Aerodynamic effects on the accuracy of an end-over-end kick of an American football

Abstract

The effects of initial conditions on the flight of an American football kicked in an end-over-end manner were investigated using a dynamic simulation employing the quaternion method. The effects of initial tilt and spin about the longitudinal axis of the ball were considered. For the most accurate kick, our simulations show that the ball should be vertical leaving the kicker’s foot, and have no angular velocity about the longitudinal axis of the ball. A case study was performed for which experimental data were available, showing the trends of the flight of the ball captured in our simulations in actual game situations.

Appendices

Appendix 1

Explanation of Signum Function Used for Quaternion Setup.

Accounting for the signum functions described above, the following plot depicts the “effective” aerodynamic coefficients over a full rotation of the football about its transverse axis with respect to a fixed velocity vector as depicted above. Positive lift is defined upward, positive drag is defined to the left, positive yaw force is defined out of the page, and positive moment is defined about an axis aligned out of the page. It is evident that the numerical formulation is continuous and able to represent the aerodynamic response over the full rotation.

Appendix 2

Generation of the initial state vector requires an initial transformation matrix, $^{B} \left[ C \right]_{0}^{O}$. For the simulations in this investigation, alignment of the coordinate systems requires two simple rotations to align the two systems. Starting with the two systems aligned, the first is a rotation of +90° about $\vec{j}_{B}$. Secondly, for tilted simulations, a rotation about $\vec{k}_{B}$ of the specified tilt angle, denoted $\theta_{tilt}$, is necessary. Therefore,

$$ {}_{{}}^{B} \left[ C \right]_{0}^{O} = \left[ {\begin{array}{*{20}c} {\cos \left( {\theta_{tilt} } \right)} & {\sin \left( {\theta_{tilt} } \right)} & 0 \\ { - \sin \left( {\theta_{tilt} } \right)} & {\cos \left( {\theta_{tilt} } \right)} & 0 \\ 0 & 0 & 1 \\ \end{array} } \right]\,\left[ {\begin{array}{*{20}c} 0 & 0 & { - 1} \\ 0 & 1 & 0 \\ 1 & 0 & 0 \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} 0 & {\sin \left( {\theta_{tilt} } \right)} & { - \cos \left( {\theta_{tilt} } \right)} \\ 0 & {\cos \left( {\theta_{tilt} } \right)} & {\sin \left( {\theta_{tilt} } \right)} \\ 1 & 0 & 0 \\ \end{array} } \right] $$

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The quaternion describing the initial orientation of the ball is obtained from the following:

$$ \begin{gathered} \Uptheta = \cos^{ - 1} \left( {\frac{1}{2}\left( {C_{11} + C_{22} + C_{33} - 1} \right)} \right) = \cos^{ - 1} \left( {\frac{1}{2}\left( {\cos \left( {\theta_{tilt} } \right) - 1} \right)} \right) = \cos^{ - 1} \left( { - \sin^{2} \frac{{\theta_{tilt} }}{2}} \right) \hfill \\ E_{x} = \frac{1}{2\sin \left( \Uptheta \right)}\left( {\sin \left( {\theta_{tilt} } \right)} \right) = \frac{1}{{2\left( {1 - \sin^{4} \left( {\frac{{\theta_{tilt} }}{2}} \right)} \right)^{1/2} }}\left( {\sin \left( {\theta_{tilt} } \right)} \right) \hfill \\ E_{y} = \frac{1}{2\sin \left( \Uptheta \right)}\left( {1 + \cos \left( {\theta_{tilt} } \right)} \right) = \frac{1}{{2\left( {1 - \sin^{4} \left( {\frac{{\theta_{tilt} }}{2}} \right)} \right)^{1/2} }}\left( {1 + \cos \left( {\theta_{tilt} } \right)} \right) \hfill \\ E_{z} = \frac{1}{2\sin \left( \Uptheta \right)}\left( {\sin \left( {\theta_{tilt} } \right)} \right) = \frac{1}{{2\left( {1 - \sin^{4} \left( {\frac{{\theta_{tilt} }}{2}} \right)} \right)^{1/2} }}\left( {\sin \left( {\theta_{tilt} } \right)} \right) \hfill \\ \end{gathered} $$

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$$ \begin{aligned} e_{0} &= \cos \left( {\frac{\Uptheta }{2}} \right) = \cos \left( {\frac{{\cos^{ - 1} \left( { - \sin^{2} \frac{{\theta_{tilt} }}{2}} \right)}}{2}} \right) \hfill \\ e_{1} &= E_{x} \sin \left( {\frac{\Uptheta }{2}} \right) = \frac{1}{{2\left( {1 - \sin^{4} \left( {\frac{{\theta_{tilt} }}{2}} \right)} \right)^{1/2} }}\left( {\sin \left( {\theta_{tilt} } \right)} \right)\left( {\sin \left( {\frac{{\cos^{ - 1} \left( { - \sin^{2} \frac{{\theta_{tilt} }}{2}} \right)}}{2}} \right)} \right) \hfill \\ e_{2} &= E_{y} \sin \left( {\frac{\Uptheta }{2}} \right) = \frac{1}{{2\left( {1 - \sin^{4} \left( {\frac{{\theta_{tilt} }}{2}} \right)} \right)^{1/2} }}\left( {1 + \cos \left( {\theta_{tilt} } \right)} \right)\left( {\sin \left( {\frac{{\cos^{ - 1} \left( { - \sin^{2} \frac{{\theta_{tilt} }}{2}} \right)}}{2}} \right)} \right) \hfill \\ e_{3} &= E_{z} \sin \left( {\frac{\Uptheta }{2}} \right) = \frac{1}{{2\left( {1 - \sin^{4} \left( {\frac{{\theta_{tilt} }}{2}} \right)} \right)^{1/2} }}\left( {\sin \left( {\theta_{tilt} } \right)} \right)\left( {\sin \left( {\frac{{\cos^{ - 1} \left( { - \sin^{2} \frac{{\theta_{tilt} }}{2}} \right)}}{2}} \right)} \right) \hfill \\ \end{aligned} $$

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Initial velocity and angular velocity components in the B-frame are obtained by transforming the vectors defined in the O-frame by multiplying the transformation matrix (Eq. 25) and a row vector of velocity or angular velocity components in the O-frame.