A Model for the Turning Snow Ski

A model is developed to investigate the steady-state solutions of a turning snow ski. A rigid body skier executes a constant radius turn on a horizontal snow surface. Two types of forces are assumed along the ski bottom: skidding forces, which arise when shearing a new snow surface, and carving forces, which maintain the curvature of the ski as part of the running surface of the ski slides along its own track without shearing the snow. Using the equations of conservation of linear and angular momentum, dynamic equilibrium is established for the three forces and accelerations acting on the skier/ski system: the force generated along the flexing ski, gravity, and the centripetal acceleration of the turn. Turn radius and speed are insufficient to specify a unique solution to these equations. To quantify the effects of different ski designs on the model, a range of possible dynamic equilibrium solutions is calculated for each turn radius and speed and compared using two indices, one measuring the mechanical efficiency of the turn resulting from the energy dissipation of shearing the snow surface, and the second measuring the equilibrium solution sensitivity to human control and regulation. When possible solutions are mapped onto the plane of these two indices, a characteristic solution behavior emerges, which has been corroborated with experimental results found using a model ski turning on Astroturf.® This behavior provides a quantitative basis for comparing different ski designs and proposing new design strategies.