Summary. Generalized A(c~)-stable Runge-Kutta methods of order four with stepsize control are studied. The equations of condition for this class of semiimplicit methods are solved taking the truncation error into consideration. For application an A-stable and an A(89.3~ method with small truncation error are proposed and test results for 25 stiff initial value problems for different tolerances are discussed.

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.

The coefficient of friction of skis on snow appears to be influenced by several factors, for example, speed, contact area, snow type, and ski properties. The purpose of this study was to determine simultaneously the coefficient of kinetic friction and the drag area in straight running on a slope with varying inclination and in traversing on an inclined plane. Experimental measurements were taken using photo cells for straight running and by film analysis for traversing. The skier was modeled as a particle that moves on the surface of a slope. The equation of motion with the algebraic constraints of the track of the skier represents a differential-algebraic equation which was solved numerically. The coefficient of friction and the drag area were calculated by minimizing the sum of the square errors between computed and measured time data. For straight running, the computed coefficient of friction and the drag area were in the same range as obtained by other methods. For traversing, the coefficient of friction could be determined but not the drag area. The skier traversed in an upright position at a speed from 0 to 17 m/s. In this range of velocity the drag area is not constant. It corresponds to critical Reynolds numbers where a sudden drop in the drag coefficient occurs if the body segments are approximated by cylinders. The results indicate that in both cases the applied method is adequate for determining simultaneously the coefficient of kinetic friction and the drag area if these parameters are independent of the velocity.

Summary. This paper deals with the solution of nonlinear stiff ordinary differential equations. The methods derived here are of Rosenbrock-type. This has the advantage that they are A-stable (or stiffly stable) and nevertheless do not require the solution of nonlinear systems of equations. We derive methods of orders 5 and 6 which require one evaluation of the Jacobian and one LU decomposition per step. We have written programs for these methods which use Richardson extrapolation for the step size control and give numerical results.

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 ski's 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.