Optimal discus trajectories - Hubbard, M.; Cheng, K. B. (Journal of Biomechanics, 2007)
A general 3-D dynamic model for men's and women's discus flight is presented including precession of spin angular momentum induced by aerodynamic pitching moment. Dependence of pitching moment coefficient on angle of attack alpha is estimated from experiment. Numerical integration of 11 equations of motion for nominal release speed v(0)=25 m/s and axial spin p(0)=42 rad/s also requires 3 other release conditions; initial discus flight path angle beta(0), pitch attitude theta(0), and roll angle phi(0). Optimal values for these release conditions are calculated iteratively to maximize range and are similar for both men and women. The optimal men's trajectory and range R=69.39 m is produced by the strategy beta(0)=38.4 degrees, theta(0)=30.7 degrees, and phi(0)=54.4 degrees. Initial angular velocities except spin are chosen to minimize wobble but an optimal initial spin rate p(0)=25.2 rad/s exists that also maximizes range. Optimal 3-D range exceeds that predicted by 2-D models because, although angle of attack and lift are negative initially, 3-D motion allows advantageous orientation of lift later in flight, with tilt of the axis of symmetry from vertical becoming much smaller at landing. Optimal strategies are discontinuous with wind speed, resulting in slicing and kiting strategies in large head and tail winds, respectively. Sensitivity of optimal range is largest to initial beta(0) and least to phi(0). Present calculations do not account for dependence of initial release angle or spin on release velocity or among other release conditions.