I think we may finally have this issue of "lightening" nailed down.
In response to a suggestion to consider the motion of the CM towards and away from the surface of the snow, Cgeib and Arcmeister responded:
“I would have to go with "c)". I would agree that minimizing the vertical rise and fall of the CM is one of the goals. … ”
“… You certainly do not "need" to soften the legs, but it sure helps if you want to aggressively ski arc2arc. If you do not soften, or retract, the legs the Cm must gets pole vaulted up and over into the next turn. This not only usually produces some unweighting, which then delays re-engaging, but the high trajectory of the Cm limits how far the path of the feet can laterally move away from it and achieve any early edge angle. …
As for the b) above, if skiing terrain with any pitch, and the legs angularly extended, retraction at the end of the turn will unlikely be extreme enough to release the CM parallel to the slope and most likely produce a more horizontal release trajectory, which actually moves the Cm away from a slope with any pitch at all. In a high energy situation, without relaxation or retraction you get the proverbial huckover launching the Cm into the on-no zone and disconnected from any pressure management options for the start of the next turn while the legs desperately reach for the ground ”
Other than a couple of technical questions (which I’ll address below), both responses seem to prefer my option (c), i.e., minimizing unnecessary vertical motion of the CM. Too stiff (ie, option “a”) , and you do a huckover, while if you do too energetic of an active retraction (ie, option “b”), and you will either wind up too low or, in some situations, possibly have the problems that Arc suggested.
In light of the above, I think that the whole question of lightening is essentially solved. Basically, as the skier goes through the transition, he (she) has to adjust the extension of each leg independently so that:
(a) The total combined force from both legs provides an appropriate amount of force acting on the skier’s torso to get his CM to "fly" through the transition smoothly, with as few vertical motion glitches as possible (consistent with terrain undulations, more strategic levels of “intent”, etc.); and,
(b) The difference in force between his L and R legs provides an appropriate amount of torque on the skier’s torso to get it to fly through the transition with as few glitches in its smooth horizontal motion as possible, as well as minimal glitching in the slower angular changes of the torso during the turn (again, consistent with terrain, intent, desired degree of counter, … blah, blah).
Over the past couple of days, I have been thinking about other ways to express the above "maximum smoothness" concept, perhaps working up some computer simulations of the force on each leg over time, or the glitch-like effects on the path of the CM caused by non-optimal individual force histories for each leg. Unfortunately, because all of the different possible combinations of speed, turn radius, slope angle, simultaneously occurring terrain undulations, etc., I don’t think that showing the results of a simulation or two would assist understanding any more than simply stating the above “control policy” and simply giving a couple of text examples of what happens if the skier doesn’t adhere to the above “policy”.
For example, we have already talked about the effects of too much or too little total combined force from both legs. This essentially was the (a), (b), and (c) possibilities of my previous post.
The effects of too much force differential between the L and R legs at the transition were talked about earlier in this thread. For example, too large of a force differential could be caused by too active of a retraction of the old outside leg, and would lead to passive “falling into the next turn” (assuming no compensatory L-R balance adjustments). With compensatory balance adjustments it could lead to “stepping up”. Too large of a force differential between the L and R legs at the transition could also result from skiing the way a veteran, narrow-stance, legs-locked powder skier might ski if put onto hardpack for the first time. He might enter the turn banked, retract both legs simultaneously to get his CM and skis to cross paths and then bank into the next turn. This wouldn’t be disastrous in a narrow stance (ie, like mono-boarding), but in a wider stance, if he persisted in simultaneous L-R retraction and extension, he could start generating some very odd L-R weight distributions like temporarily putting all of his weight on the uphill edges of his (old) uphill skis.
I don’t think that too little of a weight differential between the skier’s L and R skis at the transition (ie, no relaxation of the DH leg at the transition in a wide stance) was explicitly discussed, but would almost certainly hinder the cross over of the CM and skis, and thereby put a glitch in the path of the CM as viewed from above.
Finally, let me address CGeib's technical concerns:
“…However, wont this change with the velocity and amplitude? At some combinations of these, wont the skier end up extremely low at the apex of the turn and physically need to rise in order to pass thru transition (assuming a transition in balance)? Seems to me that vertical movement of the CM would be acceptable to the extent that it results from the pressure management and geometry of the skier. For "a)" & "b)", I don't see how they can exist in the sine wave turns you presented. Wouldn't you get spikes and dips, before and after the transition, in the forces on your illustration, instead of the smooth flow shown, if this were to happen? …”
You are absolutely right that as a result of some particular combination of skiing and terrain variables, you don’t want to wind up too low at the apex. My option (c) should have said to "let the CM fly through the transition in as straight a line as possible, while moving generally parallel to the surface of the snow" (ie, adding the phrase after the word, "while"). Basically, I am thinking that the issue of removing glitches from the transition itself is dealt with by the “fast” control policy that we’ve been talking about, while the issue of not winding up over extended or over flexed through the rest of the turn is dealt with by a slower control policy.
You are also absolutely right that (a) and (b) could not exist in the sine wave turns I presented, certainly not when viewed from the side. I felt this was true, but wanted people either to find a flaw in my argument, or else be forced to the same conclusion that I came to, thereby leaving only one viable option, (c).
Tom / PM
PS (in edit) - You might notice that appears to be a possible omission in the above argument - I never once mentioned edge angles. There is a reason for this. In exact analogy to my post timestamped November 08, 2003 11:38 PM in this thread, if you know exactly the course that the object is taking through space (for example, "the skier's CM is going in a straight line"), then you know the total force acting on that object. There are no "if's, and's, or but's" about this because it is just a restatement of F = ma. So, for example, in the case of the skier's CM, this knowledge immediately leads me to my conclusions about the total combined force of both legs.
What I haven't made any statement about is what the skis are doing, only what the CM is doing. We know that at the transition, we want the skis to cross under (cross-over / cross-under symantics not withstanding) the skier, also going in a straight line, but at an angle to the line that the skier's CM is taking.
So, with the downward force on each ski essentially established by previous arguments, and neglecting fore-aft motion, the only real option left for manipulating the skis is edge angle. The angle from the skis up to the CM will be dictated by the convergence of these two lines, so this is not a variable. Fortunately, however, a skier can contort himself into varying degrees of the classical "bananna" shaped position of angulation, so that he (she) can adjust this variable (within limits) as the skis come under him. These motions are almost totally independent of the downward forces on the skis that were discussed above, so that changes in edging at the transition can be discussed quite separately from the forces, and that's what I decided to do in the previous material. It doesn't mean edging is unimportant, just that it can be separated out from a discussion of (say) vertical forces.
PS #2 - Geeze, this got long. Sorry folks. Thank God none of us actually ski thinking about all this stuff.
[ November 12, 2003, 01:19 PM: Message edited by: PhysicsMan ]