Computation of the trajectory and the reaction force for a turn in luge - Mössner, M.; Kaps, P.; Hasler, M.; Nachbauer, W. (2010)
To simulate luge running we solved the 1d equation of motion for an entire run [1]. Although the model proved to be suitable to predict speed and normal acceleration of the luger, it might still be too simple. Therefore, the aim of this study was to compute the trajectory and the ground reaction forces for a luger in a turn by solving the 3d equation of motion on the track surface. Method A 172° turn combination was implemented consisting o f entrance, main turn, and exit, extending 60 m along the baseline. In the entrance the baseline follows a clothoid with a final radius R = 15 m which equals the radius of the turn. The cross sections were sampled every meter from a real track. The track surface data of the entrance were fitted by a 2d-polynomial in s and r, where s denotes the distance along the baseline and r the distance normal to the baseline. The main turn has a constant cross section, which coincides with the closing cross section of the entrance. Finally, the exit was constructed as reversal of the entrance. After back-transformation to the global x,y,z-coordinate system we got a differentiable constraint g = h(x,y) - z = 0 for the track surface and its spatial derivative G = (¶ h /¶ x,¶ h /¶ y, -1). The next step was to specify the 3d equation of motion for the luger. The forces considered were 1) the weight force Fw = m g · (sin a, 0, cos a), 2) the drag Fd = - ½ r Cd A |v| v, 3) the friction force Ff = - m Fr v / |v|, and 4) the ground reaction force Fr. Note, we use this representation to have the x and y coordinates parallel to the slope plane. The luger itself was modelled as a point mass moving along the surface of the track. Therefore the 3d equations of motion are: m a = Fw + Fd + Ff + Fr , where Fr is given by Fr = - GTl . l denotes the Lagrange multiplier. Together with the constraint g we get a differential-algebraic equation (DAE), which was solved by the numerical method PHEM56 [2]. When solving DAE`s, one does not need to know Fr. It is determined by the Lagrange multiplier l which is computed during the numerical integration. Further, PHEM56 computes precise estimates not only for the coordinates q and velocities v but also for the Lagrange multiplier l. Results/Discussion The equation of motion for a luger could successfully be solved on a surface which was derived from a real track. The computed trajectory of a centered and straight in-run for the three initial speeds 10, 14, and 18 m/s are given in Fig.1. For the low speed (light color) the luger passes the turn combination near the baseline and for the large speed (dark color) the path oscillates up and down. This oscillat on feature is, due to the opinion of professional luge coaches, immanent for passive luge running. In praxis a luger would try to steer the luge in such a way that the luge stays at constant height in the track. A first step to improve the model is to replace the point mass of the luger by a single rigid body. With this modification the runner`s center of mass is separated from the track surface. Shearing forces can be applied and the orientation of the luge with respect to the path can be computed. Due to the oscillating path the ground reaction forces varied between 1300-1500, 1400-2100, and 2200-3200 N for the speeds of 10, 14, and 18 m/s, respectively. The commonly used estimate for the ground reaction force Fr e = (Fw 2+Fc 2)½ with Fc = m v2 / R amounts to 1340, 1840, and 2680 N, respectively. Finally, we have to mention that the results of the 3d simulation are in good agreement with the 1d-simulation of [1].