A dynamic model of Nordic diagonal stride skiing, with a literature review of cross country skiing

Abstract

The forces during the kicking phase in Nordic diagonal stride skiing are described by differential equations and the results are compared with experiments. The difference between static and dynamic friction, interacting with characteristics of the skier such as weight, velocity and the kicking force's angle with the terrain, are essential for high-velocity diagonal striding. Analytical results for relationships between glide length, friction and kicking force are shown. Aerodynamic drag and gravity are accounted for. A propulsion force based on the Hill (1970) equation for muscle contraction velocity and activation is constructed. The model shows a feasible tool for studying the effects of ski stiffness, the kicking force and the amount of waxing during diagonal stride skiing.

Keywords:

• classical cross-country skiing
• diagonal stride technique
• forces
• friction

Acknowledgements

We thank an anonymous referee of this journal for useful comments.

Notes

^1. [email protected]

^2. E.g. Roald Amundsen's 1910–1912 South Pole expedition (Huntford 1999). In Norway, two military battalions were set up on skies from 1747 to 1826 to control the border against Sweden. Actually, these were the elite soldiers of the Norwegian army (Steen 1996; Rasch-Engh 1997; Huntford 1999).

^3. Christensen and Høgberg (1950), Åstrand and Saltin (1961), Jette et al. (1976), Stromme et al. (1977), Thomas et al. (1989), Mygind et al. (1994) and Larsson and Henriksson-Larsen (2005).

^4. Skiing cost or ski economy is defined as the aerobic power divided by the velocity of the skier.

^5. MacDougall et al. (1979), Bilodeau et al. (1991), Bilodeau et al. (1992), Mognoni et al. (2001) and Saibene and Minetti (2003).

^6. Moxnes and Hausken (2008a) classify cross-country skiing as consisting of the classical style and skating. The techniques for the classical style (also called Nordic style) are diagonal stride, double stride with kick and double pole. The techniques for skating are paddling (V1), double dance (V2, also called 1-skate) and single dance (V2-alternate).

^7. One of the authors is an enthusiastic cross country skier. He uses two different types of skis, one with a long chamber and one with a short chamber. Certainly, the feel is very different when using the two types of skies.

^8. See Spring et al. (1988) for a discussion of the drag area of a skier.

^9. The net efficiency is the specific work power divided with the net aerobic power. The net aerobic power is the aerobic power minus the basal aerobic power (when the body is at rest).

^10. The work efficiency is the specific work power divided with the aerobic power minus the power used to move the legs and arms without the external load. The gross efficiency is the specific power divided with the aerobic power.

^11. The phosphorylative coupling efficiency is the amount of Gibbs free energy in the ATP divided with the amount of free energy in the substrate (glucose/glycogen or fat).

^12. The contracting-coupling efficiency is the work power during use of a muscle divided with the Gibbs free energy of the ATP used during the contraction.

^13. Work here is used as a biological work requirement, i.e. biosynthesis and active transport muscle contractions.

^14. One of the authors is both working as a scientist and as a coach. This author usually now moves around 2000 km on ski per year. The other author varies between multiple sports disciplines.

^15. The skier plants the left pole more or less simultaneously with kicking with the right foot, which provides arm thrust and foot kick on the left and right sides, respectively, of the skier's body. These two propulsion forces, from the ski and the pole, provide virtually no torque around the vertical axis.

^16. Kuzmin and Tinnsten (2005, 2006) argue that the ski base properties are essential, and provide good wear resistance, a low friction coefficient and a good capacity for self-lubrication for today's skis made of Ultra High Molecular Weight Polyethylene, which is five times more wear-resistant than carbon steel. They thus recommend against an inferior material like ski glide wax, to which dirt and other substances get attached, which decreases gliding. We thank an anonymous referee for this reference.

^17. Cavanagh and Kram (1985) explained optimal phenomena in human motion for multijoint movement by referring to the Hill force–velocity curve for isolated muscles.

^18. The differential equations are solved by defining the time derivative as and solving numerically for Δt = 0.0001.

^19. All experimental data in this article is read off visually from the curves of Vähäsöyrinki et al. (2008) without accounting for all the points that seems to be spread somewhat randomly for the EMG signal.

Additional information

Notes on contributors

John F. Moxnes

 1.  1. [email protected]