Simulation of the speed and the normal acceleration in luge - Mössner, M.; Hasler, M.; Schindelwig, K.; Müller, P.; Becker, M.; Kaps, P; Nachbauer, W. (2009)
Introduction Luge track design requires detailed information on the expected driving dynamics during the entire run. Any misjudgment in the development phase of a track may lead to inadmissible high speeds and, consequently, to dangerously high accelerations acting on the luge runner. Therefore, the purpose of this paper was to develop a simulation tool, to predict the speed and the acceleration acting on a luge runner during the whole run. Method For the simulation of the speed and the accelerations we solved the one-dimensional equation of motion along the real trajectory of a luge runner in the track. In a first step altitude h, inclination a, turn radius r, and distance s along the baseline of the track were taken from construction plans of the Whistler Sliding Centre. In the schussing parts the track was considered to be flat and in the turns the cross section was assumed to be elliptical with half axes taken from construction plans, again. In the crossings from schussing to turn sections the baseline followed a clothoid with cross sections smoothly changing from flat to elliptical. The forces considered were 1) the weight force Fw = m g, which was decomposed in the propulsive force Fp = Fw sin a and the normal force Fn = Fw cos a, 2) the drag Fd = ½ r CdA v2, 3) the ground reaction force Fr, and 4) the friction force Ff = m Fr. During schussing the ground reaction force was given by the normal force: Fr = Fn and in the turns, including entrance and exit, by the norm of the vector sum of normal force Fn and centrifugal force Fz = m v2 / r, hence Fr = (Fn 2+Fz 2)½. Consequently, the equation of motion along the real trajectory of the luge runner was given by m a = Fp - Fd - Ff, leaving the problem of establishing equations for the location of the real trajectory. The trajectory had to be defined from the force balance in transversal direction and therefore was given in implicit form, only. In the simulation we assumed that the luge runner did not perform any steering movements, meaning that there was not a transversal force component present or, vice versa, that the ground reaction force was normal on the track surface. In schussing parts the location of the trajectory was the midpoint of the cross section. In the turns the location was that point of the elliptical cross section for which the vector of the ground reaction force, with angle c = atan(Fn / Fz), was normal to the ellipse. The solution of the differential equation was obtained iteratively by applying Euler steps. In every step the location of the trajectory was obtained by solving the transversal force equilibrium using a, r, and v of the previous integration step. To ensure convergence we used sufficiently small time steps, causing a movement of the luge of 1 mm per integration step along the baseline of the track. Finishing this process we obtained the trajectory in the track as well as the runtime, the speed, and the forces as function of the length parameter s along the baseline of the track. Test runs were performed in an official training with an elite luge runner (mass m = 113 kg with his luge) to collect run-times for five sections. In a parameter identification process these interim times were used to calculate the coefficient of friction m and the drag coefficient CdA by minimizing the deviation of the five interim times to the runtime of the luge runner in the simulation. For verification purposes the velocity at the entrance of the 180° turn was measured. Additionally one luge was equipped with a 3d accelerometer and was used in the runs to record accelerations in normal and transversal directions during the runs of an elite luge runner. Results/Discussion The simulation tool proved to be suitable to simulate the trajectory, speed, and normal acceleration of a luge runner in the track of the Whistler Sliding Centre. With the parameter identification process we calculated the coefficient of friction to be m = 0.0142 and the rag coefficient to be CdA = 0.050 m2. Due to table values the coefficient of friction for steel on ice is between 0.010 and 0.015 and the drag coefficient is, due to unpublished wind tunnel experiments, equal to 0.0495 m2. Both values agree very well with our findings. With these values the overall runtime could be reproduced exactly and the runtime differences for the five track sections were below 0.45 s. The calculated speed at the entrance of the 180° turn was 40.5 m/s, whereas the measured velocity was 40.7 m/s. Finally, in Fig 1 the simulated versus the measured acceleration normal to the ice surface is given. Because of the very good agreement of measured and computed accelerations the tool can hopefully be used as aid for planning new luge tracks.