On the Multi-Scale Nature of Ski-Snow Friction : Field Testing, Characterisation and Modelling of Cross-Country Ski Performance

Sammanfattning: The goal of most endurance sports is to get from point A to B in the shortest time possible. Throughout the course, athletes need to overcome the resistive forces that are present in their specific sport. Cross-country skiing is no exception to this, and there are mainly three resistive forces acting on a skier, the aerodynamic drag force, the friction force, and the gravitational force in the inclined parts of a track. The present work focuses on the resistive force of friction between the ski and the snow. At the highest level of ski sports, a large effort is made to reduce the ski-snow friction, and a small reduction in friction can have a large impact on the race outcome. Several aspects are considered when skis are chosen and prepared to minimise friction, ski-camber profile, ski-base material, ski-base texture, and ski-base preparation.  Depending on the prevailing snow and weather conditions, different friction mechanisms are thought to be dominant, so the choices of skis and preparations must be carefully considered and tested accordingly. The present work focuses on the multi-scale nature of ski-snow friction from a contact mechanical point of view. Where the ski-camber profile is separated into the macro- and meso-scale, and the larger deformations i.e. the macro scale are incorporated, thou ski-camber profile measurements. The smaller deformations in the snow due to the contact pressure are considered as the meso scale, where the contact mechanical response is evaluated and characterised in terms of apparent contact area and pressure. The ski-base texture is categorised as the micro-scale; on this scale, the elastic modulus is modelled as ice which is several times stiffer than the meso-scale snow. The contact mechanical response is characterised in terms of the parameters, real contact area, average interfacial separation, and average reciprocal interfacial separation. The multi-scale nature of the ski-snow contact is coupled through the apparent pressure, which acts as a load condition for the micro-scale contact simulation and considering both scales, the micro-scale parameters can be evaluated along the entire ski. To correlate the characteristics obtained from the multi-scale simulation to ski-snow friction, a full-scale ski-snow tribometer was developed. The tribometer was built to mimic an athlete on skis while performing the G7, in terms of load magnitude, positioning and transfer interface. To do so an athlete’s plantar pressure distribution was measured and analysed in different variations of the G7 position. The neutral position, resembling a load position of 55% of the athlete’s foot measured for the toe, was chosen for the tribometer. A replica of a ski boot was developed for the tribometer, herein called the measurement boot, to make it possible to use skis equipped with a regular NNN-binding system on the ski tribometer. The impact on the ski-camber profile from using the measurement boot was also studied, results showed that since a ski boot transfers the load on a larger area, the ski will collapse more, compared to the conventionally used block that is designed to fit the binding system. The tribometer from here on called the sled can be equipped with a pair of skis, where the width of the ski fits a classic ski track. The sled was designed to be loaded with regular Olympic weights to enable a large variety of different loads. During field measurements, the sled is accelerated using a downhill slope, and the velocity and position are measured using an RTK-GNSS system. The retrieved data in terms of time, altitude, and velocity, was used to calculate a mean coefficient of friction for reaching individual runs while accommodating for aerodynamic drag, centripetal force, and slope angle. During the winter period of January-February 2024, a measurement campaign was carried out to evaluate the influence of the simulated apparent and real contact area in cold conditions and hard tracks. Eight skis with different apparent contact areas were equipped with 3 different ski-base textures i.e. developing different amounts of real contact area.  Results from the friction tests indicate that there exists a different optimum combination of apparent and real contact area at the snow temperatures of -3.5, -8 and -13.5 degrees. At the warmer -3.5 degrees, a small apparent and real contact area exhibited the lowest friction, and at -13.5 degrees the opposite trend where a large apparent and contact area exhibited the lowest friction. For each condition, an empirical model was developed based on the variables apparent and real contact area.Using the developed ski-snow contact models, a pair of skis and its ski-base texture can be characterised and the frictional performance in cold conditions with hard track can be estimated using the empirical model. Employing this method makes ski and ski-base texture selection possible before testing, thus contributing to a more efficient way of conducting ski selection.