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# 4 Questions - Page 7

You mean like I said here?

Quote:
Originally Posted by borntoski683

Yes, correct, that is where the ski is able to effectively redirect the CoM.  The tip and tail are what influence how the ski itself is redirected on the snow.

Try to keep up

BTS,

you could also double eject from your bindings -- that'll "cut the string".

What is important is not that the skier can "cut the string" just that the skiers CM is trying to move on the tangent to the curve were it not for the deflecting force from the ski.

That tangent is the reference direction for steering angle.

Using the idea of a pure carve, we can assume that there is no skidding, so that the steering angle is totally responsible for all direction changes.  Nothing is lost in skidding.

That's what makes the approximation of the steering angle I offerred work.

edit.  In reality, it would be independent of velocity -- perhaps the notion of "curvature" might help. I don't know.
Quote:
Originally Posted by BigE
That tangent is the defining direction for steering angle.

yes I think so.
BTS,
You are right, I usually do ski smoothly and if I plan ahead I gradually reduce the radius so that it approaches a straight line gradually.  However if I were to suddenly untip my skis, as I have done on occassion,  the skis would straighten, steering angle at the tip and tail would become very close to zero there too,  just like the steering angle at the midsection of the ski is 0 (infinitesmally small if you prefer).

FACT: C of M momentum and skier velocity are tangential to the arc.
FACT: Skis under the boot are tangential to the arc.
No Equations needed.

There is a big difference between jamming an edge down at an angle in front of you, and arcing a turn.  The ski underfoot in the first method pushes at an angle to your direction of travel, in the second method the ski under foot pushes you straight sideways to your direction of travel.  That's why the second is so efficient.
Quote:
Originally Posted by Ghost

BTS,
You are right, I usually do ski smoothly and if I plan ahead I gradually reduce the radius so that it approaches a straight line gradually.  However if I were to suddenly untip my skis, as I have done on occassion,  the skis would straighten, steering angle at the tip and tail would become very close to zero there too,  just like the steering angle at the midsection of the ski is 0 (infinitesmally small if you prefer).

FACT: C of M momentum and skier velocity are tangential to the arc.
FACT: Skis under the boot are tangential to the arc.
No Equations needed.

There is a big difference between jamming an edge down at an angle in front of you, and arcing a turn.  The ski underfoot in the first method pushes at an angle to your direction of travel, in the second method the ski under foot pushes you straight sideways to your direction of travel.  That's why the second is so efficient.

two problems:

1) zero is not infinitesimally small.
2) The part that does the turning cannot also be pointing in the same direction as the CM.
Quote:
Originally Posted by BigE

two problems:

1) zero is not infinitesimally small.
2) The part that does the turning cannot also be pointing in the same direction as the CM.
Which of these statements do you disagree with?
1) the portion of the skis under the boots are pointed tangential to the arc.
2) the CM momentum is tangential to the arc.
3) the part of the ski that is under the boot is supplying a lot of the turning force
4) the turning force is applied in a direction that is 90 degrees to the edge.
Quote:
Originally Posted by Ghost
There is a big difference between jamming an edge down at an angle in front of you, and arcing a turn.  The ski underfoot in the first method pushes at an angle to your direction of travel, in the second method the ski under foot pushes you straight sideways to your direction of travel.  That's why the second is so efficient.

You lost me there, jamming down in front of you vs arcing?  "pushes you straight sideways"?  Huh?

The principle of an arcing ski or scarving ski are exactly the same.  Steering angle provides the change of direction, the only difference is the efficiency.  An arcing ski is actually using steering angle very efficiently.

Quote:
Originally Posted by BigE
1) zero is not infinitesimally small.

Indeed!.  That is the point.  It is not zero.  There is no turning without it.

Quote:

2) The part that does the turning cannot also be pointing in the same direction as the CM.

Exactly.
I would like to continue the discussion back to what happens when we increase steering angle.

In my world, you can increase steering angle by bending the ski more, by moving your weight forward a bit, by tipping more and lastly by actually pivoting the tail out a bit.

In all but the last thing, its possible to increase the steering angle in such a way that arc purity is not compromised so much, with the turn radius becoming tighter and tighter as steering angle is increased.  I personally think that a lot of people who talk about "steering" their turns are really making use of a variety of skills to coax a bit more performance out of their ski by essentially doing just that.

Beyond a certain point however, the steering angle cannot be increased without also introducing some tail swing out, ie, skidding.  Perhaps this is where the idea of skid angle can be contemplated.

In some sense, if you want to seperate steering angle and skid angle, I think you would have the steering angle defined as has been done on this thread, which has to do with the angle of your ski is pointed compared to the direction of your CoM is going.  That angle would be facing somewhere between directly in front of the skier and directly to the outside of the skier.

Skid angle on the other hand has more to do, perhaps with the angle of the tail skidding, regardless of what the steering angle happens to be at the time.

When skidding happens, some speed is lost.  But also, when there is skidding, then even though the steering angle continues to increase as the skid angle fans out, at some point the steering angle of the edges starts to lose its effectiveness of redirecting the CoM in an increasingly tighter arc.

I am not sure right now whether that turn tightness effectiveness is lost the minute the tail starts to fan out, or whether there is a period where the gain in steering angle continues to tighten the radius more than the skidding takes away.  As there is more and more tail fanning, the steering angle becomes bigger and bigger and less effective in directing the CoM on a curved path until eventually it would just become a washed out side slip.  Somewhere along that continuum there is a point where the tail fanning will simply not get a tighter turn radius at some speed you're going.  You would have to slow down in order go tighter.
BTS,
Do you disagree with any of these statements about an edge-locked carved arc turn?
1) the portion of the skis under the boots are pointed tangential to the arc.
2) the CM momentum is tangential to the arc.
3) the part of the ski that is under the boot is supplying a lot of the turning force.
If so which one?
Ghost I already told you don't want to go to physics class with you.  I've said all I'm gonna say on it.
All right then.  Carry on.
I want to also comment a bit on what LeMaster had to say about starting turns with more steering angle.  He was basically talking about pivot entries.  Essentially he was saying you pivot into an exxagerated steering angle, then the ski starts to skid and feather scarve and redirect your CoM into the new direction.  As your CoM moves into the new direction the steering angle becomes less and less until eventually you end up finishing out the turn using more carving with very very small steering angle.

That is another way to think about steering angle also.  Some skidding happening there yes, but the INTENTION is not skidding, the intention is to turn sharper and go for more steering angle than the skis can support in a carve for the first part of the turn.

Whereas, in my view, skid angle is the intentional fanning out of the tail in order to brush off speed.
{this post may be out of sync with the current state of this rapidly expanding thread, so bear with me as I catch up...}

Quote:
Originally Posted by Rick
LeMaster talks not only about steering angles under foot, but also about steering angles within various parts of the ski, including both tail and forebody.  ...

...Interesting too, he has a term he uses for the steering angle created solely via the sidecut of the ski.  He calls it the "local steering angle".
Thanks Rick!   That's exactly what I would have expected from him and why I wanted to know what context he was in while discussing the matter.  I note the specific way it's presented - "...steering angles within various parts of the ski" and "local steering angle".

This suggests a specific kind of analytical examination to determine what is happening at precise points along the ski (as Ghost mentioned earlier, related to the calculus of integration) to discuss net outcomes

That kind of analysis is very useful figuring out exactly what's causing the skier to turn as a summation of forces over all local points in the system. It doesn't appear to have been intended to be taken as a global "Steering Angle" as seems to have happened.

---
Skidude,

I'm always happy to discuss these things too!   And FYI... I'd put 'substantial thought' into these ideas and developed analytical spreadsheets on it many years ago.  I'm glad we're able to indulge such detail now, as in the past I tended to get burned at the stake for trying 'go there' in these threads.

I'd say "Steering Angle" is certainly a concept related to how a turn comes about, however it's also a specific measurement (Steering Angle = Angle of Steering - a specific, measure angle).  If it were not meant to be interpreted as 'the angle measured' then there would be no point in calling it "Steering Angle" would there?

The analytical value of knowing the actual Steering Angle is that we better understand what contributes to 'turning' vs. 'braking' (as has been described by several people above).   It's also a comparative measure where actual numbers are not needed - we just need to know Bigger vs. Smaller Steering Angles to understand the relative contribution a given scenario has to turning and braking.

I disagree with your basis in point # 2, and your final statement provides the clue to explain why I think that definition of Steering Angle is meaningless: "..."regardless of the ski length, we will get the same turn."  We'd be far better off talking about the Radius or Arc of a given turn rather than trying to incorporate a fuzzy version of "Steering Angle" into the discussion which simply confuses the whole communication.

Your point # 3 appears to meander quite a bit bringing up partially related ideas largely  unconnected to what I'm talking about (and doesn't really support your concluding statement in that point).  Not sure how to respond to it.

---
Ghost,
I think you and I are probably thinking along the same geometric and mechanical lines. Considering LeMaster's use of "Local Steering Angle" at many points along the ski (as Rick pointed out) it seems he was probably thinking about things the same way.

I seriously doubt he wanted to establish the "The Angle of Travel at the Ski Tip" as any kind of overall metric since he too would have realized this is meaningless.  It's far more likely he was bringing it up to discuss some specific effect at that point on the ski.

Rick, since you've got the book... would you mind re-reading that section and examining that material in light of this potential interpretation for where he was going with the idea?  Does he appear to be defining a specific metric - or is he simply analyzing the nature of accumulating turning/braking forces at specific parts of the ski?

BTS,
In posts 161 and 163 you further this discussion greatly, thanks!  That quote furthers my belief that LeMaster was describing what Ghost and I have been discussing - accumulated net steering angle rather than a specific Steering Angle as defined by where the Tip alone is pointing.

On the issue of Momentum and steering angle - be aware that the Steering Angle itself is defined as a particular direction as measured away from Momentum.  We can (and do) have momentum downhill while any Steering Angle tends to direct us away from that current direction we would otherwise continue traveling.  We could show a very large Steering Angle at our very-flat skis while Momentum continues to carry us nearly straight downhill.  It's not about the Force contributed by that Angle (which is dependent on many factors), it's about that Angle itself.

BigE,
Thanks for the mathematical approach.  I get where you're coming from now, but it's a bit different than what I'm trying to get across.

Yes, it takes some force to redirect our path of travel away from where Momentum would carry us, and yes, Steering Angle contributes to that redirection in relation to the forces it can generate based on other factors.    As just mentioned to BTS above, it's not about the Forces, but about the directional metric.   The question being examined (at least by me) is what exactly is "Steering Angle" itself.

When you quote LeMaster with, "At every moment, the groove has an infinitesimal steering angle under the skiers foot..."  ... this adds further evidence that when carving we have a Steering Angle that is virtually 0.0 (or infinitesimal - perhaps I should have said 0.000001 degrees).

.ma
Quote:
Originally Posted by michaelA

Thanks Rick!   That's exactly what I would have expected from him and why I wanted to know what context he was in while discussing the matter.  I note the specific way it's presented - "...steering angles within various parts of the ski" and "local steering angle".

This suggests a specific kind of analytical examination to determine what is happening at precise points along the ski (as Ghost mentioned earlier, related to the calculus of integration) to discuss net outcomes

LeMaster doesn't really get into that much detail in his book.  I'm certain ski manufacturers do.  Mainly he says the following things

1. Most of the turning force on the skier is underfoot, the tip and tail don't have enough support to directly influence the direction the skier is going
2. Due to sidecut, the tip will have more steering angle then then tail, progressively so, this has the effect of pivoting the ski itself.
3. The local steering angle is what he refers to as around the tip, and he doesn't go into a lot of detail but generally I think the point is that when you first tip a ski on edge, the sidecut on the tip causes some local steering angle to cause the ski to pivot a tiny bit, which has the effect of increasing the steering angle under foot, etc.  Its like an initiator.
michaelA,

Infinitesimally small is not zero.  Zero steering angle means no steering. Virtrually zero is not really zero.
Quote:
Originally Posted by michaelA
On the issue of Momentum and steering angle - be aware that the Steering Angle itself is defined as a particular direction as measured away from Momentum.  We can (and do) have momentum downhill while any Steering Angle tends to direct us away from that current direction we would otherwise continue traveling.  We could show a very large Steering Angle at our very-flat skis while Momentum continues to carry us nearly straight downhill.  It's not about the Force contributed by that Angle (which is dependent on many factors), it's about that Angle itself.

This is exactly why many skiers have a hard time establishing high-C early arc-to-arc engagement in the top of the turn, because they have to generate their own steering angle and gravity is not helping their CoM to move that direction.
See, I did say this would be fun...

And - as luck would have it while searching for more supporting material for my take on the concept of Steering Angle ... I found that Google/Amazon has posted the exact material we've been discussion Right HERE. (You might have to scroll up a bit to start)      It's just a portion of the book but it covers his use of Steering Angle.  The section shown also talks about the Virtual Bump if anyone is interested in that.

Going over it myself, I find the following...

LeMaster's discussion of "Steering Angle" arises as part of his discussion of "the Ski's self-steering effect"  In that context he describes it in relation to a slightly edged ski and describes how the sidecut itself places different portions of the edge at differing angles against the snow along the length of the ski.  This, combined with the differing lengths of forebody and tail produce torque on the ski, which then causes it to "turn itself".

He goes on to discuss how as the ski is bent more, the Tip's relative Steering Angle against the snow in relation to the smaller relative Steering Angle of the Tail produce a turning force.   This is because the forebody of the ski is angled more across our direction of momentum (deflecting us more) while the tail (behind the center of rotation) becomes more in-line with our momentum (thus deflecting us less)

Much of his material describes Steering Angle at specific parts of the ski in relation to the skier's current momentum (using Current Momentum as a specific direction from which to compare and measure other angles).

On page 40, he starts to describe 'carving' and brings back the term Steering Angle to elaborate the idea, going on to state clearly:
"In a pure carved turn, the ski cuts a groove into the snow, which it then slides along much like the ball rolls along the side of the bowl.  At every moment, the groove has an infinitesimal Steering Angle under the skier's foot and so the turn is extremely efficient."

This is what BigE referred to, and goes back to the diagram I posted earlier showing no meaningful Steering Angle exists underfoot in a carved turn.  To me, infinitesimal means small enough to be "Nearly, but not quite Zero." ( I really should have used 0.00001 instead of 0.0 - just consider it a 'rounding error' ))

Probably the most definitive statement he makes on this is,
"The middle of the ski engages the snow at a Steering Angle to the skier's momentum, and the snow pushes back on the skier to turn or slow her.  In both cases, the force that the snow exerts on the skier has a turning component.  In the case of an ideal, perfectly carved turn, there is no slowing component."

To compliment this he follows with a paragraph about carving that says,
"This is the sort of turn depicted in figure 4.15a.  The bowls that correspond to the skis's natural carving radius match up perfectly, and the ball never encounters a Steering Angle larger than the infinitesimally small steering angle of the bowl's curve."
{NOTE: figure "4.15a" depicts three linked "bowls" about which a marble traverses the side of each - see it in the link above}

All of this would suggest he was not trying to establish the term "Steering Angle" to represent only the angle described by where the Tip of a ski happens to be pointing during a carved turn.  Unfortunately, that seems to be the interpretation many have taken from what he's written.

.ma
Well infinitesimally small is not zero.  And I don't think he was pointing that out in order to infer that its not significant.  The steering angle is what causes the ski to deflect and turn.

I think by calling it that he was trying to contrast it from bigger skidding moves.  He was trying to show that steering angle does not need to be a big skid.  it can actually be hidden in a carve and a casual observer may not even realize its happening, but its still there and still causing the ski to make a ski turn at some micro level....the ski is being deflected.  The closer you can get to that low steering angle, the more efficiently you are skiing in terms of achieving carvage.  However, he points out amply, that realistically we have to stray from that ideal by choice quite often and deal with larger steering angles.

Isn't that the point of this thread after all, how to handle a variety of steering angle scenarios?  This has launched us into a big debate about what steering angle actually is or whether it exists or whether there is snow on the moon.

pp22-26 are important too ma if you can get your hands on a real copy.  You seem like the kind of guy that would want this book in his library FWIW.
Quote:
Originally Posted by borntoski683

Well infinitesimally small is not zero.  And I don't think he was pointing that out in order to infer that its not significant.  The steering angle is what causes the ski to deflect and turn.

Re bold: it is not clear what is deflecting/turning....

The "steering angle of the ski" or "self steering angle" causes the ski to turn itself.

It is another "steering angle" between the direction that the ski underfoot is pointing and the inertial path of the CM that causes the skier to be turned/deflected.

I think much confusion revolves around the mashup of these two different definitions both being called "steering angle".

But what is intersting here are Skidude72's questions of skid angle vs steering angle.

Does skid angle actually bring anything to the table that steering angle does not?  Does steering angle help understand how the skier and skis work together to turn the skier?  Does skid angle help understand how the skier and skis work together to turn the skier?
Quote:
Originally Posted by Ghost

BTS,
Do you disagree with any of these statements about an edge-locked carved arc turn?
1) the portion of the skis under the boots are pointed tangential to the arc.
2) the CM momentum is tangential to the arc.
3) the part of the ski that is under the boot is supplying a lot of the turning force.
If so which one?

Great quality discussion.

Here are some more questions...that hopefully will lead to answers for the above questions...and the significance of the 0.00000 or 0.00001.

Q1:  Does the COM take the same path as the skis?
Q2:  Is the COM in the fall-line at the exact same time as the ski under foot?
Q3:  If the answer in 2 is no...what gets there first...or put another way what should get their first?
Q4:  Following on from 3...what does that tell us about fore/aft balance?  Or is it not a factor?
Quote:
Originally Posted by BigE
I think much confusion revolves around the mashup of these two different definitions both being called "steering angle"...

..Does skid angle actually bring anything to the table that steering angle does not?

These two thoughts are exactly what caused me to explore the discussion tangents that I did.  I think the concept of Steering Angle was first divided into parts by Rick when he talked about them as two separate concepts: Steering Angle and Skid Angle.

While he (and others) talked about these two terms as being genuinely separate things, he later combined them as Steering/Skid Angle which I took to mean his earlier posts were more about differing perspectives for discussion rather than talking about numerically different metrics.  Thereafter, we saw some unsupported expansion in the artificial gulf between the two terms which is why I went down the exploratory path that I did.

If we use the terms to mean the same thing then everything works OK. If we try to define them as being very different things then we need to propose some meaningful way to differentiate them.  So far, that hasn't happened.

---
You're right BTS, Skier's edge would be a welcome addition to my library.  Couldn't find any copies anywhere though.

.ma
I don't think Rick meant for them to be the same thing, but I probably should not speak for him.
Skidude,

Those questions you post have nothing at all to do with what's being discussed here, and have no bearing at all on 'Steering Angle' as a concept.

The motion of our CM and our Fore/Aft Balance have nothing to do with how we measure a large, small or infinitesimal Steering Angle.  Those things are related only to the outcome of a given Steering Angle in given snow conditions/terrain (and other factors).

We're maybe talking about Apples and Oranges here - but you've switched to talking about the Kenworth Truck that hauls them!

.ma
Just thinking out loud outside of the box for a minute.

There is no question in my mind that steering and skidding for speed control are two separate intentions.  Sometimes over steering can lead to skidding, wanted or not.  I'm not so sure the reverse can be said.  Even though they both involve increasing the so called "steering angle", one results in steering, with lots or little skidding, and the other may result in skidding without steering.  Somewhere, at some level, something different is happening.  Lots of overlap.  I think a lot of us are just used to combining the two.

Ok...just thinking out loud here....humor me for a sec...

In a turn, LeMaster says the front of the ski will have bigger steering angle then the back of the ski.  Makes sense.  Wait, so if the foot as it or near zero steering angle, what is the tail of the ski?  Less than zero?  Just askin'.  What about toes vs heels?

But in any case, let's just imagine that the steering effect is gotten from the area just in front of the boot and skidding is gotten from the area just behind the boot.  In any given turn where the ski is redirected out of a perfect track, that steering angle at either the edge of the ski just in front of the boot or the edge of the ski just behind the boot will be different.  The front would be more than the back.

Is that significant?  I dunno, just asking...

What if we want to increase the skidding without changing our steering angle per say.  Is that possible?

Is our pivot point around which the ski steers itself always located at exactly the same spot under our foot?  Or will it be different depending on whether our intention is to steer or smear?  Or both.
Quote:
Originally Posted by michaelA

Skidude,

Those questions you post have nothing at all to do with what's being discussed here, and have no bearing at all on 'Steering Angle' as a concept.

The motion of our CM and our Fore/Aft Balance have nothing to do with how we measure a large, small or infinitesimal Steering Angle.  Those things are related only to the outcome of a given Steering Angle in given snow conditions/terrain (and other factors).

We're maybe talking about Apples and Oranges here - but you've switched to talking about the Kenworth Truck that hauls them!

.ma

Interesting   but I do think they are very relevant.

At the moment there seems to be confusion associated with this idea of there being a very limited steering angle under the foot in a carved turn.  Why is that?

Well I think people are still missing a few points.

I'll try to explain with an example:

The steering angle from the ski tip to the foot in a pure carved turn is lets just say 10degrees.
In practice this means that when the ski travels on an arc, as the foot gets to where the tip was, the foot will have turned 10 degrees.
Again for simplicity lets say the tip is 100cm in front of the foot.  In practice that means after the foot travels the fist 10cm it turned one dgree...another dgree after the next ten, and so on until it travlled the full 100cm and turned the full 10 degrees.  This is what RL means when he referest to multiple steering angles along the skis length.  In practice of course it is irrelevant becuase they all add up to the total of the tip to the foot....

The take away from that is the steering angle, is a function of how far you measure it out from the foot...ie at 0.0001cm in front of the foot the steering angle is only 0.00001 degrees...but as the ski travels along its path...all those 0.00001 degrees every 0.0001cm add up fast!  They will add up to the total max steering angle from the tip.

Hence the point of my questions is....since the ski turns the skier...the ski only needs to have a 0.0001 steering angle at the foot...relative to the path of the COM

ie the path of the ski under foot only needs to deviate from the path of the COM by a small amount becuase the steering part fo the ski is jsut slightly ahead of the COM.

The point of the questions was to get people to step back for a second, think wholistically again...skis vs. COM, BOS vs COM....then dive back into the detail.

SkiDude72,

I get what you're saying but we went a bit beyond that a while back.   The problem as covered earlier arises when we try to use points along the curve as you are doing in what you just posted as example.  In your example, what would be the steering angle for a 8-foot radius turn as measured 20 feet from the point underfoot?  Does that result make any sense?

Also, similar to your example and based on the way you're calculating things ... What is the Angle of the Circle as measured from any given Point on the circle? (feel free to select any point on the circle from which to calculate it).  By attempting this problem, you'll see why I didn't like the method you're suggesting just above.

'Holistically' virtually everything is somehow related to everything else in skiing... but we're trying to keep the Scope of this particular discussion on the nature of Steering Angle rather than possible global outcomes.

---
BTS,
Not sure if this will help but here are some drawings similar to many you've no doubt seen in the past. The first shows a rectangular 'ski' much like a 2x4.  The dashed Green line shows our direction of travel (momentum driven perhaps).  Assume each pictured 'ski' is tipped just slightly to the left.

The straight Black Arrows show the perpendicular direction of Force (friction) hitting the ski trying to deflect it. (I made no attempt to show Magnitude, but will talk about it in a moment.)

The key is to notice where the Center of Mass is on the ski (Red Dot), rather than the Center of the Ski (Blue Dot).  The ski will rotate about its CM and the Yellow line shows the division of the ski that matters.

If the CM (point of rotation) were exactly in the middle of this rectangular ski then it wouldn't actually rotate - because the Forces pressing against the tail section would exactly counterbalance the Forces pushing against the forebody - no self-steering.

Since the CM is just back of center, we now have more Black Arrows pressing against the Front edge than against the Back edge.  More importantly, the Arrows further away from the CM (Red Dot) contribute more Force than those closer to the CM (due to increased leverage) and we have those two extra Black Arrows way out at the point of highest leverage in the forebody.  This leaves the pressure/deflection at the Tail woefully lacking as compared to the Forebody, so this 2x4 will rotate with no sidecut needed.

This next image adds in the missing sidecut and I've rotated the Black Arrows to be perpendicular to our newly curved edges.  We now have different Steering Angles for our skidding ski at each point along the ski (Forebody AND Tail).

Here, I also drew in a Purple and Blue line to show the relative difference in angle of engagement at the Tip vs. at the Tail.  Because the curved edge at the Tip is now more across our direction of travel, it produces more friction/deflection Force - and at the point of greatest leverage as well.

On the Tail side of things notice the curve of the Tail does the opposite.  That part of the edge curves in such a way that it actually presents less of a frictional face against our direction of travel.  This is why Shaped Skis also turn better when just skidding - no carving is needed to gain a turning advantage from that sidecut..!

---
Getting back to your musings BTS, the Purple and Blue lines represent the degree of deflection at the very Tip and Tail.  The Steering Angle at the Tip is quite large, while at the Tail quite small.  If you check each point along the curved edge you'll detect a specific Steering Angle relevant only to that exact point.

All these microscopic Steering Angles (and the relevant Leverage + Friction at each specific point) get summed together producing a Forebody Result.  All points behind the point of rotation get summed together also for our Tail Result.  The larger the Forebody Result as compared to the Tail Result, the larger our turning force will be (torque).

The bottom line here is that our Steering Force (not Angle) is the Net Total for the front and back rather than just the front.  The more the back end of the ski drags, the less the ski will want to turn.  To prove it, just attach a 20 pound anchor to the very Tail of your ski and try to make some turns.

When you ask, "What if we want to increase the skidding without changing our steering angle per say.  Is that possible?"  I would say yes.  We simply reduce our existing edge-angle.  The Turning Torque will remain the same since the 'current ratio of deflection' between our Tip & Tail remains the same.  Friction is reduced over both Tip and Tail equally (all else being equal) so no Net Change in Torque occurs, we just skid faster due to less total friction.

"Is our pivot point around which the ski steers itself always located at exactly the same spot under our foot?  Or will it be different depending on whether our intention is to steer or smear?  Or both?"  I'd say we can easily move the pivot point at least a bit.

That Pivot Point is controllable via Fore/Aft leverage and by moving our own CM forward and back.  This affects the ski's pivot point because our own (body) Mass location influences the ski's effective CM.  This gets complicated in a hurry as now we're talking about the Mass of different Segments attached by Joints with differing Degrees of Freedom... (let's not go there)  Anyway, with more weight on our toes we move the pivot point forward a bit.  If on our heels, we move it back.  If we move *way* back... it gets really complicated as the ski also has flexibility and torsional weakness to mess up the clean images presented above.

.ma
Quote:
Originally Posted by michaelA

SkiDude72,

I get what you're saying but we went a bit beyond that a while back.   The problem as covered earlier arises when we try to use points along the curve as you are doing in what you just posted as example.  In your example, what would be the steering angle for a 8-foot radius turn as measured 20 feet from the point underfoot?  Does that result make any sense?

Also, similar to your example and based on the way you're calculating things ... What is the Angle of the Circle as measured from any given Point on the circle? (feel free to select any point on the circle from which to calculate it).  By attempting this problem, you'll see why I didn't like the method you're suggesting just above.

The method suggested above by skidude72 is fine.  It is your idea that a point NOT on the ski can say something about the steering angle of the ski that is flawed.

As to the remainder of the pics, the integration of all forces ahead of the foot vs the forces at the tail of the ski will show if the SKI turns, not if the skier turns.
Quote:
Originally Posted by michaelA

When you ask, "What if we want to increase the skidding without changing our steering angle per say.  Is that possible?"  I would say yes.  We simply reduce our existing edge-angle.  The Turning Torque will remain the same since the 'current ratio of deflection' between our Tip & Tail remains the same.  Friction is reduced over both Tip and Tail equally (all else being equal) so no Net Change in Torque occurs, we just skid faster due to less total friction.

Reducing the edge angle also reduces the turn radius, since you've reduced the deflecting force operating on the ski underfoot -- the steering angle will have less of an effect.

If you wish to track the same radius turn, you will have to adjust for lack of grip by increasing the steering angle.
Quote:
Originally Posted by BigE

As to the remainder of the pics, the integration of all forces ahead of the foot vs the forces at the tail of the ski will show if the SKI turns, not if the skier turns.

Somewhat BigE.  If the ski turns, then more steering angle is created under foot, which turns the skier.

BTS,

That depends on edge control. You can do 360's down the fall line just by playing with the forces on the tip and tail of the ski.  As michaelA says, the ski will supply torque to the skier, but that does not necessarily change the turn radius.
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