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?? [ more to be added ]
For the leg-push of skating, propulsive work is created in a couple
of ways:
(a) Most obviously by moving some part(s) of the body roughly
opposite to the intended propulsive pushing direction of the foot
against the ground. This "pushing direction" is usually along the
surface of the ground, at a right angle to the aiming direction of the
skate or ski, pointed partly sideways and partly backwards. So to move a
body part opposite to direction that means sideways the other way and/or
forward.
Sometimes the parts getting moved away are
small (e.g. the forearms, when making an arm-swing move), and other
times large, sometimes almost the whole body (e.g. all body parts above
the ankle, when making a toe-push move).
Most real push moves are mixtures of
side-side, front-back, up-down, and usually the proportions of the
mixture change during the move. So a move which is primarily upward
might have a small propulsive contribution because it has some side-side
mixed in with it.
(b) Slowing or stopping a body part is sometimes propulsive. If a
body part is already moving roughly toward the pushing direction, then
stopping that part will add force in the pushing direction (by Newton's
Third Law about action + reaction).
The positive propulsive effect from stopping
the motion of a body part can be just as real and large as from starting
a motion.
The negative propulsive effect from
stopping the motion of a body part could be also be just as real and
large as the positive starting -- which is a problem which must be
carefully managed in the cycle sequence of skating motions, because
every body-part move that gets started must also be stopped sometime
during the cycle. Have to be clever to avoid having the propulsion
effects from the stopping and the starting exactly cancel each other.
(c) Sometimes can make a move which stores energy which can be
converted into a propulsive work of type (a) or (b) later.
The classic case is to lift parts of the body
early when the foot is underneath the hip, then let those parts fall
later when the foot is out to the side, to add force to the push through
the foot outward and backward. It's really only propulsive when the
overall average vertical position ("center of mass") of the whole body
moves upward -- not when one part moves up at the cost of another part
moving down.
Another important case is the
"side-weight-shift" energy which starts the mass of the body moving away from the push but
later is found to be moving toward the next push, so then it can add
work of type (b) to that future push. In this case the same move both
adds work to the current push and stores energy to be used in the next
push.
"direct" versus "reactive force" / "inertial" moves?
Question: Are some propulsive moves "direct"
and others "reactive force" or "inertial force"?
Answer: There's no clear distinction like that
which is helpful. Virtually all "currently propulsive" moves (type (a)
and (b)) in skating have some sideways component -- and for the moves
making the larger contributions to power, the sideways component is a
rather substantial proportion of the force applied. The sideways
component works by pushing "against" the inertia of the mass of some
body part(s) and thus generating a reaction force which gets added to
the total force through the skater's foot to the ground. So all the
important currently propulsive moves in skating are substantially
"reactive force".
The joint for some moves is much closer to the
foot (e.g. knee-extension) than for others (e.g. arm-swing). The closer
moves tend to move a larger mass (e.g. everything hips and higher) away
from the pushing-direction, and tend to have less power-transmission
losses. Maybe that last is what "direct" could mean, but it's a
continuous scale, not a clear qualitatitive distinction.
Whether the joint is closer or farther from
the foot, the skater still needs to worry about timing its move with
aiming-angle changes and leg-switches, so that starting and stopping do
not cancel each other, and design the recovery move so it does not
cancel the primary push.
A move is currently propulsive if it applies a
force with a component toward the current intended pushing-direction. It
is true that one component the propulsive force through the foot pushes
"against" the overall frictional and air resistance against the forward
motion of the skater, while the other component pushes "against" the
inertia of the skater's side-to-side motion. But the proportion between
those two components is determined by the aiming-angle of the skate or
ski, not by which body-part moves are being performed.
Physics more formally:
Really type (a) and type (b) are pretty much
the same thing in the formal physics: Applying a force to an object to
produce acceleration and a reactive force. Type (b) is just a negative
acceleration.
Rotational motion: Though it's easier to think
about forces and linear motion, because of how bones and joints work,
it's usually more accurate to think of most cases where a muscle moves a
body part as applying a torque through a joint to produce rotational
motion.
Formulas:
[
see symbol
definitions ]
n = unit vector of the intended
propulsive pushing direction.
rj = position of the joint through which
this push is being applied, relative to the foot's contact with the
ground.
τj
= torque applied through this joint by this simple push.
f = force from push through that joint
which gets applied at
foot-ground-connection position (ignoring losses in transmission
through the skater's body structures).
Then
f = (rj
× τj) / |rj|2
where rj
× τj
is the vector cross-product, and |rj| is the magnitude of rj.
"Currently Propulsive" push: A simple push is "currently
propulsive" if the force through the foot has a significantly positive
component in the intended propulsive pushing direction; i.e.
f ∙
n > 0 or (rj
× τj)
∙ n > 0
Here we address how positive and negative work is
quantified, how the positive at one time in the stroke-cycle tends to
get cancelled out negative at another time, and the clever tricks use in
skating to create an overall net positive.
In physics, "work" is a very specific concept with a
very specific quantification. The basic concept is that "work" is the
increase of a kind of "energy" which is useful for some desired purpose
(or the prevention of a decrease in useful "energy"). In this analysis
of skating, the desired purpose is to maintain speed of the skater's
body moving forward against resistive forces such as friction and air
resistance.
"Energy" is also a very specific carefully-defined
concept, with a very specific quantification -- the same as for "work".
But "energy" is a quantification of the state of a system at a point in
time, while "work" is a quantification of the difference ot the system
state between two times. The skater's body moving at speed has a useful
energy, which fits the rigorous definition of "energy" in physics, a
kind called "kinetic energy".
The kind of "work" being done on this energy is a bit
tricky. It is mainly not an increase in this "kinetic energy",
but rather the difference between what the kinetic energy level was at
one point in time versus what it would have decreased to at a
later time, if this propulsive work had not been done, if the resistive
forces of friction and air resistance had operated without opposition.
Usually an easier way to calculate work for linear
motion is to use this formula for the Work performed between two points
in time:
Work = (Force externally applied) * (Distance
thru which force is applied)
This avoids direct reference to energy levels, but it
can be proven that the resulting quantification is the equal. This
simple formula assumes force is constant. If it varies, then instead
need to express the formula as an "integral", which is complicated for
non-physicists, but a straightforward extension of the definition.
Physics more formally: Since the push through
a skeletal joint is rotational, it is more accurate to calculate Work =
Torque * Angular-difference. The practical implications for skating
technique are similar (except in special cases where they're different
-- physicists can tell which cases are "special"). Anyway when the push
moves get applied at the key interface between the skater's foot and the
ground, it's the linear-motion concepts which are more helpful.
Limited Range-of-Motion
This calculation of Work has some implications for
moves of body parts which are restricted to a limited range of motion --
which for skating is all of the body parts. Rockets avoid this
restriction by propelling the mass of their fuel backward infinitely
into space. But human skaters feel a need to keep all their body parts
and equipment together with them for many repetitions of the
stroke-cycle.
So sometime after start of a move, that body part must
also stop relative to the skater's center-of-mass. That stopping takes
up time and uses part of the limited range-of-motion distance for that
body part. But this required stopping does not generate propulsive work
in the way that the starting did (indeed normally it opposes the
starting, but that's not the immediate point), so the propulsive Work
done is normally less than could be gained if the skater could use the
whole range-of-motion positively (and just throw away the body part).
If the magnitude of the stopping force is constant and
the same as the starting force (but in the opposite direction), then
propulsive work from starting the body part move is the same as the
kinetic energy at maximum velocity which occurs at the half-way point:
Work = ½
* (Force) * (Range-of-Motion distance).
Limited range-of-motion also has some implications for:
maximum Speed of body part =
= square-root { (Range-of-Motion) * (Force)
∕ (Mass-of-Body-Part) }
Time duration of move =
= square-root { (Range-of-Motion) *
(Mass-of-Body-Part) ∕ (Force) }
These formulas for Speed and Time have implications for
the "rhythm" of the moves. Note that a move of a body part with smaller
mass tends to require higher speed and quicker time, which can cause
problems -- see
below under "move body part with small mass".
Although we think of propulsive push as away from the
intended pushing-direction, any push move through a skeletal joint
actually pushes some body part(s) toward the pushing-direction.
Because any human joint is between some body parts, and
a push move through the joint tends to push the parts on each side in
different and somewhat opposing directions. In the case of skating, some
other part(s) are on the side of the joint away from the
foot-ground-connection (and tend to get pushed away from the
pushing-direction, positive for Work), but some other part(s) are
between the pushing joint and the foot-ground-connection (and tend to
get pushed toward the pushing-direction, which could be negative
for Work). So the positive push-move generates its own (potential)
negative.
Q: How prevent (or at least reduce) the potential negative Work?
A: Do not allow the joints and bones and other
bodily structures which are toward the foot-ground-cancellation to
actually move through any significant distance. The "Force" part
of the "Work" calculation is required in this case, but the "Distance"
is not. If the skater can hold the "Distance" to (near) zero, then the
negative Work will be (near) zero. This requires holding some bones and
joints and structures in a stable configuration, not collapsing.
A problem with generating forces by moving body parts
is that starting and stopping tend to produce forces in opposite
directions, by Newton's Third Law.
Note that sometimes it's the stopping that's
positive and the starting which is negative for propulsion. Other times
the roles are reversed.
We'll consider several cases:
So the self-cancellation problem is big in this
case, but the solution is straightforward: Change the direction of the
intended propulsive pushing direction in between the starting and the
stopping, by changing the aim-angle of the ski or skate, from pointing
toward one side of the overall forward motion to pointing toward the
other side. Since the magnitude and direction of the force transmitted
to the ground depends heavily on the aim-angle (see
details below), this is way to get positive propulsive work out of
both the starting and stopping of a sideways motion.
So the key is to carefully coordinate the
timing of starting and stopping of sideways moves around changes in the
aim-angle of the skate or ski.
In
normal
single-push stroking, this switch in directions is easily
accomplished by switching with foot is on the ground. In
double-push
stroking it's trickier, done with a pivot of the same foot in the
Aim-switch phase.
It does not matter whether the body parts
being pushed sideways are big or small, the principle is the same. Even
when nearly the entire mass of the body is being pushed, as by the
ankle-extension move at the end of Phase 3, much of the benefit of that
move would be cancelled if the skater somehow kept weight on the same
skate or ski, pointing toward the same side.
-
moving body part(s) mainly up-and-down. Here
there's no contribution to currently propulsive work, so
self-cancellation of starting and stopping is not a concern. As long
as they overall raising of the center-of-mass is accomplished in
time for when its needed in the next repetition of the stroke-cycle,
the details of timing and acceleration/deceleration are usually not
significant for propulsive performance results.
-
moving large-mass, large-cross-section-area sets of
body parts forward and backward. If the skater is succeeding in
maintaining a steady cruising speed, starting-versus-stopping
cancellation is not as much of a problem as in side-side motion.
Because sets of body parts which include a
large percentage of the skater's body usually cannot be starting and
stopping during the stroke-cycle -- otherwise the skater's
center-of-mass would be coming close to starting and stopping, and the
skater would not be said to be cruising steadily.
For moves of large portions of the skater's
body forward-backward, the push is mainly "against" the
forward-motion-resistive forces of air resistance, friction, hill-slope
-- not "against" inertia. Those resistive forces are generally fairly
steady, so the accelerations and decelerations of the skater's speed are
not a large percentage of the average speed -- completely unlike the big
swings and reversals of side-weight-shift moves.
Two examples are extra backward-then-forward
move in leg-recovery and set-down, and backward-forward arm-swing. To
some extent a backward-to-forward move pushes "against" the
forward-motion-resistive forces (instead of "against" inertia), but
trying to take advantage of this concept runs into a different
self-cancellation problem of Recovery move versus primary (see
below).
The problem with forward-backward moves is
that the trick with side-weight-shift moves, of just switching sides of
the intended pushing-direction, doesn't work for forward-backward moves,
because a forward-backward move does not have a positive or
negative "side".
What makes a difference for transmission of
forward-backward move is not the direction or sign of the
aim-angle, but rather
the magnitude of the aim-angle
α.
Whenever
α is in the range 0°
≤ α
≤ 45°, larger
α yields greater
transmission of forward-backward force into both currently-propulsive
and future-propulsive work -- see the "x → x"
and "x → y" columns in the
table of transmission and
conversion ratios, further
below.
So the trick is change the magnitude of the
aim-angle in between the positive part of the move and the negative
part. If the angle is larger during the positive part and smaller in the
negative part, then the total effect will be net positive for propulsive
work.
Example -- extra backward-then-forward move in
leg-recovery and set-down: The trick in this case is to set-down in
Phase 0 with the
aim-angle smaller and the skate or ski pointed closer to straight in the
overall
forward motion direction. Continue that smaller angle into the early
part of Phase 1.
Then pivot the skate to aim more outward, so the aim-angle is larger by
the the leg-push is finishing in
Phase 3. The
timing coordination is roughly like this: Reach the farthest backward
position in the leg-recovery move during Phase 3 of the other leg's
push, and start the set-down move also during Phase 3 of the other leg,
so the positive reactive force from accelerating the mass of the
recovering leg is transmitted well into the push of the other leg. The
deceleration and stopping of the forward motion then occur early in
Phase 1 of this leg's next push, but the negative reactive force
is not transmitted well into the ground, because the aim-angle is
smaller in Phase 1. Result: the carefully timed difference in aim-angle
magnitude and transmission effectiveness yields a net positive in
propulsive work.
[ Running:
Interesting that a slightly different trick is available to make the
backward-to-forward leg-recovery and set-down move into a net positive
for Running. The recovering leg starts its forward set-down move while
the other foot is on the ground, so the positive reactive force is
transmitted effectively. But much of the deceleration is accomplished
while both feet are up in the air. So some of the negative
reactive force does not get transmitted into the ground at all, so
there's an overall net positive for propulsion. A similar timing trick
works for getting net positive work from forward-backward arm-swing in
Running.
[ Walking:
but this timing trick is not available with Walking, because at least
one foot is always on the ground, so all reactive forces both positive
and negative are transmitted fully. Arm-swing is still used for balance
in fast walking, but experienced walkers climbing up a hill generally do
not swing their arms -- just let them hang down at the side -- because
arm-swing doesn't help propulsion in walking. Nor does extra
backward-forward motion in leg-recovery, so experienced Walkers do not
do that either. ]
Recovery move problem for type (a) and (b):
* must subtract the negative work of the reactive force and the
recovery move through the whole stroke-cycle.
* "locking in": above hips versus hips and below.
?? [ more to be added ]
* Muscles not good at generating high force + high velocity at the
same time.
* Time duration of move gets small, tricky to coordinate timing of
start versus stop with changing the intended-pushing-direction.
(especially if the rhythm of the pushing-direction changes is driven by
moving larger-mass parts).
?? [ more to be added ]
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What matters for propulsion is not the force or torque
from any particular move, but what all of them together combine into as
a force applied at the position of the foot-ground-connection.
So it is not necessary to align the force direction of
each move with the intended pushing-direction.
As long as a given move has some substantial
component in the intended pushing-direction, that move can add
propulsive work. Often the components which are off from the
pushing-direction are accidentally cancelled by off-pushing-direction
components of other moves, or can be cancelled deliberately by
non-propulsive control moves or by modifications of certain other
propulsive moves.
"All of them" includes not just forces from muscle
moves, but also forces of resistance and gravity.
So the total force through the foot-ground-connection
("fgc") is
ffgc
=
forces from all muscle moves on
body parts
+ weight of the skater's body
+ air resistance
+ gliding friction
+ hill slope resistance
Here's more specifics on each . . .
The force applied by a muscle move at the
foot-ground-connection ("fgc") is dependent on
-
the position and orientation of the joint through
which the push is made (symbol rj)
-
the orientation of the joint and the magnitude of
the push (described by the symbol Tj)
-
how much force is lost in transmission, by being
absorbed by body structures or the unintended collapsing of other
joints "along the way" in between the joint and fgc.
Formula:
The force through the
foot-ground-connection from the push through joint j is:
f = (rj
× τj) / |rj|2
for more detail and explanation,
see above under
ways to create propulsive work.
"Gearing"
This formula implies that the further away the joint is
from fgc (larger magnitude of |rj|,
the smaller the force (but the higher the linear speed). So the
push-moves through joints further away from the foot tend to be better
for "high gear" situations (higher speed, lower required force
intensity), while the joints closer to the foot position rfgc
tend to be better for "low gear" situations (higher required force
intensitey, lower speed).
On the other hand, you use what you got. Even if a move
is not the more effective "gear" for your situation, if it helps more
than it hurts, makes sense to use it. For example, arm-swing is a "high
gear" kind of move, but it's helpful for climbing up a steep hill.
Anyway it's more complicated than that, because the
main "gearing" control in skating is the
aim-angle of the skate
or ski. By reducing the aim-angle, the relative speed at fgc of the foot
pushing out from the skater's body might be slower, even the absolute
speed over the ground is higher.
Gravity always has a downward force based on the total
mass of the skater's body. Whether it changes current propulsion work
depends on the positional configuration of the skater's body -- and not
on upward or downward accelerations (parallel to the z
direction).
If the skater is in balance with
center-of-mass directly vertically above the foot-ground-connection,
then there is no propulsive force resulting from gravity.
But if the skater's body mass is out of balance, then
there is a non-vertical component of propulsive force into the
foot-ground-connection, based on how far the angle from the
center-of-mass to foot-ground-connection is from straight vertical. The
amount of propulsive work added in a period of time depends on how far
the center-of-mass drops during that period (which depends partly on
previous moves in the stroke-cycle). As the foot moves out further away
from underneath the hip, the CoM-foot-slant angle increases, and it's
harder for the leg to support the skater's upper body, so it's normal
for the skater's overall body and center-of-mass to fall during the
later Phase 3 of the leg-push.
Of course in order for the stroke-cycle to be
repeatable, the center-of-mass must get raised up again, and that takes
real work of type (c), usually done in Phase 1 just after set-down. It's
real work being done, but it's not currently propulsive.
Actually it has a negative effect on current propulsive
work since moving the center-of-mass upward is also somewhat away from
the foot-ground-connection, but not as a result of force in the
pushing-direction, so it somewhat "softens" the force in the pushing
direction. But it's usually worth accepting that cost in order to make
such an effective use of the strong knee-extension muscles.
Formulas: [
see symbol
definitions ]
The CoM-foot-slant angle
β is calculated by ignoring
any component of the position of the Center-of-Mass relative to
foot-ground-connection which is parallel to the aiming-angle of the
skate or ski, the angle between that and vertical:
β
= arccos { (r ∙ z)
∕ square-root[ (r
∙ n)2
+ (r ∙ z)2
] }
Then the magnitude of currently propulsive
component of force, in the pushing-direction is:
|fn|
= M g cos β sin
β =
½ M g sin 2β
The configuration for maximum propulsive force
is at β = 45°,
at which point the force is half the skater's body weight. But the
skater's body is dropping fast by then, so that force is only brief and
temporary until the skater sets the other foot down to stop the falling.
For most of our analysis of skating we assume that
gliding friction is negligible in comparison with air resistance, as on
ice or very smooth pavement or hard snow.
If there is significant gliding friction in the
aiming-direction, it makes the analysis (and practical implications)
more complicated, but the analysis is already plenty complicated and
interesting without it.
Friction is proportional mainly to
"coefficient of gliding friction" and to the component of force at the
foot-ground-connection which is perpendicular to the surface of the
ground.
Since this friction is directed opposite to
the aiming-direction, it usually has a significant sideways component.
This means that if we get precise in situations with significant gliding
friction, the sideways component of propulsive force is partly "against"
resistive force, and partly currently propulsive. Although it's still
mainly "against" inertial force and substantially future-propulsive.
Air resistance is the main force opposing forward
motion of the skater on flat ground. It is roughly proportional to the
square of the skater's velocity, so it is very important for determining
the skater's speed.
Magnitude and direction of air resistance
force is largely based on current forward velocity and current sideways
velocity of skater's center-of-mass, and on cross-section area,
turbulence, and specific-body-part-velocity deviations due to specific
body-part moves
On flat terrain, the main desired purpose of propulsive work is in
counter-acting the slowing effect of air resistance. On flat terrain, a
major consideration limiting use of certain moves and positions is that
they increase the amount of air resistance force.
The direction of air resistance is roughly
opposite to the direction of the skater's center-of-mass. Since the
center-of-mass moves somewhat side-to-side, the main resistive force
also often has a sideways component. This means that if we get precise,
the sideways component of propulsive force is partly "against" resistive
force, and partly currently propulsive. Although it's still mainly
"against" inertial force and substantially future-propulsive.
Modifying the analysis and formulas for the situation
of skating up a hill makes everything more complicated in the details.
First the positional coordinate frame needs to be
re-defined:
Because the "upward" direction given by the
unit-vector z is no longer straight vertical. It is defined as
perpendicular to the ground surface, but the ground is no longer
horizontal. On a hill slope, we can define the upward direction as:
z = ( − (cos
γ) g
− (sin γ)
(g × y) )
∕ |g|
where
y is the sideways direction which
remains pure horizontal, unchanged by the slope, and
(g × y)
∕ |g| is the vector-cross-product
which yields the pure-horizontal forward unit vector which is x
projected onto the pure-horizontal plane (which is perpendicular to g).
So then we get the expected result: cos
γ =
− z ∙
g ∕ |g|
and the "forward-motion" direction given by
the unit-vector x is not longer pure horizontal, but now includes a
vertical component. On a hill slope, we can define the forward direction
as:
x = ( (cos
γ) (g
× y) - (sin
γ) (g) )
∕ |g|
So then we get the expected result: sin
γ =
− x ∙
g ∕ |g|
The slope of the hill also has a big impact on at least
one key force:
Resistive force opposing forward motion =
= (air resistance) +
(gliding friction) + g sin
γ
?? [ more to be added ]
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The total net force into the foot-ground-connection
("fgc") is split into directional components, in two conceptual stages:
The total force through the foot, ffgc
is split into three orthogonal directions: upward-downward, longitudinal, and
current propulsion, so ffgc = fn
+ fa + fz:
up-down fz component
fz up-down component goes downward parallel to
z, and
perpendicular to the ground surface plane.
Formula: fz
= (ffgc
∙ z) z.
To calculate
the magnitude of the downward contribution from skater's body weight
and body configuration, see formulas above for
weight + gravity
under Force gets "totaled" at
foot-ground-connection. (There needs to be a further adjustment
if hill-slope angle is non-zero).
Effect: If its downward magnitude is greater than body-weight,
the excess goes into building future
potential energy and the skater's
center-of-mass
(CoM) rises. If its downward magnitude is less than body-weight,
there is a deduction from future
potential energy as the skater's CoM falls.
Grip -- The amount of downward force is important for
sideways friction to maintain "grip" -- i.e. to prevent the blade or
edge or wheels from skidding out sideways perpendicular to the aiming
direction.
Characteristics of downward: Contributions to vertical force are not currently propulsive, only future-propulsive.
And they're based only on the current body configuration of the skater
(how far the foot is out toward the side away from underneath CoM), not
on accelerations or decelerations of body parts. Gravitational force is
what it is regardless of which way and how fast various body parts are
moving.
Managing the downward component: So there's no point in worrying about
timing-coordination or quickness of up-down moves. The only thing that
matters is if there's sufficient total upward-pushing work done any time
during the stroke-cycle to get the average position of the total mass of
the body raised up to as high off the ground as it was in the previous
stroke-cycle -- otherwise the stroke-cycle sequence of motions is not
repeatable.
longitudinal fa
component
fa
points longitudinally along aiming-direction a. of the skate or
ski, and lies in the ground surface plane.
Formula: fa
= (ffgc
∙ a) a.
Effect: Moves the skate
or ski relative to the skater's center-of-mass. Large uncanceled
components in this direction are a problem for control -- could result
in the skate or ski jetting out from underneath the skater and dumping
the skater on the ground either to the front or the rear.
No propulsive significance unless gliding
friction is signficant (which we will ignore for now).
?? Further investigation:
- How are forces in this direction cancelled or controlled?
- Is there a muscle-move available which could generate
substantial propulsive work, but its use is greatly restricted
because of its side-effects on this fa
component?
currently propulsive fn
component
fn
is propulsive in the intended pushing direction, in the ground surface
plane, perpendicular ("normal") to the aiming-direction of the skate or
ski
Formula: fn
= (ffgc
∙ n) n
Effect: This is the main push out from the skater's
body. But it's not all currently propulsive. It gets split two ways in
the second conceptual stage of de-composition of forces.
It is helpful conceptually to further de-compose fn
into two components which get through the "other end" of the foot-ground
interface -- definitely transmitted into the ground:
-
fny is
the sideways component of fully-transmitted force. This component is
not currently propulsive, but it does effective work to add to
future propulsion by first stopping the kinetic energy toward the
current pushing-direction, then creating side-weight-shift kinetic energy
toward the next pushing-direction. [ fny
= (fn
∙ y) y ]
It is interesting to calculate the magnitude of these
components in the fully-transmitted horizontal force fn
based on the corresponding components of a push-force applied at fgc
from created from a muscle move at joint j. Let
ffx
= (ffgc
∙ x) x be the
forward-backward component applied at fgc.
ffy
= (ffgc
∙ y) y be the sideways
component applied at fgc.
fx =
magnitude of | ffx |
fy =
magnitude of | ffy |
Then
fn
= (( fx x + fy
y + fz z )
∙ n) n = (
fx x
∙ n + fy
y
∙ n) n
and so
fnx
= (fn
∙ x) x
= (( fx x
∙ n + fy
y
∙ n)) (n
∙ x) x
magnitude | fnx
| = fx (x
∙ n)2
+ fy (x
∙ n)(y
∙ n)
Since x
∙ n = sin
α and y
∙ n = cos
α (where
α is the aim-angle of the
skate or ski),
?? something might be not quite right with the
signs in these formulas ??
| fnx
| = fx sin2
α +
fy sin
α cos
α
| fnx
| = ½ (1
− cos 2α)
fx +
½ (sin 2α)
fy
| fny
| = ½ (sin 2α)
fx +
½ (1 + cos 2α)
fy
ratios of force components applied in ffgc
→ transmitted + converted
→ fn
| aim-angle
α |
x → x |
x → y |
y → x |
y → y |
| 0° |
0.00 |
0.00 |
0.00 |
1.00 |
| 7.5° |
0.02 |
0.13 |
0.13 |
0.98 |
| 15° |
0.07 |
0.25 |
0.25 |
0.93 |
| 22.5° |
0.15 |
0.35 |
0.35 |
0.85 |
| 30°
|
0.25 |
0.43 |
0.43 |
0.75 |
| 45° |
0.50 |
0.50 |
0.50 |
0.50 |
| 60° |
0.75 |
0.43 |
0.43 |
0.25 |
| 90° |
1.00 |
0.00 |
0.00 |
0.00 |
observations from these transmission and
direction-conversion ratios:
-
Magic: Column "y → x"
shows that sideways force can produce currently propulsive backward
force, when transmitted through an angled skate or ski -- even if
there is no forward-backward component at all in the applied force.
-
Super-Magic: even if the applied backward
component fx is slightly
negative for propulsion, the transmitted force can have a positively
propulsive backward component, provided that the applied sideways
component fy is relatively large and
the aim-angle
α
is not too large.
-
up to 45 degrees, aiming the
skate or ski out further to the side improves transmission and
conversion into currently propulsive backward-directed force.
Power = Force * Speed = Work / Time
* main Power formula
Power - Force - Velocity trade-offs.
* Isometric force transmission.
Absorption of work (thru counter-productive counter-moves)
* Sample formulas (linear Force-Velocity, quadratic Power-Velocity)
?? [ more to be added ]
?? [ more to be added ]
back to Top |
overview of phases | d-p
phases | definitions | more Leg
?? [ more to be added ]
strategies for managing these limits
?? [ more to be added ]
back to Top |
overview of phases | d-p
phases | definitions | more Leg
back to Top |
overview of phases | d-p
phases | definitions | more Leg
|
