# Bending Math's

(Hans Gijsen 2009 - now) A mathematical approach to skate bending, for an illustrated explanation
see page Bending.

# How much?

Imagine a speedskates blade as a segment of 40cm from a large circular plate of steel with a thickness of 1,1mm and a radius
of 23m. (8 floors) 2cm Of the outer border of that segment resembles our blade.

On page Bending. you'll find how to make a cone of such a circle.

In our imagination this time, otherwise we'll need a soccer stadium.

We make a cone with a 140 degrees top-angle out of the circular plate.

To accomplish this the ends of the cut in plate should overlap for 22 degrees. See Bending. again.
The cones base-circle has got a radius of 21,6m.

23cos20 = 23*0,94 = 21,6 m

We have accomplished that our blade keeps its rocker of 23m when
perpendicular to the bottom.

The rocker is the horizontal projection of the cones base-circle on the perpendicular.

Blade 90 degrees relative to the ice.

Cone base 70 degrees relative to the ice.

Cone base 20 degrees relative to the perpendicular.

21,6m / cos20 = 23 m the same as the original rocker in the flat circular plate.

The optimal properties for minimum friction on the straight-end aren't changed.

It should be noticed however, that the blade perpendular to the ice, doesn't go straight ahead any longer.
The base-circle of the cone makes an angle of 70 degrees with the ice,
so the blade wil make a self chosen circular turn with a radius of 63,1m.

The vertical projection of the cones base-circle on the ice.

21.6 / cos70 = 63,1 m

If we really want the blade to go straight ahead, the cones base must be perpendicular on the ice.

The angle of the blade itself is 110 degrees then. Performing your skating skills you wil notice that.
But, you'll get a lot in return.

# Behavior in the turn

Forced by its angle to the ice, our skates blade makes a self chosen circular turn.

We compare the bended blade -Cone model-

with the straight blade -Circular plate model-

and give the radius of the corresponding turn.

Table: a = inner angle skate relative to the ice.

a = 90 degrees -bended- 63,1 m -straight- straight ahead

a = 80 degrees -bended- 43,2 m -straight- 132,5 m

a = 70 degrees -bended- 33,6 m -straight- 67,2 m

a = 60 degrees -bended- 28,2 m -straight- 46 m

a = 50 degrees -bended- 24,9 m -straight- 35,8 m

a = 40 degrees -bended- 22,9 m -straight- 30 m

a = 30 degrees -bended- 21,9 m -straight- 26,5 m

a = 20 degrees -bended- 21,6 m -straight- 24,5 m

In our model the cone's base always makes a 20 degrees smaller angle to the ice than the skate itself.

So if the blade reaches an angle of 20 degrees, the base wil rest entirely on the ice.
(See page Bending, final picture)
The blade follows the cone's 21,6m base circle, and touches the ice from front to rear.
Any smaller angle then 20 degrees wil force the front and rear end of the blade in the ice,
the centerpiece is lifted off forming a "bridge". Rather not!

Mind, in the table, the angles between 70 en 30 degrees.

These are important for speedskaters.

The speed in the turn determines the angle of the skaters body to the ice. This wil be about 45 degrees
for a fast skater. (500m in 38sec.)

The blades wil then make an angle between 70 degrees just after touching the ice
and 30 degrees at the end of the push-off, with a corresonding turn between 33,6 and 21,9m.

Averaged this is about 25m corresponding wel with the inner turn of a 400m track.
Compare this to the turns of the straight blade, going from 67,2 to 26,5m. With an average of 35m.
These won't follow the inner turn, and give the skater a feeling of drifting out.

# Actual Bending

Now the angle of 90 degrees from the table becomes important.

Here the perpendicular standing cone-segment is projected on the ice.
(cone-base 70 degrees)

This projection, in this case a circle with a radius of 63,1m comes very close to the shape we should bend our blade to.

Mathematically it's a conic-section perpendicular on the cones surface, a parabola.

But our conic-section resides very near the cones base,
and the segment is very small compared to the circumference of the base circle.
This makes the circle approximation quite accurate.

With a gauge of 80mm you'll get a reading of -12,5um (anti clockwise)

which should be equal allover the blade.

To consider: New blades, the best you can get,

often give readings between -20 en +20um for a single blade!

# Bending speedskates is straightening with a small

# preoccupation anti-clockwise

And a fantastic result.