MacDonald, J.S., Cardales, R.O., & Stockie, J.M. (2024). A Mathematical Model for Nordic Skiing. arXiv preprint arXiv:2410.02767v1.
This research paper aims to develop a more realistic and accurate mathematical model for simulating the dynamics of a Nordic skier traversing a 2D course, focusing on incorporating smooth course representation and accurate force calculations.
The authors employ a system of ODEs derived from Newton's laws of motion to model the skier's speed and position as a function of time. They utilize Hermite spline interpolation to represent the course elevation profile smoothly, overcoming limitations of previous models that used piecewise linear approximations. The model considers various forces acting on the skier, including propulsion, gravity, snow friction, and aerodynamic drag, with parameters derived from experimental data. Numerical simulations are performed using MATLAB's ode15s solver, which offers adaptive time-stepping and event detection for enhanced accuracy.
The study demonstrates that using a smooth Hermite spline interpolant for the course geometry leads to more realistic simulation results compared to piecewise linear approximations. The model accurately captures the interplay of forces affecting the skier's speed and acceleration, particularly on varying terrain. The authors highlight the importance of accurate course representation and force calculations in achieving realistic simulations, especially in competitive scenarios where small time differences are significant.
The proposed model, combining ODEs and Hermite spline interpolation, provides a robust and accurate framework for simulating Nordic skiing dynamics on 2D courses. The use of a smooth course representation and detailed force calculations enhances the model's realism and applicability to real-world scenarios, including race strategy analysis and performance prediction.
This research contributes to the field of sports science by providing a sophisticated yet accessible mathematical tool for analyzing and understanding the complex dynamics of Nordic skiing. The model's ability to accurately simulate skier performance on varying terrain makes it valuable for athletes, coaches, and researchers seeking to optimize training regimens, evaluate course designs, and gain deeper insights into the sport's biomechanics.
The current model focuses on 2D course representation, neglecting the effects of track curvature and skier turning dynamics. Future research could extend the model to incorporate 3D course geometry and simulate more complex skiing maneuvers, such as cornering and braking. Additionally, incorporating physiological factors like fatigue and varying snow conditions could further enhance the model's realism and predictive capabilities.
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