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,
$$ {}_{{}}^{B} \left[ C \right]_{0}^{O} = \left[ {\begin{array}{*{20}c} {\cos \left( {\theta_{tilt} } \right)} & {\sin \left( {\theta_{tilt} } \right)} & 0 \\ { - \sin \left( {\theta_{tilt} } \right)} & {\cos \left( {\theta_{tilt} } \right)} & 0 \\ 0 & 0 & 1 \\ \end{array} } \right]\,\left[ {\begin{array}{*{20}c} 0 & 0 & { - 1} \\ 0 & 1 & 0 \\ 1 & 0 & 0 \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} 0 & {\sin \left( {\theta_{tilt} } \right)} & { - \cos \left( {\theta_{tilt} } \right)} \\ 0 & {\cos \left( {\theta_{tilt} } \right)} & {\sin \left( {\theta_{tilt} } \right)} \\ 1 & 0 & 0 \\ \end{array} } \right] $$
(25)
The quaternion describing the initial orientation of the ball is obtained from the following:
$$ \begin{gathered} \Uptheta = \cos^{ - 1} \left( {\frac{1}{2}\left( {C_{11} + C_{22} + C_{33} - 1} \right)} \right) = \cos^{ - 1} \left( {\frac{1}{2}\left( {\cos \left( {\theta_{tilt} } \right) - 1} \right)} \right) = \cos^{ - 1} \left( { - \sin^{2} \frac{{\theta_{tilt} }}{2}} \right) \hfill \\ E_{x} = \frac{1}{2\sin \left( \Uptheta \right)}\left( {\sin \left( {\theta_{tilt} } \right)} \right) = \frac{1}{{2\left( {1 - \sin^{4} \left( {\frac{{\theta_{tilt} }}{2}} \right)} \right)^{1/2} }}\left( {\sin \left( {\theta_{tilt} } \right)} \right) \hfill \\ E_{y} = \frac{1}{2\sin \left( \Uptheta \right)}\left( {1 + \cos \left( {\theta_{tilt} } \right)} \right) = \frac{1}{{2\left( {1 - \sin^{4} \left( {\frac{{\theta_{tilt} }}{2}} \right)} \right)^{1/2} }}\left( {1 + \cos \left( {\theta_{tilt} } \right)} \right) \hfill \\ E_{z} = \frac{1}{2\sin \left( \Uptheta \right)}\left( {\sin \left( {\theta_{tilt} } \right)} \right) = \frac{1}{{2\left( {1 - \sin^{4} \left( {\frac{{\theta_{tilt} }}{2}} \right)} \right)^{1/2} }}\left( {\sin \left( {\theta_{tilt} } \right)} \right) \hfill \\ \end{gathered} $$
(26)
$$ \begin{aligned} e_{0} &= \cos \left( {\frac{\Uptheta }{2}} \right) = \cos \left( {\frac{{\cos^{ - 1} \left( { - \sin^{2} \frac{{\theta_{tilt} }}{2}} \right)}}{2}} \right) \hfill \\ e_{1} &= E_{x} \sin \left( {\frac{\Uptheta }{2}} \right) = \frac{1}{{2\left( {1 - \sin^{4} \left( {\frac{{\theta_{tilt} }}{2}} \right)} \right)^{1/2} }}\left( {\sin \left( {\theta_{tilt} } \right)} \right)\left( {\sin \left( {\frac{{\cos^{ - 1} \left( { - \sin^{2} \frac{{\theta_{tilt} }}{2}} \right)}}{2}} \right)} \right) \hfill \\ e_{2} &= E_{y} \sin \left( {\frac{\Uptheta }{2}} \right) = \frac{1}{{2\left( {1 - \sin^{4} \left( {\frac{{\theta_{tilt} }}{2}} \right)} \right)^{1/2} }}\left( {1 + \cos \left( {\theta_{tilt} } \right)} \right)\left( {\sin \left( {\frac{{\cos^{ - 1} \left( { - \sin^{2} \frac{{\theta_{tilt} }}{2}} \right)}}{2}} \right)} \right) \hfill \\ e_{3} &= E_{z} \sin \left( {\frac{\Uptheta }{2}} \right) = \frac{1}{{2\left( {1 - \sin^{4} \left( {\frac{{\theta_{tilt} }}{2}} \right)} \right)^{1/2} }}\left( {\sin \left( {\theta_{tilt} } \right)} \right)\left( {\sin \left( {\frac{{\cos^{ - 1} \left( { - \sin^{2} \frac{{\theta_{tilt} }}{2}} \right)}}{2}} \right)} \right) \hfill \\ \end{aligned} $$
(27)
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.