
 



Ken Roberts   Skating what's here
see also: other Years  public discussion  Skating home simple model of propulsive force in skating legpush08dec what's here
_____________________________________ simplest modelI'll start with the simplest model of propulsive force and work
where
concepts so far:
components of force at ground contactGetting into some more useful detail, the force applied at the point of ground contact can be thought of as having three components:
where
substituting these components into the propulsive force and work model:
and the magnitude of the component of f_{G} in the direction of n is
and get the result
and the propulsive component of f_{G} is then
where
sideways weight shiftBut it's more helpful for understanding skating propulsion to refine this further and add two further concepts:
This allows us to write:
where
concepts
Therefore a key determinant of the propulsive contribution if sideways weight shift is the quickness of the sideways move, not the size of the sidetoside motion.
propulsive portion of force currently being transmitted to the ground is:
future propulsive workForce applied thru the foot also makes a contribution to propulsive work in the next legpush, by building sideways kinetic energy.
where
The portion of current force which goes into this future propulsive energy and work is:
The portion which is successfully transmitted into propulsive force in the next legpush must by multiplied by k_{y} cos α. And if we take k_{KE} = k_{y} (which is not unreasonable), we get this as the as the portion of f_{n} which is effective for future propulsive work:
adding together the current and future contribution to propulsion:
concepts



vectors defined in xyz coordinate framev = (cos η, sin η, 0) v dr_{B} = (cos η, sin η, 0) dr_{B} "aiming" direction of foot gliding = (cos α, sin α, 0) f^{aim} = (cos α, sin α, 0) f^{aim} f^{ext} = (sin α sin λ, cos α sin λ, −cos λ) f^{ext} f^{swe} = (sin α cos λ, cos α cos λ, sin λ) f^{swe} n = (−sin α, cos α, 0) f_{n} = (−sin α, cos α, 0) f_{n} g = (−sin γ, 0, −cos γ) g (so if γ = 0, then g = −gz) component of gravitational force in the direction of v
effective slope angle in the direction of v angle = cos^{−1}[(g × y) • v / (g v)] = cos^{−1}(cos γ cos η) (note that overall slope angle is defined in the direction of x) where g × y = (cos γ, 0, −sin γ) g (vector product of g with y, or "cross" product) dynamic equationHere's the basic dynamic equation for acceleration of the skater's body mass:
where
From the perspective of propulsive work and power, we're interested mainly in the components of force in the direction of current forward motion dr = v dt.
calculating in more detail
where
next
and
where
Expressing everything as incremental Work we get (roughly) dW = m a • dr = {
concept
where does the power get "wasted"?first written: April 2008 I keep reading posts on various forums about "efficiency", which makes me think about how to analyze it. Here's a try: aerobic power is generated mainly from carbohydrate fuel and oxygen. about 75% of the energy in the chemical bonds in the carbohydrate fuel is lost in this process  the resulting work of the muscle fibers contracting is only 25% of the energy available. (Also carbon dioxide is released). Therefore if "not wasting energy" is the goal, just stay home and watch TV. Active life is inefficient.
What's "left over" after that is available for power from "skeletal" or "peripheral" muscles. Here's some of the things it gets used for:
Unlike some machines, we don't just "lock" our bones and joints into a desired configuration  just holding a stable "isometric" configuration requires muscular effort. It burns calories and uses up some oxygen. Different motion patterns for propulsion might require more or less energy to be used to maintain the "posture" configurations which are appropriate for that motion. Typically motions in which the posture is more "bent over" require more energy to hold that posture stable. Some postural configurations which require more energy to sustain, also offer more effective leverage or "gearing" of key muscles or a higher percentage of direct transmission of muscular moves into propulsive work  (so there are tradeoffs) Some people use more muscular tension than necessary to maintain posture. Some energy to maintain postural configuration is required for the desire propulsive motion, other energy might be avoidable.
People with wellpracticed specific balance use smaller more accurate corrective moves, which take less work. Also sometimes a person with better specific balance can sustain a configuration which offers more effective leverage or "gearing" of key muscles or a higher percentage of direct transmission of muscular moves into propulsive work. Some energy to maintain balance is required for the desire propulsive motion, other energy might be avoidable.
Some recovery moves themselves add positive propulsive work. Often this depends on timing and other tricky points, since often the attempted propulsive work in a recovery move is selfcancelling. For example, recovering the leg forward is partly propulsive in running, but not in normal walking (with continual ground contact). Some propulsive muscle moves are "selfrecovering". For example pelvis  lower spine rotation in walking and running. If pushing with the Right leg, the rotation of the pelvis about the spinal axis to bring the nonpushing Left hip forward to add propulsive work to the push thru the Right leg also finishes with the Right hip backward. Which is exactly the configuration needed to make a propulsive pelvis  lower spine rotation move for the push of the Left leg.
An example might be classic striding in crosscountry skiing, especially when climbing a steep hill. Extra downward pushing force is needed at the same time forwardbackward force is being applied  in order to prevent the ski from losing grip and slipping back. This extra force causes the mass of the upper body to move upward in reaction, and can sometimes it has so much upward momentum that both feet are lifted off the ground simultaneously after the legpush finishes  so the next legpush cannot start until time has elapsed for the mass of the upper body to fall down again. In most situations for most propulsive motions, body weight and the propulsive forces themselves are sufficient to provide sufficient static friction. Some performers might apply more downward force than necessary, while other more skillful performers might take it closer to the "edge" of only the minimum required.
Human muscles and joints are useful in this static transmission role. Especially smaller muscles can sometimes make a larger contribution to overall power by statically transmitting the work from larger muscles, rather than attempting to add active work of their own. This function of "isometric" static transmission does not do propulsive work, but it does consume fuel and oxygen.
I guess this is what lots of people mean by "wasted energy" or "inefficient" motion. But it's only one way in which the power generated from fuel and oxygen is not available to add to propulsion work. Also moves with lots of people think are just "wasted" actually do make a contribution to propulsion (thru a less obvious exploitation of the physics of propulsion), or are necessary to support propulsive contribution of other muscles (in one of the ways described above). What remains after those uses can be applied to propulsive work. But it's not so simple . . . mainly because the moves are not in exactly the right direction for propulsion. The ideal propulsive force pushes straight backward against the ground and straight forward against the mass of the body. But most actual propulsive moves by the human body push diagonally. Either a combination of backward and downward against the ground (as in striding or polepushing). Or a combination of backward and sideways against the ground (as in skating)  well actually the skating legpush is a diagonal combination of three directions: backward and sideways and downward against the ground. Only the directional "component" of force which is in the forwardbackward direction is directly + immediately added to propulsive work. Other directional components (downward and sideways) of the pushing force can also contribute to propulsion, but only indirectly, deferred to a later phase of the strokecycle, or to the next strokecycle. These deferred forces may result in additional losses of power:
If some of these muscles are used in static "isometric" mode, that's a loss, as described above under "transmit".
Again and again and again.
Since one of the three main drivers of propulsive power is strokecycle time, delay means loss of power.
Or some other moves could be pushed into a more favorable power point on their curve. So even the muscle moves and forces which are not "wasted" still have power losses. Why does power matter so much?Because the goal of human propulsion is to move some distance in some amount of time  so some positive speed is required. The rate of speed is determined by this equation: [Resistance Force] * [Foreward Speed] = [Direct Propulsive Power from muscles] where [Resistance Force] is the sum of the forces opposing forward motion: like air resistance, or friction against the ground, or if climbing up a hill then gravity is included. Especially for air resistance, the intensity of resistance depends on the overall body posture and to some extent which muscle moves are being used  (so a change to postural configuration which takes less energy to maintain might also increase air resistance  there are tradeoffs.) Actually it's more complicated, since each of those quantities varies at different times in the strokecycle, so the equation has to be sort of "integrated" over the whole strokecycle. It gets more complicated, because [Resistance Force] depends on [Foreward Speed] because air resistance gets disproportionately larger at higher speeds. [Direct Propulsive Power from muscles] depends on [Foreward Speed] because muscles produce different power levels depending on the speed, sometimes higher, sometimes lower  but above some speed (different for each muscle move) the power capability only declines with speed. Yet another complication is that how much power each muscles can deliver depends on the duration of time it needs to sustain that power level. At higher speeds, as the performer tries to raise the speed rise even higher, [Resistance Force] tends to only rise and [Direct Propulsive Power] tends only to fall. So there's an upper limit. But despite all that complexity, more Power generally means higher speed  in getting from one place to another.
Chasing efficiency is a complicated game. Trying to use "efficiency" to increase speed is a very complicated game. more . . .see also
concept words: ski skiing snow roberts report reports learn learning skating: skate skates skater skaters push glide inline inlines ice speed speedskate speedskating speedskater speedskaters roller technique: techniques technical theory theories theoretical physics physical biomechanics biomechanical mechanics mechanical model models concept concepts idea ideas 








