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# Sidecut vs. Radius and CM-Angle Calculations

A recent thread in the Racing Forum about the FIS changing sidecut standards (again) caused me to develop calculations comparing the turning performance of the old vs. new sidecut standards.

The change of most interest to me was from 27 meters to 40 meters of sidecut - just about back to that of classic skis. Turns out this seemingly huge change didn't actually have all that much skier-positioning impact at typical racing speeds due to the very high edge-angles and CM inclinations already found in that realm. At lower speeds though, such a change might impact the typical recreational skier a great deal.

Unable to leave well enough alone, I decided to create several spreadsheets calculating the impact of sidecut changes on CM positioning (for lateral balance) at a variety of speeds. Figuring all these things out for multiple speeds resulted in a complex set of spreadsheets but the results were interesting enough to trigger a rewrite consolidating everything into a single workbook and hiding all that complexity behind a simple form-based UI.

The top section has a number of fields for data entry. The bottom section (Chart) updates immediately when fields above are changed. Specific information on using the sheet is listed below this image.

You can enter values in most fields at the top. Data entered is validated to make sure weird data doesn't cause problems. I've also established some practical Minimum and Maximum values for each field for the same reason.

Sidecut for Ski #1
- Enter in meters or feet (the other UM field will adjust itself)

Sidecut for Ski #2
- Enter in meters or feet (the other UM field will adjust itself)

Linear Speed (how fast the skier is traveling)
- Enter in kph, mph or feet per second (the other UM fields will adjust themselves)

Max Skier's CM-Angle (limit on CM inclination)
- This setting establishes the maximum CM inclination considered possible for the skier. The CM-Angle is NOT the angle of the legs off vertical. Remember, when angulated a skier's CM is further away from the surface - and may be well above the belly button at high angulation angles.

Max Ski Edge Angle (limit on ski-tipping angle)
- This setting establishes the maximum tipping angle of the ski considered possible or practical. Beyond this angle boot-out will likely occur. Edge-grip also fails when a ski starts riding on its sidewall rather than it's edge. Consider a practical limitation for this setting.

Note: Both of these settings are also useful when set artificially lower to represent the angles a beginner, intermediate or advanced skier are likely to achieve. How much do you really tip yourself (or your skis) over? Set those angles and see what your minimum carved turn radius will really be on those "13.5 Meter" skis!

Locked fields for "Calculated Minimum Radius Possible"
- Lined up with each Ski's Sidecut, these are the fields that report the actual Minimum Turn Radius a skier is likely to achieve at the specified Speed for the specified Sidecut with respect to the limitations of Max CM Angle and Max Edge-Angle you set above.
- These fields report the smallest Turn Radius possible before a skier is stymied by either the Max CM-Angle or the Max Edge-Angle - whichever causes problems first.

The Orange and Blue fields go with the Orange and Blue graph lines. These show the diminishing radius (left scale) against the increasing Edge-Angle of the ski against a "flat" surface (bottom scale).

The Green and Light-Green graph lines represent the increasing CM inclination angle required (right scale) for the given Speed at the Turn-Radius delivered by the increasing Ski Edge-Angle (bottom scale again).

Assumptions
These calculations assume a number of situational characteristics as follows:

1) Evenly weighted contact along the entire ski's edge. No provision is made for 'levering forward' or 'levering backward' to modify expected turn radius.

2) No provision exists for uneven pressure distribution due to ski flex in soft or semi-firm snow. In reality, the midsection of a ski nearly always presses deeper into the snow than the tip thereby bending the ski more and decreasing turn radius from that normally predicted. The softer the ski, the greater the impact.

3) No provision is made for rotary induced Steering of the ski. (though that would be an interesting characteristic to include in the model)

4) All calculations were done against a 'flat' surface. In the real world we ski on uneven terrain, but in racing there is a specific notable issue - trenches (aka: Ruts). A racer turning in concert with a Rut is skiing against a curved and banked surface - which greatly alters the equations. Something could be modeled if Rut Characteristics were specified but we'll save that for later. Perhaps much later...

Let me know what you think.

.ma

PS: Can't be sure I've worked out all the bugs yet so consider this beta testing.
Hmmm, bug #1 seems to be the image Z-Order for that text frame holding the calculated minimum radii fields. Clicking on things in a newer version of excel sometimes hides the data fields, or hides the text of the label. Seems to be a sizing thing.

Lemme know of anything else that pops up. Would like to do cleanup all at once.

.ma

I won't even pretend to understand the math, however, one thing I learned  reading the Federolf Study  http://e-collection.library.ethz.ch/eserv/eth:28070/eth-28070-02.pdf  is that once your edge angle gets past 40 degrees, the math changes greatly, especially if you are on soft snow.  Your chart looks too linear at the higher edge angles.  Not sure, just thought I'd mention it.

Ken

L&AirC,

If you look at Figure 84 on page 106 of that analysis you'll seem my radius graph is similar to that produced by Howe's non-penetrating results. As mentioned in my Assumption #2 (above) no effort was made in the spreadsheet to model results based on surface penetration due to pressure distribution.

I also didn't model a two-footed skier which changes things greatly. In fact, I think I see some empirical results cited in that paper you link which are probably the result of two-footedness. In particular, a typical skier has weight on the inside-ski which changes the angle of pressure applied to the outside-ski. because this skier isn't actually balanced (perfectly) on that outside-edge (as my own calculations model) the outside-ski likely ends up with a larger Gravity component on the outside-ski, increasing pressure on that ski's sidewall but decreasing pressure against the base.

I'd have to think about it for a while to estimate probable impacts but I suspect any real-world measurements taken will be consistently skewed by two-footedness unless the mathematical model incorporates pressure redistribution brought about by that other foot and ski. In fact, I'd say the impact of two-footed distribution is probably what skews the results above 45 degrees since that's where base-to-sidewall pressure distribution can begin to change greatly.

Also, my spreadsheet is only intended to create a general comparison of sidecut performance, not to perfectly model exactly what occurs in this exceptionally indefinable world...

---

1) When a ski is 'flat' to a flat surface and perfectly balanced side-to-side (neither edge penetrating the surface) then it would have an infinite turn radius and be going straight. This is really difficult to show on a chart as it wastes a lot of white space on the scale that must include infinity. As any ski untips from its last 1-degree of edge-angle down to 0-degrees of edge-angle the turn radius rapidly grows to infinity. Rather than mess with clever solutions for infinity I simply 'accepted' the sidecut of the ski as the maximum turn radius.

2) As just described, Two-Footed weight distribution and the resulting pressure distribution effects on turn radius are not considered in the calculations.

3) In all calculations, the ski's base-bevel is considered to be 'flat' or 0-degrees of base-bevel. Depending on surface penetration depth(s), base-bevel can have a variable impact on turn radius.

.ma

Interesting calculations Michael, but I think you have neglected a very important factor.

The thing is that at a certain speed you can only incline a certain amount before you fall to the inside.

To achieve equilibrium the gravity force must be balanced by the centripetal force. The centripetal force in turn depends on the radius which in turn depends on the edge angle. You have vicious circle for the larger radius ski. The 40 m ski causes a larger turning radius, which means you cannot incline as much, which means the turning radius is even larger because of smaller edge angles etc.

For example if you have a speed of 15 m/s (54 km/h) you can incline 58 degrees with the 27 ski, and this gives you a theoretical turning radius of about 14 meter. The 40 meter ski however, can only be tipped to 35 degrees to get equilibrium, and this gives a turning radius of 33 meters. This is indeed a HUGE difference, even larger than the 40/27 ratio.

So to turn more with the 40 meter ski you have to add angulation. If you add 10 degrees of angulation you get the 40 meter ski down to 27 meters.

I suspect it is extremely difficult to add enough angulation to bring it down to 14 meters.

I think we will see a lot more non-carved turns. Maybe some of the older skiers which started out on straighter skis have an advantage now.

Jamt,

I'm aware of the dynamic feedback loop you mention. The original spreadsheet (in the racing thread) made no effort to consider CM angles at all. This one includes calculations to identify the CM-Angle necessary for (static) lateral balance but makes no attempt to calculate Angulation requirements let alone validate the reasonableness of those requirements.
.
Speed = 54 kph (or 15 meter/sec)

Sidecut 27 ski: 58 degree edge-angle required for a 14.31 meter radius turn forcing 58.07 degrees of CM inclination )
Sidecut 40 ski: 69 degree edge-angle required for a 14.33 meter radius turn forcing 58.07 degrees of CM inclination )

The question is how does our skier gets those Sidecut-40 ski edges to a 69-degree tipping angle? This is where another calculation would be needed, one that requires specific knowledge of the skier's own characteristics.

Things like...
- How tall are they?
- Are they long-bodied with short legs or short-bodied with long legs?
- How massive is their upper-body vs. their lower-body?
- How exactly does this skier angulate? (knee-angulation? Hips? Waist? Head?)
- Where are the skier's arms during this time?
- How much do they rotate the upper-body to face outsude the turn? (also shifts relative mass)

The only way to properly determine actual angulation possibilities would be to quantify exactly where each component of the skier's mass is located while executing the given radius. And even this isn't the final determinant since it assumes a desire for an ongoing state of lateral balance.

In reality a skier is always somewhat out of lateral balance in the 'dynamic feedback loop' described above. In a typical turn a skier intentionally violates lateral balance requirements knowing they'll be tossed out of that radius of turn in a moment because their CM isn't properly positioned to retain that specific turn radius. In this case the skier isn't trying for ongoing lateral balance - they're aiming for a temporary state of planned imbalance to achieve a future intent. For instance, near Apex a skier wants a state of imbalance that will propel their CM out of the current turn to where it needs to be for the next turn.

Looking at it this way we might detect the "safety concerns" cited by FIS and discover a legitimate and definable basis after all since a sharper sidecut can generate a good deal more "unbalanced force" driving the skier more quickly and forcefully out of the current turn.

My spreadsheet isn't intended for that much depth. A really good analysis would include not only the calculated states of static balance (positions vs speed and radius) but also take intentionally unbalanced lateral forces into consideration during each phase of a turn at likely speeds. ( And if you'd like to put those calculations in I'm happy to send you an unprotected version of the sheet... )

So no, I didn't neglect the scenario you describe. I just didn't put my time into the calculation effort. Instead, I sidestepped the issue by providing the ability to enter "limiting factors" for both CM-Angle and Edge-Angle...

.ma

I does not really matter how you angulate, the only important thing is the relative angle of the CM and the ski bases.

I also agree with you that you have to take the dynamics into account to get the whole truth, but the equilibrium angles still give some indication.

In order to calculate how much angulation you need for 69 degrees you must solve the equation that balances the centripetal moment against the CM moment.

The funny thing is that with enough angulation angle this equation has two solutions. For low inclination angles the centripetal force will be the strongest and have an uprighting moment on the body, then for higher angles the moment caused by the CM will take over and it will tip the body over. However, Here comes the interesting thing becuase as the angle continues to increase the radius will drop quickly due to its cosine nature. This will cause a rapid increase in the centripetal force and cause the body to be righted again.

Of course this is all in theory, at very high angles the ski is physically limited by its bending properties, but IMO it is still an interesting observation that explains what happens for normal edge angles (<70)

From a dynamic skiing perspective this means that as long as you have enough angulation you can lounge the body into the turn and rely on the forces later in the turn to right you again.

This does not happen if you only incline. It explains why it is much more stable to use angulation than inclination, and why you can achieve very high edge angles with angulation.

Maybe it explains the incline early, angulate through the rest of the turn idea Jamt.

Good point jasp
I don't understand the rule and how it will be measured. Will the sidecut go from 36 mm to 24 mm? Does the sidecut have to be a smooth arc or are only minimum and maximum widths measured? Does the ski designer have any leeway to play with?
Eric
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