as a function of blade thickness came up.  Rather than responding there, I've decided to launch a new thread devoted to the discussion of blade edge parameters.  The attached Fig. 1 (you can click on it to enlarge it) shows the blade edge parameters I will be discussing.  The figure shows a reference model for a parallel edge geometry.  Actual blade geometry will deviate from the reference model.  In particular, the reference model assumes the blade is straight, the edges are parallel, the blade thickness is constant, the hollow is a section of a cylindrical surface, and the hollow is centered with respect to the inside and outside blade edges.   The drawing is not to scale; in particular, the hollow has been exaggerated to make it more visible.

The key blade edge parameters in this discussion are the blade thickness [t], radius of hollow (ROH) [r], depth of hollow (DOH) [d], blade edge included angle [α], and blade edge bite angle [β].  The blade thickness can be measured with a micrometer or caliper; the depth of hollow can be measured with a hollow depth indicator gauge.  The radius of hollow is primarily set by the radius of the grinding wheel (or other grinding tool); although, in practice,  the actual radius of hollow ground into the blade will deviate from the radius of the grinding wheel.

The relevant equations are given in (E1) – (E4).  Equation (E1) allows you to calculate the depth of hollow if the values of radius of hollow and blade thickness are known.  Equation (E2) allows you to calculate the radius of hollow if the values of depth of hollow and blade thickness are known.  Equations (E3) and (E4) allow you to calculate the blade edge angles if the values of radius of hollow and blade thickness are known.  Most discussions of blade edge angles I've read focus on the bite angle.  I'm not sure why; I think logically the discussion should focus on the included angle.  I will defer discussion of blade edge angles to a follow-up post.  Since the blade edge angles are related by the equation [included angle + bite angle = 90 deg], however, if one blade edge angle is known, the other blade edge angle is also known.  Note that in an actual blade, the blade edge angles are more complex since the outside surfaces of the blade edges are usually hand finished with a honing stone to remove burrs:  to avoid scratching the polished faces, the honing stone is typically held at a slight angle to the polished faces.

I will post tables showing values of depth of hollow and blade edge angles for a range of blade thickness and radius of hollow.

[tstop4me]

Pt. 2.  Depth of Hollow.

Shown below are values of depth of hollow as a function of blade thickness and radius of hollow.  The same values are presented in two tables.

In Table 1, the blade thickness is fixed, and the depth of hollow is shown as a function of the radius of hollow.  This table will be of general interest to most skaters who want to check out what their skate sharpener is doing.   For a fixed blade thickness, the depth of hollow increases as the radius of hollow decreases.  But note the relatively small variation in the depth of hollow as a function of the radius of hollow, especially for thin blades.  If you are checking your radius of hollow by measuring the blade thickness and the depth of hollow, you need high accuracy.

In Table 2, the radius of hollow is fixed, and the depth of hollow is shown as a function of blade thickness.  This table will be of interest to skaters with different types of blades; for example, freestyle, dance, and figures.  For a fixed radius of hollow, the depth of hollow decreases as the blade thickness decreases.  Note the relatively small depth of hollow for thin blades.


Table 1.  Depth of Hollow as a Function of Radius of Hollow for Fixed Blade Width.

t:  blade thickness (decimal inch)
r:  radius of hollow (fractional inch)
d:  depth of hollow (decimal inch)

__t______r______d__

0.100     5/16   0.0040
0.100     3/8     0.0033
0.100     7/16   0.0029
0.100     1/2     0.0025
0.100     9/16   0.0022
0.100     5/8     0.0020
0.100     3/4     0.0017
0.100   1          0.0013
      
0.125     5/16   0.0063
0.125     3/8     0.0052
0.125     7/16   0.0045
0.125     1/2     0.0039
0.125     9/16   0.0035
0.125     5/8     0.0031
0.125     3/4     0.0026
0.125   1          0.0020
      
0.150     5/16   0.0091
0.150     3/8     0.0076
0.150     7/16   0.0065
0.150     1/2     0.0057
0.150     9/16   0.0050
0.150     5/8     0.0045
0.150     3/4     0.0038
0.150   1          0.0028
      
0.175     5/16   0.0125
0.175     3/8     0.0104
0.175     7/16   0.0088
0.175     1/2     0.0077
0.175     9/16   0.0068
0.175     5/8     0.0062
0.175     3/4     0.0051
0.175   1          0.0038
      
0.200     5/16   0.0164
0.200     3/8     0.0136
0.200     7/16   0.0116
0.200     1/2     0.0101
0.200     9/16   0.0090
0.200     5/8     0.0081
0.200     3/4     0.0067
0.200   1          0.0050


Table 2.  Depth of Hollow as a Function of Blade Width for Fixed Radius of Hollow.

r:  radius of hollow (fractional inch)
t:  blade thickness (decimal inch)
d:  depth of hollow (decimal inch)

__r_____t______d__

5/16   0.100   0.0040
5/16   0.125   0.0063
5/16   0.150   0.0091
5/16   0.175   0.0125
5/16   0.200   0.0164
      
  3/8   0.100   0.0033
  3/8   0.125   0.0052
  3/8   0.150   0.0076
  3/8   0.175   0.0104
  3/8   0.200   0.0136
      
7/16   0.100   0.0029
7/16   0.125   0.0045
7/16   0.150   0.0065
7/16   0.175   0.0088
7/16   0.200   0.0116
      
  1/2   0.100   0.0025
  1/2   0.125   0.0039
  1/2   0.150   0.0057
  1/2   0.175   0.0077
  1/2   0.200   0.0101
      
9/16   0.100   0.0022
9/16   0.125   0.0035
9/16   0.150   0.0050
9/16   0.175   0.0068
9/16   0.200   0.0090
      
  5/8   0.100   0.0020
  5/8   0.125   0.0031
  5/8   0.150   0.0045
  5/8   0.175   0.0062
  5/8   0.200   0.0081
      
  3/4   0.100   0.0017
  3/4   0.125   0.0026
  3/4   0.150   0.0038
  3/4   0.175   0.0051
  3/4   0.200   0.0067
      
     1   0.100   0.0013
     1   0.125   0.0020
     1   0.150   0.0028
     1   0.175   0.0038
     1   0.200   0.0050

[tstop4me]

Pt. 3.  Blade Edge Angles.

As I wrote in Pt. 1, most discussions of blade edge angles focus on the bite angle:  as the bite angle increases, the blade bites more deeply into the ice.  I believe it's more intuitive to focus on the included angle, as this corresponds to the cutting angle of a tool (such as a knife blade):  the smaller the included angle, the sharper the tool.  For a knife blade, the included angle is typically 40 deg or less, depending on the application.  For tools designed for chopping, such as a cleaver or axe, the included angle is typically larger, since a more robust edge is needed.  For a skate blade, the included angle is relatively large, on the order of 80 deg. Since the sum of the included angle and the bite angle is equal to 90 deg, the included angle decreases as the bite angle increases.  Hence, the scenario is consistent:  as the bite angle increases, the included angle decreases, and the tool (skate blade) becomes sharper and cuts deeper into the ice.

Shown below are values of the blade edge angles as a function of blade thickness and radius of hollow. 

In Table 3, the blade thickness is fixed, and the blade edge angles are shown as functions of the radius of hollow.  For a fixed blade thickness, as the radius of hollow decreases, the included angle decreases and the bite angle increases.

In Table 4, the radius of hollow is fixed, and the blade edge angles are shown as functions of the blade thickness.  For a fixed radius of hollow, as the blade thickness decreases, the included angle increases and the bite angle decreases; this is the basis for the recommendation to sharpen (thinner) dance blades to a smaller radius of hollow than (thicker) freestyle blades.  For example, consider a freestyle blade with a thickness t = 0.150 inch and a dance blade with a thickness t = 0.100 inch.  If they are both sharpened to a 7/16 inch radius of hollow, then the blade edge angles are:

included angle:  80.13 (t = 0.150 inch ); 83.44 deg (t = 0.100 inch)
bite angle:  9.87 (t = 0.150 inch ); 6.56 deg (t = 0.100 inch).

On the basis of blade edge angles, the dance blade is less sharp than the freestyle blade, for the same radius of hollow.

If you wish the dance blade to have approximately the same blade edge angles as the freestyle blade, you need a radius of hollow of approximately 5/16 inch:

included angle:  80.79 deg (t = 0.100 inch)
bite angle:  9.21 deg (t = 0.100 inch).



Table 3.  Blade Edge Angles as a Function of Radius of Hollow for Fixed Blade Width.

t:  blade thickness (decimal inch)
r:  radius of hollow (fractional inch)
α:  blade edge included angle (deg)
β:  blade edge bite angle (deg)

__t______r_____α_______β_

0.100     5/16   80.79       9.21
0.100     3/8     82.34       7.66
0.100     7/16   83.44       6.56
0.100     1/2     84.26       5.74
0.100     9/16   84.90       5.10
0.100     5/8     85.41       4.59
0.100     3/4     86.18       3.82
0.100   1          87.13       2.87
         
0.125     5/16   78.46     11.54
0.125     3/8     80.41       9.59
0.125     7/16   81.79       8.21
0.125     1/2     82.82       7.18
0.125     9/16   83.62       6.38
0.125     5/8     84.26       5.74
0.125   3/4       85.22       4.78
0.125   1          86.42       3.58
         
0.150     5/16   76.11     13.89
0.150     3/8     78.46     11.54
0.150     7/16   80.13       9.87
0.150     1/2     81.37       8.63
0.150     9/16   82.34       7.66
0.150     5/8     83.11       6.89
0.150     3/4     84.26       5.74
0.150   1          85.70       4.30
         
0.175     5/16   73.74     16.26
0.175     3/8     76.51     13.49
0.175     7/16   78.46     11.54
0.175     1/2     79.92     10.08
0.175     9/16   81.05       8.95
0.175     5/8     81.95       8.05
0.175     3/4     83.30       6.70
0.175   1          84.98       5.02
         
0.200     5/16   71.34     18.66
0.200     3/8     74.53     15.47
0.200     7/16   76.79     13.21
0.200     1/2     78.46     11.54
0.200     9/16   79.76     10.24
0.200     5/8     80.79       9.21
0.200     3/4     82.34       7.66
0.200   1          84.26       5.74 




Table 4.  Blade Edge Angles as a Function of Blade Width for Fixed Radius of Hollow.

r:  radius of hollow (fractional inch)
t:  blade thickness (decimal inch)
α:  blade edge included angle (deg)
β:  blade edge bite angle (deg)

__r______t_____α_____β__

  5/16   0.100   80.79     9.21
  5/16   0.125   78.46   11.54
  5/16   0.150   76.11   13.89
  5/16   0.175   73.74   16.26
  5/16   0.200   71.34   18.66
         
  3/8     0.100   82.34     7.66
  3/8     0.125   80.41     9.59
  3/8     0.150   78.46   11.54
  3/8     0.175   76.51   13.49
  3/8     0.200   74.53   15.47
         
  7/16   0.100   83.44     6.56
  7/16   0.125   81.79     8.21
  7/16   0.150   80.13     9.87
  7/16   0.175   78.46   11.54
  7/16   0.200   76.79   13.21
         
  1/2     0.100   84.26     5.74
  1/2     0.125   82.82     7.18
  1/2     0.150   81.37     8.63
  1/2     0.175   79.92   10.08
  1/2     0.200   78.46   11.54
         
  9/16   0.100   84.90     5.10
  9/16   0.125   83.62     6.38
  9/16   0.150   82.34     7.66
  9/16   0.175   81.05     8.95
  9/16   0.200   79.76   10.24
         
  5/8     0.100   85.41     4.59
  5/8     0.125   84.26     5.74
  5/8     0.150   83.11     6.89
  5/8     0.175   81.95     8.05
  5/8     0.200   80.79     9.21
         
  3/4     0.100   86.18     3.82
  3/4     0.125   85.22     4.78
  3/4     0.150   84.26     5.74
  3/4     0.175   83.30     6.70
  3/4     0.200   82.34     7.66
         
1          0.100   87.13     2.87
1          0.125   86.42     3.58
1          0.150   85.70     4.30
1          0.175   84.98     5.02
1          0.200   84.26     5.74

[Query]

You could include side-honing - for vertically side honed blades, the sides of the blade should not show as vertical in your diagram - e.g., when the bottom of the blade is bevelled to be wider than the area just above it, the included angle (which is often called the "edge angle" in knife and razor blade discussions) will be less, making the blade more sharp. This is done on a number of high end blades.

In any event, I remain amazed at how such a small deviation in edge angle from 90 degrees can make such a large difference.

I have wondered whether a large part of what actually happens is that the tilt of the surfaces affects how rapidly water is pushed out of the way - and thereby affects the thickness of the water layer available for hydroplaning.

Another issue is that the shapes shown in your diagram, and the one I cited, are not really there on a small scale. If you look at edges under a microscope with a magnification of about 100, you will see that

(1) Depending on how the sharpener works (e.g., coarseness of sharpening wheel, whether and how he or she deburrs), there are often foil edges - very thin almost planar sheets that can extend vertically on the surface - on the sides of the blade.

(2) Even if there is no foil edge, the edge is quite ragged. At most points along the edge, it doesn't look all that much like the intersection of the hollow curve with the side of the blade. I'm not all that sure how this affects the way blades interact with the ice and surface water layers. Given that the depth of the surface water layer has been estimated to be around 40 - 60 nm at ice rink temperatures, very small variation and roughness might be significant. I would love to see blade edges magnified sufficiently for that scale to be visible - like the razor blades shown in these electron micrographs.