Cross-country skiing motion equations, locomotive forces and mass scaling laws

Abstract

This article presents differential equations for locomotive force and velocity during cross-country skiing. A muscle's work power is modelled. Thereafter, a locomotive force that depends on the skier's velocity is constructed. The external forces aerodynamic drag, friction forces and the force of gravity are incorporated in order to provide the equation of motion. Some allometric mass scaling relations are established and used to analyse the effect of a skier's mass on velocity. The model is tested by using a GPS instrument. We compare analytically and experimentally determined skiing distances and velocities as functions of time, and under different conditions. The article provides tools useful for practising athletes and coaches.

Numerical optimization of pacing strategy in cross-country skiing

“…van Ingen Schenau and Cavanagh (1990) and Moxnes and Hausken (2008) described the equations of motion in endurance sports and cross-country skiing respectively. Carlsson et al (2011) came up with a model which can simulate a skier along a predetermined course profile by utilizing these motion equations.…”

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

“…Thus, slow muscle fibers exert higher oxidative power than fast muscle fibers. The literature has also provided a model for a muscle's force during contraction accounting for slow and fast isoforms (Petrofsky and Phillips 1980;Chow and Darling 1999;Camilleri and Hull 2004;Moxnes and Hausken 2008a), hereafter referred to as the lumped Hill force, expressed as…”

“…BW is the body weight in kg, UBS 60 is the upper body torque in N m at 608/s (around 130 J), QE 60 (around 200 J) is the quadriceps extensor torque in N m at 608/s and QE 240 (around 100 J) is the quadriceps extensor torque in N m at 2408/s. See Moxnes and Hausken (2008a) for further studies on mass dependency and mathematical models.…”

Comparison of Power Output Estimates in Treadmill Roller-Skiing

“…Furthermore, Bergh [3] introduced a power balance model for cross-country skiing which he used for allometric scaling analysis of cross-country skiing performance. Moxnes et al [4] derived a motion equation, similar to Bergh's power balance model [3], utilized for simulation and compared these results to GNSSpositioning data for an athlete performing cross-country skiing. In this study, physiological measurements and modeling were used to estimate metabolic power and power output for the simulations.…”