# Ground Reaction Forces Generated by Runners--Harmonic Analyses and Modelling
## 4 Discussion
2-15 is more even than in the case of forefoot-strike running. A large number of harmonics reach amplitudes in the range of 4.0-9.0% of the amplitude of the 1 st harmonic. The 2 nd harmonic reaches a mean value of around 9% of the amplitude of the 1 st harmonic. The 3 rd harmonic reaches a mean value of around 7% of the amplitude of the 1 st harmonic. The mean amplitudes of the 6 th and 7 th harmonics reach around 4.6-5.6% of the amplitude of the 1 st harmonic. The mean amplitudes of the mean amplitudes of the 6 th and 7 th harmonics reach about 4.6-5.6% of the amplitude of the 1 st harmonic. The mean amplitudes of The mean amplitudes of harmonics nos. 8-14 reach values in the range of 1.0-4.0% of the amplitude of the 1 st harmonic of the 1 st harmonic. Analysing the charts showing the phase angles of the \(F_{VCRF}\) harmonics (\(\phi_{i}\)) presented in Figures 5-8, a relatively large scatter of the \(\phi_{i}\) of individual harmonics can be noticed. This feature of the \(F_{VCRF}\) harmonics causes major problems in \(F_{VCRF}\) modelling using the Fourier series, especially in the case of the \(F_{VGFR}\) generated by heel-strike runners. The shape of the \(F_{VGFR}\) generated by forefoot-strike runners is dominated by the 1 st harmonic and is slightly dependent on the values of the phase angles of the higher harmonics. Proper assumption of the phase angles for the first three harmonics is sufficient to correct modelling of the \(F_{VCRF}\) generated by forefoot-strike runners. \begin{table} \begin{tabular}{c c c c c c c c c} \hline \hline
**Frequency** & \(t_{ca,he}\) **[s]** & \(t_{ca,fe}\) **[s]** & \(t_{cc}\) **= 0.5** & \(A_{pp,he}\) **(for** \(t_{ca,h}\))** & \(A_{pp,fe}\) **(for** \(t_{ca,h}\))** & \(t_{c}\) & \(t_{c,he}\) **(for** \(t_{ca,h}\))** & \(t_{c,he}\) **(for** \(t_{ca,h}\))** & \(t_{c}\) **(for** \(t_{ca,h}\))** \\ \hline
2.4 & 0.322 & 0.283 & 0.208 & 2.035 & 2.315 & 3.142 & 0.768 & 0.679 \\
2.6 & 0.282 & 0.251 & 0.192 & 2.145 & 2.407 & 3.142 & 0.731 & 0.658 \\
2.8 & 0.257 & 0.228 & 0.179 & 2.186 & 2.462 & 3.142 & 0.711 & 0.633 \\
3.0 & 0.241 & 0.211 & 0.167 & 2.172 & 2.481 & 3.142 & 0.720 & 0.639 \\
3.2 & 0.231 & 0.199 & 0.156 & 2.123 & 2.469 & 3.142 & 0.752 & 0.640 \\
3.4 & 0.225 & 0.190 & 0.147 & 2.052 & 2.433 & 3.142 & 0.755 & 0.646 \\ \hline \hline \end{tabular} 
\begin{tabular}{c} where: \(T_{r}=1/f_{r}\); \(A_{pp,he}\), \(A_{pp,he}\), \(A_{pp}\)—estimated amplitudes of normalised ground reaction forces \(F_{VCRF}\): \(A_{pp,he}=\pi/(2t_{ca,he}f_{r})\); \(A_{pp,he}=\pi/(2t_{ca,fe}f_{r})\); \(A_{pp}=\pi/(2t_{ca,fe}f_{r})\) according to [7]. The amplitudes of the real \(VGFR\)s should be calculated as follows, e.g., \(A_{pp,he}=A_{pp,he}\); \(A_{pp,he}=A_{pp,he}\); \(A_{pp,he}=A_{pp,he}\); \(A_{pp,he}=A_{pp,he}\); \(A_{pp}=A_{pp,he}\); \(A_{pp}\);In the case of analysed running pace with a step frequency of \(f_{r}\) = 2.80 Hz the mean phase angles for the 1 st harmonics of the \(F_{VCRF}\) are \(\phi_{1,h}\) = 5.48 \(\pm\) 0.13 rad (\(\sim\)1.75\(\pi\)) and \(\phi_{1,f}\) = 5.83 \(\pm\) 0.19 rad (\(\sim\)1.86\(\pi\)) for heel-strike and forefoot-strike running respectively. In the case of forefoot-strike running, the constant value of the phase angles \(\phi_{i}\) \(\approx\) 4.0 rad (\(\sim\)1.30\(\pi\)) for \(i\) \(\geq\) 2 can be assumed without significant loss of accuracy in determining the \(F_{VCRF}\) (compare Figure 8b). In the case of the \(F_{VCRF}\) generated during heel-strike running, the characteristic is a large scatter of the phase angel values for the 2 nd harmonic and almost linear decrease of the phase angles for harmonics 3-15 (Figure 7b). It should be noted that the reader should remember about the periodicity of the phase angles of \(\pm\)2\(\pi\) (6.28 rad), which allows a large value of phase angles determined for higher harmonics (shown in Figure 7b for harmonics 10-15) to be converted to a small or negative value. The mean value and the standard deviation of the phase angle for the 2 nd harmonic is \(\phi_{2}\) = 4.0 \(\pm\) 1.15 rad.