Optimal discus trajectories

Abstract 

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 α is estimated from experiment. Numerical integration of 11 equations of motion for nominal release speed v₀=25m/s and axial spin p₀=42rad/s also requires 3 other release conditions; initial discus flight path angle β₀, pitch attitude θ₀, and roll angle φ₀. 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=69.39m is produced by the strategy β₀=38.4°, θ₀=30.7°, and φ₀=54.4°. Initial angular velocities except spin are chosen to minimize wobble but an optimal initial spin rate p₀=25.2rad/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 β₀ and least to φ₀. Present calculations do not account for dependence of initial release angle or spin on release velocity or among other release conditions.

Keywords: Discus, Optimization, Flight, Release conditions, Aerodynamics