analysis of the physics of a spreadsheet model 
of motions and power in Skating

Ken Roberts

what's here

• Derivation of the model 

• some Findings and Analysis  

• more notes 

see also

• main Skate Power model page 

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Derivation of the model

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 leg-push force F_n and sideways force F_y are determined by q, the angle of the ski (or ice skate blade or inline skate wheel-frame).

By symmetry, the sideways velocity at the end of one leg-push stroke must be the same as the velocity at the start of the next leg-push stroke, so the force-mass-acceleration 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 leg-push stroke can be calculated by Work = Force * Distance.  Finally Power is calculated by dividing the Work by the time for a single leg-push, 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. 

Over-simplifications

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 non-uniform 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 higher-than-average 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 side-to-side oscillations of the skater (e.g. narrower angle of the ski or skate blade or wheel-frame). 

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Some Findings and Analysis

[ to be added ]

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more notes

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 forward-force Fx and sideways-force 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 side-to-side 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 leg-stroke, 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.