# Ground Reaction Forces Generated by Runners--Harmonic Analyses and Modelling
## 4 Discussion  
Section 3 presents the results of research on the vertical component of the ground reaction forces generated by running people (\(VGRF\)). The research took account of two different running styles: heel-strike running and forefoot-strike running. To the best of the author's knowledge, previous work  
Figure 8: The charts of (**a**) normalised amplitudes (log scale for the \(\phi_{I}\)-axis) and (**b**) phase angles (linear scale for both axis) of the fifteen harmonics determined for forty \(F_{VGRF}\) samples generated by four forefoot-strike runners (\(f_{r}=2.80\) Hz) (The data presented can be found in Supplementary Materials in the form of text files containing numerical data. See section Supplementary Materials). Figure 7: The charts of (**a**) normalised amplitudes (log scale for the \(\phi_{I}\)-axis) and (**b**) phase angles (linear scale for both axis) of the fifteen harmonics determined for forty \(F_{VGRF}\) samples generated by four heel-strike runners (\(f_{r}=2.80\) Hz) (The data presented can be found in Supplementary Materials in the form of text files containing numerical data. See section Supplementary Materials). presenting the results of a similar study do not take into account the division of _VGRF_ into different running styles. The presented analyses enabled the identification of the main features and differences between the _VGRF_ generated as a result of two different running styles. Comparing the amplitudes of the propulsive peaks \(A_{pp,h}\) and \(A_{ppf}\) (Table 1, Figures 3a and 4a) and the values of the contact time of the foot with the ground \(t_{ca,h}\), \(t_{caf}\) (Table 2, Figures 3c and 4c), two main differences between heel-strike and forefoot-strike \(F_{VGRF}\) can be observed. Firstly, the mean \(F_{VGRF}\) amplitudes generated during forefoot-strike running (\(A_{pp,f}\)) reach values around 10-15% greater than the mean \(F_{VGRF}\) amplitudes generated during heel-strike running (\(A_{pp,h}\)). Secondly, the approximated contact time of the foot with the ground for forefoot-strike running (\(t_{ca,f}\)) is around 10-15% less than the approximated contact time for heel-strike running (\(t_{ca,h}\)). As confirmed by the results of analyses performed by the author, the use of accurate values of the contact time of the foot with the ground is crucial for proper modelling of the \(F_{VGRF}\) using the half-sine model presented in [7; 13]. A similar relationship between the \(F_{VGRF}\) amplitudes generated during the normal pace of running (with a velocity of around 3.0 m/s) by forefoot-strike and heel-strike runners can be found in [28; 29]. It was considered that results obtained by the author, which are consistent with the results presented in [28; 29], indicate the relatively high importance of running technique for the \(F_{VGRF}\) amplitude. Further analyses of this topic are required. The values of \(t_{ca,h}\) and \(t_{ca,f}\) can be estimated by empirical Formula (1) and (2):  
\[t_{ca,he}\ =\ 0.215\ +\ 30\text{-}e^{-2.35\text{-}f} \tag{1}\]  
\[t_{ca,fe}\ =\ 0.166\ +\ 5.25\text{-}e^{-1.586\text{-}f} \tag{2}\]  
Equations (1) and (2) were determined by the author using an exponential regression line best fitted to the data set presented in Table 2 using the weighted nonlinear least-squares method. Fitted regression lines are shown in Figures 3c and 4c. The use of Formula (1) and (2) allows the approximated contact times of the foot with the ground (\(t_{ca,he}\), \(t_{ca,fe}\)) to be correctly estimated, and thus, the \(F_{VGRF}\) amplitudes for heel-strike and forefoot-strike running, respectively, can be correctly estimated. To compare the estimated \(F_{VGRF}\) amplitudes with the experimental values, in Table 4 the estimated values of amplitudes \(A_{pp,he}\), \(A_{pp,fe}\) determined for the estimated contact times \(t_{ca,he}\), \(t_{ca,fe}\) are presented.