analysis of the physics of a spreadsheet model of motions and power in Skating
Ken Roberts
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Some important choices are the simplifying assumptions (see under
"Limitations" on the
main Skate Power page) and which variables to use as primary inputs on
the spreadsheet.
Once I had decided that forward velocity v_{x} and required forward force
F_{x} should be inputs, then total legpush force F_{n} and sideways force
F_{y}
are determined by q, the angle of the
ski (or ice skate blade or inline skate wheelframe).
By symmetry, the sideways velocity at the end of one legpush stroke
must be the same as the velocity at the start of the next legpush stroke,
so the forcemassacceleration relationship (F = ma) requires:
v_{y} =  v_{y} + (F_{y} / Mass) * Time_{h}
which is easily solved for v_{y}  and that determines the
distance of the maximum sideways motion of the skater's body center.
b_{yq} = v_{y} * (0.5 * Time_{h})
Then the forces and the distances in each orthogonal direction are
known, so the work for a single legpush stroke can be calculated by Work
= Force * Distance. Finally Power is calculated by dividing the Work
by the time for a single legpush, Time_{h}.
One key step for making this realistic for the biomechanics of a human
skater is to introduce Sy_max_h or s_{yh}^{max} as a constraint on the sideways motion
of the ski (or skate) s_{yh} being pushed.
There's a couple of points I'm aware of where the spreadsheet model does
not include some of the exact correct approach to calculate the physical
quantities even based on the simplified assumptions given in the Limitations.

The model does not account for the fact that introducing a Passive
glide phase deviates from the assumption of constant Velocity. The
formulas would need to be much more complicated to handle this
deviation, without generating much insight. I think being more
accurate in handling nonuniform velocity with passive glide would increase the power expenditure for
using a positive Passive glide phase with same average forward
velocity  because of increase in work required to overcome air
resistance due to sometimes higherthanaverage velocity.

The model does not account for the additional air resistance due to
the fact that the skater's body follows a path which is not exactly
straight, so both the actual distance and velocity are slightly higher
than used in this model. If included correctly, it would tend to
increase Power expenditure for situations that cause larger
sidetoside oscillations of the skater (e.g. narrower angle of the
ski or skate blade or wheelframe).
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Some Findings and Analysis
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Not all power is created equal?
I wonder if some kinds of power expenditure are more critical than
others for limiting a skater's performance. Then it would be
important to focus mostly on improving the efficiency of those critical
kinds, perhaps even at the cost of becoming less efficient with other
kinds of power.
Perhaps the most critical kinds of power might be what is delivered by
the big muscles of the legs (including the quads and glutes). Two
kinds of expenditure that draw heavily on the big leg muscles are using
forwardforce Fx and sidewaysforce Fy.
One kind that draws much less on the big muscles is the work of lifting
the ski (or skate) and foot off the ground to prepare to step it forward. If we
added to the model the possible rotation of the body mass sideways about
the vertical axis  seems like that would expend some power, but not much
from the big leg muscles.
Wy_h  another way to think of it
The spreadsheet formula for Work in the sidetoside direction is based
on the basic principle that Work = Force * Distance: :
Wy_h = 2 * Fy * By_q
A different way to see it is that the Work must be equal to the
"sideways kinetic energy" lost in bringing the initial sideways
motion (Vy) to a stop during the first part of the legstroke, plus the
kinetic energy gained in getting the skater's Body mass back up to same
velocity going back the opposite way to the other side:
Wy_h = 0.5 * Mass * (Vy)^2 + 0.5 * Mass * (Vy)^2 = Mass * Vy^2
The danger of thinking this way is that splitting the total kinetic
energy of an object into directional "components" is not really
a legitimate way to use the kinetic energy concept, and can lead to
fallacies.
