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Besides the roughly parabolic path shape, the next most apparent feature of discus flight is slow but uneven rolling of the discus symmetry plane (counterclockwise viewed behind a right-handed thrower). Although this motion has not been measured accurately, tens of degrees of roll occur in typical flights and the women’s discus appears more susceptible to it. This motion changes the lift vector direction and consequently the trajectory from an otherwise vertical plane.

Although previous studies accounted for aerodynamic forces and variations in angle of attack, all these models held discus pitch attitude constant and maintained the symmetry axis in the vertical flight plane. Very few researchers have considered the possibility of general three-dimensional (3-D) motion allowing the discus spin axis and the velocity vector not to lie in the same plane.

The objectives of this work are to derive general 3-D equations of motion, and to numerically calculate discus flight trajectories including the assumption that discus attitude changes (mainly in roll) due to precession of spin angular momentum by nonzero pitching moments. Range (distance traveled before ground contact) is viewed as an implicit function of release parameters. Optimization in initial condition parameter space allows iterative calculation of release conditions (initial pitch and roll attitude and release angle) that produce maximum range and that may be used as training goals by throwers.

Fig. 1 Discus orientation and the angle of attack

The 3-D discus flight model (Fig. 1) is first aligned with an inertial xNyNzN frame with xN horizontal, zN vertically downward, and the xN direction chosen so that the release velocity vector has zero yN component. A 2-1-3 set of Euler rotations (θ,

, and ψ) defines discus orientation relative to xNyNzN; first about yN through θ, next about the new x1 through

, and finally about the axis of symmetry z2 through ψ (not shown due to symmetry). A right-handed thrower gives positive spin

about the generally downward z2 axis.

With discus velocity v and wind velocity vw, the relative wind vector vr = vw – v, and associated unit vector ur = vr/|vr|. Define unit vector ua (in z2 direction) upward along the axis of symmetry. Let vf be the projection of -vr (discus velocity into the wind) on the plane of symmetry; vf = –vr + (vr • ua)ua, with associated unit vector uf = vf/|vf|. The angle of attack

is the angle between –ur and uf and is positive when –ur has negative ua component (Fig. 1). Further define the pitching moment direction as the unit vector um = (ua × ur)/|ua × ur|.

During flight, aerodynamic drag and lift forces D and L and pitching moment M act on the discus in the ur , u1 = -um×ur, and um directions, respectively, with magnitudes given by:

where ρ is atmospheric density, d discus diameter, A=πd²/4 planform area, and CD, CL, and CM dimensionless drag, lift, and pitching moment coefficients depending on αThe functions CD and CL used in these flight simulations are averaged values from the literature and CM is estimated from our experimental data (Fig. 2). Since axial angular momentum

and spin-down torque about ua (due to shear stresses) are so large and small, respectively, axial spin changes during flight are negligible and the spin-down torque is neglected.

Fig. 2 Dependence of aerodynamic coefficients CD, CL, CM on angle of attack. Solid lines denote coefficients used herein.

Equations of motion are derived using a Newton-Euler formulation. The state vector for flight contains 11 state variables; 3 xNyNzN positions, 3 velocities, 2 orientation angles θ,

, and 3 angular velocities

, and p=

(spin angle ψ is omitted due to symmetry). For initial conditions the discus is assumed to be released above the origin at a typical height (xo=0; yo=0; zo=-1.8 m), with initial velocity vector in the xNzN plane at release angle βo from horizontal and with fixed speed (vo=25 m/s) so that vxo=vocosβo; vyo=0; vzo=-vosinβo. Nominal spin angular velocity po=42 rad/s was chosen. The initial angular velocities

and

are chosen to yield wobble-free initial flight. Initial discus orientation angles θo and

o, and release angle βo are variables to be optimized to achieve maximum range.

Numerical integration of the state equations with optimizing the initial release conditions shows that maximum range R=69.39 m is achieved with βo=38.4, θo=30.7, and

o=54.4 deg, after flight time t=3.77 s. A gentle lateral acceleration gives impact slightly (y=6.88 m) right of the initial throw direction. It is optimal to release the discus rolled significantly (

o=54.4 deg) so that the lift vector can remain near vertical throughout much, and especially near the end, of flight. Three 2-D sections at the optimal solution illustrate range sensitivity to each control variable βo, θo, and φo (Fig. 3). Range is most sensitive to βo and least sensitive to φo.

Fig. 3 Two-dimensional sectional contours of range plotted with fixing one optimal initial releasing condition while varying the other two.

For both men’s and women’s discuses (differing in size and weight), optimal βo, θo, and φo and resultant range R vary substantially with wind speed (Fig. 4). When winds are small (-6 < vw < 10 m/s) optimal strategies vary little from the wind-free strategy. Advantages of headwinds are clearly evident, when throwing into a 5 m/s headwind compared to a 10 m/s tailwind.

Fig. 4 Effects of head and tail winds on range and optimal throwing strategies for men’s and women’s discuses.

For larger tail or head winds different interesting optimal strategies emerge (Fig. 4). In strong headwinds (-20 < vw < -6 m/s) it gradually becomes optimal to initially orient the men’s discus symmetry plane nearly vertically and nearly aligned with the velocity vector, by increasing initial pitch and roll substantially (θo>110 deg; φo>70 deg), although the release angle remains relatively constant near βo=35 deg. We call this the “slicing strategy.” With strong tailwinds (vw>>0), however, optimal strategies change discontinuously at vw=10.786 m/s, where both strategies (βo=44.3, θo=36.7, and

o=58.6 deg) and (βo=46.0, θo=46.3, and

o=19.1 deg) produce equal range R=65.54 m. Similar discontinuities have been found in optimal javelin release conditions as functions of release velocity.

For large tailwinds vw>10.786 m/s (Figs. 4), optimal strategies take advantage of the fact that wind speed can be greater than discus horizontal velocity and choose an initially positive angle of attack (θo>βo), eventually increasing both βo and θo to near 75 and 90 deg, respectively, and changing the sign of φo. We call this the “kiting strategy.” Many optimal initial conditions for high winds would become extremely difficult, if not impossible, to achieve in actual throws.

Although we are not aware of any experimental studies of discus throw with measuring the initial roll angle

o, a value considerably smaller than 50 deg is apparent by our casual observations. It may be that throwers employ a sub-optimum strategy because a near-50 deg roll angle detracts from their ability to maximize release velocity or to achieve other optimal release parameters. 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.

Both men’s and women’s nominal optimal ranges (69.39 and 72.13 m, respectively) are larger than the vacuum range (65.49 m). Increasing release velocity by 50% from 20 to 30 m/s increases ranges by 104.18/43.06=239% and 109.94/44.14=249%, respectively. Both factors are greater than 225%=1.5² expected if range were exactly quadratic in velocity. This shows again that, for both discuses, the increases in range due to lift are larger than the decreases due to drag, and the discus can always be thrown further in air than in a vacuum.

These optimal release conditions neglect the likely dependence of release velocity and spin, not yet documented experimentally. Release velocity may also depend on βo as in the shot put and even on αo and φo, motivating further study of these dependences. This will require a biomechanical model of release mechanics or experimental measurements of sensitivities.