Martin Mössner

Calculation of the contact pressure between ski and snow during a carved turn in Alpine skiing

The macroscopic contact area between ski and snow and the contact pressure are crucial influencing factors for carved turns in Alpine skiing. In the present paper, a simulation model is developed to quantify these factors. The ski is modelled as an Euler–Bernoulli beam with variable cross section, camber, bending and torsional stiffness using measured data from skis. The reaction forces of the snow are decomposed in penetration and shear forces. For the penetration forces a hypoplastic constitutive law is applied incorporating elastic and plastic deformation of the snow at the contact area. For the shear forces metal cutting theory is used. Ski deformation, contact area and contact pressure are computed based on quasi‐static equilibrium between forces exerted by the skier and snow reaction forces. Parameter studies are performed to investigate the influence of edging and distributing the load between the inner and outer ski. Higher edging angles as well as loading both skis affected the contact pressure positively by increasing the resistance against shearing. The results of our study agree well with measurement data taken from literature. Based on the results, the importance of actions of the skier during carved turns is concluded.

Safety assessment of jumps in ski racing

The influence of important parameters on the flight trajectory for jumps in downhill World Cup races was investigated. To quantify the impact injury risk at landing, the parameter equivalent landing height (ELH) was introduced, which considered a variable slope inclination during the landing movement. Altogether, 145 runs at four different jumps in World Cup races and trainings were recorded and analyzed. A simulation model was developed to predict the flight phase of the skier. Drag and lift areas were selected by parameter identification to fit the simulation trajectory to the two‐dimensional data from the video analysis. The maximum values of the ELH which can be absorbed with muscle force was taken from the study of Minetti et al. for elite female and male ski racers. A sensitivity analysis based on the four jumps showed that ELH is mainly influenced by takeoff angle, takeoff speed, and the steepness of the landing surface. With the help of the developed simulation software, it should be possible to predict the ELH for jumps in advance. In case of an excessive ELH, improvements can be made by changing the takeoff inclination or the approach speed.

Modeling of the Ski-Snow Contact for a Carved Turn

Carved turns with alpine skis are investigated. During the movement of a ski, snow is loaded and unloaded. Compacted snow is not elastic, i.e. deformations remain. Such effects are modeled by a hypoplastic constitutive equation. During a turn the shovel digs into the snow and the tail maintains nearly the same penetration depth as the part under maximum load. This results in a higher resistance against shearing for the afterbody of the ski. In the present work we investigated the benefits of the hypoplastic against the elastic forcepenetration relationship. Simulation results for a sledge on two skis are compared to experimental track data.

Influence of Ski Bending Stiffness on the Turning Radius of Alpine Skis at Different Edging Angles and Velocities

Carved turns with Alpine Skis were investigated using a computer simulation model. Varied input data to the model were the bending stiffness of the skis, the edging angle, and the velocity. Results include the turn radius and the force distribution along the running surface of the skis.

Modeling the ski–snow contact in skiing turns using a hypoplastic vs an elastic force–penetration relation

A ski–snow interaction model is presented. The force between ski and snow is decomposed into a penetration force normal to the snow surface, a shear force parallel to it, and friction. The purpose of this study was to investigate the benefits of a hypoplastic vs an elastic contact for penetration in the simulation of skiing turns. To reduce the number of influencing factors, a sledge equipped with skis was considered. A forward dynamic simulation model for the sledge was implemented. For the evaluation of both contact models, the deviation between simulated trajectories and experimental track data was computed for turns of 67 and 42 m. Maximum deviations for these turns were 0.44 and 0.14 m for the hypoplastic contact, and 0.6 and 7.5 m for the elastic contact, respectively. In the hypoplastic contact, the penetration depth of the skis afterbody maintained nearly the same value as the part under maximum load, whereas it decreased in the elastic contact. Because the shear force is proportional to the penetration depth, the hypoplastic contact resulted in a higher shearing resistance. By replacing the sledge with a skier model, one may investigate more complex skier actions, skiing performance, or accident‐prone skiing maneuvers.

An approximate simulation model for initial luge track design.

Competitive and recreational sport on artificial ice tracks has grown in popularity. For track design one needs knowledge of the expected speed and acceleration of the luge on the ice track. The purpose of this study was to develop an approximate simulation model for luge in order to support the initial design of new ice tracks. Forces considered were weight, drag, friction, and surface reaction force. The trajectory of the luge on the ice track was estimated using a quasi-static force balance and a 1d equation of motion was solved along that trajectory. The drag area and the coefficient of friction for two runs were determined by parameter identification using split times of five sections of the Whistler Olympic ice track. The values obtained agreed with experimental data from ice friction and wind tunnel measurements. To validate the ability of the model to predict speed and accelerations normal to the track surface, a luge was equipped with an accelerometer to record the normal acceleration during the entire run. Simulated and measured normal accelerations agreed well. In a parameter study the vertical drop and the individual turn radii turned out to be the main variables that determine speed and acceleration. Thus the safety of a new ice track is mainly ensured in the planning phase, in which the use of a simulation model similar to this is essential.

Calculation of Friction and Reaction Forces During an Alpine World Cup Downhill Race

Understanding friction and reaction forces involved in Alpine Skiing is of great theoretical importance for sport science. We have developed a method to analyze a skier’s motion during a downhill race from video data taken by a single camera. This may help to compare the technical equipment and the skills of different skiers.

A Method for Obtaining 3-D Data in Alpine Skiing Using Pan-and-Tilt Cameras with Zoom Lenses

The direct linear transformation (DLT) was applied for three-dimensional reconstruction in Alpine skiing. Because of the large field of view it is necessary to follow the skier with the cameras to receive frames with a suciently large image of the skier. In our implementation, the DLT parameters are computed for each frame of each camera separately. Consequently it is possible to rotate the cameras and to zoom the lenses. The locations of the cameras need not be known. Control points were distributed on the slope. Their coordinates were determined by geode- tic surveying. The method requires at least six control points visible in each frame. Using more than six control points reduces the reconstruction error considerably. Under optimal conditions the mean errors are less than 5 cm in each coordinate. This accuracy is sucient for inverse dynamics if the data are smoothed and the

Friction Between Ski and Snow

The first skis were developed to improve locomotion across the natural, wind packed snow surface in the European northern countries. The skis were made of flat planks with shovels at the tips. Under load, the tips and ends of the skis bended up causing resistance against forward movement. An improvement of the gliding of skis was the invention of the bow-shaped cambered ski, arched up towards his center. Under load, the ski lies flat on the snow surface and the load is more evenly distributed along the ski. With the appearance of downhill skiing, the turning properties of skis became more important. In 1928, Lettner (AT) invented steel edges to give the skis more grip. During the first half of the twentieth century, the technique was developed to produce laminated skis composed of a wooden core with different bottom and upper layers. In 1955, Kofler (AT) introduced the first ski with a polyethylene base, which remarkably improved the gliding properties. In addition, the repair of minor scratches was easily possible. In the recent past, the gliding properties of skis were further developed by special grinding techniques for the ski base and by the development of special waxes.

Reaction Forces and Moments in Carved Turns

We computed reaction forces and moments acting on a skier during a carved turn. We performed an inverse and a forward dynamic analysis. For a run of an elite skier, marker positions on skier and skis were obtained as functions of time from a video analysis and smoothed by splines. Linear velocities and accelerations were computed by differentiating the splines, angular velocities, and accelerations via rotation matrices. The forces acting at the right ski were measured with two Kistler force plates. For the inverse dynamics, we used an adapted Hanavan model for a skier consisting of upper body, left and right thighs, shanks, and skis. Applied forces considered were weight and ski-snow friction. Drag was neglected. By prescribing a lateral weight distribution from the outer to the inner ski during the turn, reaction forces and moments at the left and right ankle, knee and hip joints were computed from the Newton–Euler equations of motion for constrained rigid multibody systems. The forward dynamics was performed with a three-segment model of a mono-skier consisting of trunk, thigh, and shank. Rotational joints were assumed in knee and hip. The track and the joint angles were prescribed. The inward lean angle was determined by a balance condition that led to nonholonomic constraints. After formulating the equations of motion in descriptor form, the resulting differential-algebraic system was solved with the numerical code RADAU5. Computed and measured reaction forces and moments agreed well within the accuracy of the measurements. The calculated joint loads are consistent with results from the literature. The forward dynamics model can be used to simulate consecutive ski turns. With parameter studies, the effects of slope, tracks, segment properties, ski-snow friction, and velocity of the skier on joint loads and performance of a run can be investigated. Further, injury mechanisms can be analyzed.