# Ground Reaction Forces Generated by Runners--Harmonic Analyses and Modelling
## 4 Discussion
It can be seen that the estimated amplitudes are consistent with the experimental values of \(A_{pp,h}\), \(A_{pp,f}\) presented in Table 1. It is worth noting that the estimated amplitudes \(A_{pp,he}\), \(A_{pp,fe}\) are also consistent with the dynamic load factor \(k_{p}\) defined in [13]. However, it should be noted that according to the results obtained for the step frequency during running \(f\), in a range of 2.4-3.4 Hz, the mean value of the dynamic impact factor \(k_{p}\) defined in [13] as a function of \(t_{c}/T_{r}\) should be determined for the \(t_{c}/T_{r}\) ratio in a range of 0.63-0.77 (see the last two columns in Table 4, note: in the formula \(t_{c}/T_{r}\) use \(t_{ca,he}\) or \(t_{ca,fe}\) instead of \(t_{c}\) for heel-strike and forefoot-strike runners respectively). In addition, it is worth noting that the values of the contact duration of the foot with the ground presented in Table 2 are consistent with the recommendations presented in [13], after [30], only in the step frequency range \(f_{r}\) = 2.4-3.0 Hz. For \(f_{r}\) > 3.0 Hz, the values of the contact durations of the foot with the ground recommended in [13; 28] are less than the experimental values presented in Table 2. Normalised force amplitudes determined for the shorter contact times of the foot with the ground, e.g., using the half-sine model, reach values of up to \(F_{VGRF}\approx\) 3.0 that do not occur during the normal pace of the run (recreational running) [28; 29]. These values can be considered the upper limit of the force amplitudes which can only occur during fast running at a speed of 5-6 m/s [28]. It is also worth noting that assuming the contact time \(t_{ca}\) = 0.5 \(T_{r}\), recommended in [7], leads to a constant and overestimated value of the \(F_{VGRF}\) amplitude for all step frequencies during running (see values of \(A_{pp}\) in Table 4). Analysing the results presented in Tables 1 and 2 and in Figures 3 and 4, it can be observed that the \(F_{VCRF}\) amplitudes reach the highest values in the step frequency range of \(f_{r}\) = 2.70-3.10 Hz, which corresponds to a typical running pace. Analysing the results presented in Figure 3d, the lack of an explicit trend of the impact peak indicator \(A_{ip}\) can be identified (large scatter of \(A_{ip}\) values). A linear regression was used to approximate the \(A_{ip}\) values. The mean value of the normalised amplitude of the impact peak is \(A_{ip}=1.30\). The impact peak \(A_{ip}\) occurs on average about 42.0 ms after the contact of the heel with the ground. Analysing Figures 5-8, which show the normalised amplitudes of the \(F_{VCRF}\) harmonics (\(\alpha\)/G, where: \(\alpha_{i}\)--amplitude of the harmonic, \(G\)--weight of running person) generated during heel-strike and forefoot-strike running, it can be observed that the amplitude spectra for both running techniques are dominated by the amplitudes of the 1 st harmonics with mean values and standard deviations of \(1.08\pm 0.09\) and \(1.36\pm 0.07\) for heel-strike and forefoot-strike running respectively. It can be also seen that the scatter of amplitudes of individual harmonics in the case of forefoot-strike running is greater than in the case of heel-strike running especially for higher harmonics (for \(i\geq 4\), where: \(i\)--number of harmonic). In the case of forefoot-strike running the 2 nd harmonic reaches the mean value of around 24% of the amplitude of the 1 st harmonic. The 3 rd harmonic reaches a mean value of about 8% of the amplitude of the 1 st harmonic. The mean amplitudes of the 4 th and 5 th harmonics reach around 2.5-2.7% of the amplitude of the 1 st harmonic. The mean amplitudes of the 6 th and 7 th harmonics reach around 1.4-1.8% of the amplitude of the 1 st harmonic. Higher harmonics (for \(i\geq 8\)) reach amplitudes of less than 1.0% of the amplitude of the 1 st harmonic. A relatively fast decrease in the mean amplitude value of individual harmonics can be seen. In the case of heel-strike running the distribution of normalised amplitudes of the analysed harmonics nos.