a mathematical spreadsheet model of motions and power in skating

Ken Roberts

what's here

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What can this model be used for? 

  • compare the effectiveness of some choices in skating technique -- differences in power and peak force required for different selections of turnover frequency, angle of ski (or ice skate blade or inline skate wheel-frame), passive glide. 

  • see some of the basic physics that delivers the "magic" of skating 

  • estimate the impact on your own effective power output of some non-technique factors -- speed, hills, wind, body weight  

  • see "Questions to play with" below 

  • see other "advanced" uses below


  • no use of ski poles or arm swing motion -- leg-push only

  • constant forward speed 

  • uniform leg-push force during the active push phase (which is unlikely to be close to correct).

  • the ski (or skate blade or wheel-frame) must travel in a straight line on the surface of the snow (or ice or pavement) -- no curve.

  • assumes no force or power is expended on moving the skater's body mass up and down (which is far from correct). 

  • assumes no torque or power is expended on rotating or turning the skater's body about the vertical axis. 

  • requires "informed guesses" about some of the input variables that are impossible for most of us ever to measure accurately.

  • see also "over-simplications" on the Physics of Skate Power page

  • Because of these limitations, the model is not very useful for estimating the actual power expended by any skater, but rather for getting a sense of the relative impact magnitude of different factors and for playing with the concepts of the physics.   

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How to use it

Open the spreadsheet

The first thing is to open the spreadsheet, using spreadsheet software on your computer (such as Microsoft Excel): 

spreadsheet in Microsoft Excel version 5

We hope this older version of Microsoft Excel can easily be imported into newer versions of Excel, and also other spreadsheet programs. 

You may want to save a copy on your local computer, so you can input your own personal factors like body mass and velocity, and save some of your results.

Change the values of the input factors

The cells that are are outlined like a box hold the values of the usual "input" factors that you will want to change and play with. 

You can type in a new value in the box and see it changes some of the other values. 

In the version as given, some of the input values are calculated by formulas from other inputs, but you can over-ride that by just typing in the value you want.

This spreadsheet gives you the freedom to enter combinations of input values that make no sense physically or biomechanically for a real human skater.  The "Constraints" try to give some help in avoiding this, and so do some of the formulas given in some input boxes.  

But in the end the "reality" of the combination of inputs is controlled by your own judgment and understanding.  Some of the notes in the "Comments" column try to provide some guidance. 

Supporting Variables:  These are in a separate column, but some of them are very important for the calculations.  Just leaving them alone could be OK for answering some questions, but being careful with them could be critical for other questions.  Especially the "Friction_Coefficient" could vary widely depending on  condition of the snow (or pavement).  And if you think skating technique changes a lot for going up a steep hill, the "Hill_Grade" value will be important for you.

Check the output results

These are under "Calculation of Results"

Some key output results are: 

  • Power -- Total power (P_total), and also check some of its components like Power going into forward motion (Px_total) versus side-to-side motion (Py_total)

  • Force Fn = "force perpendicular to the ski" (or skate blade or wheel-frame) is the peak leg-push force.  Leg muscles get stressed and tired quickly when this force gets large. 

  • Passive Glide distance:  this is actually above under "Technique factors", but if you select a large enough angle of the ski (or blade or wheel-frame), the spreadsheet formulas might show some passive glide even if that is not what you intended.

Some other ones to watch: 

  • sideways distance of the skater's Body center:  By_q  

Constraints and Warnings

Near the bottom of the spreadsheet are some "Constraints".

The spreadsheet cell to the right of each constraint should normally say "OK".  

If they say something other than "OK", like an Error or Warning message, you may be  trying to make your model skater do something physically impossible. 

Questions to play with

  • Angle of ski (or skate blade or wheel-frame): wider or narrower. What's the impact of this factor just on its own? 

  • Is it better for the ski and skater to track closer together (compare Sy_q with By_q) -- or for the ski to go wide while the skater stays narrow?  What do you have to do with the input variables to make the skater go wider side-to-side (i.e. how to increase By_q)? 

  • Passive Glide phase:  more or less or none?  (watch for resulting peak leg-push Force (Fn) required, not just power expenditure) 

  • cadence:  higher or lower turnover frequency? 

  • combinations:  What happens when you combine changes in ski angle with other factors like cadence and passive glide phase? 

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What some of the variables mean

The key things we are modeling is the motion of the skater's body center of mass and the angle and motion of the pushing through the ski -- and the forces between the snow (or ice or pavement), the ski (or skate), and the skater.  The motions of the skater's body center and the  pushes on the ski (or skate) are like this: 

[click on it to view larger image]

At the start of each new leg-push stroke, the body center of mass is already moving sideways toward the new pushing ski.  For simplicity we assume that the ski or skate moves forward and outward at a constant angle.  The ski or skate pushes outward and backward against the snow (or ice or pavement), and the snow (or ice or pavement) "reactively" pushes the skater's body inward and forward.  At first the inward component of this push slows the sideways motion of the skater's body center, then stops it and turns it around back toward the opposite side.

Here's more detail for a single leg-stroke:  total leg-push force vector shown as red arrow, forward and sideways components of force shown as pink arrows: 

[click on it to view larger image]

Passive glide

A "passive glide phase" is when the skater's weight is on the ski, and the ski or skate is gliding on the snow (or ice or pavement), but the ski (or skate) is not pushing sideways against the snow (or ice or pavement).  A passive glide phase is not shown in the two diagrams above. 

As given, the spreadsheet calculates the length of "passive glide" -- zero if the ski angle q (Theta) is small enough so that the sideways move of the center of the ski or skate (Sy_h or syh) during a single leg-stroke is less than or equal to the maximum permitted by the skater's biomechanics (Sy_max_h or syhmax).  If the angle of the ski (or skate blade or wheel-frame) is too large for that, the spreadsheet selects the length of the active push phase so that Sy_h is equal to Sy_max_h, and takes the rest of the forward motion during the leg-stroke as the passive glide phase. 

But you can over-ride this by entering your own formula or value into the cell for "Forward motion distance (single-leg-stroke)", Bx_h or bxh.  (Directly changing the value or formula in the "Passive glide distance" cell has no effect on anything else in the spreadsheet).

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"Advanced" uses

  • Add your own formulas to the spreadsheet to make the physics and biomechanics more sophisticated; e.g., make the maximum leg-push range (Smax_h) dependent on peak leg-push force (Fn).  Or add some assumptions about the skater's body mass moving up and down, the forces and power used for that, and what portion of that power can be converted to contribute to forward motion. 
  • Set up your spreadsheet software so you can hold power fixed, and "solve" for impact of changing various factors on forward velocity. (or perhaps some other variable currently used as an input). 

more . . . 

see also

back to Top | more Skating | FAQ | Resources | XCski index