Turns in Alpine Skiing

In this project we want to develop a multibody model for a skier who performs consecutive turns on compacted snow. The skier is modeled by rigid segments. The equipment cannot be treated as rigid. One has to consider real skis with side cut, camber and given flexural and torsional stiffness. Additionally, one has to take into account the stiffness of ski boots, binding plates and bindings. In order to establish the equations of motion they are written in descriptor form as differential algebraic equations (DAEs). As variables generalized Cartesian coordinates are used, i.e. the position of the center of gravity of the body segments and Euler parameters for their orientation. The joints lead to holonomic algebraic constraints depending on the coordinates only. The group of W. Nachbauer is leading in modeling turns and has been using the software Virtual.Lab [LMS]. For realistic turns one has to solve two main problems: firstly one has to model the snow reaction forces and secondly one needs a strategy to keep the skier`s balance. The snow reaction forces are treated in a proven way as in [MHK]: a hypoplastic constitutive equation for the normal component and metal cutting theory for the shear forces. For low velocities the model is validated by a sledge with skis instead of runners [MHS1], [MHS2]. Such a sledge can be considered as a one body model of a skier. At higher velocities one might have to consider the snow jet according to Hirano-Tada [HT1]. The balance problem is difficult. To keep balance the skier has to lean inward such that the resultant force between gravity and centrifugal force is directed to the area between the skis. When the lateral component of that resultant force exceeds the ultimate shear force of snow skidding occurs. The ultimate shear force linearly depends on the penetration depth of the skis into the snow. Thus, the penetration depth is a crucial factor. Compacted snow is hardly elastic. Deformations remain. In a carved turn the front parts of the ski dig a track into the snow and the rear parts remain in this track. Depending on the treatment of the centrifugal force the balance problem leads to nonholonomic constraints which may depend on velocity, acceleration or Lagrange parameters. Also, inequalities might occur. Such constraints cannot be handled by Virtual.Lab. Therefore, we want to develop own methods to solve DAEs with nonholonomic constraints. The numerical analysts participating at the project have a considerable experience with DAEs.

During turning a skier steers his run by edging and loading the skis. The shape of the turns depends on the ski and the snow properties, too. The aim of the present project was the development of a model for simulating turning as realistic as possible. It consists of three submodels: a multibody model for the skier, an Euler-Bernoulli beam as ski model and a ski- snow interaction model. The multibody model for the skier consists of 8 rigid segments: upper trunk together with head, arms and ski poles, lower trunk, left and right thigh, shank and ski. The shanks comprise the ski boot leg. The ski comprises the foot, the lower part of the ski boot, the binding and the damping plate. Ankle and knee joints as well as the joint between upper and lower trunk were modeled as rotational joints, the hip joints as spherical joints. Muscle forces are considered by a realistic choice of the joint angles. With this skier model the edging angles and the loading of the skis by the skier were calculated. The skis were modeled as Euler-Bernoulli beams with real geometric properties (length, width, thickness, side cut, camber) and real mechanical properties (mass, inertia, flexural and torsional stiffness). The loading consists of the forces and torques which skier and snow exert on the ski. With this model penetration depth and torsion are computed. The forces between ski and snow are decomposed into 3 components: normal force perpendicular to the hill snow surface, shear force and friction force. In a carved turn, the front part of the ski digs a trace into the snow and the rear part moves on in this trace. The penetration of the rear part remains approximately as large as under the binding, the region with the largest loading. This behavior is modeled by a hypoplastic constitutive equation. The motion of the body segments is computed by solving the equation of motion for the skier model numerically. The equation of motion is established in descriptor form as differential- algebraic equation. Drag is considered, skidding is admitted when the shear force becomes too large. For validation a video analysis was provided for a run of an expert skier with two and a half turns. By an adequate choice of the joint angles the skier model could follow the motion of the skier accurately. The distances of the bindings between video and simulation were smaller than 50 cm. This is a remarkable accuracy if one considers that the length of the test run was 36.45 m. For future research it is planned to investigate ski vibrations and muscle forces. For transversal vibrations of skis not edged we have obtained promising results. The presented model could serve as basis for investigations on reducing the injury risk in skiing and on the construction of skiing equipment. It is relevant for a general understanding of skiing mechanics.