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# Time to expand the Cosine (theta) model

OK, most of us are familiar with the approximate equation for turn radius on the snow as a function of tipping angle and side-cut radius, On snow radius = side-cut radius times cosine (tipping angle).   We all know that there is bend in the ski, give in the snow, and that it requires a hard surface for the ski to ride against as it cuts its groove and due to all these factors the equation is only approximate (indeed the best fit circle to the ski often is quoted as two, one for the front and one for the back of the ski).  Still it is a pretty good approximation for hard-packed groomers.

The approximate equation does a wonderful job of showing how longer radius skis are required to produce high g turns that are clean arc-2-arc turns at speed, but it breaks down at higher tipping angles.  The problem is that at fun speeds and tipping angles (say greater than about 60 degrees from the horizontal ) the skis stiffness prevents that radius from being attained.

Any ideas on how to modify this basic equation (which is the equation of intersection between a cylinder and an ellipse), to that range of real world experience?

Inquiring minds want to know.

A little thought experiment:

We require a given tipping angle at a given speed in order to produce a given g-force turn.  If we tip the skis to that tipping angle we will according to the equation dial up a turn radius.  If that radius is bigger than the turn we are trying to make it is easy to counter-balance/angulate and tip the skis more to make the turn.   If that radius is smaller than the turn we are trying to make, the ski won't hold because it will not be at the critical angle for the smaller turn and there is nothing we can do about it, except try and tip past the critical angle (which may make things worse or not).

Example:  35 m sidecut gs ski, tipped at an angle equal to the critical angle required to Hold against forces producing a given g-force for speeds of 20, 25, and 40 mph, and the g force that would be produced by that turn.  Red numbers indicate that the turn being dialed up would require a bigger tipping angle to hold on, but increasing that angle would result in a tighter turn with greater g-force.  As can be seen this may be helpful (eg- see 25 mph column), or not.

I would like to have a better equation though, as experience indicates something is off at the higher tipping angles  (Maybe an error in one of the equations in the spreadsheet!).  Any nerds want to correct it, go ahead

 gs resisted at this angle Tipping Angle 35 m side-cut radius Radians Degrees Turn Radius at tipping angle gs produced by turn 20 25 40 0.25 0.245 14.0 34.0 0.23998283 0.37497318 0.95993134 0.5 0.464 26.6 31.3 0.26029793 0.40671552 1.04119172 0.75 0.644 36.9 28.0 0.29102193 0.45472177 1.16408773 1 0.785 45.0 24.7 0.32925373 0.51445896 1.31701493 1.25 0.896 51.3 21.9 0.37268992 0.582328 1.49075968 1.5 0.983 56.3 19.4 0.4197178 0.65580906 1.6788712 1.75 1.052 60.3 17.4 0.46925877 0.73321682 1.87703507 2 1.107 63.4 15.7 0.52059586 0.81343103 2.08238344 2.25 1.153 66.0 14.2 0.57324673 0.89569801 2.29298691 2.5 1.190 68.2 13.0 0.62688043 0.97950067 2.50752172 2.75 1.222 70.0 12.0 0.68126488 1.06447637 2.72505952
This is what happens when summer lasts too long. God help us if it doesn't snow soon.

BK

Hope that helps.

I'm actually tempted to figure it out. I am pretty sure I did it once but I can't find what I wrote.
Quote:
Originally Posted by mdf

I'm actually tempted to figure it out. I am pretty sure I did it once but I can't find what I wrote.
You should just go find something to drink
It's probably better for you anyways

Or ....
Grab a flight to Colorado cause they seem to have lots of snow to keep your mind occupied
Quote:
Originally Posted by Ghost

We all know that there is bend in the ski, give in the snow, and that it requires a hard surface for the ski to ride against as it cuts its groove and due to all these factors the equation is only approximate

You can stop here.

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