Mechanics of alpine skiing: turn with side-slip and ski judder on carving skis

In the works published to date, considering the question of mathematical modeling of a ski sliding turn, the parameters of a side cutout of a certain radius characteristic of the design of modern alpine skis are not taken into account. The presence of such a cutout is one of the main factors of the behavior of the ski when performing any turn, including a side-slip turn. The presented work is devoted to the creation of a numerical model of the movement of the mechanical system "skier-skier" when performing a turn on carving skis with lateral slippage and ski judder. In this case, the turn is considered under conditions of relative equilibrium in the reference frame associated with the moving center of mass and taking into account the angulation controlled by the skier. As the main assumption of the model, it is assumed that the movement in a sliding turn can be decomposed into orthogonal directions: lateral sliding and movement along the skis according to the algorithm corresponding to a clean-cut turn (carving). The model takes into account the forces of resistance to movement - the sliding friction force and the aerodynamic drag force. For greater generality and universality, the developed numerical model is implemented for dimensionless motion parameters, namely, to describe the dimensionless kinematic characteristics of the motion of the center of mass of the mechanical system under consideration. A system of integral equations is obtained, with the help of which it is possible to find the speed, time and trajectory of the center of mass, as well as to estimate the angles characterizing the position of the skier's body and his skis relative to the slope with various stylistic angulations set by the athlete. The created model is illustrated by solutions of problems for cases with various dependences of the sliding angle on the angle of movement controlled by the skier.